Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
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import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 92 | 97 | theorem evariance_lt_top_iff_memℒp [IsFiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
evariance X μ < ∞ ↔ Memℒp X 2 μ := by |
refine ⟨?_, MeasureTheory.Memℒp.evariance_lt_top⟩
contrapose
rw [not_lt, top_le_iff]
exact evariance_eq_top hX
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,927 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 100 | 103 | theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
ENNReal.ofReal (variance X μ) = evariance X μ := by |
rw [variance, ENNReal.ofReal_toReal]
exact hX.evariance_lt_top.ne
| 2 | 7.389056 | 1 | 1.857143 | 7 | 1,927 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 106 | 113 | theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ := by |
rw [evariance]
congr
ext1 ω
rw [pow_two, ← ENNReal.coe_mul, ← nnnorm_mul, ← pow_two]
congr
exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm
| 6 | 403.428793 | 2 | 1.857143 | 7 | 1,927 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 116 | 125 | theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
(hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)] <;>
simp_rw [hXint, sub_zero]
· rfl
· convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top with ω
simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toRe... | 8 | 2,980.957987 | 2 | 1.857143 | 7 | 1,927 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 128 | 139 | theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
variance X μ = μ[(X - fun _ => μ[X] :) ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)]
· rfl
· -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,
-- and `convert` cannot disambiguate based on typeclass inference failure.
convert (hX.sub... | 10 | 22,026.465795 | 2 | 1.857143 | 7 | 1,927 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 27 | 32 | theorem add_eq_zero_iff {n m : WithBot ℕ} : n + m = 0 ↔ n = 0 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat (· exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.1⟩)
· exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.2⟩
repeat erw [WithBot.coe_eq_coe]
exact add_eq_zero_iff' (zero_le _) (zero_le _)
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 35 | 40 | theorem add_eq_one_iff {n m : WithBot ℕ} : n + m = 1 ↔ n = 0 ∧ m = 1 ∨ n = 1 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_one_iff
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 43 | 49 | theorem add_eq_two_iff {n m : WithBot ℕ} :
n + m = 2 ↔ n = 0 ∧ m = 2 ∨ n = 1 ∧ m = 1 ∨ n = 2 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_two_iff
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 52 | 58 | theorem add_eq_three_iff {n m : WithBot ℕ} :
n + m = 3 ↔ n = 0 ∧ m = 3 ∨ n = 1 ∧ m = 2 ∨ n = 2 ∧ m = 1 ∨ n = 3 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_three_iff
| 5 | 148.413159 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 61 | 63 | theorem coe_nonneg {n : ℕ} : 0 ≤ (n : WithBot ℕ) := by |
rw [← WithBot.coe_zero]
exact WithBot.coe_le_coe.mpr (Nat.zero_le n)
| 2 | 7.389056 | 1 | 1.857143 | 7 | 1,928 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 70 | 74 | theorem one_le_iff_zero_lt {x : WithBot ℕ} : 1 ≤ x ↔ 0 < x := by |
refine ⟨fun h => lt_of_lt_of_le (WithBot.coe_lt_coe.mpr zero_lt_one) h, fun h => ?_⟩
induction x
· exact (not_lt_bot h).elim
· exact WithBot.coe_le_coe.mpr (Nat.succ_le_iff.mpr (WithBot.coe_lt_coe.mp h))
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 81 | 85 | theorem add_one_le_of_lt {n m : WithBot ℕ} (h : n < m) : n + 1 ≤ m := by |
cases n
· exact bot_le
cases m
exacts [(not_lt_bot h).elim, WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 h)]
| 4 | 54.59815 | 2 | 1.857143 | 7 | 1,928 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 52 | 60 | theorem exists_multiset (x : M ⊗[R] N) :
∃ S : Multiset (M × N), x = (S.map fun i ↦ i.1 ⊗ₜ[R] i.2).sum := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨{(x, y)}, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
exact ⟨Sx + Sy, by rw [Multiset.map_add, Multiset.sum_add, hx, hy]⟩
| 7 | 1,096.633158 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 65 | 75 | theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
| 9 | 8,103.083928 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 80 | 84 | theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by |
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
| 3 | 20.085537 | 1 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 88 | 93 | theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by |
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
| 4 | 54.59815 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 98 | 116 | theorem exists_finite_submodule_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N), Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (mapIncl M' N') := by |
simp_rw [Module.Finite.iff_fg]
refine hs.induction_on ⟨_, _, fg_bot, fg_bot, Set.empty_subset _⟩ ?_
rintro a s - - ⟨M', N', hM', hN', h⟩
refine TensorProduct.induction_on a ?_ (fun x y ↦ ?_) fun x y hx hy ↦ ?_
· exact ⟨M', N', hM', hN', Set.insert_subset (zero_mem _) h⟩
· refine ⟨_, _, hM'.sup (fg_span_sin... | 16 | 8,886,110.520508 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 121 | 126 | theorem exists_finite_submodule_left_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ M' : Submodule R M, Module.Finite R M' ∧ s ⊆ LinearMap.range (M'.subtype.rTensor N) := by |
obtain ⟨M', _, hfin, _, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨M', hfin, ?_⟩
rw [mapIncl, ← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| 4 | 54.59815 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 131 | 136 | theorem exists_finite_submodule_right_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ N' : Submodule R N, Module.Finite R N' ∧ s ⊆ LinearMap.range (N'.subtype.lTensor M) := by |
obtain ⟨_, N', _, hfin, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨N', hfin, ?_⟩
rw [mapIncl, ← LinearMap.lTensor_comp_rTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| 4 | 54.59815 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 140 | 152 | theorem exists_finite_submodule_of_finite' (s : Set (M₁ ⊗[R] N₁)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N) (hM : M' ≤ M₁) (hN : N' ≤ N₁),
Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (TensorProduct.map (inclusion hM) (inclusion hN)) := by |
obtain ⟨M', N', _, _, h⟩ := exists_finite_submodule_of_finite s hs
have hM := map_subtype_le M₁ M'
have hN := map_subtype_le N₁ N'
refine ⟨_, _, hM, hN, .map _ _, .map _ _, ?_⟩
rw [mapIncl,
show M'.subtype = inclusion hM ∘ₗ M₁.subtype.submoduleMap M' by ext; simp,
show N'.subtype = inclusion hN ∘ₗ N₁... | 9 | 8,103.083928 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 85 | 105 | theorem reduce_to_p_prime {P : Prop} :
(∀ p > 0, MixedCharZero R p → P) ↔ ∀ p : ℕ, p.Prime → MixedCharZero R p → P := by |
constructor
· intro h q q_prime q_mixedChar
exact h q (Nat.Prime.pos q_prime) q_mixedChar
· intro h q q_pos q_mixedChar
rcases q_mixedChar.charP_quotient with ⟨I, hI_ne_top, _⟩
-- Krull's Thm: There exists a prime ideal `P` such that `I ≤ P`
rcases Ideal.exists_le_maximal I hI_ne_top with ⟨M, hM_... | 19 | 178,482,300.963187 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 112 | 136 | theorem reduce_to_maximal_ideal {p : ℕ} (hp : Nat.Prime p) :
(∃ I : Ideal R, I ≠ ⊤ ∧ CharP (R ⧸ I) p) ↔ ∃ I : Ideal R, I.IsMaximal ∧ CharP (R ⧸ I) p := by |
constructor
· intro g
rcases g with ⟨I, ⟨hI_not_top, _⟩⟩
-- Krull's Thm: There exists a prime ideal `M` such that `I ≤ M`.
rcases Ideal.exists_le_maximal I hI_not_top with ⟨M, ⟨hM_max, hM_ge⟩⟩
use M
constructor
· exact hM_max
· cases CharP.exists (R ⧸ M) with
| intro r hr =>
... | 23 | 9,744,803,446.248903 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 161 | 169 | theorem of_algebraRat [Algebra ℚ R] : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) := by |
intro I hI
constructor
intro a b h_ab
contrapose! hI
-- `↑a - ↑b` is a unit contained in `I`, which contradicts `I ≠ ⊤`.
