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.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {α β : Type*}
def cmpLE {α} [LE α] [@DecidableRel α (· ≤ ·)] (x y : α) : Ordering :=
if x ≤ y then if y ≤ x then Ordering.eq ... | Mathlib/Order/Compare.lean | 78 | 79 | theorem swap_eq_iff_eq_swap {o o' : Ordering} : o.swap = o' ↔ o = o'.swap := by |
rw [← swap_inj, swap_swap]
| 1 | 2.718282 | 0 | 1 | 4 | 1,096 |
import Mathlib.Data.Fin.VecNotation
#align_import data.fin.tuple.monotone from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Fin Matrix Function
variable {α : Type*}
| Mathlib/Data/Fin/Tuple/Monotone.lean | 21 | 24 | theorem liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :
((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f := by |
simp only [liftFun_iff_succ r, forall_fin_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,
castSucc_zero]
| 2 | 7.389056 | 1 | 1 | 1 | 1,097 |
import Mathlib.Topology.Category.LightProfinite.Limits
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
universe u
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
namespace LightProfinite
noncomputable
def EffectiveEpi.struct {B X : LightProfinite.{u}} (... | Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean | 54 | 58 | theorem effectiveEpi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
EffectiveEpi f ↔ Function.Surjective f := by |
refine ⟨fun h ↦ ?_, fun h ↦ ⟨⟨EffectiveEpi.struct f h⟩⟩⟩
rw [← epi_iff_surjective]
infer_instance
| 3 | 20.085537 | 1 | 1 | 1 | 1,098 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 52 | 55 | theorem _root_.Filter.cardinalInterFilter_aleph0 (l : Filter α) : CardinalInterFilter l aleph0 where
cardinal_sInter_mem := by |
simp_all only [aleph_zero, lt_aleph0_iff_subtype_finite, setOf_mem_eq, sInter_mem,
implies_true, forall_const]
| 2 | 7.389056 | 1 | 1 | 6 | 1,099 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 90 | 94 | theorem cardinal_iInter_mem {s : ι → Set α} (hic : #ι < c) :
(⋂ i, s i) ∈ l ↔ ∀ i, s i ∈ l := by |
rw [← sInter_range _]
apply (cardinal_sInter_mem (lt_of_le_of_lt Cardinal.mk_range_le hic)).trans
exact forall_mem_range
| 3 | 20.085537 | 1 | 1 | 6 | 1,099 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 96 | 100 | theorem cardinal_bInter_mem {S : Set ι} (hS : #S < c)
{s : ∀ i ∈ S, Set α} :
(⋂ i, ⋂ hi : i ∈ S, s i ‹_›) ∈ l ↔ ∀ i, ∀ hi : i ∈ S, s i ‹_› ∈ l := by |
rw [biInter_eq_iInter]
exact (cardinal_iInter_mem hS).trans Subtype.forall
| 2 | 7.389056 | 1 | 1 | 6 | 1,099 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 102 | 105 | theorem eventually_cardinal_forall {p : α → ι → Prop} (hic : #ι < c) :
(∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_iInter_mem hic
| 2 | 7.389056 | 1 | 1 | 6 | 1,099 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 107 | 111 | theorem eventually_cardinal_ball {S : Set ι} (hS : #S < c)
{p : α → ∀ i ∈ S, Prop} :
(∀ᶠ x in l, ∀ i hi, p x i hi) ↔ ∀ i hi, ∀ᶠ x in l, p x i hi := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_bInter_mem hS
| 2 | 7.389056 | 1 | 1 | 6 | 1,099 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 123 | 127 | theorem EventuallyLE.cardinal_bUnion {S : Set ι} (hS : #S < c)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_› := by |
simp only [biUnion_eq_iUnion]
exact EventuallyLE.cardinal_iUnion hS fun i => h i i.2
| 2 | 7.389056 | 1 | 1 | 6 | 1,099 |
import Mathlib.Analysis.NormedSpace.ContinuousAffineMap
import Mathlib.Analysis.Calculus.ContDiff.Basic
#align_import analysis.calculus.affine_map from "leanprover-community/mathlib"@"839b92fedff9981cf3fe1c1f623e04b0d127f57c"
namespace ContinuousAffineMap
variable {𝕜 V W : Type*} [NontriviallyNormedField 𝕜]
va... | Mathlib/Analysis/Calculus/AffineMap.lean | 30 | 33 | theorem contDiff {n : ℕ∞} (f : V →ᴬ[𝕜] W) : ContDiff 𝕜 n f := by |
rw [f.decomp]
apply f.contLinear.contDiff.add
exact contDiff_const
| 3 | 20.085537 | 1 | 1 | 1 | 1,100 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 42 | 44 | theorem verschiebungFun_coeff (x : 𝕎 R) (n : ℕ) :
(verschiebungFun x).coeff n = if n = 0 then 0 else x.coeff (n - 1) := by |
simp only [verschiebungFun, ge_iff_le]
| 1 | 2.718282 | 0 | 1 | 6 | 1,101 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 47 | 48 | theorem verschiebungFun_coeff_zero (x : 𝕎 R) : (verschiebungFun x).coeff 0 = 0 := by |
rw [verschiebungFun_coeff, if_pos rfl]
| 1 | 2.718282 | 0 | 1 | 6 | 1,101 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 58 | 61 | theorem ghostComponent_zero_verschiebungFun (x : 𝕎 R) :
ghostComponent 0 (verschiebungFun x) = 0 := by |
rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,
verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul]
| 2 | 7.389056 | 1 | 1 | 6 | 1,101 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 65 | 71 | theorem ghostComponent_verschiebungFun (x : 𝕎 R) (n : ℕ) :
ghostComponent (n + 1) (verschiebungFun x) = p * ghostComponent n x := by |
