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 |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.AbsMax
#align_import analysis.complex.open_mapping from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
open Set Filter Metric Complex
open scoped Topology
vari... | Mathlib/Analysis/Complex/OpenMapping.lean | 44 | 70 | theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball z₀ r)) (hr : 0 < r)
(hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖) (hz₀ : ∃ᶠ z in 𝓝 z₀, f z ≠ f z₀) :
ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r := by |
/- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
the function `fun z ↦ ‖f z - v‖`: it is bounded below on the circle, and takes a small value
at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
hence that `v` is in t... | 24 | 26,489,122,129.84347 | 2 | 2 | 2 | 2,017 |
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.AbsMax
#align_import analysis.complex.open_mapping from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
open Set Filter Metric Complex
open scoped Topology
vari... | Mathlib/Analysis/Complex/OpenMapping.lean | 77 | 106 | theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt ℂ f z₀) :
(∀ᶠ z in 𝓝 z₀, f z = f z₀) ∨ 𝓝 (f z₀) ≤ map f (𝓝 z₀) := by |
/- The function `f` is analytic in a neighborhood of `z₀`; by the isolated zeros principle, if `f`
is not constant in a neighborhood of `z₀`, then it is nonzero, and therefore bounded below, on
every small enough circle around `z₀` and then `DiffContOnCl.ball_subset_image_closedBall`
provides an explicit... | 28 | 1,446,257,064,291.475 | 2 | 2 | 2 | 2,017 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.preadditive.of_biproducts from "leanprover-community/mathlib"@"061ea99a5610cfc72c286aa930d3c1f47f74f3d0"
noncomputable section
universe v u
op... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 54 | 68 | theorem isUnital_leftAdd : EckmannHilton.IsUnital (· +ₗ ·) 0 := by |
have hr : ∀ f : X ⟶ Y, biprod.lift (0 : X ⟶ Y) f = f ≫ biprod.inr := by
intro f
ext
· aesop_cat
· simp [biprod.lift_fst, Category.assoc, biprod.inr_fst, comp_zero]
have hl : ∀ f : X ⟶ Y, biprod.lift f (0 : X ⟶ Y) = f ≫ biprod.inl := by
intro f
ext
· aesop_cat
· simp [biprod.lift_snd... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,018 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.preadditive.of_biproducts from "leanprover-community/mathlib"@"061ea99a5610cfc72c286aa930d3c1f47f74f3d0"
noncomputable section
universe v u
op... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 71 | 85 | theorem isUnital_rightAdd : EckmannHilton.IsUnital (· +ᵣ ·) 0 := by |
have h₂ : ∀ f : X ⟶ Y, biprod.desc (0 : X ⟶ Y) f = biprod.snd ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, BinaryBicone.inr_snd_assoc]
have h₁ : ∀ f : X ⟶ Y, biprod.desc f (0 : X ⟶ Y) = biprod.fst ≫ f := by
intro f
ext
· aesop_cat
· simp only [biprod.inr_desc, Bin... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,018 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.preadditive.of_biproducts from "leanprover-community/mathlib"@"061ea99a5610cfc72c286aa930d3c1f47f74f3d0"
noncomputable section
universe v u
op... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 88 | 96 | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by |
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp [diag]
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp [diag]
have h₁ : biprod.lift (f +ᵣ g) (h +ᵣ k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag := by
ext ... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,018 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 49 | 67 | theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_cosZeta x (?_ : 1 < re (2 * k))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_cos hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc,
... | 16 | 8,886,110.520508 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 76 | 97 | theorem sinZeta_two_mul_nat_add_one (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_sinZeta x (?_ : 1 < re (2 * k + 1))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_sin hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc]
... | 19 | 178,482,300.963187 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 100 | 110 | theorem cosZeta_two_mul_nat' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) / (2 * k) / Gammaℂ (2 * k) *
((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [cosZeta_two_mul_nat hk hx]
congr 1
have : (2 * k)! = (2 * k) * Complex.Gamma (2 * k) := by
rw [(by { norm_cast; omega } : 2 * (k : ℂ) = ↑(2 * k - 1) + 1), Complex.Gamma_nat_eq_factorial,
← Nat.cast_add_one, ← Nat.cast_mul, ← Nat.factorial_succ, Nat.sub_add_cancel (by omega)]
simp_rw [this, Gammaℂ... | 8 | 2,980.957987 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 113 | 124 | theorem sinZeta_two_mul_nat_add_one' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) / (2 * k + 1) / Gammaℂ (2 * k + 1) *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [sinZeta_two_mul_nat_add_one hk hx]
congr 1
have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by
rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)),
Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul,
← Nat.cast_add_one, ← Nat.cast_m... | 9 | 8,103.083928 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 126 | 146 | theorem hurwitzZetaEven_one_sub_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZetaEven x (1 - 2 * k) =
-1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
have h1 (n : ℕ) : (2 * k : ℂ) ≠ -n := by
rw [← Int.cast_ofNat, ← Int.cast_natCast, ← Int.cast_mul, ← Int.cast_natCast n, ← Int.cast_neg,
Ne, Int.cast_inj, ← Ne]
refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))
exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)
have... | 18 | 65,659,969.137331 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 211 | 217 | theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (2 * k)! := by |
convert congr_arg ((↑) : ℝ → ℂ) (hasSum_zeta_nat hk).tsum_eq
· rw [← Nat.cast_two, ← Nat.cast_mul, zeta_nat_eq_tsum_of_gt_one (by omega)]
simp only [push_cast]
· norm_cast
| 4 | 54.59815 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 220 | 224 | theorem riemannZeta_two : riemannZeta 2 = (π : ℂ) ^ 2 / 6 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
· rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp only [push_cast]
· norm_cast
| 4 | 54.59815 | 2 | 2 | 8 | 2,019 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 227 | 232 | theorem riemannZeta_four : riemannZeta 4 = π ^ 4 / 90 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_four.tsum_eq
· rw [← Nat.cast_one, show (4 : ℂ) = (4 : ℕ) by norm_num,
zeta_nat_eq_tsum_of_gt_one (by norm_num : 1 < 4)]
simp only [push_cast]
· norm_cast
| 5 | 148.413159 | 2 | 2 | 8 | 2,019 |
import Mathlib.CategoryTheory.Linear.LinearFunctor
import Mathlib.CategoryTheory.Monoidal.Preadditive
#align_import category_theory.monoidal.linear from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
namespace CategoryTheory
open CategoryTheory.Limits
open CategoryTheory.MonoidalCateg... | Mathlib/CategoryTheory/Monoidal/Linear.lean | 58 | 70 | theorem monoidalLinearOfFaithful {D : Type*} [Category D] [Preadditive D] [Linear R D]
[MonoidalCategory D] [MonoidalPreadditive D] (F : MonoidalFunctor D C) [F.Faithful]
[F.toFunctor.Additive] [F.toFunctor.Linear R] : MonoidalLinear R D :=
{ whiskerLeft_smul := by |
intros X Y Z r f
apply F.toFunctor.map_injective
rw [F.map_whiskerLeft]
simp
smul_whiskerRight := by
intros r X Y f Z
apply F.toFunctor.map_injective
rw [F.map_whiskerRight]
simp }
| 9 | 8,103.083928 | 2 | 2 | 1 | 2,020 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
suppress_compilation
open Bornology Metric Set Real
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special cases about linear (as o... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 70 | 86 | theorem tendsto_of_tendsto_pointwise_of_cauchySeq {f : ℕ → E' →SL[σ₁₂] F} {g : E' →SL[σ₁₂] F}
(hg : Tendsto (fun n x => f n x) atTop (𝓝 g)) (hf : CauchySeq f) : Tendsto f atTop (𝓝 g) := by |
/- Since `f` is a Cauchy sequence, there exists `b → 0` such that `‖f n - f m‖ ≤ b N` for any
`m, n ≥ N`. -/
rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, hb₀, hfb, hb_lim⟩
-- Since `b → 0`, it suffices to show that `‖f n x - g x‖ ≤ b n * ‖x‖` for all `n` and `x`.
suffices ∀ n x, ‖f n x - g x‖ ≤ b n * ‖x... | 15 | 3,269,017.372472 | 2 | 2 | 2 | 2,021 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
suppress_compilation
open Bornology Metric Set Real
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special cases about linear (as o... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 246 | 263 | theorem opNorm_extend_le :
‖f.extend e h_dense (uniformEmbedding_of_bound _ h_e).toUniformInducing‖ ≤ N * ‖f‖ := by |
-- Add `opNorm_le_of_dense`?
