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.Algebra.Lie.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
#align_import algebra.lie.semisimple from "leanprover-community/mathlib"@"356447fe00e75e54777321045cdff7c9ea212e60"
namespace LieAlgebra
variable (R L : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
variable {R L} in
theorem Has... | Mathlib/Algebra/Lie/Semisimple/Basic.lean | 71 | 77 | theorem hasTrivialRadical_iff_no_abelian_ideals :
HasTrivialRadical R L ↔ ∀ I : LieIdeal R L, IsLieAbelian I → I = ⊥ := by |
rw [hasTrivialRadical_iff_no_solvable_ideals]
constructor <;> intro h₁ I h₂
· exact h₁ _ <| LieAlgebra.ofAbelianIsSolvable R I
· rw [← abelian_of_solvable_ideal_eq_bot_iff]
exact h₁ _ <| abelian_derivedAbelianOfIdeal I
| 5 | 148.413159 | 2 | 2 | 1 | 2,360 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 58 | 62 | theorem ultrafilter_isClosed_basic (s : Set α) : IsClosed { u : Ultrafilter α | s ∈ u } := by |
rw [← isOpen_compl_iff]
convert ultrafilter_isOpen_basic sᶜ using 1
ext u
exact Ultrafilter.compl_mem_iff_not_mem.symm
| 4 | 54.59815 | 2 | 2 | 5 | 2,361 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 67 | 77 | theorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :
↑u ≤ 𝓝 x ↔ x = joinM u := by |
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro h a ⟨xi, a, rfl⟩
exact h ... | 9 | 8,103.083928 | 2 | 2 | 5 | 2,361 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 110 | 117 | theorem ultrafilter_comap_pure_nhds (b : Ultrafilter α) : comap pure (𝓝 b) ≤ b := by |
rw [TopologicalSpace.nhds_generateFrom]
simp only [comap_iInf, comap_principal]
intro s hs
rw [← le_principal_iff]
refine iInf_le_of_le { u | s ∈ u } ?_
refine iInf_le_of_le ⟨hs, ⟨s, rfl⟩⟩ ?_
exact principal_mono.2 fun a => id
| 7 | 1,096.633158 | 2 | 2 | 5 | 2,361 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 122 | 126 | theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) := by |
intro x y h
have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure
rw [h] at this
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
| 4 | 54.59815 | 2 | 2 | 5 | 2,361 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 138 | 143 | theorem induced_topology_pure :
TopologicalSpace.induced (pure : α → Ultrafilter α) Ultrafilter.topologicalSpace = ⊥ := by |
apply eq_bot_of_singletons_open
intro x
use { u : Ultrafilter α | {x} ∈ u }, ultrafilter_isOpen_basic _
simp
| 4 | 54.59815 | 2 | 2 | 5 | 2,361 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 89 | 106 | theorem mdifferentiableAt_atlas (h : e ∈ atlas H M) {x : M} (hx : x ∈ e.source) :
MDifferentiableAt I I e x := by |
rw [mdifferentiableAt_iff]
refine ⟨(e.continuousOn x hx).continuousAt (e.open_source.mem_nhds hx), ?_⟩
have mem :
I ((chartAt H x : M → H) x) ∈ I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range I := by
simp only [hx, mfld_simps]
have : (chartAt H x).symm.trans e ∈ contDiffGroupoid ∞ I :=
HasGroup... | 16 | 8,886,110.520508 | 2 | 2 | 6 | 2,362 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 113 | 129 | theorem mdifferentiableAt_atlas_symm (h : e ∈ atlas H M) {x : H} (hx : x ∈ e.target) :
MDifferentiableAt I I e.symm x := by |
rw [mdifferentiableAt_iff]
refine ⟨(e.continuousOn_symm x hx).continuousAt (e.open_target.mem_nhds hx), ?_⟩
have mem : I x ∈ I.symm ⁻¹' (e.symm ≫ₕ chartAt H (e.symm x)).source ∩ range I := by