simp only [ghostComponent_apply, aeval_wittPolynomial]
rw [Finset.sum_range_succ', verschiebungFun_coeff, if_pos rfl,
zero_pow (pow_ne_zero _ hp.1.ne_zero), mul_zero, add_zero, Finset.mul_sum, Finset.sum_congr rfl]
rintro i -
simp only [pow_succ', verschiebungFun_coeff_succ, Nat.succ_sub_succ_eq_sub, mul_a... | 5 | 148.413159 | 2 | 1 | 6 | 1,101 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 86 | 92 | theorem aeval_verschiebung_poly' (x : 𝕎 R) (n : ℕ) :
aeval x.coeff (verschiebungPoly n) = (verschiebungFun x).coeff n := by |
cases' n with n
· simp only [verschiebungPoly, Nat.zero_eq, ge_iff_le, tsub_eq_zero_of_le, ite_true, map_zero,
verschiebungFun_coeff_zero]
· rw [verschiebungPoly, verschiebungFun_coeff_succ, if_neg n.succ_ne_zero, aeval_X,
add_tsub_cancel_right]
| 5 | 148.413159 | 2 | 1 | 6 | 1,101 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 139 | 143 | theorem map_verschiebung (f : R →+* S) (x : 𝕎 R) :
map f (verschiebung x) = verschiebung (map f x) := by |
ext ⟨-, -⟩
· exact f.map_zero
· rfl
| 3 | 20.085537 | 1 | 1 | 6 | 1,101 |
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
universe u₁ u₂ w
namespace CategoryTheory
open Limits Functor
variable {C : Type u₁} [Category.{u₂} C]
namespace PreGaloisCategory
variable [GaloisCategory C]
section Decomposition
private lemma... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 111 | 115 | theorem has_decomp_connected_components (X : C) :
∃ (ι : Type) (f : ι → C) (g : (i : ι) → f i ⟶ X) (_ : IsColimit (Cofan.mk X g)),
(∀ i, IsConnected (f i)) ∧ Finite ι := by |
let F := GaloisCategory.getFiberFunctor C
exact has_decomp_connected_components_aux F (Nat.card <| F.obj X) X rfl
| 2 | 7.389056 | 1 | 1 | 2 | 1,102 |
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
universe u₁ u₂ w
namespace CategoryTheory
open Limits Functor
variable {C : Type u₁} [Category.{u₂} C]
namespace PreGaloisCategory
variable [GaloisCategory C]
section Decomposition
private lemma... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 118 | 121 | theorem has_decomp_connected_components' (X : C) :
∃ (ι : Type) (_ : Finite ι) (f : ι → C) (_ : ∐ f ≅ X), ∀ i, IsConnected (f i) := by |
obtain ⟨ι, f, g, hl, hc, hf⟩ := has_decomp_connected_components X
exact ⟨ι, hf, f, colimit.isoColimitCocone ⟨Cofan.mk X g, hl⟩, hc⟩
| 2 | 7.389056 | 1 | 1 | 2 | 1,102 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 42 | 47 | theorem frobenius_verschiebung (x : 𝕎 R) : frobenius (verschiebung x) = x * p := by |
have : IsPoly p fun {R} [CommRing R] x ↦ frobenius (verschiebung x) :=
IsPoly.comp (hg := frobenius_isPoly p) (hf := verschiebung_isPoly)
have : IsPoly p fun {R} [CommRing R] x ↦ x * p := mulN_isPoly p p
ghost_calc x
ghost_simp [mul_comm]
| 5 | 148.413159 | 2 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 51 | 52 | theorem verschiebung_zmod (x : 𝕎 (ZMod p)) : verschiebung x = x * p := by |
rw [← frobenius_verschiebung, frobenius_zmodp]
| 1 | 2.718282 | 0 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 57 | 61 | theorem coeff_p_pow [CharP R p] (i : ℕ) : ((p : 𝕎 R) ^ i).coeff i = 1 := by |
induction' i with i h
· simp only [Nat.zero_eq, one_coeff_zero, Ne, pow_zero]
· rw [pow_succ, ← frobenius_verschiebung, coeff_frobenius_charP,
verschiebung_coeff_succ, h, one_pow]
| 4 | 54.59815 | 2 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 64 | 71 | theorem coeff_p_pow_eq_zero [CharP R p] {i j : ℕ} (hj : j ≠ i) : ((p : 𝕎 R) ^ i).coeff j = 0 := by |
induction' i with i hi generalizing j
· rw [pow_zero, one_coeff_eq_of_pos]
exact Nat.pos_of_ne_zero hj
· rw [pow_succ, ← frobenius_verschiebung, coeff_frobenius_charP]
cases j
· rw [verschiebung_coeff_zero, zero_pow hp.out.ne_zero]
· rw [verschiebung_coeff_succ, hi (ne_of_apply_ne _ hj), zero_pow... | 7 | 1,096.633158 | 2 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 74 | 77 | theorem coeff_p [CharP R p] (i : ℕ) : (p : 𝕎 R).coeff i = if i = 1 then 1 else 0 := by |
split_ifs with hi
· simpa only [hi, pow_one] using coeff_p_pow p R 1
· simpa only [pow_one] using coeff_p_pow_eq_zero p R hi
| 3 | 20.085537 | 1 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 81 | 83 | theorem coeff_p_zero [CharP R p] : (p : 𝕎 R).coeff 0 = 0 := by |
rw [coeff_p, if_neg]
exact zero_ne_one
| 2 | 7.389056 | 1 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 87 | 87 | theorem coeff_p_one [CharP R p] : (p : 𝕎 R).coeff 1 = 1 := by | rw [coeff_p, if_pos rfl]
| 1 | 2.718282 | 0 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 90 | 92 | theorem p_nonzero [Nontrivial R] [CharP R p] : (p : 𝕎 R) ≠ 0 := by |
intro h
simpa only [h, zero_coeff, zero_ne_one] using coeff_p_one p R
| 2 | 7.389056 | 1 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 95 | 96 | theorem FractionRing.p_nonzero [Nontrivial R] [CharP R p] : (p : FractionRing (𝕎 R)) ≠ 0 := by |
simpa using (IsFractionRing.injective (𝕎 R) (FractionRing (𝕎 R))).ne (WittVector.p_nonzero _ _)