refine opNorm_le_bound _ ?_ (isClosed_property h_dense (isClosed_le ?_ ?_) fun x ↦ ?_)
· cases le_total 0 N with
| inl hN => exact mul_nonneg hN (norm_nonneg _)
| inr hN =>
have : Unique E := ⟨⟨0⟩, fun x ↦ norm_le_zero_iff.mp <|
(h_e x).trans (mul_nonpos_of_nonp... | 16 | 8,886,110.520508 | 2 | 2 | 2 | 2,021 |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Data.Set.Lattice
#align_import topology.algebra.module.character_space from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
namespace ... | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 128 | 134 | theorem union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A ∪ {0}) := by |
simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)
| 5 | 148.413159 | 2 | 2 | 1 | 2,022 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fin.tuple.nat_antidiagonal from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
namespace List.Nat
def antidiagona... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 79 | 92 | theorem mem_antidiagonalTuple {n : ℕ} {k : ℕ} {x : Fin k → ℕ} :
x ∈ antidiagonalTuple k n ↔ ∑ i, x i = n := by |
induction x using Fin.consInduction generalizing n with
| h0 =>
cases n
· decide
· simp [eq_comm]
| h x₀ x ih =>
simp_rw [Fin.sum_cons]
rw [antidiagonalTuple] -- Porting note: simp_rw doesn't use the equation lemma properly
simp_rw [List.mem_bind, List.mem_map,
List.Nat.mem_antidia... | 12 | 162,754.791419 | 2 | 2 | 4 | 2,023 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fin.tuple.nat_antidiagonal from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
namespace List.Nat
def antidiagona... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 96 | 119 | theorem nodup_antidiagonalTuple (k n : ℕ) : List.Nodup (antidiagonalTuple k n) := by |
induction' k with k ih generalizing n
· cases n
· simp
· simp [eq_comm]
simp_rw [antidiagonalTuple, List.nodup_bind]
constructor
· intro i _
exact (ih i.snd).map (Fin.cons_right_injective (α := fun _ => ℕ) i.fst)
induction' n with n n_ih
· exact List.pairwise_singleton _ _
· rw [List.Nat.an... | 23 | 9,744,803,446.248903 | 2 | 2 | 4 | 2,023 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fin.tuple.nat_antidiagonal from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
namespace List.Nat
def antidiagona... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 131 | 139 | theorem antidiagonalTuple_one (n : ℕ) : antidiagonalTuple 1 n = [![n]] := by |
simp_rw [antidiagonalTuple, antidiagonal, List.range_succ, List.map_append, List.map_singleton,
tsub_self, List.append_bind, List.bind_singleton, List.map_bind]
conv_rhs => rw [← List.nil_append [![n]]]
congr 1
simp_rw [List.bind_eq_nil, List.mem_range, List.map_eq_nil]
intro x hx
obtain ⟨m, rfl⟩ := Na... | 8 | 2,980.957987 | 2 | 2 | 4 | 2,023 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fin.tuple.nat_antidiagonal from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
namespace List.Nat
def antidiagona... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 142 | 147 | theorem antidiagonalTuple_two (n : ℕ) :
antidiagonalTuple 2 n = (antidiagonal n).map fun i => ![i.1, i.2] := by |
rw [antidiagonalTuple]
simp_rw [antidiagonalTuple_one, List.map_singleton]
rw [List.map_eq_bind]
rfl
| 4 | 54.59815 | 2 | 2 | 4 | 2,023 |
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.CategoryTheory.Iso
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.IsTensorProduct
#align_import ring_theory.ring_hom_properties from "leanprover-community/mathlib"@"a7c017d75051... | Mathlib/RingTheory/RingHomProperties.lean | 65 | 91 | theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
(R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
[Algebra S S'] (f : R →+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :
P (Localization.awayMap f r) ↔ P (I... |
let e₁ : R' ≃+* Localization.Away r :=
(IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv
let e₂ : Localization.Away (f r) ≃+* S' :=
(IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv
refine (hP.cancel_left_isIso e₁.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans ?_
r... | 23 | 9,744,803,446.248903 | 2 | 2 | 1 | 2,024 |
import Mathlib.MeasureTheory.Integral.Asymptotics
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#align_import measure_theory.integral.exp_decay from "leanprover-community/mathlib"@"d4817f8867c368d6c5571f7379b3888aaec1d95a"
noncomputable section
op... | Mathlib/MeasureTheory/Integral/ExpDecay.lean | 30 | 36 | theorem exp_neg_integrableOn_Ioi (a : ℝ) {b : ℝ} (h : 0 < b) :
IntegrableOn (fun x : ℝ => exp (-b * x)) (Ioi a) := by |
have : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b)) := by
refine Tendsto.div_const (Tendsto.neg ?_) _
exact tendsto_exp_atBot.comp (tendsto_id.const_mul_atTop_of_neg (neg_neg_iff_pos.2 h))
refine integrableOn_Ioi_deriv_of_nonneg' (fun x _ => ?_) (fun x _ => (exp_pos _).le) this
simpa [h.ne'] ... | 5 | 148.413159 | 2 | 2 | 1 | 2,025 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 25 | 44 | theorem Complex.subfield_eq_of_closed {K : Subfield ℂ} (hc : IsClosed (K : Set ℂ)) :
K = ofReal.fieldRange ∨ K = ⊤ := by |
suffices range (ofReal' : ℝ → ℂ) ⊆ K by
rw [range_subset_iff, ← coe_algebraMap] at this
have :=
(Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
(Subfield.toIntermediateField K this).toSubalgebra
simp_rw [← SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] a... | 18 | 65,659,969.137331 | 2 | 2 | 2 | 2,026 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 50 | 116 | theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype := by |
letI : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
letI : TopologicalRing K.topologicalClosure :=
Subring.instTopologicalRing K.topologicalClosure.toSubring
set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure)
have ui : UniformInducing ι :=
⟨by
erw [unifor... | 65 | 16,948,892,444,103,336,000,000,000,000 | 2 | 2 | 2 | 2,026 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 27 | 72 | theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by |
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`.
It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/
let S : Set (Set M) :=
{ N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }
rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul... | 43 | 4,727,839,468,229,346,000 | 2 | 2 | 2 | 2,027 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 82 | 95 | theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) (s : Set M) (snemp : s.Nonempty)
(s_compact : IsCompact s) (s_add : ∀ᵉ (x ∈ s) (y ∈ s), x * y ∈ s) :
∃ m ∈ s, m * m = m := by |
let M' := { m // m ∈ s }
letI : Semigroup M' :=
{ mul := fun p q => ⟨p.1 * q.1, s_add _ p.2 _ q.2⟩
mul_assoc := fun p q r => Subtype.eq (mul_assoc _ _ _) }
haveI : CompactSpace M' := isCompact_iff_compactSpace.mp s_compact
haveI : Nonempty M' := nonempty_subtype.mpr snemp
have : ∀ p : M', Continuou... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,027 |
import Mathlib.Algebra.Algebra.Basic
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.Data.Real.Archimedean
#align_import number_theory.class_number.admissible_abs from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
namespace AbsoluteValue
open Int
| Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean | 31 | 52 | theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb : b ≠ 0) (A : Fin n → ℤ) :
∃ t : Fin n → Fin ⌈1 / ε⌉₊,
∀ i₀ i₁, t i₀ = t i₁ → ↑(abs (A i₁ % b - A i₀ % b)) < abs b • ε := by |
have hb' : (0 : ℝ) < ↑(abs b) := Int.cast_pos.mpr (abs_pos.mpr hb)
have hbε : 0 < abs b • ε := by
rw [Algebra.smul_def]
exact mul_pos hb' hε
have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=
fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg.mpr (emod_nonneg _ hb)) hbε.le)