refine I.eq_top_of_isUnit_mem ?_ (IsUnit.map (algebraMap ℚ R) (IsUnit.mk0 (a - b : ℚ) ?_))
· simpa only [← Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]
simpa on... | 8 | 2,980.957987 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 178 | 189 | theorem PNat.isUnit_natCast [h : Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))]
(n : ℕ+) : IsUnit (n : R) := by |
-- `n : R` is a unit iff `(n)` is not a proper ideal in `R`.
rw [← Ideal.span_singleton_eq_top]
-- So by contrapositive, we should show the quotient does not have characteristic zero.
apply not_imp_comm.mp (h.elim (Ideal.span {↑n}))
intro h_char_zero
-- In particular, the image of `n` in the quotient shoul... | 10 | 22,026.465795 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 204 | 209 | theorem pnatCast_one [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] : ((1 : ℕ+) : Rˣ) = 1 := by |
apply Units.ext
rw [Units.val_one]
change ((PNat.isUnit_natCast (R := R) 1).unit : R) = 1
rw [IsUnit.unit_spec (PNat.isUnit_natCast 1)]
rw [PNat.one_coe, Nat.cast_one]
| 5 | 148.413159 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 214 | 217 | theorem pnatCast_eq_natCast [Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))] (n : ℕ+) :
((n : Rˣ) : R) = ↑n := by |
change ((PNat.isUnit_natCast (R := R) n).unit : R) = ↑n
simp only [IsUnit.unit_spec]
| 2 | 7.389056 | 1 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 250 | 260 | theorem of_not_mixedCharZero [CharZero R] (h : ∀ p > 0, ¬MixedCharZero R p) :
∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) := by |
intro I hI_ne_top
suffices CharP (R ⧸ I) 0 from CharP.charP_to_charZero _
cases CharP.exists (R ⧸ I) with
| intro p hp =>
cases p with
| zero => exact hp
| succ p =>
have h_mixed : MixedCharZero R p.succ := ⟨⟨I, ⟨hI_ne_top, hp⟩⟩⟩
exact absurd h_mixed (h p.succ p.succ_pos)
| 9 | 8,103.083928 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 264 | 270 | theorem to_not_mixedCharZero (h : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I)) :
∀ p > 0, ¬MixedCharZero R p := by |
intro p p_pos
by_contra hp_mixedChar
rcases hp_mixedChar.charP_quotient with ⟨I, hI_ne_top, hI_p⟩
replace hI_zero : CharP (R ⧸ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)
exact absurd (CharP.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos)
| 5 | 148.413159 | 2 | 1.875 | 8 | 1,930 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 69 | 77 | theorem polynomial_comp_attachBound (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound =
Polynomial.aeval f g := by |
ext
simp only [ContinuousMap.coe_comp, Function.comp_apply, ContinuousMap.attachBound_apply_coe,
Polynomial.toContinuousMapOn_apply, Polynomial.aeval_subalgebra_coe,
Polynomial.aeval_continuousMap_apply, Polynomial.toContinuousMap_apply]
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2... | 6 | 403.428793 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 88 | 91 | theorem polynomial_comp_attachBound_mem (A : Subalgebra ℝ C(X, ℝ)) (f : A) (g : ℝ[X]) :
(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (f : C(X, ℝ)).attachBound ∈ A := by |
rw [polynomial_comp_attachBound]
apply SetLike.coe_mem
| 2 | 7.389056 | 1 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 94 | 113 | theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
(p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure := by |
-- `p` itself is in the closure of polynomials, by the Weierstrass theorem,
have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure :=
continuousMap_mem_polynomialFunctions_closure _ _ p
-- and so there are polynomials arbitrarily close.
have frequently_mem_polynomials := mem_c... | 18 | 65,659,969.137331 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 116 | 121 | theorem abs_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A) :
|(f : C(X, ℝ))| ∈ A.topologicalClosure := by |
let f' := attachBound (f : C(X, ℝ))
let abs : C(Set.Icc (-‖f‖) ‖f‖, ℝ) := { toFun := fun x : Set.Icc (-‖f‖) ‖f‖ => |(x : ℝ)| }
change abs.comp f' ∈ A.topologicalClosure
apply comp_attachBound_mem_closure
| 4 | 54.59815 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 124 | 134 | theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by |
rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.sub_mem
(A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)
(A.le_topologicalClosure g.property))
?_)
_
exact mod_cast abs_mem_subalgebra_closure A _
| 9 | 8,103.083928 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 137 | 143 | theorem inf_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A := by |
convert inf_mem_subalgebra_closure A f g
apply SetLike.ext'
symm
erw [closure_eq_iff_isClosed]
exact h
| 5 | 148.413159 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 146 | 156 | theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure := by |
rw [sup_eq_half_smul_add_add_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.add_mem
(A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)
(A.le_topologicalClosure g.property))
?_)
_
exact mod_cast abs_mem_subalgebra_closure A _
| 9 | 8,103.083928 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 159 | 165 | theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A := by |
convert sup_mem_subalgebra_closure A f g
apply SetLike.ext'
symm
erw [closure_eq_iff_isClosed]
exact h
| 5 | 148.413159 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 355 | 377 | theorem Subalgebra.SeparatesPoints.rclike_to_real {A : StarSubalgebra 𝕜 C(X, 𝕜)}
(hA : A.SeparatesPoints) :
((A.restrictScalars ℝ).comap
(ofRealAm.compLeftContinuous ℝ continuous_ofReal)).SeparatesPoints := by |
intro x₁ x₂ hx
-- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx
let F : C(X, 𝕜) := f - const _ (f x₂)
-- Subtract the constant `f x₂` from `f`; this is still an element of the subalgebra
have hFA : F ∈ A := by
refine A.sub_mem hfA (@Eq.subst _ (· ∈... | 19 | 178,482,300.963187 | 2 | 1.888889 | 9 | 1,931 |
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.RingTheory.TensorProduct.Basic
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f4335... | Mathlib/FieldTheory/Adjoin.lean | 54 | 60 | theorem mem_adjoin_iff (x : E) :
x ∈ adjoin F S ↔ ∃ r s : MvPolynomial S F,
x = MvPolynomial.aeval Subtype.val r / MvPolynomial.aeval Subtype.val s := by |
simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
Algebra.adjoin_eq_range, AlgHom.mem_range, exists_exists_eq_and]
tauto
| 4 | 54.59815 | 2 | 2 | 2 | 1,932 |
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.RingTheory.TensorProduct.Basic