| 1 | 2.718282 | 0 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 103 | 111 | theorem verschiebung_mul_frobenius (x y : 𝕎 R) :
verschiebung (x * frobenius y) = verschiebung x * y := by |
have : IsPoly₂ p fun {R} [Rcr : CommRing R] x y ↦ verschiebung (x * frobenius y) :=
IsPoly.comp₂ (hg := verschiebung_isPoly)
(hf := IsPoly₂.comp (hh := mulIsPoly₂) (hf := idIsPolyI' p) (hg := frobenius_isPoly p))
have : IsPoly₂ p fun {R} [CommRing R] x y ↦ verschiebung x * y :=
IsPoly₂.comp (hh := mu... | 7 | 1,096.633158 | 2 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 114 | 116 | theorem mul_charP_coeff_zero [CharP R p] (x : 𝕎 R) : (x * p).coeff 0 = 0 := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_zero,
zero_pow hp.out.ne_zero]
| 2 | 7.389056 | 1 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 119 | 121 | theorem mul_charP_coeff_succ [CharP R p] (x : 𝕎 R) (i : ℕ) :
(x * p).coeff (i + 1) = x.coeff i ^ p := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_succ]
| 1 | 2.718282 | 0 | 1 | 13 | 1,103 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 124 | 127 | theorem verschiebung_frobenius [CharP R p] (x : 𝕎 R) : verschiebung (frobenius x) = x * p := by |
ext ⟨i⟩
· rw [mul_charP_coeff_zero, verschiebung_coeff_zero]
· rw [mul_charP_coeff_succ, verschiebung_coeff_succ, coeff_frobenius_charP]
| 3 | 20.085537 | 1 | 1 | 13 | 1,103 |
import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 78 | 85 | theorem union_left {s'} (h : IsMetricSeparated s t) (h' : IsMetricSeparated s' t) :
IsMetricSeparated (s ∪ s') t := by |
rcases h, h' with ⟨⟨r, r0, hr⟩, ⟨r', r0', hr'⟩⟩
refine ⟨min r r', ?_, fun x hx y hy => hx.elim ?_ ?_⟩
· rw [← pos_iff_ne_zero] at r0 r0' ⊢
exact lt_min r0 r0'
· exact fun hx => (min_le_left _ _).trans (hr _ hx _ hy)
· exact fun hx => (min_le_right _ _).trans (hr' _ hx _ hy)
| 6 | 403.428793 | 2 | 1 | 3 | 1,104 |
import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 106 | 109 | theorem finite_iUnion_left_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set X}
{t : Set X} : IsMetricSeparated (⋃ i ∈ I, s i) t ↔ ∀ i ∈ I, IsMetricSeparated (s i) t := by |
refine Finite.induction_on hI (by simp) @fun i I _ _ hI => ?_
rw [biUnion_insert, forall_mem_insert, union_left_iff, hI]
| 2 | 7.389056 | 1 | 1 | 3 | 1,104 |
import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 115 | 117 | theorem finite_iUnion_right_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : Set X}
{t : ι → Set X} : IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) := by |
simpa only [@comm _ _ s] using finite_iUnion_left_iff hI
| 1 | 2.718282 | 0 | 1 | 3 | 1,104 |
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.PowerSeries.Order
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 54 | 81 | theorem coeff_inv_aux (n : ℕ) (a : R) (φ : R⟦X⟧) :
coeff R n (inv.aux a φ) =
if n = 0 then a
else
-a *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (inv.aux a φ) else 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, inv.aux, MvPowerSeries.coeff_inv_aux]
simp only [Finsupp.single_eq_zero]
split_ifs; · rfl
congr 1
symm
apply Finset.sum_nbij' (fun (a, b) ↦ (single () a, single () b))
fun (f, g) ↦ (f (), g ())
· aesop
· aesop
· ... | 21 | 1,318,815,734.483215 | 2 | 1 | 2 | 1,105 |
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.MvPowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.RingTheory.PowerSeries.Order
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 100 | 102 | theorem constantCoeff_invOfUnit (φ : R⟦X⟧) (u : Rˣ) :
constantCoeff R (invOfUnit φ u) = ↑u⁻¹ := by |
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
| 1 | 2.718282 | 0 | 1 | 2 | 1,105 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 42 | 44 | theorem log_eq_zero_iff {b n : ℕ} : log b n = 0 ↔ n < b ∨ b ≤ 1 := by |
rw [log, dite_eq_right_iff]
simp only [Nat.add_eq_zero_iff, Nat.one_ne_zero, and_false, imp_false, not_and_or, not_le, not_lt]
| 2 | 7.389056 | 1 | 1 | 6 | 1,106 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 56 | 57 | theorem log_pos_iff {b n : ℕ} : 0 < log b n ↔ b ≤ n ∧ 1 < b := by |
rw [Nat.pos_iff_ne_zero, Ne, log_eq_zero_iff, not_or, not_lt, not_le]
| 1 | 2.718282 | 0 | 1 | 6 | 1,106 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 64 | 66 | theorem log_of_one_lt_of_le {b n : ℕ} (h : 1 < b) (hn : b ≤ n) : log b n = log b (n / b) + 1 := by |
rw [log]
exact if_pos ⟨hn, h⟩
| 2 | 7.389056 | 1 | 1 | 6 | 1,106 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 89 | 101 | theorem pow_le_iff_le_log {b : ℕ} (hb : 1 < b) {x y : ℕ} (hy : y ≠ 0) :
b ^ x ≤ y ↔ x ≤ log b y := by |
induction' y using Nat.strong_induction_on with y ih generalizing x
cases x with
| zero => dsimp; omega
| succ x =>
rw [log]; split_ifs with h
· have b_pos : 0 < b := lt_of_succ_lt hb
rw [Nat.add_le_add_iff_right, ← ih (y / b) (div_lt_self