refine ⟨fun ... | 19 | 178,482,300.963187 | 2 | 2 | 1 | 2,028 |
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 114 | 122 | theorem poly_eq_of_wittPolynomial_bind_eq' [Fact p.Prime] (f g : ℕ → MvPolynomial (idx × ℕ) ℤ)
(h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by |
ext1 n
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [← Function.funext_iff] at h
replace h :=
congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
bind₁_wi... | 7 | 1,096.633158 | 2 | 2 | 3 | 2,029 |
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 125 | 133 | theorem poly_eq_of_wittPolynomial_bind_eq [Fact p.Prime] (f g : ℕ → MvPolynomial ℕ ℤ)
(h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by |
ext1 n
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [← Function.funext_iff] at h
replace h :=
congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
bind₁_wi... | 7 | 1,096.633158 | 2 | 2 | 3 | 2,029 |
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 172 | 195 | theorem ext [Fact p.Prime] {f g} (hf : IsPoly p f) (hg : IsPoly p g)
(h : ∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R) (n : ℕ),
ghostComponent n (f x) = ghostComponent n (g x)) :
∀ (R : Type u) [_Rcr : CommRing R] (x : 𝕎 R), f x = g x := by |
obtain ⟨φ, hf⟩ := hf
obtain ⟨ψ, hg⟩ := hg
intros
ext n
rw [hf, hg, poly_eq_of_wittPolynomial_bind_eq p φ ψ]
intro k
apply MvPolynomial.funext
intro x
simp only [hom_bind₁]
specialize h (ULift ℤ) (mk p fun i => ⟨x i⟩) k
simp only [ghostComponent_apply, aeval_eq_eval₂Hom] at h
apply (ULift.ringEq... | 20 | 485,165,195.40979 | 2 | 2 | 3 | 2,029 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 75 | 84 | theorem Submodule.isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (⊤ : Submodule R M).annihilator).toFinset =>
torsionBy R M
(IsPrincipal.generator (p : Ideal R) ^
(factors (⊤ : Submodule R M).annihilator).coun... |
convert isInternal_prime_power_torsion hM
ext p : 1
rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ← Ideal.span_singleton_pow,
Ideal.span_singleton_generator]
| 4 | 54.59815 | 2 | 2 | 5 | 2,030 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 89 | 98 | theorem Submodule.exists_isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
∃ (ι : Type u) (_ : Fintype ι) (_ : DecidableEq ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i)
(e : ι → ℕ), DirectSum.IsInternal fun i => torsionBy R M <| p i ^ e i := by |
refine ⟨_, ?_, _, _, ?_, _, Submodule.isInternal_prime_power_torsion_of_pid hM⟩
· exact Finset.fintypeCoeSort _
· rintro ⟨p, hp⟩
have hP := prime_of_factor p (Multiset.mem_toFinset.mp hp)
haveI := Ideal.isPrime_of_prime hP
exact (IsPrincipal.prime_generator_of_isPrime p hP.ne_zero).irreducible
| 6 | 403.428793 | 2 | 2 | 5 | 2,030 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 110 | 121 | theorem _root_.Ideal.torsionOf_eq_span_pow_pOrder (x : M) :
torsionOf R M x = span {p ^ pOrder hM x} := by |
dsimp only [pOrder]
rw [← (torsionOf R M x).span_singleton_generator, Ideal.span_singleton_eq_span_singleton, ←
Associates.mk_eq_mk_iff_associated, Associates.mk_pow]
have prop :
(fun n : ℕ => p ^ n • x = 0) = fun n : ℕ =>
(Associates.mk <| generator <| torsionOf R M x) ∣ Associates.mk p ^ n := by
... | 10 | 22,026.465795 | 2 | 2 | 5 | 2,030 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 124 | 148 | theorem p_pow_smul_lift {x y : M} {k : ℕ} (hM' : Module.IsTorsionBy R M (p ^ pOrder hM y))
(h : p ^ k • x ∈ R ∙ y) : ∃ a : R, p ^ k • x = p ^ k • a • y := by |
-- Porting note: needed to make `smul_smul` work below.
letI : MulAction R M := MulActionWithZero.toMulAction
by_cases hk : k ≤ pOrder hM y
· let f :=
((R ∙ p ^ (pOrder hM y - k) * p ^ k).quotEquivOfEq _ ?_).trans
(quotTorsionOfEquivSpanSingleton R M y)
· have : f.symm ⟨p ^ k • x, h⟩ ∈
... | 23 | 9,744,803,446.248903 | 2 | 2 | 5 | 2,030 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 153 | 165 | theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^ pOrder hM z))
{k : ℕ} (f : (R ⧸ R ∙ p ^ k) →ₗ[R] M ⧸ R ∙ z) :
∃ x : M, p ^ k • x = 0 ∧ Submodule.Quotient.mk (p := span R {z}) x = f 1 := by |
have f1 := mk_surjective (R ∙ z) (f 1)
have : p ^ k • f1.choose ∈ R ∙ z := by
rw [← Quotient.mk_eq_zero, mk_smul, f1.choose_spec, ← f.map_smul]
convert f.map_zero; change _ • Submodule.Quotient.mk _ = _
rw [← mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]
exact Submodule.mem_span_si... | 10 | 22,026.465795 | 2 | 2 | 5 | 2,030 |
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noetherian from "leanprover-community/mathlib"@"7c4c90f422a4a4477e4d8bc4dc9f1e634e6b2349"
namespace CategoryTheory
open CategoryTheory.Limits
variable... | Mathlib/CategoryTheory/Noetherian.lean | 82 | 87 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by |
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
| 4 | 54.59815 | 2 | 2 | 1 | 2,031 |
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.henselian from "leanprover-community/mathlib"@"d1accf4f9cddb3666c6e8e4da0ac2d19c4ed73f0"
noncomputable section
universe u v
open Polynomial LocalRing Polyno... | Mathlib/RingTheory/Henselian.lean | 65 | 82 | theorem isLocalRingHom_of_le_jacobson_bot {R : Type*} [CommRing R] (I : Ideal R)
(h : I ≤ Ideal.jacobson ⊥) : IsLocalRingHom (Ideal.Quotient.mk I) := by |
constructor
intro a h
have : IsUnit (Ideal.Quotient.mk (Ideal.jacobson ⊥) a) := by
rw [isUnit_iff_exists_inv] at *
obtain ⟨b, hb⟩ := h
obtain ⟨b, rfl⟩ := Ideal.Quotient.mk_surjective b
use Ideal.Quotient.mk _ b
rw [← (Ideal.Quotient.mk _).map_one, ← (Ideal.Quotient.mk _).map_mul, Ideal.Quotie... | 16 | 8,886,110.520508 | 2 | 2 | 2 | 2,032 |
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.henselian from "leanprover-community/mathlib"@"d1accf4f9cddb3666c6e8e4da0ac2d19c4ed73f0"
noncomputable section
universe u v
open Polynomial LocalRing Polyno... | Mathlib/RingTheory/Henselian.lean | 121 | 155 | theorem HenselianLocalRing.TFAE (R : Type u) [CommRing R] [LocalRing R] :
TFAE
[HenselianLocalRing R,
∀ f : R[X], f.Monic → ∀ a₀ : ResidueField R, aeval a₀ f = 0 →
aeval a₀ (derivative f) ≠ 0 → ∃ a : R, f.IsRoot a ∧ residue R a = a₀,
∀ {K : Type u} [Field K],
∀ (φ : R →+* K... |
tfae_have 3 → 2
· intro H
exact H (residue R) Ideal.Quotient.mk_surjective
tfae_have 2 → 1
· intro H
constructor
intro f hf a₀ h₁ h₂
specialize H f hf (residue R a₀)
have aux := flip mem_nonunits_iff.mp h₂
simp only [aeval_def, ResidueField.algebraMap_eq, eval₂_at_apply, ←
Ideal.Q... | 27 | 532,048,240,601.79865 | 2 | 2 | 2 | 2,032 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 64 | 112 | theorem step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x := by |
obtain ⟨y, -, hy⟩ : ∃ y ∈ ⊤, y ∉ f.domain := SetLike.exists_of_lt (lt_top_iff_ne_top.2 hdom)
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x : E) - y ∈ s → f x ≤ c) ∧
∀ x : f.domain, (x : E) + y ∈ s → c ≤ f x := by
set Sp := f '' { x : f.domain | (x : E) + y ∈ s }
set Sn := f '' { x : f.do... | 46 | 94,961,194,206,024,480,000 | 2 | 2 | 2 | 2,033 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 115 | 139 | theorem exists_top (p : E →ₗ.[ℝ] ℝ) (hp_nonneg : ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x)
(hp_dense : ∀ y, ∃ x : p.domain, (x : E) + y ∈ s) :
∃ q ≥ p, q.domain = ⊤ ∧ ∀ x : q.domain, (x : E) ∈ s → 0 ≤ q x := by |
set S := { p : E →ₗ.[ℝ] ℝ | ∀ x : p.domain, (x : E) ∈ s → 0 ≤ p x }
have hSc : ∀ c, c ⊆ S → IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub := by
intro c hcs c_chain y hy
clear hp_nonneg hp_dense p
have cne : c.Nonempty := ⟨y, hy⟩
have hcd : DirectedOn (· ≤ ·) c := c_chain.directedOn
ref... | 22 | 3,584,912,846.131591 | 2 | 2 | 2 | 2,033 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