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f4335... | Mathlib/FieldTheory/Adjoin.lean | 62 | 67 | theorem mem_adjoin_simple_iff {α : E} (x : E) :
x ∈ adjoin F {α} ↔ ∃ r s : F[X], x = aeval α r / aeval α s := by |
simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
Algebra.adjoin_singleton_eq_range_aeval, AlgHom.mem_range, exists_exists_eq_and]
tauto
| 4 | 54.59815 | 2 | 2 | 2 | 1,932 |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 86 | 92 | theorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)
(hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by |
by_cases h : S = ∅
· rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]
exact h𝒜.univ_mem
· rw [← ne_eq, ← Finset.nonempty_iff_ne_empty] at h
exact h𝒜.isSetRing.biInter_mem S h hs
| 5 | 148.413159 | 2 | 2 | 4 | 1,933 |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 122 | 134 | theorem generateFrom_generateSetAlgebra_eq :
generateFrom (generateSetAlgebra 𝒜) = generateFrom 𝒜 := by |
refine le_antisymm (fun s ms ↦ ?_) (generateFrom_mono self_subset_generateSetAlgebra)
refine @generateFrom_induction _ _ (generateSetAlgebra 𝒜) (fun t ht ↦ ?_)
(@MeasurableSet.empty _ (generateFrom 𝒜))
(fun t ↦ MeasurableSet.compl)
(fun f hf ↦ MeasurableSet.iUnion hf)
s ms
induction ht with
|... | 11 | 59,874.141715 | 2 | 2 | 4 | 1,933 |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 138 | 145 | theorem generateSetAlgebra_mono {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ) :
generateSetAlgebra 𝒜 ⊆ generateSetAlgebra ℬ := by |
intro s hs
induction hs with
| base t t_mem => exact self_subset_generateSetAlgebra (h t_mem)
| empty => exact isSetAlgebra_generateSetAlgebra.empty_mem
| compl t _ t_mem => exact isSetAlgebra_generateSetAlgebra.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact isSetAlgebra_generateSetAlgebra.union_mem ... | 6 | 403.428793 | 2 | 2 | 4 | 1,933 |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 151 | 158 | theorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
(hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ := by |
intro s hs
induction hs with
| base t t_mem => exact h t_mem
| empty => exact hℬ.empty_mem
| compl t _ t_mem => exact hℬ.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact hℬ.union_mem t_mem u_mem
| 6 | 403.428793 | 2 | 2 | 4 | 1,933 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 48 | 70 | theorem RespectsIso.basicOpen_iff (hP : RespectsIso @P) {X Y : Scheme.{u}} [IsAffine X] [IsAffine Y]
(f : X ⟶ Y) (r : Y.presheaf.obj (Opposite.op ⊤)) :
P (Scheme.Γ.map (f ∣_ Y.basicOpen r).op) ↔
P (@IsLocalization.Away.map (Y.presheaf.obj (Opposite.op ⊤)) _
(Y.presheaf.obj (Opposite.op <| Y.basicOpen ... |
rw [Γ_map_morphismRestrict, hP.cancel_left_isIso, hP.cancel_right_isIso,
← hP.cancel_right_isIso (f.val.c.app (Opposite.op (Y.basicOpen r)))
(X.presheaf.map (eqToHom (Scheme.preimage_basicOpen f r).symm).op), ← eq_iff_iff]
congr
delta IsLocalization.Away.map
refine IsLocalization.ringHom_ext (Submono... | 16 | 8,886,110.520508 | 2 | 2 | 6 | 1,934 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 73 | 80 | theorem RespectsIso.basicOpen_iff_localization (hP : RespectsIso @P) {X Y : Scheme.{u}} [IsAffine X]
[IsAffine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (Opposite.op ⊤)) :
P (Scheme.Γ.map (f ∣_ Y.basicOpen r).op) ↔ P (Localization.awayMap (Scheme.Γ.map f.op) r) := by |
refine (hP.basicOpen_iff _ _).trans ?_
-- Porting note: was a one line term mode proof, but this `dsimp` is vital so the term mode
-- one liner is not possible
dsimp
rw [← hP.is_localization_away_iff]
| 5 | 148.413159 | 2 | 2 | 6 | 1,934 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 86 | 102 | theorem RespectsIso.ofRestrict_morphismRestrict_iff (hP : RingHom.RespectsIso @P) {X Y : Scheme.{u}}
[IsAffine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (Opposite.op ⊤)) (U : Opens X.carrier)
(hU : IsAffineOpen U) {V : Opens _}
(e : V = (Scheme.ιOpens <| f ⁻¹ᵁ Y.basicOpen r) ⁻¹ᵁ U) :
P (Scheme.Γ.map (Scheme.ιO... |
subst e
refine (hP.cancel_right_isIso _
(Scheme.Γ.mapIso (Scheme.restrictRestrictComm _ _ _).op).inv).symm.trans ?_
haveI : IsAffine _ := hU
rw [← hP.basicOpen_iff_localization, iff_iff_eq]
congr 1
simp only [Functor.mapIso_inv, Iso.op_inv, ← Functor.map_comp, ← op_comp, morphismRestrict_comp]
rw [← ... | 11 | 59,874.141715 | 2 | 2 | 6 | 1,934 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 105 | 123 | theorem StableUnderBaseChange.Γ_pullback_fst (hP : StableUnderBaseChange @P) (hP' : RespectsIso @P)
{X Y S : Scheme} [IsAffine X] [IsAffine Y] [IsAffine S] (f : X ⟶ S) (g : Y ⟶ S)
(H : P (Scheme.Γ.map g.op)) : P (Scheme.Γ.map (pullback.fst : pullback f g ⟶ _).op) := by |
-- Porting note (#11224): change `rw` to `erw`
erw [← PreservesPullback.iso_inv_fst AffineScheme.forgetToScheme (AffineScheme.ofHom f)
(AffineScheme.ofHom g)]
rw [op_comp, Functor.map_comp, hP'.cancel_right_isIso, AffineScheme.forgetToScheme_map]
have :=
_root_.congr_arg Quiver.Hom.unop
(Preser... | 16 | 8,886,110.520508 | 2 | 2 | 6 | 1,934 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 145 | 156 | theorem sourceAffineLocally_respectsIso (h₁ : RingHom.RespectsIso @P) :
(sourceAffineLocally @P).toProperty.RespectsIso := by |
apply AffineTargetMorphismProperty.respectsIso_mk
· introv H U
rw [← h₁.cancel_right_isIso _ (Scheme.Γ.map (Scheme.restrictMapIso e.inv U.1).hom.op), ←
Functor.map_comp, ← op_comp]
convert H ⟨_, U.prop.map_isIso e.inv⟩ using 3
rw [IsOpenImmersion.isoOfRangeEq_hom_fac_assoc, Category.assoc,
... | 10 | 22,026.465795 | 2 | 2 | 6 | 1,934 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 163 | 205 | theorem affineLocally_iff_affineOpens_le
(hP : RingHom.RespectsIso @P) {X Y : Scheme.{u}} (f : X ⟶ Y) :
affineLocally.{u} (@P) f ↔
∀ (U : Y.affineOpens) (V : X.affineOpens) (e : V.1 ≤ (Opens.map f.1.base).obj U.1),
P (Scheme.Hom.appLe f e) := by |
apply forall_congr'
intro U
delta sourceAffineLocally
simp_rw [op_comp, Scheme.Γ.map_comp, Γ_map_morphismRestrict, Category.assoc, Scheme.Γ_map_op,
hP.cancel_left_isIso (Y.presheaf.map (eqToHom _).op)]
constructor
· intro H V e
let U' := (Opens.map f.val.base).obj U.1
have e'' : (Scheme.Hom.ope... | 38 | 31,855,931,757,113,756 | 2 | 2 | 6 | 1,934 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 90 | 98 | theorem pi_univ_Ioc_update_left {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : x i₀ ≤ m) :
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) =
{ z | m < z i₀ } ∩ pi univ fun i ↦ Ioc (x i) (y i) := by |
have : Ioc m (y i₀) = Ioi m ∩ Ioc (x i₀) (y i₀) := by