(Nat.pos_iff_ne_zero.2 hy) hb) (Nat.div_pos h.1 b... | 11 | 59,874.141715 | 2 | 1 | 6 | 1,106 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 108 | 111 | theorem pow_le_of_le_log {b x y : ℕ} (hy : y ≠ 0) (h : x ≤ log b y) : b ^ x ≤ y := by |
refine (le_or_lt b 1).elim (fun hb => ?_) fun hb => (pow_le_iff_le_log hb hy).2 h
rw [log_of_left_le_one hb, Nat.le_zero] at h
rwa [h, Nat.pow_zero, one_le_iff_ne_zero]
| 3 | 20.085537 | 1 | 1 | 6 | 1,106 |
import Mathlib.Data.Nat.Defs
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.log from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Nat
--@[pp_nodot] porting note: unknown attribute
def log (b : ℕ) : ℕ → ℕ
| n => i... | Mathlib/Data/Nat/Log.lean | 114 | 116 | theorem le_log_of_pow_le {b x y : ℕ} (hb : 1 < b) (h : b ^ x ≤ y) : x ≤ log b y := by |
rcases ne_or_eq y 0 with (hy | rfl)
exacts [(pow_le_iff_le_log hb hy).1 h, (h.not_lt (Nat.pow_pos (Nat.zero_lt_one.trans hb))).elim]
| 2 | 7.389056 | 1 | 1 | 6 | 1,106 |
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.CategoryTheory.Preadditive.Yoneda.Limits
import Mathlib.CategoryTheory.Preadditive.Yoneda.Injective
#align_import category_theory.abelian.injective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9ed22... | Mathlib/CategoryTheory/Abelian/Injective.lean | 45 | 48 | theorem injective_of_preservesFiniteColimits_preadditiveYonedaObj (J : C)
[hP : PreservesFiniteColimits (preadditiveYonedaObj J)] : Injective J := by |
rw [injective_iff_preservesEpimorphisms_preadditive_yoneda_obj']
infer_instance
| 2 | 7.389056 | 1 | 1 | 1 | 1,107 |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
| Mathlib/Data/Complex/Determinant.lean | 24 | 26 | theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by |
rw [← LinearMap.det_toMatrix basisOneI, toMatrix_conjAe, Matrix.det_fin_two_of]
simp
| 2 | 7.389056 | 1 | 1 | 2 | 1,108 |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by
rw [← LinearMap.det_toMat... | Mathlib/Data/Complex/Determinant.lean | 31 | 33 | theorem linearEquiv_det_conjAe : LinearEquiv.det conjAe.toLinearEquiv = -1 := by |
rw [← Units.eq_iff, LinearEquiv.coe_det, AlgEquiv.toLinearEquiv_toLinearMap, det_conjAe,
Units.coe_neg_one]
| 2 | 7.389056 | 1 | 1 | 2 | 1,108 |
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Types
namespace CategoryTheory.FunctorToTypes
open CategoryTheory.Limits
universe w v₁ v₂ u₁ u₂
variable {J : Type u₁} [Category.{v₁} J] {K : Type u₂} [Category.{v₂} K]
variable (F : J ⥤ K ⥤ TypeMax.{u₁, w})
| Mathlib/CategoryTheory/Limits/FunctorToTypes.lean | 25 | 29 | theorem jointly_surjective (k : K) {t : Cocone F} (h : IsColimit t) (x : t.pt.obj k) :
∃ j y, x = (t.ι.app j).app k y := by |
let hev := isColimitOfPreserves ((evaluation _ _).obj k) h
obtain ⟨j, y, rfl⟩ := Types.jointly_surjective _ hev x
exact ⟨j, y, by simp⟩
| 3 | 20.085537 | 1 | 1 | 1 | 1,109 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 41 | 48 | theorem lift_rank_mul_lift_rank :
Cardinal.lift.{w} (Module.rank F K) * Cardinal.lift.{v} (Module.rank K A) =
Cardinal.lift.{v} (Module.rank F A) := by |
let b := Module.Free.chooseBasis F K
let c := Module.Free.chooseBasis K A
rw [← (Module.rank F K).lift_id, ← b.mk_eq_rank, ← (Module.rank K A).lift_id, ← c.mk_eq_rank,
← lift_umax.{w, v}, ← (b.smul c).mk_eq_rank, mk_prod, lift_mul, lift_lift, lift_lift, lift_lift,
lift_lift, lift_umax.{v, w}]
| 5 | 148.413159 | 2 | 1 | 5 | 1,110 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 55 | 58 | theorem rank_mul_rank (A : Type v) [AddCommGroup A]
[Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A] :
Module.rank F K * Module.rank K A = Module.rank F A := by |
convert lift_rank_mul_lift_rank F K A <;> rw [lift_id]
| 1 | 2.718282 | 0 | 1 | 5 | 1,110 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 63 | 66 | theorem FiniteDimensional.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by |
simp_rw [finrank]
rw [← toNat_lift.{w} (Module.rank F K), ← toNat_lift.{v} (Module.rank K A), ← toNat_mul,
lift_rank_mul_lift_rank, toNat_lift]
| 3 | 20.085537 | 1 | 1 | 5 | 1,110 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 88 | 90 | theorem _root_.FiniteDimensional.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
finrank R M = Fintype.card (ChooseBasisIndex R M) := by |
simp [finrank, rank_eq_card_chooseBasisIndex]
| 1 | 2.718282 | 0 | 1 | 5 | 1,110 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 111 | 118 | theorem nonempty_linearEquiv_of_lift_rank_eq
(cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :
Nonempty (M ≃ₗ[R] M') := by |
obtain ⟨⟨α, B⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
obtain ⟨⟨β, B'⟩⟩ := Module.Free.exists_basis (R := R) (M := M')
have : Cardinal.lift.{v', v} #α = Cardinal.lift.{v, v'} #β := by
rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']
exact (Cardinal.lift_mk_eq.{v, v', 0}.1 this).map (B.equiv B')
| 5 | 148.413159 | 2 | 1 | 5 | 1,110 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 79 | 81 | theorem invOf_eq [Invertible A.det] [Invertible A] : ⅟ A = ⅟ A.det • A.adjugate := by |