#align_import analysis.special_functions.trigonometric.bounds from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Set
namespace Real
variable {x : ℝ}
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 39 | 49 | theorem sin_lt (h : 0 < x) : sin x < x := by |
cases' lt_or_le 1 x with h' h'
· exact (sin_le_one x).trans_lt h'
have hx : |x| = x := abs_of_nonneg h.le
have := le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx])
rw [sub_le_iff_le_add', hx] at this
apply this.trans_lt
rw [sub_add, sub_lt_self_iff, sub_pos, div_eq_mul_inv (x ^ 3)]
refine mul_lt_mul... | 10 | 22,026.465795 | 2 | 2 | 1 | 2,034 |
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Set.Finite
import Mathlib.Order.Category.FinPartOrd
import Mathlib.Order.Category.LinOrd
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
import Mathlib.Data.Set.Subsingleton
#align_import order.category.No... | Mathlib/Order/Category/NonemptyFinLinOrd.lean | 150 | 163 | theorem mono_iff_injective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :
Mono f ↔ Function.Injective f := by |
refine ⟨?_, ConcreteCategory.mono_of_injective f⟩
intro
intro a₁ a₂ h
let X := NonemptyFinLinOrd.of (ULift (Fin 1))
let g₁ : X ⟶ A := ⟨fun _ => a₁, fun _ _ _ => by rfl⟩
let g₂ : X ⟶ A := ⟨fun _ => a₂, fun _ _ _ => by rfl⟩
change g₁ (ULift.up (0 : Fin 1)) = g₂ (ULift.up (0 : Fin 1))
have eq : g₁ ≫ f = g... | 12 | 162,754.791419 | 2 | 2 | 2 | 2,035 |
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Set.Finite
import Mathlib.Order.Category.FinPartOrd
import Mathlib.Order.Category.LinOrd
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
import Mathlib.Data.Set.Subsingleton
#align_import order.category.No... | Mathlib/Order/Category/NonemptyFinLinOrd.lean | 171 | 209 | theorem epi_iff_surjective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :
Epi f ↔ Function.Surjective f := by |
constructor
· intro
dsimp only [Function.Surjective]
by_contra! hf'
rcases hf' with ⟨m, hm⟩
let Y := NonemptyFinLinOrd.of (ULift (Fin 2))
let p₁ : B ⟶ Y :=
⟨fun b => if b < m then ULift.up 0 else ULift.up 1, fun x₁ x₂ h => by
simp only
split_ifs with h₁ h₂ h₂
any_g... | 37 | 11,719,142,372,802,612 | 2 | 2 | 2 | 2,035 |
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.Norm
#align_import linear_algebra.free_module.norm from "leanprover-community/mathlib"@"90b0d53ee6ffa910e5c2a977ce7e2fc704647974"
open Ideal Polynomial
open scoped Polynomial
variable {R S ι : Type*} [CommRing R] [IsDomain R] [IsPri... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 30 | 50 | theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i) := by |
have hI := span_singleton_eq_bot.not.2 hf
let b' := ringBasis b (span {f}) hI
classical
rw [← Matrix.det_diagonal, ← LinearMap.det_toLin b']
let e :=
(b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans
((LinearEquiv.coord S S f hf).restrictScalars R)
refine (LinearMap.associated_det_of_e... | 19 | 178,482,300.963187 | 2 | 2 | 1 | 2,036 |
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Rat
#align_import data.rat.order from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
assert_not_exists Field
assert_not_exists Finset
assert_not_exists Set.Icc
assert_not_exists GaloisConnection
namespace Rat
variable {a... | Mathlib/Algebra/Order/Ring/Rat.lean | 50 | 59 | theorem ofScientific_nonneg (m : ℕ) (s : Bool) (e : ℕ) :
0 ≤ Rat.ofScientific m s e := by |
rw [Rat.ofScientific]
cases s
· rw [if_neg (by decide)]
refine num_nonneg.mp ?_
rw [num_natCast]
exact Int.natCast_nonneg _
· rw [if_pos rfl, normalize_eq_mkRat]
exact Rat.mkRat_nonneg (Int.natCast_nonneg _) _
| 8 | 2,980.957987 | 2 | 2 | 1 | 2,037 |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 122 | 127 | theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
(star s).topologicalClosure = star s.topologicalClosure := by |
suffices ∀ t : Subalgebra R A, (star t).topologicalClosure ≤ star t.topologicalClosure from
le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))
exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)
(isClosed_closure.preimage continu... | 4 | 54.59815 | 2 | 2 | 2 | 2,038 |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 146 | 163 | theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
{φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ)
(h :
φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) :
φ = ψ := by |
rw [DFunLike.ext'_iff]
have : Dense (Set.range <| inclusion (le_topologicalClosure S)) := by
refine embedding_subtype_val.toInducing.dense_iff.2 fun x => ?_
convert show ↑x ∈ closure (S : Set A) from x.prop
rw [← Set.range_comp]
exact
Set.ext fun y =>
⟨by
rintro ⟨y, rfl⟩
... | 13 | 442,413.392009 | 2 | 2 | 2 | 2,038 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.function.special_functions.inner from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
variable {α : Type*} {𝕜 : Type*} {E : Type*}
variable [RCLike ... | Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean | 41 | 47 | theorem AEMeasurable.inner {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E]
[SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f g : α → E}
(hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : AEMeasurable (fun x => ⟪f x, g x⟫) μ := by |
refine ⟨fun x => ⟪hf.mk f x, hg.mk g x⟫, hf.measurable_mk.inner hg.measurable_mk, ?_⟩
refine hf.ae_eq_mk.mp (hg.ae_eq_mk.mono fun x hxg hxf => ?_)
dsimp only
congr
| 4 | 54.59815 | 2 | 2 | 1 | 2,039 |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import number_theory.frobenius_number from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open Nat
def FrobeniusNumber (n : ℕ) (s : Set ℕ) : Pro... | Mathlib/NumberTheory/FrobeniusNumber.lean | 55 | 82 | theorem frobeniusNumber_pair (cop : Coprime m n) (hm : 1 < m) (hn : 1 < n) :
FrobeniusNumber (m * n - m - n) {m, n} := by |
simp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]
have hmn : m + n ≤ m * n := add_le_mul hm hn
constructor
· push_neg
intro a b h
apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
simp only [Nat.sub_sub, smul_eq_mul] at h
zify [hmn] at h ⊢
rw [← sub_eq_zero] at h ⊢... | 26 | 195,729,609,428.83878 | 2 | 2 | 1 | 2,040 |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertNth
import Mathlib.Init.Data.List.Lemmas
#align_import data.list.lemmas from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β γ : Type*}
namespace List
| Mathlib/Data/List/Lemmas.lean | 23 | 41 | theorem injOn_insertNth_index_of_not_mem (l : List α) (x : α) (hx : x ∉ l) :
Set.InjOn (fun k => insertNth k x l) { n | n ≤ l.length } := by |
induction' l with hd tl IH
· intro n hn m hm _
simp only [Set.mem_singleton_iff, Set.setOf_eq_eq_singleton,
length] at hn hm
simp_all [hn, hm]
· intro n hn m hm h
simp only [length, Set.mem_setOf_eq] at hn hm
simp only [mem_cons, not_or] at hx
cases n <;> cases m
· rfl
· simp [h... | 17 | 24,154,952.753575 | 2 | 2 | 3 | 2,041 |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertNth
import Mathlib.Init.Data.List.Lemmas
#align_import data.list.lemmas from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β γ : Type*}
namespace List
theorem injOn_insertNth_index_of_not_mem (l : List... | Mathlib/Data/List/Lemmas.lean | 44 | 52 | theorem foldr_range_subset_of_range_subset {f : β → α → α} {g : γ → α → α}
(hfg : Set.range f ⊆ Set.range g) (a : α) : Set.range (foldr f a) ⊆ Set.range (foldr g a) := by |
rintro _ ⟨l, rfl⟩
induction' l with b l H
· exact ⟨[], rfl⟩
· cases' hfg (Set.mem_range_self b) with c hgf
cases' H with m hgf'
rw [foldr_cons, ← hgf, ← hgf']
exact ⟨c :: m, rfl⟩
| 7 | 1,096.633158 | 2 | 2 | 3 | 2,041 |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertNth
import Mathlib.Init.Data.List.Lemmas
#align_import data.list.lemmas from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β γ : Type*}