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, ← inter_assoc,
inter_eq_self_of_subset_left (Ioi_subset_Ioi hm)]
simp_rw [univ_pi_update i₀ _ _ fun i z ↦ Ioc z (y i), ← pi_inter_compl ({i₀} : Set ι),
singleton_pi', ← inter_assoc, this]
rfl
| 6 | 403.428793 | 2 | 2 | 3 | 1,935 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 101 | 109 | theorem pi_univ_Ioc_update_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : m ≤ y i₀) :
(pi univ fun i ↦ Ioc (x i) (update y i₀ m i)) =
{ z | z i₀ ≤ m } ∩ pi univ fun i ↦ Ioc (x i) (y i) := by |
have : Ioc (x i₀) m = Iic m ∩ Ioc (x i₀) (y i₀) := by
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic, inter_left_comm,
inter_eq_self_of_subset_left (Iic_subset_Iic.2 hm)]
simp_rw [univ_pi_update i₀ y m fun i z ↦ Ioc (x i) z, ← pi_inter_compl ({i₀} : Set ι),
singleton_pi', ← inter_assoc, this]
rfl
| 6 | 403.428793 | 2 | 2 | 3 | 1,935 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
-- Porting note: Added, since dot nota... | Mathlib/Order/Interval/Set/Pi.lean | 112 | 118 | theorem disjoint_pi_univ_Ioc_update_left_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} :
Disjoint (pi univ fun i ↦ Ioc (x i) (update y i₀ m i))
(pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) := by |
rw [disjoint_left]
rintro z h₁ h₂
refine (h₁ i₀ (mem_univ _)).2.not_lt ?_
simpa only [Function.update_same] using (h₂ i₀ (mem_univ _)).1
| 4 | 54.59815 | 2 | 2 | 3 | 1,935 |
import Mathlib.Algebra.Module.MinimalAxioms
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Topolo... | Mathlib/Topology/ContinuousFunction/Bounded.lean | 158 | 162 | theorem dist_set_exists : ∃ C, 0 ≤ C ∧ ∀ x : α, dist (f x) (g x) ≤ C := by |
rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self
| 4 | 54.59815 | 2 | 2 | 1 | 1,936 |
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theory.covering.density_theorem from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filt... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 71 | 109 | theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
(rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x := by |
let R := scalingScaleOf μ (max (4 * K + 3) 3)
simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,
isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
measurableSet_closedBall]
/- The measure is doubling on scales smaller than ... | 37 | 11,719,142,372,802,612 | 2 | 2 | 2 | 1,937 |
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theory.covering.density_theorem from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filt... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 112 | 132 | theorem tendsto_closedBall_filterAt {K : ℝ} {x : α} {ι : Type*} {l : Filter ι} (w : ι → α)
(δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0)) (xmem : ∀ᶠ j in l, x ∈ closedBall (w j) (K * δ j)) :
Tendsto (fun j => closedBall (w j) (δ j)) l ((vitaliFamily μ K).filterAt x) := by |
refine (vitaliFamily μ K).tendsto_filterAt_iff.mpr ⟨?_, fun ε hε => ?_⟩
· filter_upwards [xmem, δlim self_mem_nhdsWithin] with j hj h'j
exact closedBall_mem_vitaliFamily_of_dist_le_mul μ hj h'j
· rcases l.eq_or_neBot with rfl | h
· simp
have hK : 0 ≤ K := by
rcases (xmem.and (δlim self_mem_nhds... | 18 | 65,659,969.137331 | 2 | 2 | 2 | 1,937 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Dynamics.FixedPoints.Basic
#align_import data.set.pointwise.iterate from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Pointwise
open Set Function
@[to_additive
"Let `n : ℤ`... | Mathlib/Data/Set/Pointwise/Iterate.lean | 31 | 42 | theorem smul_eq_self_of_preimage_zpow_eq_self {G : Type*} [CommGroup G] {n : ℤ} {s : Set G}
(hs : (fun x => x ^ n) ⁻¹' s = s) {g : G} {j : ℕ} (hg : g ^ n ^ j = 1) : g • s = s := by |
suffices ∀ {g' : G} (_ : g' ^ n ^ j = 1), g' • s ⊆ s by
refine le_antisymm (this hg) ?_
conv_lhs => rw [← smul_inv_smul g s]
replace hg : g⁻¹ ^ n ^ j = 1 := by rw [inv_zpow, hg, inv_one]
simpa only [le_eq_subset, set_smul_subset_set_smul_iff] using this hg
rw [(IsFixedPt.preimage_iterate hs j : (zp... | 10 | 22,026.465795 | 2 | 2 | 1 | 1,938 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Symmetric
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Algebra.DirectSum.Decomposition
#align_import analysis.inner_product_space.proje... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 70 | 177 | theorem exists_norm_eq_iInf_of_complete_convex {K : Set F} (ne : K.Nonempty) (h₁ : IsComplete K)
(h₂ : Convex ℝ K) : ∀ u : F, ∃ v ∈ K, ‖u - v‖ = ⨅ w : K, ‖u - w‖ := fun u => by
let δ := ⨅ w : K, ‖u - w‖
letI : Nonempty K := ne.to_subtype
have zero_le_δ : 0 ≤ δ := le_ciInf fun _ => norm_nonneg _
have δ_le : ... |
have hδ : ∀ n : ℕ, δ < δ + 1 / (n + 1) := fun n =>
lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat
have h := fun n => exists_lt_of_ciInf_lt (hδ n)
let w : ℕ → K := fun n => Classical.choose (h n)
exact ⟨w, fun n => Classical.choose_spec (h n)⟩
rcases exists_seq with ⟨w, hw⟩
have norm_tendst... | 97 | 1,338,334,719,204,269,500,000,000,000,000,000,000,000,000 | 2 | 2 | 1 | 1,939 |
import Mathlib.RingTheory.Finiteness
import Mathlib.Logic.Equiv.TransferInstance
universe u v w
open Function
variable (R : Type u) [Semiring R]
@[mk_iff]
class OrzechProperty : Prop where
injective_of_surjective_of_submodule' : ∀ {M : Type u} [AddCommMonoid M] [Module R M]
[Module.Finite R M] {N : Submod... | Mathlib/RingTheory/OrzechProperty.lean | 69 | 82 | theorem injective_of_surjective_of_injective
{N : Type w} [AddCommMonoid N] [Module R N]
(i f : N →ₗ[R] M) (hi : Injective i) (hf : Surjective f) : Injective f := by |
obtain ⟨n, g, hg⟩ := Module.Finite.exists_fin' R M
haveI := small_of_surjective hg
letI := Equiv.addCommMonoid (equivShrink M).symm
letI := Equiv.module R (equivShrink M).symm
let j : Shrink.{u} M ≃ₗ[R] M := Equiv.linearEquiv R (equivShrink M).symm
haveI := Module.Finite.equiv j.symm
let i' := j.symm.toL... | 11 | 59,874.141715 | 2 | 2 | 1 | 1,940 |
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-community/mathlib"@"cca40788df1b8755d5baf17ab2f27dacc2e17acb"
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 56 | 67 | theorem Gamma_integrand_isLittleO (s : ℝ) :
(fun x : ℝ => exp (-x) * x ^ s) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
refine isLittleO_of_tendsto (fun x hx => ?_) ?_
· exfalso; exact (exp_pos (-(1 / 2) * x)).ne' hx
have : (fun x : ℝ => exp (-x) * x ^ s / exp (-(1 / 2) * x)) =
(fun x : ℝ => exp (1 / 2 * x) / x ^ s)⁻¹ := by
ext1 x
field_simp [exp_ne_zero, exp_neg, ← Real.exp_add]
left
ring
rw [this]
exac... | 10 | 22,026.465795 | 2 | 2 | 2 | 1,941 |