letI := invertibleOfDetInvertible A
convert (rfl : ⅟ A = _)
| 2 | 7.389056 | 1 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 103 | 105 | theorem det_invOf [Invertible A] [Invertible A.det] : (⅟ A).det = ⅟ A.det := by |
letI := detInvertibleOfInvertible A
convert (rfl : _ = ⅟ A.det)
| 2 | 7.389056 | 1 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 120 | 129 | theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1 :=
suffices ∀ A B : Matrix n n α, A * B = 1 → B * A = 1 from ⟨this A B, this B A⟩
fun A B h => by
letI : Invertible B.det := detInvertibleOfLeftInverse _ _ h
letI : Invertible B := invertibleOfDetInvertible B
calc
B * A = B * A * (B * ⅟ B) := by | rw [mul_invOf_self, Matrix.mul_one]
_ = B * (A * B * ⅟ B) := by simp only [Matrix.mul_assoc]
_ = B * ⅟ B := by rw [h, Matrix.one_mul]
_ = 1 := mul_invOf_self B
| 4 | 54.59815 | 2 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 151 | 152 | theorem isUnit_iff_isUnit_det : IsUnit A ↔ IsUnit A.det := by |
simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivDetInvertible A).nonempty_congr]
| 1 | 2.718282 | 0 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 205 | 207 | theorem isUnit_det_transpose (h : IsUnit A.det) : IsUnit Aᵀ.det := by |
rw [det_transpose]
exact h
| 2 | 7.389056 | 1 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 221 | 222 | theorem nonsing_inv_apply_not_isUnit (h : ¬IsUnit A.det) : A⁻¹ = 0 := by |
rw [inv_def, Ring.inverse_non_unit _ h, zero_smul]
| 1 | 2.718282 | 0 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 225 | 226 | theorem nonsing_inv_apply (h : IsUnit A.det) : A⁻¹ = (↑h.unit⁻¹ : α) • A.adjugate := by |
rw [inv_def, ← Ring.inverse_unit h.unit, IsUnit.unit_spec]
| 1 | 2.718282 | 0 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 385 | 392 | theorem vecMul_surjective_iff_exists_left_inverse [Fintype m] [Finite n] {A : Matrix m n R} :
Function.Surjective A.vecMul ↔ ∃ B : Matrix n m R, B * A = 1 := by |
cases nonempty_fintype n
refine ⟨fun h ↦ ?_, fun ⟨B, hBA⟩ y ↦ ⟨y ᵥ* B, by simp [hBA]⟩⟩
choose rows hrows using (h <| Pi.single · 1)
refine ⟨Matrix.of rows, Matrix.ext fun i j => ?_⟩
rw [mul_apply_eq_vecMul, one_eq_pi_single, ← hrows]
rfl
| 6 | 403.428793 | 2 | 1 | 9 | 1,111 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 394 | 401 | theorem mulVec_surjective_iff_exists_right_inverse [Finite m] [Fintype n] {A : Matrix m n R} :
Function.Surjective A.mulVec ↔ ∃ B : Matrix n m R, A * B = 1 := by |
cases nonempty_fintype m
refine ⟨fun h ↦ ?_, fun ⟨B, hBA⟩ y ↦ ⟨B *ᵥ y, by simp [hBA]⟩⟩
choose cols hcols using (h <| Pi.single · 1)
refine ⟨(Matrix.of cols)ᵀ, Matrix.ext fun i j ↦ ?_⟩
rw [one_eq_pi_single, Pi.single_comm, ← hcols j]
rfl
| 6 | 403.428793 | 2 | 1 | 9 | 1,111 |
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368cac4abd5a5cbe44317ba7e87379d51ed88"
open Finset
namespace Nat
| Mathlib/Data/Nat/Squarefree.lean | 28 | 30 | theorem squarefree_iff_nodup_factors {n : ℕ} (h0 : n ≠ 0) : Squarefree n ↔ n.factors.Nodup := by |
rw [UniqueFactorizationMonoid.squarefree_iff_nodup_normalizedFactors h0, Nat.factors_eq]
simp
| 2 | 7.389056 | 1 | 1 | 1 | 1,112 |
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Pointwise
universe u v
namespace MonoidHom
o... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 35 | 36 | theorem ker_eq_bot_of_cancel {f : A →* B} (h : ∀ u v : f.ker →* A, f.comp u = f.comp v → u = v) :
f.ker = ⊥ := by | simpa using _root_.congr_arg range (h f.ker.subtype 1 (by aesop_cat))
| 1 | 2.718282 | 0 | 1 | 2 | 1,113 |
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Pointwise
universe u v
namespace MonoidHom
o... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 47 | 56 | theorem range_eq_top_of_cancel {f : A →* B}
(h : ∀ u v : B →* B ⧸ f.range, u.comp f = v.comp f → u = v) : f.range = ⊤ := by |
specialize h 1 (QuotientGroup.mk' _) _
· ext1 x
simp only [one_apply, coe_comp, coe_mk', Function.comp_apply]
rw [show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _, QuotientGroup.eq, inv_one,
one_mul]
exact ⟨x, rfl⟩
replace h : (QuotientGroup.mk' f.range).ker = (1 : B →* B ⧸ f.range)... | 8 | 2,980.957987 | 2 | 1 | 2 | 1,113 |
import Mathlib.Data.Part
import Mathlib.Data.Nat.Upto
import Mathlib.Data.Stream.Defs
import Mathlib.Tactic.Common
#align_import control.fix from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v
open scoped Classical
variable {α : Type*} {β : α → Type*}
class Fix (α : Typ... | Mathlib/Control/Fix.lean | 111 | 113 | theorem fix_def' {x : α} (h' : ¬∃ i, (Fix.approx f i x).Dom) : Part.fix f x = none := by |
dsimp [Part.fix]
rw [assert_neg h']
| 2 | 7.389056 | 1 | 1 | 1 | 1,114 |
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
#align_import algebra.category.Module.kernels from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_option linter.uppercaseLean3 false