namespace List
theorem injOn_insertNth_index_of_not_mem (l : List... | Mathlib/Data/List/Lemmas.lean | 55 | 66 | theorem foldl_range_subset_of_range_subset {f : α → β → α} {g : α → γ → α}
(hfg : (Set.range fun a c => f c a) ⊆ Set.range fun b c => g c b) (a : α) :
Set.range (foldl f a) ⊆ Set.range (foldl g a) := by |
change (Set.range fun l => _) ⊆ Set.range fun l => _
-- Porting note: This was simply `simp_rw [← foldr_reverse]`
simp_rw [← foldr_reverse _ (fun z w => g w z), ← foldr_reverse _ (fun z w => f w z)]
-- Porting note: This `change` was not necessary in mathlib3
change (Set.range (foldr (fun z w => f w z) a ∘ r... | 9 | 8,103.083928 | 2 | 2 | 3 | 2,041 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section LinearInde... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 44 | 49 | theorem disjoint_span_sum : Disjoint (span R (range (u ∘ Sum.inl)))
(span R (range (u ∘ Sum.inr))) := by |
rw [huv, disjoint_comm]
refine Disjoint.mono_right (span_mono (range_comp_subset_range _ _)) ?_
rw [← LinearMap.range_coe, span_eq (LinearMap.range S.f), hS.moduleCat_range_eq_ker]
exact range_ker_disjoint hw
| 4 | 54.59815 | 2 | 2 | 5 | 2,042 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section LinearInde... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 62 | 68 | theorem linearIndependent_leftExact : LinearIndependent R u := by |
rw [linearIndependent_sum]
refine ⟨?_, LinearIndependent.of_comp S.g hw, disjoint_span_sum hS hw huv⟩
rw [huv, LinearMap.linearIndependent_iff S.f]; swap
· rw [LinearMap.ker_eq_bot, ← mono_iff_injective]
infer_instance
exact hv
| 6 | 403.428793 | 2 | 2 | 5 | 2,042 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section LinearInde... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 72 | 78 | theorem linearIndependent_shortExact {w : ι' → S.X₃} (hw : LinearIndependent R w) :
LinearIndependent R (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w)) := by |
apply linearIndependent_leftExact hS'.exact hv _ hS'.mono_f rfl
dsimp
convert hw
ext
apply Function.rightInverse_invFun ((epi_iff_surjective _).mp hS'.epi_g)
| 5 | 148.413159 | 2 | 2 | 5 | 2,042 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section Span
| Mathlib/Algebra/Category/ModuleCat/Free.lean | 94 | 125 | theorem span_exact {β : Type*} {u : ι ⊕ β → S.X₂} (huv : u ∘ Sum.inl = S.f ∘ v)
(hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range (S.g ∘ u ∘ Sum.inr))) :
⊤ ≤ span R (range u) := by |
intro m _
have hgm : S.g m ∈ span R (range (S.g ∘ u ∘ Sum.inr)) := hw mem_top
rw [Finsupp.mem_span_range_iff_exists_finsupp] at hgm
obtain ⟨cm, hm⟩ := hgm
let m' : S.X₂ := Finsupp.sum cm fun j a ↦ a • (u (Sum.inr j))
have hsub : m - m' ∈ LinearMap.range S.f := by
rw [hS.moduleCat_range_eq_ker]
simp... | 28 | 1,446,257,064,291.475 | 2 | 2 | 5 | 2,042 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section Span
the... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 129 | 138 | theorem span_rightExact {w : ι' → S.X₃} (hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range w)) (hE : Epi S.g) :
⊤ ≤ span R (range (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w))) := by |
refine span_exact hS ?_ hv ?_
· simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inl]
· convert hw
simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inr]
rw [ModuleCat.epi_iff_surjective] at hE
rw [← Function.comp.assoc, Function.RightInverse.comp_eq_id (Funct... | 7 | 1,096.633158 | 2 | 2 | 5 | 2,042 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 93 | 102 | theorem valuationOfNeZeroToFun_eq (x : Kˣ) :
(v.valuationOfNeZeroToFun x : ℤₘ₀) = v.valuation (x : K) := by |
rw [show v.valuation (x : K) = _ * _ by rfl]
rw [Units.val_inv_eq_inv_val]
change _ = ite _ _ _ * (ite _ _ _)⁻¹
simp_rw [IsLocalization.toLocalizationMap_sec, SubmonoidClass.coe_subtype,
if_neg <| IsLocalization.sec_fst_ne_zero le_rfl x.ne_zero,
if_neg (nonZeroDivisors.coe_ne_zero _),
valuationOfNe... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,043 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 120 | 131 | theorem valuation_of_unit_eq (x : Rˣ) :
v.valuationOfNeZero (Units.map (algebraMap R K : R →* K) x) = 1 := by |
rw [← WithZero.coe_inj, valuationOfNeZero_eq, Units.coe_map, eq_iff_le_not_lt]
constructor
· exact v.valuation_le_one x
· cases' x with x _ hx _
change ¬v.valuation (algebraMap R K x) < 1
apply_fun v.intValuation at hx
rw [map_one, map_mul] at hx
rw [not_lt, ← hx, ← mul_one <| v.valuation _, va... | 10 | 22,026.465795 | 2 | 2 | 3 | 2,043 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 150 | 155 | theorem valuation_of_unit_mod_eq (n : ℕ) (x : Rˣ) :
v.valuationOfNeZeroMod n (Units.map (algebraMap R K : R →* K) x : K/n) = 1 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [valuationOfNeZeroMod, MonoidHom.comp_apply, ← QuotientGroup.coe_mk',
QuotientGroup.map_mk' (G := Kˣ) (N := MonoidHom.range (powMonoidHom n)),
valuation_of_unit_eq, QuotientGroup.mk_one, map_one]
| 4 | 54.59815 | 2 | 2 | 3 | 2,043 |
import Mathlib.CategoryTheory.EffectiveEpi.RegularEpi
import Mathlib.CategoryTheory.EffectiveEpi.Comp
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
universe u
open CategoryTheory Limits
namespace TopCat
noncomputable
def effectiveEpiStructOfQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) (hπ : QuotientMap ... | Mathlib/Topology/Category/TopCat/EffectiveEpi.lean | 53 | 75 | theorem effectiveEpi_iff_quotientMap {B X : TopCat.{u}} (π : X ⟶ B) :
EffectiveEpi π ↔ QuotientMap π := by |
/- The backward direction is given by `effectiveEpiStructOfQuotientMap` above. -/
refine ⟨fun _ ↦ ?_, fun hπ ↦ ⟨⟨effectiveEpiStructOfQuotientMap π hπ⟩⟩⟩
/- Since `TopCat` has pullbacks, `π` is in fact a `RegularEpi`. This means that it exhibits `B` as
a coequalizer of two maps into `X`. It suffices to prove ... | 21 | 1,318,815,734.483215 | 2 | 2 | 1 | 2,044 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Logic.Equiv.Fin
#align_import category_theory.limits.... | Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean | 106 | 110 | theorem hasFiniteProducts_of_has_binary_and_terminal : HasFiniteProducts C := by |
refine ⟨fun n => ⟨fun K => ?_⟩⟩
letI := hasProduct_fin n fun n => K.obj ⟨n⟩
let that : (Discrete.functor fun n => K.obj ⟨n⟩) ≅ K := Discrete.natIso fun ⟨i⟩ => Iso.refl _
apply @hasLimitOfIso _ _ _ _ _ _ this that
| 4 | 54.59815 | 2 | 2 | 1 | 2,045 |
import Mathlib.Algebra.Group.Defs
variable {α β δ : Type*} [AddZeroClass δ] [Min δ]
namespace Levenshtein
structure Cost (α β δ : Type*) where
delete : α → δ
insert : β → δ
substitute : α → β → δ
@[simps]
def defaultCost [DecidableEq α] : Cost α α ℕ where
delete _ := 1
insert _ := 1
substi... | Mathlib/Data/List/EditDistance/Defs.lean | 125 | 135 | theorem impl_length (d : {r : List δ // 0 < r.length}) (w : d.1.length = xs.length + 1) :
(impl C xs y d).1.length = xs.length + 1 := by |
induction xs generalizing d with
| nil => rfl
| cons x xs ih =>
dsimp [impl]
match d, w with
| ⟨d₁ :: d₂ :: ds, _⟩, w =>
dsimp
congr 1
exact ih ⟨d₂ :: ds, (by simp)⟩ (by simpa using w)
| 9 | 8,103.083928 | 2 | 2 | 1 | 2,046 |
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import field_theory.normal from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
noncomputable section
open scoped Classical Polynomial
open Polynomial ... | Mathlib/FieldTheory/Normal.lean | 66 | 81 | theorem Normal.exists_isSplittingField [h : Normal F K] [FiniteDimensional F K] :
∃ p : F[X], IsSplittingField F K p := by |
let s := Basis.ofVectorSpace F K
refine
⟨∏ x, minpoly F (s x), splits_prod _ fun x _ => h.splits (s x),
Subalgebra.toSubmodule.injective ?_⟩
rw [Algebra.top_toSubmodule, eq_top_iff, ← s.span_eq, Submodule.span_le, Set.range_subset_iff]