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-community/mathlib"@"cca40788df1b8755d5baf17ab2f27dacc2e17acb"
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 71 | 82 | theorem GammaIntegral_convergent {s : ℝ} (h : 0 < s) :
IntegrableOn (fun x : ℝ => exp (-x) * x ^ (s - 1)) (Ioi 0) := by |
rw [← Ioc_union_Ioi_eq_Ioi (@zero_le_one ℝ _ _ _ _), integrableOn_union]
constructor
· rw [← integrableOn_Icc_iff_integrableOn_Ioc]
refine IntegrableOn.continuousOn_mul continuousOn_id.neg.rexp ?_ isCompact_Icc
refine (intervalIntegrable_iff_integrableOn_Icc_of_le zero_le_one).mp ?_
exact intervalInt... | 10 | 22,026.465795 | 2 | 2 | 2 | 1,941 |
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.PreservesLocallyBijective
import Mathlib.CategoryTheory.Sites.Whiskering
universe u
namespace CategoryTheory
open Functor Limits GrothendieckTopology
variable {C : Type*} [C... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 51 | 65 | theorem locallyCoverDense : LocallyCoverDense J e.inverse := by |
intro X T
convert T.prop
ext Z f
constructor
· rintro ⟨_, _, g', hg, rfl⟩
exact T.val.downward_closed hg g'
· intro hf
refine ⟨e.functor.obj Z, (Adjunction.homEquiv e.toAdjunction _ _).symm f, e.unit.app Z, ?_, ?_⟩
· simp only [Adjunction.homEquiv_counit, Functor.id_obj, Equivalence.toAdjunctio... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 1,942 |
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.PreservesLocallyBijective
import Mathlib.CategoryTheory.Sites.Whiskering
universe u
namespace CategoryTheory
open Functor Limits GrothendieckTopology
variable {C : Type*} [C... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 67 | 82 | theorem coverPreserving : CoverPreserving J (e.locallyCoverDense J).inducedTopology e.functor where
cover_preserve {U S} h := by |
change _ ∈ J.sieves (e.inverse.obj (e.functor.obj U))
convert J.pullback_stable (e.unitInv.app U) h
ext Z f
rw [← Sieve.functorPushforward_comp]
simp only [Sieve.functorPushforward_apply, Presieve.functorPushforward, exists_and_left, id_obj,
comp_obj, Sieve.pullback_apply]
constructor
... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 1,942 |
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Tactic.Generalize
#align_import analysis.box_integral.integrability from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open scoped Classical NNReal ENNReal Topology
universe u v
... | Mathlib/Analysis/BoxIntegral/Integrability.lean | 39 | 99 | theorem hasIntegralIndicatorConst (l : IntegrationParams) (hl : l.bRiemann = false)
{s : Set (ι → ℝ)} (hs : MeasurableSet s) (I : Box ι) (y : E) (μ : Measure (ι → ℝ))
[IsLocallyFiniteMeasure μ] :
HasIntegral.{u, v, v} I l (s.indicator fun _ => y) μ.toBoxAdditive.toSMul
((μ (s ∩ I)).toReal • y) := by |
refine HasIntegral.of_mul ‖y‖ fun ε ε0 => ?_
lift ε to ℝ≥0 using ε0.le; rw [NNReal.coe_pos] at ε0
/- First we choose a closed set `F ⊆ s ∩ I.Icc` and an open set `U ⊇ s` such that
both `(s ∩ I.Icc) \ F` and `U \ s` have measure less than `ε`. -/
have A : μ (s ∩ Box.Icc I) ≠ ∞ :=
((measure_mono Set.inte... | 56 | 2,091,659,496,012,996,000,000,000 | 2 | 2 | 2 | 1,943 |
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Tactic.Generalize
#align_import analysis.box_integral.integrability from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open scoped Classical NNReal ENNReal Topology
universe u v
... | Mathlib/Analysis/BoxIntegral/Integrability.lean | 104 | 155 | theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
{μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
(hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by |
/- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
refine hasIntegral_iff.2 fun ε ε0 => ?_
lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
rcases N... | 49 | 1,907,346,572,495,099,800,000 | 2 | 2 | 2 | 1,943 |
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.extend_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Filter Set Metric Contin... | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 37 | 106 | theorem has_fderiv_at_boundary_of_tendsto_fderiv {f : E → F} {s : Set E} {x : E} {f' : E →L[ℝ] F}
(f_diff : DifferentiableOn ℝ f s) (s_conv : Convex ℝ s) (s_open : IsOpen s)
(f_cont : ∀ y ∈ closure s, ContinuousWithinAt f s y)
(h : Tendsto (fun y => fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
HasFDerivWithinAt f... |
classical
-- one can assume without loss of generality that `x` belongs to the closure of `s`, as the
-- statement is empty otherwise
by_cases hx : x ∉ closure s
· rw [← closure_closure] at hx; exact hasFDerivWithinAt_of_nmem_closure hx
push_neg at hx
rw [HasFDerivWithinAt, hasFDerivAtFilter_... | 65 | 16,948,892,444,103,336,000,000,000,000 | 2 | 2 | 2 | 1,944 |
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.extend_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Filter Set Metric Contin... | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 111 | 140 | theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
(f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
(f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by |
/- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
obtain ⟨b, ab : a < b, ... | 27 | 532,048,240,601.79865 | 2 | 2 | 2 | 1,944 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 72 | 81 | theorem affineIndependent_iff_of_fintype [Fintype ι] (p : ι → P) :
AffineIndependent k p ↔
∀ w : ι → k, ∑ i, w i = 0 → Finset.univ.weightedVSub p w = (0 : V) → ∀ i, w i = 0 := by |
constructor
· exact fun h w hw hs i => h Finset.univ w hw hs i (Finset.mem_univ _)
· intro h s w hw hs i hi
rw [Finset.weightedVSub_indicator_subset _ _ (Finset.subset_univ s)] at hs
rw [← Finset.sum_indicator_subset _ (Finset.subset_univ s)] at hw
replace h := h ((↑s : Set ι).indicator w) hw hs i
... | 7 | 1,096.633158 | 2 | 2 | 3 | 1,945 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 86 | 134 | theorem affineIndependent_iff_linearIndependent_vsub (p : ι → P) (i1 : ι) :
AffineIndependent k p ↔ LinearIndependent k fun i : { x // x ≠ i1 } => (p i -ᵥ p i1 : V) := by |
classical
constructor
· intro h
rw [linearIndependent_iff']
intro s g hg i hi
set f : ι → k := fun x => if hx : x = i1 then -∑ y ∈ s, g y else g ⟨x, hx⟩ with hfdef
let s2 : Finset ι := insert i1 (s.map (Embedding.subtype _))
have hfg : ∀ x : { x // x ≠ i1 }, g x = f x := by
... | 47 | 258,131,288,619,006,750,000 | 2 | 2 | 3 | 1,945 |
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 139 | 158 | theorem affineIndependent_set_iff_linearIndependent_vsub {s : Set P} {p₁ : P} (hp₁ : p₁ ∈ s) :