simp only [hx, mfld_simps]
have : e.symm.trans (chartAt H (e.symm x)) ∈ contDiffGroupoid ∞ I :=
HasGroupoid.com... | 15 | 3,269,017.372472 | 2 | 2 | 6 | 2,362 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 146 | 153 | theorem tangentMap_chart {p q : TangentBundle I M} (h : q.1 ∈ (chartAt H p.1).source) :
tangentMap I I (chartAt H p.1) q =
(TotalSpace.toProd _ _).symm
((chartAt (ModelProd H E) p : TangentBundle I M → ModelProd H E) q) := by |
dsimp [tangentMap]
rw [MDifferentiableAt.mfderiv]
· rfl
· exact mdifferentiableAt_atlas _ (chart_mem_atlas _ _) h
| 4 | 54.59815 | 2 | 2 | 6 | 2,362 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 159 | 169 | theorem tangentMap_chart_symm {p : TangentBundle I M} {q : TangentBundle I H}
(h : q.1 ∈ (chartAt H p.1).target) :
tangentMap I I (chartAt H p.1).symm q =
(chartAt (ModelProd H E) p).symm (TotalSpace.toProd H E q) := by |
dsimp only [tangentMap]
rw [MDifferentiableAt.mfderiv (mdifferentiableAt_atlas_symm _ (chart_mem_atlas _ _) h)]
simp only [ContinuousLinearMap.coe_coe, TangentBundle.chartAt, h, tangentBundleCore,
mfld_simps, (· ∘ ·)]
-- `simp` fails to apply `PartialEquiv.prod_symm` with `ModelProd`
congr
exact ((char... | 7 | 1,096.633158 | 2 | 2 | 6 | 2,362 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 200 | 210 | theorem symm_comp_deriv {x : M} (hx : x ∈ e.source) :
(mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) =
ContinuousLinearMap.id 𝕜 (TangentSpace I x) := by |
have : mfderiv I I (e.symm ∘ e) x = (mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) :=
mfderiv_comp x (he.mdifferentiableAt_symm (e.map_source hx)) (he.mdifferentiableAt hx)
rw [← this]
have : mfderiv I I (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id I
rw [← this]
apply Filter.Even... | 8 | 2,980.957987 | 2 | 2 | 6 | 2,362 |
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 263 | 273 | theorem trans (he' : e'.MDifferentiable I' I'') : (e.trans e').MDifferentiable I I'' := by |
constructor
· intro x hx
simp only [mfld_simps] at hx
exact
((he'.mdifferentiableAt hx.2).comp _ (he.mdifferentiableAt hx.1)).mdifferentiableWithinAt
· intro x hx
simp only [mfld_simps] at hx
exact
((he.symm.mdifferentiableAt hx.2).comp _
(he'.symm.mdifferentiableAt hx.1)).m... | 10 | 22,026.465795 | 2 | 2 | 6 | 2,362 |
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Common
#align_import data.nat.bitwise from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Function
names... | Mathlib/Data/Nat/Bitwise.lean | 75 | 81 | theorem binaryRec_of_ne_zero {C : Nat → Sort*} (z : C 0) (f : ∀ b n, C n → C (bit b n)) {n}
(h : n ≠ 0) :
binaryRec z f n = bit_decomp n ▸ f (bodd n) (div2 n) (binaryRec z f (div2 n)) := by |
rw [Eq.rec_eq_cast]
rw [binaryRec]
dsimp only
rw [dif_neg h, eq_mpr_eq_cast]
| 4 | 54.59815 | 2 | 2 | 1 | 2,363 |
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 93 | 98 | theorem isSatisfiable_onTheory_iff {L' : Language.{w, w'}} {φ : L →ᴸ L'} (h : φ.Injective) :
(φ.onTheory T).IsSatisfiable ↔ T.IsSatisfiable := by |
classical
refine ⟨isSatisfiable_of_isSatisfiable_onTheory φ, fun h' => ?_⟩
haveI : Inhabited h'.some := Classical.inhabited_of_nonempty'
exact Model.isSatisfiable (h'.some.defaultExpansion h)
| 4 | 54.59815 | 2 | 2 | 5 | 2,364 |