refine fun x =>
Algebra.subset_adjoin
(Multiset.mem_toF... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,047 |
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import field_theory.normal from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
noncomputable section
open scoped Classical Polynomial
open Polynomial ... | Mathlib/FieldTheory/Normal.lean | 107 | 111 | theorem Normal.of_algEquiv [h : Normal F E] (f : E ≃ₐ[F] E') : Normal F E' := by |
rw [normal_iff] at h ⊢
intro x; specialize h (f.symm x)
rw [← f.apply_symm_apply x, minpoly.algEquiv_eq, ← f.toAlgHom.comp_algebraMap]
exact ⟨h.1.map f, splits_comp_of_splits _ _ h.2⟩
| 4 | 54.59815 | 2 | 2 | 3 | 2,047 |
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import field_theory.normal from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
noncomputable section
open scoped Classical Polynomial
open Polynomial ... | Mathlib/FieldTheory/Normal.lean | 120 | 142 | theorem Normal.of_isSplittingField (p : F[X]) [hFEp : IsSplittingField F E p] : Normal F E := by |
rcases eq_or_ne p 0 with (rfl | hp)
· have := hFEp.adjoin_rootSet
rw [rootSet_zero, Algebra.adjoin_empty] at this
exact Normal.of_algEquiv
(AlgEquiv.ofBijective (Algebra.ofId F E) (Algebra.bijective_algebraMap_iff.2 this.symm))
refine normal_iff.mpr fun x ↦ ?_
haveI : FiniteDimensional F E := IsS... | 22 | 3,584,912,846.131591 | 2 | 2 | 3 | 2,047 |
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
#align_import algebraic_topology.dold_kan.n_comp_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
noncomputable section
open CategoryTheory CategoryTheory.Category Categor... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 38 | 78 | theorem PInfty_comp_map_mono_eq_zero (X : SimplicialObject C) {n : ℕ} {Δ' : SimplexCategory}
(i : Δ' ⟶ [n]) [hi : Mono i] (h₁ : Δ'.len ≠ n) (h₂ : ¬Isδ₀ i) :
PInfty.f n ≫ X.map i.op = 0 := by |
induction' Δ' using SimplexCategory.rec with m
obtain ⟨k, hk⟩ := Nat.exists_eq_add_of_lt (len_lt_of_mono i fun h => by
rw [← h] at h₁
exact h₁ rfl)
simp only [len_mk] at hk
rcases k with _|k
· change n = m + 1 at hk
subst hk
obtain ⟨j, rfl⟩ := eq_δ_of_mono i
rw [Isδ₀.iff] at h₂
... | 38 | 31,855,931,757,113,756 | 2 | 2 | 2 | 2,048 |
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
#align_import algebraic_topology.dold_kan.n_comp_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
noncomputable section
open CategoryTheory CategoryTheory.Category Categor... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 83 | 124 | theorem Γ₀_obj_termwise_mapMono_comp_PInfty (X : SimplicialObject C) {Δ Δ' : SimplexCategory}
(i : Δ ⟶ Δ') [Mono i] :
Γ₀.Obj.Termwise.mapMono (AlternatingFaceMapComplex.obj X) i ≫ PInfty.f Δ.len =
PInfty.f Δ'.len ≫ X.map i.op := by |
induction' Δ using SimplexCategory.rec with n
induction' Δ' using SimplexCategory.rec with n'
dsimp
-- We start with the case `i` is an identity
by_cases h : n = n'
· subst h
simp only [SimplexCategory.eq_id_of_mono i, Γ₀.Obj.Termwise.mapMono_id, op_id, X.map_id]
dsimp
simp only [id_comp, comp_... | 38 | 31,855,931,757,113,756 | 2 | 2 | 2 | 2,048 |
import Mathlib.Data.Sym.Sym2
import Mathlib.Logic.Relation
#align_import order.game_add from "leanprover-community/mathlib"@"fee218fb033b2fd390c447f8be27754bc9093be9"
set_option autoImplicit true
variable {α β : Type*} {rα : α → α → Prop} {rβ : β → β → Prop}
namespace Prod
variable (rα rβ)
inductive Game... | Mathlib/Order/GameAdd.lean | 60 | 67 | theorem gameAdd_iff {rα rβ} {x y : α × β} :
GameAdd rα rβ x y ↔ rα x.1 y.1 ∧ x.2 = y.2 ∨ rβ x.2 y.2 ∧ x.1 = y.1 := by |
constructor
· rintro (@⟨a₁, a₂, b, h⟩ | @⟨a, b₁, b₂, h⟩)
exacts [Or.inl ⟨h, rfl⟩, Or.inr ⟨h, rfl⟩]
· revert x y
rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ (⟨h, rfl : b₁ = b₂⟩ | ⟨h, rfl : a₁ = a₂⟩)
exacts [GameAdd.fst h, GameAdd.snd h]
| 6 | 403.428793 | 2 | 2 | 1 | 2,049 |
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Logic.UnivLE
import Mathlib.Logic.Small.Basic
#align_import category_theory.essentially_small from "leanprover-community/mathlib"@"f7707875544ef1f81b32cb68c79e0e24e45a0e76"
universe w v v' u u'
open CategoryTheory
... | Mathlib/CategoryTheory/EssentiallySmall.lean | 71 | 77 | theorem essentiallySmall_congr {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
(e : C ≌ D) : EssentiallySmall.{w} C ↔ EssentiallySmall.{w} D := by |
fconstructor
· rintro ⟨S, 𝒮, ⟨f⟩⟩
exact EssentiallySmall.mk' (e.symm.trans f)
· rintro ⟨S, 𝒮, ⟨f⟩⟩
exact EssentiallySmall.mk' (e.trans f)
| 5 | 148.413159 | 2 | 2 | 1 | 2,050 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.SetTheory.Cardinal.Finite
#align_import combinatorics.configuration from "leanprover-community/mathlib"@"d2d8742b0c21426362a9dacebc6005db895ca963"
open Finset
nam... | Mathlib/Combinatorics/Configuration.lean | 125 | 166 | theorem Nondegenerate.exists_injective_of_card_le [Nondegenerate P L] [Fintype P] [Fintype L]
(h : Fintype.card L ≤ Fintype.card P) : ∃ f : L → P, Function.Injective f ∧ ∀ l, f l ∉ l := by |
classical
let t : L → Finset P := fun l => Set.toFinset { p | p ∉ l }
suffices ∀ s : Finset L, s.card ≤ (s.biUnion t).card by
-- Hall's marriage theorem
obtain ⟨f, hf1, hf2⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this
exact ⟨f, hf1, fun l => Set.mem_toFinset.mp (hf2 ... | 40 | 235,385,266,837,019,970 | 2 | 2 | 1 | 2,051 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Algebra.MulAction
#align_import topology.algebra.affine from "leanprover-community/mathlib"@"717c073262cd9d59b1a1dcda7e8ab570c5b63370"
namespace AffineMap
variable {R E F : Type*}
variable [AddC... | Mathlib/Topology/Algebra/Affine.lean | 36 | 43 | theorem continuous_iff {f : E →ᵃ[R] F} : Continuous f ↔ Continuous f.linear := by |
constructor
· intro hc
rw [decomp' f]
exact hc.sub continuous_const
· intro hc
rw [decomp f]
exact hc.add continuous_const
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,052 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Algebra.MulAction
#align_import topology.algebra.affine from "leanprover-community/mathlib"@"717c073262cd9d59b1a1dcda7e8ab570c5b63370"
namespace AffineMap
variable {R E F : Type*}
variable [AddC... | Mathlib/Topology/Algebra/Affine.lean | 61 | 67 | theorem homothety_continuous (x : F) (t : R) : Continuous <| homothety x t := by |
suffices ⇑(homothety x t) = fun y => t • (y - x) + x by
rw [this]
exact ((continuous_id.sub continuous_const).const_smul _).add continuous_const
-- Porting note: proof was `by continuity`
ext y
simp [homothety_apply]
| 6 | 403.428793 | 2 | 2 | 2 | 2,052 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 40 | 49 | theorem Fintype.isPrimePow_card_of_field {α} [Fintype α] [Field α] : IsPrimePow ‖α‖ := by |
-- TODO: `Algebra` version of `CharP.exists`, of type `∀ p, Algebra (ZMod p) α`
cases' CharP.exists α with p _
haveI hp := Fact.mk (CharP.char_is_prime α p)
letI : Algebra (ZMod p) α := ZMod.algebra _ _
let b := IsNoetherian.finsetBasis (ZMod p) α
rw [Module.card_fintype b, ZMod.card, isPrimePow_pow_iff]
... | 9 | 8,103.083928 | 2 | 2 | 4 | 2,053 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 53 | 57 | theorem Fintype.nonempty_field_iff {α} [Fintype α] : Nonempty (Field α) ↔ IsPrimePow ‖α‖ := by |
refine ⟨fun ⟨h⟩ => Fintype.isPrimePow_card_of_field, ?_⟩
rintro ⟨p, n, hp, hn, hα⟩
haveI := Fact.mk hp.nat_prime
exact ⟨(Fintype.equivOfCardEq ((GaloisField.card p n hn.ne').trans hα)).symm.field⟩
| 4 | 54.59815 | 2 | 2 | 4 | 2,053 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 66 | 76 | theorem Infinite.nonempty_field {α : Type u} [Infinite α] : Nonempty (Field α) := by |
letI K := FractionRing (MvPolynomial α <| ULift.{u} ℚ)
suffices #α = #K by
obtain ⟨e⟩ := Cardinal.eq.1 this