AffineIndependent k (fun p => p : s → P) ↔
LinearIndependent k (fun v => v : (fun p => (p -ᵥ p₁ : V)) '' (s \ {p₁}) → V) := by |
rw [affineIndependent_iff_linearIndependent_vsub k (fun p => p : s → P) ⟨p₁, hp₁⟩]
constructor
· intro h
have hv : ∀ v : (fun p => (p -ᵥ p₁ : V)) '' (s \ {p₁}), (v : V) +ᵥ p₁ ∈ s \ {p₁} := fun v =>
(vsub_left_injective p₁).mem_set_image.1 ((vadd_vsub (v : V) p₁).symm ▸ v.property)
let f : (fun p : ... | 17 | 24,154,952.753575 | 2 | 2 | 3 | 1,945 |
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Fintype.Basic
#align_import data.fintype.pi from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*}
open Finset
namespace Fintype
variable [DecidableEq α] [Fintype α] {γ δ : α → Type*} {s : ∀ a, Finset (γ a)}
def pi... | Mathlib/Data/Fintype/Pi.lean | 34 | 42 | theorem mem_piFinset {t : ∀ a, Finset (δ a)} {f : ∀ a, δ a} : f ∈ piFinset t ↔ ∀ a, f a ∈ t a := by |
constructor
· simp only [piFinset, mem_map, and_imp, forall_prop_of_true, exists_prop, mem_univ, exists_imp,
mem_pi]
rintro g hg hgf a
rw [← hgf]
exact hg a
· simp only [piFinset, mem_map, forall_prop_of_true, exists_prop, mem_univ, mem_pi]
exact fun hf => ⟨fun a _ => f a, hf, rfl⟩
| 8 | 2,980.957987 | 2 | 2 | 1 | 1,946 |
import Mathlib.Data.List.Lex
import Mathlib.Data.Char
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algebra.Order.Group.Nat
#align_import data.string.basic from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
namespace String
def ltb (s₁ s₂ : Iterator) : Bool :=
if s₂.hasNext th... | Mathlib/Data/String/Basic.lean | 60 | 74 | theorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos) :
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ = ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ := by |
apply ltb.inductionOn ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ (motive := fun ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩ ↦
ltb ⟨⟨c :: cs₁⟩, i₁ + c⟩ ⟨⟨c :: cs₂⟩, i₂ + c⟩ =
ltb ⟨⟨cs₁⟩, i₁⟩ ⟨⟨cs₂⟩, i₂⟩) <;> simp only <;>
intro ⟨cs₁⟩ ⟨cs₂⟩ i₁ i₂ <;>
intros <;>
(conv => lhs; unfold ltb) <;> (conv => rhs; unfold ltb) <;>
simp only [Iterator.... | 13 | 442,413.392009 | 2 | 2 | 1 | 1,947 |
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tactic.RSuffices
#align_import ring_theory.artinian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
open Set Filter Po... | Mathlib/RingTheory/Artinian.lean | 175 | 195 | theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [IsArtinian R M] {s : Set M}
(hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by |
refine by_contradiction fun hf => (RelEmbedding.wellFounded_iff_no_descending_seq.1
(wellFounded_submodule_lt (R := R) (M := M))).elim' ?_
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
have : ∀ n, (↑) ∘ f '' { m | n ≤ m } ⊆ s := by
rintro n x ⟨y, _, rfl⟩
exact (f y).2
have : ∀ a b : ℕ, a ≤ b ↔
... | 19 | 178,482,300.963187 | 2 | 2 | 1 | 1,948 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.ZMod.Basic
open Finset Nat
namespace ZMod
| Mathlib/Data/ZMod/Factorial.lean | 31 | 42 | theorem cast_descFactorial {n p : ℕ} (h : n ≤ p) :
(descFactorial (p - 1) n : ZMod p) = (-1) ^ n * n ! := by |
rw [descFactorial_eq_prod_range, ← prod_range_add_one_eq_factorial]
simp only [cast_prod]
nth_rw 2 [← card_range n]
rw [pow_card_mul_prod]
refine prod_congr rfl ?_
intro x hx
rw [← tsub_add_eq_tsub_tsub_swap,
Nat.cast_sub <| Nat.le_trans (Nat.add_one_le_iff.mpr (List.mem_range.mp hx)) h,
CharP.ca... | 10 | 22,026.465795 | 2 | 2 | 1 | 1,949 |
import Mathlib.Algebra.Homology.HomologicalComplex
import Mathlib.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Abelian.Basic
#align_import algebraic_topology.Moore_complex from "leanprover-community/mathlib"@"0bd2ea37bcba5769e14866170f251c9bc64e35d7"
universe v u
noncomputable section
open C... | Mathlib/AlgebraicTopology/MooreComplex.lean | 100 | 111 | theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by |
-- It's a pity we need to do a case split here;
-- after the first erw the proofs are almost identical
rcases n with _ | n <;> dsimp [objD]
· erw [Subobject.factorThru_arrow_assoc, Category.assoc,
← X.δ_comp_δ_assoc (Fin.zero_le (0 : Fin 2)),
← factorThru_arrow _ _ (finset_inf_arrow_factors Finse... | 11 | 59,874.141715 | 2 | 2 | 1 | 1,950 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 61 | 77 | theorem multiplicity_eq_card_pow_dvd {m n b : ℕ} (hm : m ≠ 1) (hn : 0 < n) (hb : log m n < b) :
multiplicity m n = ↑((Finset.Ico 1 b).filter fun i => m ^ i ∣ n).card :=
calc
multiplicity m n = ↑(Ico 1 <| (multiplicity m n).get (finite_nat_iff.2 ⟨hm, hn⟩) + 1).card := by |
simp
_ = ↑((Finset.Ico 1 b).filter fun i => m ^ i ∣ n).card :=
congr_arg _ <|
congr_arg card <|
Finset.ext fun i => by
rw [mem_filter, mem_Ico, mem_Ico, Nat.lt_succ_iff, ← @PartENat.coe_le_coe i,
PartENat.natCast_get, ← pow_dvd_iff_le_multiplicity, and_right_... | 13 | 442,413.392009 | 2 | 2 | 3 | 1,951 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 108 | 123 | theorem multiplicity_factorial {p : ℕ} (hp : p.Prime) :
∀ {n b : ℕ}, log p n < b → multiplicity p n ! = (∑ i ∈ Ico 1 b, n / p ^ i : ℕ)
| 0, b, _ => by simp [Ico, hp.multiplicity_one]
| n + 1, b, hb =>
calc
multiplicity p (n + 1)! = multiplicity p n ! + multiplicity p (n + 1) := by |
rw [factorial_succ, hp.multiplicity_mul, add_comm]
_ = (∑ i ∈ Ico 1 b, n / p ^ i : ℕ) +
((Finset.Ico 1 b).filter fun i => p ^ i ∣ n + 1).card := by
rw [multiplicity_factorial hp ((log_mono_right <| le_succ _).trans_lt hb), ←
multiplicity_eq_card_pow_dvd hp.ne_one (succ_pos _... | 10 | 22,026.465795 | 2 | 2 | 3 | 1,951 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Prime
import Mathlib.Data.Nat.Digits
import Mathlib.RingTheory.Multiplicity
#align_import data.nat.multiplicity from "l... | Mathlib/Data/Nat/Multiplicity.lean | 138 | 158 | theorem multiplicity_factorial_mul_succ {n p : ℕ} (hp : p.Prime) :
multiplicity p (p * (n + 1))! = multiplicity p (p * n)! + multiplicity p (n + 1) + 1 := by |
have hp' := hp.prime
have h0 : 2 ≤ p := hp.two_le
have h1 : 1 ≤ p * n + 1 := Nat.le_add_left _ _
have h2 : p * n + 1 ≤ p * (n + 1) := by linarith
have h3 : p * n + 1 ≤ p * (n + 1) + 1 := by omega
have hm : multiplicity p (p * n)! ≠ ⊤ := by
rw [Ne, eq_top_iff_not_finite, Classical.not_not, finite_nat_if... | 19 | 178,482,300.963187 | 2 | 2 | 3 | 1,951 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Sub... | Mathlib/RingTheory/Localization/Ideal.lean | 53 | 64 | theorem mem_map_algebraMap_iff {I : Ideal R} {z} : z ∈ Ideal.map (algebraMap R S) I ↔
∃ x : I × M, z * algebraMap R S x.2 = algebraMap R S x.1 := by |