open CategoryTheory CategoryTheory.Limits... | Mathlib/Algebra/Category/ModuleCat/Kernels.lean | 137 | 140 | theorem cokernel_π_ext {M N : ModuleCat.{u} R} (f : M ⟶ N) {x y : N} (m : M) (w : x = y + f m) :
cokernel.π f x = cokernel.π f y := by |
subst w
simpa only [map_add, add_right_eq_self] using cokernel.condition_apply f m
| 2 | 7.389056 | 1 | 1 | 1 | 1,115 |
import Mathlib.CategoryTheory.LiftingProperties.Basic
import Mathlib.CategoryTheory.Adjunction.Basic
#align_import category_theory.lifting_properties.adjunction from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
open Category
variable {C D : Type*} [Category ... | Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean | 66 | 68 | theorem right_adjoint_hasLift_iff : HasLift (sq.right_adjoint adj) ↔ HasLift sq := by |
simp only [HasLift.iff]
exact Equiv.nonempty_congr (sq.rightAdjointLiftStructEquiv adj).symm
| 2 | 7.389056 | 1 | 1 | 2 | 1,116 |
import Mathlib.CategoryTheory.LiftingProperties.Basic
import Mathlib.CategoryTheory.Adjunction.Basic
#align_import category_theory.lifting_properties.adjunction from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
open Category
variable {C D : Type*} [Category ... | Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean | 111 | 113 | theorem left_adjoint_hasLift_iff : HasLift (sq.left_adjoint adj) ↔ HasLift sq := by |
simp only [HasLift.iff]
exact Equiv.nonempty_congr (sq.leftAdjointLiftStructEquiv adj).symm
| 2 | 7.389056 | 1 | 1 | 2 | 1,116 |
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 56 | 60 | theorem pow_le_pow_right' {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m :=
let ⟨k, hk⟩ := Nat.le.dest h
calc
a ^ n ≤ a ^ n * a ^ k := le_mul_of_one_le_right' (one_le_pow_of_one_le' ha _)
_ = a ^ m := by | rw [← hk, pow_add]
| 1 | 2.718282 | 0 | 1 | 3 | 1,117 |
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 71 | 77 | theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k := by |
rcases Nat.exists_eq_succ_of_ne_zero hk with ⟨l, rfl⟩
clear hk
induction' l with l IH
· rw [pow_succ]; simpa using ha
· rw [pow_succ]
exact one_lt_mul'' IH ha
| 6 | 403.428793 | 2 | 1 | 3 | 1,117 |
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 88 | 92 | theorem pow_lt_pow_right' [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ} (ha : 1 < a)
(h : n < m) : a ^ n < a ^ m := by |
rcases Nat.le.dest h with ⟨k, rfl⟩; clear h
rw [pow_add, pow_succ, mul_assoc, ← pow_succ']
exact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)
| 3 | 20.085537 | 1 | 1 | 3 | 1,117 |
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib.CategoryTheory.MorphismProperty.Basic
universe w v v' u u'
namespace CategoryTheory
namespace MorphismProperty
variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
def IsInvertedBy (P : MorphismProperty C) (F : C ⥤ D) : Prop :=
... | Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 128 | 131 | theorem IsInvertedBy.iff_of_iso (W : MorphismProperty C) {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) :
W.IsInvertedBy F₁ ↔ W.IsInvertedBy F₂ := by |
dsimp [IsInvertedBy]
simp only [NatIso.isIso_map_iff e]
| 2 | 7.389056 | 1 | 1 | 1 | 1,118 |
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed_space.indicator_function from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
variable {α E : Type*} [SeminormedAddCommGroup E] {s t : Set α} (f : α → E) (a : α)
open Se... | Mathlib/Analysis/NormedSpace/IndicatorFunction.lean | 34 | 37 | theorem norm_indicator_le_of_subset (h : s ⊆ t) (f : α → E) (a : α) :
‖indicator s f a‖ ≤ ‖indicator t f a‖ := by |
simp only [norm_indicator_eq_indicator_norm]
exact indicator_le_indicator_of_subset ‹_› (fun _ => norm_nonneg _) _
| 2 | 7.389056 | 1 | 1 | 2 | 1,119 |
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Analysis.Normed.Group.Basic
#align_import analysis.normed_space.indicator_function from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
variable {α E : Type*} [SeminormedAddCommGroup E] {s t : Set α} (f : α → E) (a : α)
open Se... | Mathlib/Analysis/NormedSpace/IndicatorFunction.lean | 44 | 46 | theorem norm_indicator_le_norm_self : ‖indicator s f a‖ ≤ ‖f a‖ := by |
rw [norm_indicator_eq_indicator_norm]
apply indicator_norm_le_norm_self
| 2 | 7.389056 | 1 | 1 | 2 | 1,119 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.concrete_category.reflects_isomorphisms from "leanprover-community/mathlib"@"73dd4b5411ec8fafb18a9d77c9c826907730af80"
universe u
namespace CategoryTheory
instance : (forget (Type... | Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.lean | 31 | 38 | theorem reflectsIsomorphisms_forget₂ [HasForget₂ C D] [(forget C).ReflectsIsomorphisms] :
(forget₂ C D).ReflectsIsomorphisms :=
{ reflects := fun X Y f {i} => by
haveI i' : IsIso ((forget D).map ((forget₂ C D).map f)) := Functor.map_isIso (forget D) _
haveI : IsIso ((forget C).map f) := by |
have := @HasForget₂.forget_comp C D
rwa [← this]
apply isIso_of_reflects_iso f (forget C) }