exact ⟨e.field⟩
rw [← IsLocalization.card (MvPolynomial α <| ULift.{u} ℚ)⁰ K le_rfl]
apply le_antisymm
· refine
⟨⟨fun a => MvPolynomial.monomial (Finsupp.single a 1) (1 : ULift.{u} ℚ), fu... | 10 | 22,026.465795 | 2 | 2 | 4 | 2,053 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 80 | 85 | theorem Field.nonempty_iff {α : Type u} : Nonempty (Field α) ↔ IsPrimePow #α := by |
rw [Cardinal.isPrimePow_iff]
cases' fintypeOrInfinite α with h h
· simpa only [Cardinal.mk_fintype, Nat.cast_inj, exists_eq_left',
(Cardinal.nat_lt_aleph0 _).not_le, false_or_iff] using Fintype.nonempty_field_iff
· simpa only [← Cardinal.infinite_iff, h, true_or_iff, iff_true_iff] using Infinite.nonempty... | 5 | 148.413159 | 2 | 2 | 4 | 2,053 |
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.complex.conformal from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
noncomputable section
open Complex Continuous... | Mathlib/Analysis/Complex/Conformal.lean | 49 | 62 | theorem isConformalMap_complex_linear {map : ℂ →L[ℂ] E} (nonzero : map ≠ 0) :
IsConformalMap (map.restrictScalars ℝ) := by |
have minor₁ : ‖map 1‖ ≠ 0 := by simpa only [ext_ring_iff, Ne, norm_eq_zero] using nonzero
refine ⟨‖map 1‖, minor₁, ⟨‖map 1‖⁻¹ • ((map : ℂ →ₗ[ℂ] E) : ℂ →ₗ[ℝ] E), ?_⟩, ?_⟩
· intro x
simp only [LinearMap.smul_apply]
have : x = x • (1 : ℂ) := by rw [smul_eq_mul, mul_one]
nth_rw 1 [this]
rw [LinearMap... | 12 | 162,754.791419 | 2 | 2 | 2 | 2,054 |
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.complex.conformal from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
noncomputable section
open Complex Continuous... | Mathlib/Analysis/Complex/Conformal.lean | 78 | 91 | theorem IsConformalMap.is_complex_or_conj_linear (h : IsConformalMap g) :
(∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g) ∨
∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g ∘L ↑conjCLE := by |
rcases h with ⟨c, -, li, rfl⟩
obtain ⟨li, rfl⟩ : ∃ li' : ℂ ≃ₗᵢ[ℝ] ℂ, li'.toLinearIsometry = li :=
⟨li.toLinearIsometryEquiv rfl, by ext1; rfl⟩
rcases linear_isometry_complex li with ⟨a, rfl | rfl⟩
-- let rot := c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ,
· refine Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.i... | 11 | 59,874.141715 | 2 | 2 | 2 | 2,054 |
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.UnitInterval
import Mathlib.Algebra.Star.Subalgebra
#align_import topology.continuous_function.polynomial from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {R... | Mathlib/Topology/ContinuousFunction/Polynomial.lean | 76 | 82 | theorem aeval_continuousMap_apply (g : R[X]) (f : C(α, R)) (x : α) :
((Polynomial.aeval f) g) x = g.eval (f x) := by |
refine Polynomial.induction_on' g ?_ ?_
· intro p q hp hq
simp [hp, hq]
· intro n a
simp [Pi.pow_apply]
| 5 | 148.413159 | 2 | 2 | 1 | 2,055 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Adjugate
#align_import linear_algebra.matrix.nondegenerate from "leanprover-community/mathlib"@"2a32c70c78096758af93e997b978a5d461007b4f"
namespace Matrix
variable {m R A : Type*} [Fintype m... | Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | 50 | 63 | theorem nondegenerate_of_det_ne_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) :
Nondegenerate M := by |
intro v hv
ext i
specialize hv (M.cramer (Pi.single i 1))
refine (mul_eq_zero.mp ?_).resolve_right hM
convert hv
simp only [mulVec_cramer M (Pi.single i 1), dotProduct, Pi.smul_apply, smul_eq_mul]
rw [Finset.sum_eq_single i, Pi.single_eq_same, mul_one]
· intro j _ hj
simp [hj]
· intros
have :... | 12 | 162,754.791419 | 2 | 2 | 1 | 2,056 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open Set
namespace MeasureTheory
variable {α : Type*} [MeasurableSpace α] {f : α → ℝ} (... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 50 | 72 | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by |
have one_lt_p : -1 < p - 1 := by linarith
have obs : ∀ x : ℝ, ∫ t : ℝ in (0)..x, t ^ (p - 1) = x ^ p / p := by
intro x
rw [integral_rpow (Or.inl one_lt_p)]
simp [Real.zero_rpow p_pos.ne.symm]
set g := fun t : ℝ => t ^ (p - 1)
have g_nn : ∀ᵐ t ∂volume.restrict (Ioi (0 : ℝ)), 0 ≤ g t := by
filter... | 20 | 485,165,195.40979 | 2 | 2 | 1 | 2,057 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 55 | 64 | theorem projectiveSeminormAux_nonneg (p : FreeAddMonoid (𝕜 × Π i, E i)) :
0 ≤ projectiveSeminormAux p := by |
simp only [projectiveSeminormAux, Function.comp_apply]
refine List.sum_nonneg ?_
intro a
simp only [Multiset.map_coe, Multiset.mem_coe, List.mem_map, Prod.exists, forall_exists_index,
and_imp]
intro x m _ h
rw [← h]
exact mul_nonneg (norm_nonneg _) (Finset.prod_nonneg (fun _ _ ↦ norm_nonneg _))
| 8 | 2,980.957987 | 2 | 2 | 6 | 2,058 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 66 | 71 | theorem projectiveSeminormAux_add_le (p q : FreeAddMonoid (𝕜 × Π i, E i)) :
projectiveSeminormAux (p + q) ≤ projectiveSeminormAux p + projectiveSeminormAux q := by |
simp only [projectiveSeminormAux, Function.comp_apply, Multiset.map_coe, Multiset.sum_coe]
erw [List.map_append]
rw [List.sum_append]
rfl
| 4 | 54.59815 | 2 | 2 | 6 | 2,058 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 73 | 82 | theorem projectiveSeminormAux_smul (p : FreeAddMonoid (𝕜 × Π i, E i)) (a : 𝕜) :
projectiveSeminormAux (List.map (fun (y : 𝕜 × Π i, E i) ↦ (a * y.1, y.2)) p) =
‖a‖ * projectiveSeminormAux p := by |
simp only [projectiveSeminormAux, Function.comp_apply, Multiset.map_coe, List.map_map,
Multiset.sum_coe]
rw [← smul_eq_mul, List.smul_sum, ← List.comp_map]
congr 2
ext x
simp only [Function.comp_apply, norm_mul, smul_eq_mul]
rw [mul_assoc]
| 7 | 1,096.633158 | 2 | 2 | 6 | 2,058 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 84 | 90 | theorem bddBelow_projectiveSemiNormAux (x : ⨂[𝕜] i, E i) :
BddBelow (Set.range (fun (p : lifts x) ↦ projectiveSeminormAux p.1)) := by |
existsi 0
rw [mem_lowerBounds]
simp only [Set.mem_range, Subtype.exists, exists_prop, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
exact fun p _ ↦ projectiveSeminormAux_nonneg p
| 5 | 148.413159 | 2 | 2 | 6 | 2,058 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 126 | 132 | theorem projectiveSeminorm_tprod_le (m : Π i, E i) :
projectiveSeminorm (⨂ₜ[𝕜] i, m i) ≤ ∏ i, ‖m i‖ := by |
rw [projectiveSeminorm_apply]
convert ciInf_le (bddBelow_projectiveSemiNormAux _) ⟨[((1 : 𝕜), m)] ,?_⟩
· simp only [projectiveSeminormAux, Function.comp_apply, List.map_cons, norm_one, one_mul,
List.map_nil, List.sum_cons, List.sum_nil, add_zero]
· rw [mem_lifts_iff, List.map_singleton, List.sum_singleton... | 5 | 148.413159 | 2 | 2 | 6 | 2,058 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 134 | 153 | theorem norm_eval_le_projectiveSeminorm (x : ⨂[𝕜] i, E i) (G : Type*) [SeminormedAddCommGroup G]
[NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) :
‖lift f.toMultilinearMap x‖ ≤ projectiveSeminorm x * ‖f‖ := by |
letI := nonempty_subtype.mpr (nonempty_lifts x)
rw [projectiveSeminorm_apply, Real.iInf_mul_of_nonneg (norm_nonneg _), projectiveSeminormAux]
refine le_ciInf ?_
intro ⟨p, hp⟩
rw [mem_lifts_iff] at hp
conv_lhs => rw [← hp, ← List.sum_map_hom, ← Multiset.sum_coe]
refine le_trans (norm_multiset_sum_le _) ?_... | 17 | 24,154,952.753575 | 2 | 2 | 6 | 2,058 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 56 | 103 | theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
(hf : ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp (ContinuousMap.addRight n)).restrict K‖)
(m : ℤ) : fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m = 𝓕 f m := by |
-- NB: This proof can be shortened somewhat by telescoping together some of the steps in the calc
-- block, but I think it's more legible this way. We start with preliminaries about the integrand.