constructor
· change _ → z ∈ map_ideal M S I
refine fun h => Ideal.mem_sInf.1 h fun z hz => ?_
obtain ⟨y, hy⟩ := hz
let Z : { x // x ∈ I } := ⟨y, hy.left⟩
use ⟨Z, 1⟩
simp [hy.right]
· rintro ⟨⟨a, s⟩, h⟩
rw [← Ideal.unit_mul_mem_iff_mem _ (map_units S s), mul_comm]
exact h.symm ▸ Ideal... | 10 | 22,026.465795 | 2 | 2 | 4 | 1,952 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Sub... | Mathlib/RingTheory/Localization/Ideal.lean | 78 | 89 | theorem comap_map_of_isPrime_disjoint (I : Ideal R) (hI : I.IsPrime) (hM : Disjoint (M : Set R) I) :
Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) = I := by |
refine le_antisymm ?_ Ideal.le_comap_map
refine (fun a ha => ?_)
obtain ⟨⟨b, s⟩, h⟩ := (mem_map_algebraMap_iff M S).1 (Ideal.mem_comap.1 ha)
replace h : algebraMap R S (s * a) = algebraMap R S b := by
simpa only [← map_mul, mul_comm] using h
obtain ⟨c, hc⟩ := (eq_iff_exists M S).1 h
have : ↑c * ↑s * a ... | 10 | 22,026.465795 | 2 | 2 | 4 | 1,952 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Sub... | Mathlib/RingTheory/Localization/Ideal.lean | 108 | 132 | theorem isPrime_iff_isPrime_disjoint (J : Ideal S) :
J.IsPrime ↔
(Ideal.comap (algebraMap R S) J).IsPrime ∧
Disjoint (M : Set R) ↑(Ideal.comap (algebraMap R S) J) := by |
constructor
· refine fun h =>
⟨⟨?_, ?_⟩,
Set.disjoint_left.mpr fun m hm1 hm2 =>
h.ne_top (Ideal.eq_top_of_isUnit_mem _ hm2 (map_units S ⟨m, hm1⟩))⟩
· refine fun hJ => h.ne_top ?_
rw [eq_top_iff, ← (orderEmbedding M S).le_iff_le]
exact le_of_eq hJ.symm
· intro x y hxy
... | 21 | 1,318,815,734.483215 | 2 | 2 | 4 | 1,952 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommRing
variable {R : Type*} [CommRing R] (M : Submonoid R... | Mathlib/RingTheory/Localization/Ideal.lean | 171 | 204 | theorem surjective_quotientMap_of_maximal_of_localization {I : Ideal S} [I.IsPrime] {J : Ideal R}
{H : J ≤ I.comap (algebraMap R S)} (hI : (I.comap (algebraMap R S)).IsMaximal) :
Function.Surjective (Ideal.quotientMap I (algebraMap R S) H) := by |
intro s
obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective s
obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M s
by_cases hM : (Ideal.Quotient.mk (I.comap (algebraMap R S))) m = 0
· have : I = ⊤ := by
rw [Ideal.eq_top_iff_one]
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_comap] at hM
convert I.mul... | 31 | 29,048,849,665,247.426 | 2 | 2 | 4 | 1,952 |
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.Category... | Mathlib/CategoryTheory/Limits/Final.lean | 386 | 404 | theorem zigzag_of_eqvGen_quot_rel {F : C ⥤ D} {d : D} {f₁ f₂ : ΣX, d ⟶ F.obj X}
(t : EqvGen (Types.Quot.Rel.{v, v} (F ⋙ coyoneda.obj (op d))) f₁ f₂) :
Zigzag (StructuredArrow.mk f₁.2) (StructuredArrow.mk f₂.2) := by |
induction t with
| rel x y r =>
obtain ⟨f, w⟩ := r
fconstructor
swap
· fconstructor
left; fconstructor
exact StructuredArrow.homMk f
| refl => fconstructor
| symm x y _ ih =>
apply zigzag_symmetric
exact ih
| trans x y z _ _ ih₁ ih₂ =>
apply Relation.ReflTransGen.trans
... | 16 | 8,886,110.520508 | 2 | 2 | 1 | 1,953 |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Nilpotent
open scoped Classical Polynomial
open Polynomial
noncomputable section
| Mathlib/RingTheory/Polynomial/IrreducibleRing.lean | 37 | 61 | theorem Polynomial.Monic.irreducible_of_irreducible_map_of_isPrime_nilradical
{R S : Type*} [CommRing R] [(nilradical R).IsPrime] [CommRing S] [IsDomain S]
(φ : R →+* S) (f : R[X]) (hm : f.Monic) (hi : Irreducible (f.map φ)) : Irreducible f := by |
let R' := R ⧸ nilradical R
let ψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ
(haveI := RingHom.ker_isPrime φ; nilradical_le_prime (RingHom.ker φ))
let ι := algebraMap R R'
rw [show φ = ψ.comp ι from rfl, ← map_map] at hi
replace hi := hm.map ι |>.irreducible_of_irreducible_map _ _ hi
refine ⟨fun... | 22 | 3,584,912,846.131591 | 2 | 2 | 1 | 1,954 |
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 50 | 54 | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by |
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| 4 | 54.59815 | 2 | 2 | 2 | 1,955 |
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 57 | 61 | theorem turnBound_ne_zero_of_right_move {s t : S} (m : t ∈ r s) : turnBound s ≠ 0 := by |
intro h
have t := right_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| 4 | 54.59815 | 2 | 2 | 2 | 1,955 |
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology E... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 139 | 169 | theorem ofFunction_union_of_top_of_nonempty_inter {s t : Set α}
(h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ∞) :
OuterMeasure.ofFunction m m_empty (s ∪ t) =
OuterMeasure.ofFunction m m_empty s + OuterMeasure.ofFunction m m_empty t := by |
refine le_antisymm (measure_union_le _ _) (le_iInf₂ fun f hf ↦ ?_)
set μ := OuterMeasure.ofFunction m m_empty
rcases Classical.em (∃ i, (s ∩ f i).Nonempty ∧ (t ∩ f i).Nonempty) with (⟨i, hs, ht⟩ | he)
· calc
μ s + μ t ≤ ∞ := le_top
_ = m (f i) := (h (f i) hs ht).symm
_ ≤ ∑' i, m (f i) := ENNR... | 25 | 72,004,899,337.38586 | 2 | 2 | 1 | 1,956 |
import Mathlib.Topology.UniformSpace.AbsoluteValue
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.UniformSpace.Completion
#align_import topology.uniform_space.compare_reals from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
open Set... | Mathlib/Topology/UniformSpace/CompareReals.lean | 60 | 65 | theorem Rat.uniformSpace_eq :
(AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by |
ext s
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
simp only [Rat.dist_eq, AbsoluteValue.abs_apply, ← Rat.cast_sub, ← Rat.cast_abs, Rat.cast_lt,
abs_sub_comm]
| 4 | 54.59815 | 2 | 2 | 1 | 1,957 |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 58 | 79 | theorem isUnit_res_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : IsUnit (X.presheaf.germ x f)) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), IsUnit (X.presheaf.map i.op f) := by |
obtain ⟨g', heq⟩ := h.exists_right_inv
obtain ⟨V, hxV, g, rfl⟩ := X.presheaf.germ_exist x.1 g'
let W := U ⊓ V
have hxW : x.1 ∈ W := ⟨x.2, hxV⟩
-- Porting note: `erw` can't write into `HEq`, so this is replaced with another `HEq` in the
-- desired form
replace heq : (X.presheaf.germ ⟨x.val, hxW⟩) ((X.pres... | 19 | 178,482,300.963187 | 2 | 2 | 2 | 1,958 |
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 84 | 125 | theorem isUnit_of_isUnit_germ (U : Opens X) (f : X.presheaf.obj (op U))
(h : ∀ x : U, IsUnit (X.presheaf.germ x f)) : IsUnit f := by |
-- We pick a cover of `U` by open sets `V x`, such that `f` is a unit on each `V x`.