| 3 | 20.085537 | 1 | 1 | 1 | 1,120 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open scoped Int
variable {M G : Type*}
namespace Sem... | Mathlib/Algebra/Group/Semiconj/Units.lean | 48 | 51 | theorem units_inv_right {a : M} {x y : Mˣ} (h : SemiconjBy a x y) : SemiconjBy a ↑x⁻¹ ↑y⁻¹ :=
calc
a * ↑x⁻¹ = ↑y⁻¹ * (y * a) * ↑x⁻¹ := by | rw [Units.inv_mul_cancel_left]
_ = ↑y⁻¹ * a := by rw [← h.eq, mul_assoc, Units.mul_inv_cancel_right]
| 2 | 7.389056 | 1 | 1 | 2 | 1,121 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open scoped Int
variable {M G : Type*}
namespace Sem... | Mathlib/Algebra/Group/Semiconj/Units.lean | 64 | 67 | theorem units_inv_symm_left {a : Mˣ} {x y : M} (h : SemiconjBy (↑a) x y) : SemiconjBy (↑a⁻¹) y x :=
calc
↑a⁻¹ * y = ↑a⁻¹ * (y * a * ↑a⁻¹) := by | rw [Units.mul_inv_cancel_right]
_ = x * ↑a⁻¹ := by rw [← h.eq, ← mul_assoc, Units.inv_mul_cancel_left]
| 2 | 7.389056 | 1 | 1 | 2 | 1,121 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 58 | 60 | theorem zeroProdIso_inv_snd (X : C) : (zeroProdIso X).inv ≫ prod.snd = 𝟙 X := by |
dsimp [zeroProdIso, binaryFanZeroLeft]
simp
| 2 | 7.389056 | 1 | 1 | 3 | 1,122 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 89 | 91 | theorem prodZeroIso_iso_inv_snd (X : C) : (prodZeroIso X).inv ≫ prod.fst = 𝟙 X := by |
dsimp [prodZeroIso, binaryFanZeroRight]
simp
| 2 | 7.389056 | 1 | 1 | 3 | 1,122 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 115 | 117 | theorem inr_zeroCoprodIso_hom (X : C) : coprod.inr ≫ (zeroCoprodIso X).hom = 𝟙 X := by |
dsimp [zeroCoprodIso, binaryCofanZeroLeft]
simp
| 2 | 7.389056 | 1 | 1 | 3 | 1,122 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 91 | 94 | theorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :
HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by |
rw [← tsum_riemannZetaSummand hs]
apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs
| 2 | 7.389056 | 1 | 1 | 4 | 1,123 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 104 | 108 | theorem riemannZeta_eulerProduct (hs : 1 < s.re) :
Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - (p : ℂ) ^ (-s))⁻¹) atTop
(𝓝 (riemannZeta s)) := by |
rw [← tsum_riemannZetaSummand hs]
apply eulerProduct_completely_multiplicative <| summable_riemannZetaSummand hs
| 2 | 7.389056 | 1 | 1 | 4 | 1,123 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 114 | 118 | theorem dirichletLSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)
(hs : 1 < s.re) :
HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by |
rw [← tsum_dirichletSummand χ hs]
convert eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs
| 2 | 7.389056 | 1 | 1 | 4 | 1,123 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 129 | 133 | theorem dirichletLSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :
Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop
(𝓝 (L ↗χ s)) := by |
rw [← tsum_dirichletSummand χ hs]
apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs
| 2 | 7.389056 | 1 | 1 | 4 | 1,123 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 32 | 34 | theorem Coprime.dvd_of_dvd_mul_right (H1 : Coprime k n) (H2 : k ∣ m * n) : k ∣ m := by |
let t := dvd_gcd (Nat.dvd_mul_left k m) H2
rwa [gcd_mul_left, H1.gcd_eq_one, Nat.mul_one] at t
| 2 | 7.389056 | 1 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 39 | 44 | theorem Coprime.gcd_mul_left_cancel (m : Nat) (H : Coprime k n) : gcd (k * m) n = gcd m n :=
have H1 : Coprime (gcd (k * m) n) k := by |
rw [Coprime, Nat.gcd_assoc, H.symm.gcd_eq_one, gcd_one_right]
Nat.dvd_antisymm
(dvd_gcd (H1.dvd_of_dvd_mul_left (gcd_dvd_left _ _)) (gcd_dvd_right _ _))
(gcd_dvd_gcd_mul_left _ _ _)
| 4 | 54.59815 | 2 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 46 | 47 | theorem Coprime.gcd_mul_right_cancel (m : Nat) (H : Coprime k n) : gcd (m * k) n = gcd m n := by |
rw [Nat.mul_comm m k, H.gcd_mul_left_cancel m]
| 1 | 2.718282 | 0 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 49 | 51 | theorem Coprime.gcd_mul_left_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (k * n) = gcd m n := by |
rw [gcd_comm m n, gcd_comm m (k * n), H.gcd_mul_left_cancel n]
| 1 | 2.718282 | 0 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 53 | 55 | theorem Coprime.gcd_mul_right_cancel_right (n : Nat)
(H : Coprime k m) : gcd m (n * k) = gcd m n := by |
rw [Nat.mul_comm n k, H.gcd_mul_left_cancel_right n]
| 1 | 2.718282 | 0 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 57 | 59 | theorem coprime_div_gcd_div_gcd
(H : 0 < gcd m n) : Coprime (m / gcd m n) (n / gcd m n) := by |
rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H]
| 1 | 2.718282 | 0 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 65 | 75 | theorem exists_coprime (m n : Nat) :
∃ m' n', Coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n := by |
cases eq_zero_or_pos (gcd m n) with
| inl h0 =>
rw [gcd_eq_zero_iff] at h0
refine ⟨1, 1, gcd_one_left 1, ?_⟩
simp [h0]
| inr hpos =>