let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
have neK : ∀ (K : Comp... | 45 | 34,934,271,057,485,095,000 | 2 | 2 | 3 | 2,059 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 107 | 121 | theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
(h_norm :
∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
(h_sum : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) := by |
let F : C(UnitAddCircle, ℂ) :=
⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩
have : Summable (fourierCoeff F) := by
convert h_sum
exact Real.fourierCoeff_tsum_comp_add h_norm _
convert (has_pointwise_sum_fourier_series_of_summable this x).tsum_eq.symm usin... | 10 | 22,026.465795 | 2 | 2 | 3 | 2,059 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 131 | 157 | theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by |
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved in-line rather than being formulated as a separate lemma.
have claim : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y →
... | 24 | 26,489,122,129.84347 | 2 | 2 | 3 | 2,059 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Topology.TietzeExtension
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analysis.NormedSpace.RCLike
universe u u₁ v w
-- this is not an instance because Lean cannot determine `𝕜`.
theorem TietzeExtension.o... | Mathlib/Analysis/Complex/Tietze.lean | 105 | 118 | theorem exists_norm_eq_restrict_eq (f : s →ᵇ E) :
∃ g : X →ᵇ E, ‖g‖ = ‖f‖ ∧ g.restrict s = f := by |
by_cases hf : ‖f‖ = 0; · exact ⟨0, by aesop⟩
have := Metric.instTietzeExtensionClosedBall.{u, v} 𝕜 (0 : E) (by aesop : 0 < ‖f‖)
have hf' x : f x ∈ Metric.closedBall 0 ‖f‖ := by simpa using f.norm_coe_le_norm x
obtain ⟨g, hg_mem, hg⟩ := (f : C(s, E)).exists_forall_mem_restrict_eq hs hf'
simp only [Metric.mem... | 12 | 162,754.791419 | 2 | 2 | 1 | 2,060 |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Types
#align_import category_theory.limits.filtered from "leanprover-community/mathlib"@"e4ee4e30418efcb8cf304ba76ad653aeec04ba6e"
universe w' w v u
noncomputable section
open Categor... | Mathlib/CategoryTheory/Limits/Filtered.lean | 40 | 48 | theorem IsFiltered.iff_nonempty_limit : IsFiltered C ↔
∀ {J : Type v} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
∃ (X : C), Nonempty (limit (F.op ⋙ yoneda.obj X)) := by |
rw [IsFiltered.iff_cocone_nonempty.{v}]
refine ⟨fun h J _ _ F => ?_, fun h J _ _ F => ?_⟩
· obtain ⟨c⟩ := h F
exact ⟨c.pt, ⟨(limitCompYonedaIsoCocone F c.pt).inv c.ι⟩⟩
· obtain ⟨pt, ⟨ι⟩⟩ := h F
exact ⟨⟨pt, (limitCompYonedaIsoCocone F pt).hom ι⟩⟩
| 6 | 403.428793 | 2 | 2 | 2 | 2,061 |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Types
#align_import category_theory.limits.filtered from "leanprover-community/mathlib"@"e4ee4e30418efcb8cf304ba76ad653aeec04ba6e"
universe w' w v u
noncomputable section
open Categor... | Mathlib/CategoryTheory/Limits/Filtered.lean | 52 | 60 | theorem IsCofiltered.iff_nonempty_limit : IsCofiltered C ↔
∀ {J : Type v} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
∃ (X : C), Nonempty (limit (F ⋙ coyoneda.obj (op X))) := by |
rw [IsCofiltered.iff_cone_nonempty.{v}]
refine ⟨fun h J _ _ F => ?_, fun h J _ _ F => ?_⟩
· obtain ⟨c⟩ := h F
exact ⟨c.pt, ⟨(limitCompCoyonedaIsoCone F c.pt).inv c.π⟩⟩
· obtain ⟨pt, ⟨π⟩⟩ := h F
exact ⟨⟨pt, (limitCompCoyonedaIsoCone F pt).hom π⟩⟩
| 6 | 403.428793 | 2 | 2 | 2 | 2,061 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.angle.sphere from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open FiniteDimensional Complex
open scoped EuclideanGeomet... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 32 | 48 | theorem oangle_eq_two_zsmul_oangle_sub_of_norm_eq {x y z : V} (hxyne : x ≠ y) (hxzne : x ≠ z)
(hxy : ‖x‖ = ‖y‖) (hxz : ‖x‖ = ‖z‖) : o.oangle y z = (2 : ℤ) • o.oangle (y - x) (z - x) := by |
have hy : y ≠ 0 := by
rintro rfl
rw [norm_zero, norm_eq_zero] at hxy
exact hxyne hxy
have hx : x ≠ 0 := norm_ne_zero_iff.1 (hxy.symm ▸ norm_ne_zero_iff.2 hy)
have hz : z ≠ 0 := norm_ne_zero_iff.1 (hxz ▸ norm_ne_zero_iff.2 hx)
calc
o.oangle y z = o.oangle x z - o.oangle x y := (o.oangle_sub_left... | 15 | 3,269,017.372472 | 2 | 2 | 1 | 2,062 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.calculus.diff_cont_on_cl from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Metric
open scoped Topology
variable (𝕜 : Type*) {E F G : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/DiffContOnCl.lean | 64 | 70 | theorem continuousOn_ball [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (ball x r)) :
ContinuousOn f (closedBall x r) := by |
rcases eq_or_ne r 0 with (rfl | hr)
· rw [closedBall_zero]
exact continuousOn_singleton f x
· rw [← closure_ball x hr]
exact h.continuousOn
| 5 | 148.413159 | 2 | 2 | 1 | 2,063 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.Perm
#align_import data.list.prime from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
open List
section CommMonoidWithZero
variable {M : Type*} [CommMonoidWithZero M]
| Mathlib/Data/List/Prime.lean | 27 | 38 | theorem Prime.dvd_prod_iff {p : M} {L : List M} (pp : Prime p) : p ∣ L.prod ↔ ∃ a ∈ L, p ∣ a := by |
constructor
· intro h
induction' L with L_hd L_tl L_ih
· rw [prod_nil] at h
exact absurd h pp.not_dvd_one
· rw [prod_cons] at h
cases' pp.dvd_or_dvd h with hd hd
· exact ⟨L_hd, mem_cons_self L_hd L_tl, hd⟩
· obtain ⟨x, hx1, hx2⟩ := L_ih hd
exact ⟨x, mem_cons_of_mem L_hd ... | 11 | 59,874.141715 | 2 | 2 | 1 | 2,064 |
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.convex.body from "leanprover-community/mathlib"@"858a10cf68fd6c06872950fc58c4dcc68d465591"
open scoped Pointwise Topology NNReal
variable {V : Type*}
struc... | Mathlib/Analysis/Convex/Body.lean | 93 | 97 | theorem zero_mem_of_symmetric (K : ConvexBody V) (h_symm : ∀ x ∈ K, - x ∈ K) : 0 ∈ K := by |
obtain ⟨x, hx⟩ := K.nonempty
rw [show 0 = (1/2 : ℝ) • x + (1/2 : ℝ) • (- x) by field_simp]
apply convex_iff_forall_pos.mp K.convex hx (h_symm x hx)
all_goals linarith
| 4 | 54.59815 | 2 | 2 | 1 | 2,065 |
import Mathlib.Algebra.Category.GroupCat.Colimits
import Mathlib.Algebra.Category.GroupCat.FilteredColimits
import Mathlib.Algebra.Category.GroupCat.Kernels
import Mathlib.Algebra.Category.GroupCat.Limits
import Mathlib.Algebra.Category.GroupCat.ZModuleEquivalence
import Mathlib.Algebra.Category.ModuleCat.Abelian
impo... | Mathlib/Algebra/Category/GroupCat/Abelian.lean | 51 | 57 | theorem exact_iff : Exact f g ↔ f.range = g.ker := by |
rw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm
((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),
fun h => ⟨(AddMonoidHom.range_le_ker_iff _ _).mp h.le,
(QuotientAddGroup.ker_le_range_iff ... | 6 | 403.428793 | 2 | 2 | 1 | 2,066 |
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