choose V iVU m h_unit using fun x : U => X.isUnit_res_of_isUnit_germ U f x (h x)
have hcover : U ≤ iSup V := by
intro x hxU
-- Porting note: in Lean3 `rw` is sufficient
erw [Opens.mem_iSup]
exact ⟨⟨x, hxU⟩, m ⟨x,... | 40 | 235,385,266,837,019,970 | 2 | 2 | 2 | 1,958 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Basis
#align_import ring_theory.algebra_tower from "leanprover-community/mathlib"@"94825b2b0b982306be14d891c4f063a1eca4f370"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type ... | Mathlib/RingTheory/AlgebraTower.lean | 108 | 121 | theorem linearIndependent_smul {ι : Type v₁} {b : ι → S} {ι' : Type w₁} {c : ι' → A}
(hb : LinearIndependent R b) (hc : LinearIndependent S c) :
LinearIndependent R fun p : ι × ι' => b p.1 • c p.2 := by |
rw [linearIndependent_iff'] at hb hc; rw [linearIndependent_iff'']; rintro s g hg hsg ⟨i, k⟩
by_cases hik : (i, k) ∈ s
· have h1 : ∑ i ∈ s.image Prod.fst ×ˢ s.image Prod.snd, g i • b i.1 • c i.2 = 0 := by
rw [← hsg]
exact
(Finset.sum_subset Finset.subset_product fun p _ hp =>
show... | 11 | 59,874.141715 | 2 | 2 | 1 | 1,959 |
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 47 | 56 | theorem eq_condKernel_of_measure_eq_compProd' (κ : kernel α Ω) [IsSFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s) :
∀ᵐ x ∂ρ.fst, κ x s = ρ.condKernel x s := by |
refine ae_eq_of_forall_set_lintegral_eq_of_sigmaFinite
(kernel.measurable_coe κ hs) (kernel.measurable_coe ρ.condKernel hs) (fun t ht _ ↦ ?_)
conv_rhs => rw [Measure.set_lintegral_condKernel_eq_measure_prod ht hs, hκ]
simp only [Measure.compProd_apply (ht.prod hs), Set.mem_prod, ← lintegral_indicator _ ht]
... | 7 | 1,096.633158 | 2 | 2 | 2 | 1,960 |
import Mathlib.Probability.Kernel.Disintegration.Integral
open MeasureTheory Set Filter MeasurableSpace
open scoped ENNReal MeasureTheory Topology ProbabilityTheory
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] ... | Mathlib/Probability/Kernel/Disintegration/Unique.lean | 81 | 124 | theorem eq_condKernel_of_measure_eq_compProd (κ : kernel α Ω) [IsFiniteKernel κ]
(hκ : ρ = ρ.fst ⊗ₘ κ) :
∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x := by |
-- The idea is to transport the question to `ℝ` from `Ω` using `embeddingReal`
-- and then construct a measure on `α × ℝ`
let f := embeddingReal Ω
have hf := measurableEmbedding_embeddingReal Ω
set ρ' : Measure (α × ℝ) := ρ.map (Prod.map id f) with hρ'def
have hρ' : ρ'.fst = ρ.fst := by
ext s hs
rw... | 41 | 639,843,493,530,055,000 | 2 | 2 | 2 | 1,960 |
import Mathlib.Algebra.Homology.Exact
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Preserves.Finite
#align_import category_theory.preadditive.projective from "leanprover-community/mathlib"@"3974a774a707e2e06046a14c0eaef4654... | Mathlib/CategoryTheory/Preadditive/Projective.lean | 208 | 214 | theorem map_projective (adj : F ⊣ G) [G.PreservesEpimorphisms] (P : C) (hP : Projective P) :
Projective (F.obj P) where
factors f g _ := by |
rcases hP.factors (adj.unit.app P ≫ G.map f) (G.map g) with ⟨f', hf'⟩
use F.map f' ≫ adj.counit.app _
rw [Category.assoc, ← Adjunction.counit_naturality, ← Category.assoc, ← F.map_comp, hf']
simp
| 4 | 54.59815 | 2 | 2 | 1 | 1,961 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
| Mathlib/Data/List/NodupEquivFin.lean | 116 | 137 | theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
(hf : ∀ ix : ℕ, l.get? ix = l'.get? (f ix)) : l <+ l' := by |
induction' l with hd tl IH generalizing l' f
· simp
have : some hd = _ := hf 0
rw [eq_comm, List.get?_eq_some] at this
obtain ⟨w, h⟩ := this
let f' : ℕ ↪o ℕ :=
OrderEmbedding.ofMapLEIff (fun i => f (i + 1) - (f 0 + 1)) fun a b => by
dsimp only
rw [Nat.sub_le_sub_iff_right, OrderEmbedding.le... | 20 | 485,165,195.40979 | 2 | 2 | 4 | 1,962 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
... | Mathlib/Data/List/NodupEquivFin.lean | 144 | 161 | theorem sublist_iff_exists_orderEmbedding_get?_eq {l l' : List α} :
l <+ l' ↔ ∃ f : ℕ ↪o ℕ, ∀ ix : ℕ, l.get? ix = l'.get? (f ix) := by |
constructor
· intro H
induction' H with xs ys y _H IH xs ys x _H IH
· simp
· obtain ⟨f, hf⟩ := IH
refine ⟨f.trans (OrderEmbedding.ofStrictMono (· + 1) fun _ => by simp), ?_⟩
simpa using hf
· obtain ⟨f, hf⟩ := IH
refine
⟨OrderEmbedding.ofMapLEIff (fun ix : ℕ => if ix = 0 th... | 16 | 8,886,110.520508 | 2 | 2 | 4 | 1,962 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
... | Mathlib/Data/List/NodupEquivFin.lean | 168 | 205 | theorem sublist_iff_exists_fin_orderEmbedding_get_eq {l l' : List α} :
l <+ l' ↔
∃ f : Fin l.length ↪o Fin l'.length,
∀ ix : Fin l.length, l.get ix = l'.get (f ix) := by |
rw [sublist_iff_exists_orderEmbedding_get?_eq]
constructor
· rintro ⟨f, hf⟩
have h : ∀ {i : ℕ}, i < l.length → f i < l'.length := by
intro i hi
specialize hf i
rw [get?_eq_get hi, eq_comm, get?_eq_some] at hf
obtain ⟨h, -⟩ := hf
exact h
refine ⟨OrderEmbedding.ofMapLEIff (fun... | 34 | 583,461,742,527,454.9 | 2 | 2 | 4 | 1,962 |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
... | Mathlib/Data/List/NodupEquivFin.lean | 211 | 232 | theorem duplicate_iff_exists_distinct_get {l : List α} {x : α} :
l.Duplicate x ↔
∃ (n m : Fin l.length) (_ : n < m),
x = l.get n ∧ x = l.get m := by |
classical
rw [duplicate_iff_two_le_count, le_count_iff_replicate_sublist,
sublist_iff_exists_fin_orderEmbedding_get_eq]
constructor
· rintro ⟨f, hf⟩
refine ⟨f ⟨0, by simp⟩, f ⟨1, by simp⟩, f.lt_iff_lt.2 (Nat.zero_lt_one), ?_⟩
rw [← hf, ← hf]; simp
· rintro ⟨n, m, hnm, h, h'⟩
r... | 18 | 65,659,969.137331 | 2 | 2 | 4 | 1,962 |
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.Analysis.Normed.MulAction
#align_import analysis.normed_space.continuous_linear_map from "leanprover-community/mathlib"@"fe18deda804e30c594e75a6e5fe0f7d14695289f"
open Metric ContinuousLinearMap
open Set Real
open NNReal
variable {𝕜 𝕜₂ E F G : Type*}... | Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean | 198 | 205 | theorem ContinuousLinearEquiv.homothety_inverse (a : ℝ) (ha : 0 < a) (f : E ≃ₛₗ[σ] F) :
(∀ x : E, ‖f x‖ = a * ‖x‖) → ∀ y : F, ‖f.symm y‖ = a⁻¹ * ‖y‖ := by |
intro hf y
calc
‖f.symm y‖ = a⁻¹ * (a * ‖f.symm y‖) := by
rw [← mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]
_ = a⁻¹ * ‖f (f.symm y)‖ := by rw [hf]
_ = a⁻¹ * ‖y‖ := by simp
| 6 | 403.428793 | 2 | 2 | 1 | 1,963 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
#align_import group_theory.subgroup.saturated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
namespace Subgroup
variable {G : Type*} [Group G]
@[to_additive
"An additive subgroup `H` of `G` is *... | Mathlib/GroupTheory/Subgroup/Saturated.lean | 42 | 56 | theorem saturated_iff_zpow {H : Subgroup G} :
Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := by |
constructor
· intros hH n g hgn
induction' n with n n
· simp only [Int.natCast_eq_zero, Int.ofNat_eq_coe, zpow_natCast] at hgn ⊢
exact hH hgn
· suffices g ^ (n + 1) ∈ H by
refine (hH this).imp ?_ id
simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]
simpa only [inv_mem_iff, zp... | 13 | 442,413.392009 | 2 | 2 | 1 | 1,964 |
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a"
universe v u
namespace CategoryTheory
open scoped Classical
open CategoryTheory Category Limits Sieve
variable {C : Type u} [Category.{v} C]
na... | Mathlib/CategoryTheory/Sites/Canonical.lean | 61 | 113 | theorem isSheafFor_bind (P : Cᵒᵖ ⥤ Type v) (U : Sieve X) (B : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄, U f → Sieve Y)
(hU : Presieve.IsSheafFor P (U : Presieve X))
(hB : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (hf : U f), Presieve.IsSheafFor P (B hf : Presieve Y))
(hB' : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (h : U f) ⦃Z⦄ (g : Z ⟶ Y),
Presieve.IsSeparatedFor P (((... |
intro s hs
let y : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (hf : U f), Presieve.FamilyOfElements P (B hf : Presieve Y) :=
fun Y f hf Z g hg => s _ (Presieve.bind_comp _ _ hg)
have hy : ∀ ⦃Y⦄ ⦃f : Y ⟶ X⦄ (hf : U f), (y hf).Compatible := by
intro Y f H Y₁ Y₂ Z g₁ g₂ f₁ f₂ hf₁ hf₂ comm
apply hs
apply reassoc_of% comm
l... | 47 | 258,131,288,619,006,750,000 | 2 | 2 | 2 | 1,965 |
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