exact ⟨_, _, coprime_div_gcd_div_gcd hpos,
(Nat.div_mul_cancel (gcd_dvd_left m n)).symm,
(Nat.div_mul_cancel (gcd_dvd_right m n)).symm⟩
| 9 | 8,103.083928 | 2 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 87 | 91 | theorem Coprime.coprime_dvd_left (H1 : m ∣ k) (H2 : Coprime k n) : Coprime m n := by |
apply eq_one_of_dvd_one
rw [Coprime] at H2
have := Nat.gcd_dvd_gcd_of_dvd_left n H1
rwa [← H2]
| 4 | 54.59815 | 2 | 1 | 9 | 1,124 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 108 | 118 | theorem Coprime.coprime_div_left (cmn : Coprime m n) (dvd : a ∣ m) : Coprime (m / a) n := by |
match eq_zero_or_pos a with
| .inl h0 =>
rw [h0] at dvd
rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢
simp; assumption
| .inr hpos =>
let ⟨k, hk⟩ := dvd
rw [hk, Nat.mul_div_cancel_left _ hpos]
rw [hk] at cmn
exact cmn.coprime_mul_left
| 10 | 22,026.465795 | 2 | 1 | 9 | 1,124 |
import Mathlib.CategoryTheory.NatIso
import Mathlib.Logic.Equiv.Defs
#align_import category_theory.functor.fully_faithful from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ u₁ u₂ u₃
namespac... | Mathlib/CategoryTheory/Functor/FullyFaithful.lean | 125 | 127 | theorem preimageIso_mapIso (f : X ≅ Y) : F.preimageIso (F.mapIso f) = f := by |
ext
simp
| 2 | 7.389056 | 1 | 1 | 1 | 1,125 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.normed_space.extend from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable {𝕜 : Ty... | Mathlib/Analysis/NormedSpace/Extend.lean | 88 | 90 | theorem extendTo𝕜'_apply_re (fr : F →ₗ[ℝ] ℝ) (x : F) : re (fr.extendTo𝕜' x : 𝕜) = fr x := by |
simp only [extendTo𝕜'_apply, map_sub, zero_mul, mul_zero, sub_zero,
rclike_simps]
| 2 | 7.389056 | 1 | 1 | 2 | 1,126 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.normed_space.extend from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable {𝕜 : Ty... | Mathlib/Analysis/NormedSpace/Extend.lean | 93 | 99 | theorem norm_extendTo𝕜'_apply_sq (fr : F →ₗ[ℝ] ℝ) (x : F) :
‖(fr.extendTo𝕜' x : 𝕜)‖ ^ 2 = fr (conj (fr.extendTo𝕜' x : 𝕜) • x) :=
calc
‖(fr.extendTo𝕜' x : 𝕜)‖ ^ 2 = re (conj (fr.extendTo𝕜' x) * fr.extendTo𝕜' x : 𝕜) := by |
rw [RCLike.conj_mul, ← ofReal_pow, ofReal_re]
_ = fr (conj (fr.extendTo𝕜' x : 𝕜) • x) := by
rw [← smul_eq_mul, ← map_smul, extendTo𝕜'_apply_re]
| 3 | 20.085537 | 1 | 1 | 2 | 1,126 |
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
universe w' w v₁ v₂ u₁ u₂
noncomputable section
namespace CategoryTheory
open Category Limits
variable {C : Type u₁} [Catego... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 69 | 73 | theorem lift_comp_preservesLimitsIso_hom (t : Cone F) :
G.map (limit.lift _ t) ≫ (preservesLimitIso G F).hom =
limit.lift (F ⋙ G) (G.mapCone _) := by |
ext
simp [← G.map_comp]
| 2 | 7.389056 | 1 | 1 | 2 | 1,127 |
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
universe w' w v₁ v₂ u₁ u₂
noncomputable section
namespace CategoryTheory
open Category Limits
variable {C : Type u₁} [Catego... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 142 | 146 | theorem preservesColimitsIso_inv_comp_desc (t : Cocone F) :
(preservesColimitIso G F).inv ≫ G.map (colimit.desc _ t) =
colimit.desc _ (G.mapCocone t) := by |
ext
simp [← G.map_comp]
| 2 | 7.389056 | 1 | 1 | 2 | 1,127 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.hom.group from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
-- `NeZero` cannot be additivised, hence its theory should be developed outside of the
-- `Algebra.Group` folder.
assert_not_exists... | Mathlib/Algebra/Group/Hom/Basic.lean | 110 | 113 | theorem comp_mul [Mul M] [CommSemigroup N] [CommSemigroup P] (g : N →ₙ* P) (f₁ f₂ : M →ₙ* N) :
g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ := by |
ext
simp only [mul_apply, Function.comp_apply, map_mul, coe_comp]
| 2 | 7.389056 | 1 | 1 | 2 | 1,128 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.hom.group from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
-- `NeZero` cannot be additivised, hence its theory should be developed outside of the
-- `Algebra.Group` folder.
assert_not_exists... | Mathlib/Algebra/Group/Hom/Basic.lean | 252 | 254 | theorem comp_inv (φ : G →* H) (ψ : M →* G) : φ.comp ψ⁻¹ = (φ.comp ψ)⁻¹ := by |
ext
simp only [Function.comp_apply, inv_apply, map_inv, coe_comp]
| 2 | 7.389056 | 1 | 1 | 2 | 1,128 |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Filtered.Basic
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.Limits.Types.Filtered... | Mathlib/CategoryTheory/Limits/TypesFiltered.lean | 112 | 117 | theorem colimit_eq_iff_aux {i j : J} {xi : F.obj i} {xj : F.obj j} :
(colimitCocone F).ι.app i xi = (colimitCocone F).ι.app j xj ↔
FilteredColimit.Rel.{v, u} F ⟨i, xi⟩ ⟨j, xj⟩ := by |
dsimp
rw [← (equivShrink _).symm.injective.eq_iff, Equiv.symm_apply_apply, Equiv.symm_apply_apply,
Quot.eq, FilteredColimit.rel_eq_eqvGen_quot_rel]
| 3 | 20.085537 | 1 | 1 | 1 | 1,129 |
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