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 107 | 126 | theorem isSatisfiable_iff_isFinitelySatisfiable {T : L.Theory} :
T.IsSatisfiable ↔ T.IsFinitelySatisfiable :=
⟨Theory.IsSatisfiable.isFinitelySatisfiable, fun h => by
classical
set M : Finset T → Type max u v := fun T0 : Finset T =>
(h (T0.map (Function.Embedding.subtype fun x => x ∈ T)) T0.map_... |
refine ⟨fun φ hφ => ?_⟩
rw [Ultraproduct.sentence_realize]
refine
Filter.Eventually.filter_mono (Ultrafilter.of_le _)
(Filter.eventually_atTop.2
⟨{⟨φ, hφ⟩}, fun s h' =>
Theory.realize_sentence_of_mem (s.map (Function.Embedding.subtype fun x =>... | 12 | 162,754.791419 | 2 | 2 | 5 | 2,364 |
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 129 | 135 | theorem isSatisfiable_directed_union_iff {ι : Type*} [Nonempty ι] {T : ι → L.Theory}
(h : Directed (· ⊆ ·) T) : Theory.IsSatisfiable (⋃ i, T i) ↔ ∀ i, (T i).IsSatisfiable := by |
refine ⟨fun h' i => h'.mono (Set.subset_iUnion _ _), fun h' => ?_⟩
rw [isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]
intro T0 hT0
obtain ⟨i, hi⟩ := h.exists_mem_subset_of_finset_subset_biUnion hT0
exact (h' i).mono hi
| 5 | 148.413159 | 2 | 2 | 5 | 2,364 |
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 138 | 154 | theorem isSatisfiable_union_distinctConstantsTheory_of_card_le (T : L.Theory) (s : Set α)
(M : Type w') [Nonempty M] [L.Structure M] [M ⊨ T]
(h : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} #M) :
((L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s).IsSatisfiable := by |
haveI : Inhabited M := Classical.inhabited_of_nonempty inferInstance
rw [Cardinal.lift_mk_le'] at h
letI : (constantsOn α).Structure M := constantsOn.structure (Function.extend (↑) h.some default)
have : M ⊨ (L.lhomWithConstants α).onTheory T ∪ L.distinctConstantsTheory s := by
refine ((LHom.onTheory_model... | 13 | 442,413.392009 | 2 | 2 | 5 | 2,364 |
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 212 | 224 | theorem exists_elementaryEmbedding_card_eq_of_le (M : Type w') [L.Structure M] [Nonempty M]
(κ : Cardinal.{w}) (h1 : ℵ₀ ≤ κ) (h2 : lift.{w} L.card ≤ Cardinal.lift.{max u v} κ)
(h3 : lift.{w'} κ ≤ Cardinal.lift.{w} #M) :
∃ N : Bundled L.Structure, Nonempty (N ↪ₑ[L] M) ∧ #N = κ := by |
obtain ⟨S, _, hS⟩ := exists_elementarySubstructure_card_eq L ∅ κ h1 (by simp) h2 h3
have : Small.{w} S := by
rw [← lift_inj.{_, w + 1}, lift_lift, lift_lift] at hS
exact small_iff_lift_mk_lt_univ.2 (lt_of_eq_of_lt hS κ.lift_lt_univ')
refine
⟨(equivShrink S).bundledInduced L,
⟨S.subtype.comp (Eq... | 9 | 8,103.083928 | 2 | 2 | 5 | 2,364 |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 90 | 99 | theorem IsComplement'.symm (h : IsComplement' H K) : IsComplement' K H := by |
let ϕ : H × K ≃ K × H :=
Equiv.mk (fun x => ⟨x.2⁻¹, x.1⁻¹⟩) (fun x => ⟨x.2⁻¹, x.1⁻¹⟩)
(fun x => Prod.ext (inv_inv _) (inv_inv _)) fun x => Prod.ext (inv_inv _) (inv_inv _)
let ψ : G ≃ G := Equiv.mk (fun g : G => g⁻¹) (fun g : G => g⁻¹) inv_inv inv_inv
suffices hf : (ψ ∘ fun x : H × K => x.1.1 * x.2.1) ... | 9 | 8,103.083928 | 2 | 2 | 3 | 2,365 |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 124 | 128 | theorem isComplement_singleton_left {g : G} : IsComplement {g} S ↔ S = univ := by |
refine
⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => (congr_arg _ h).mpr isComplement_singleton_univ⟩
obtain ⟨⟨⟨z, rfl : z = g⟩, y, _⟩, hy⟩ := h.2 (g * x)
rwa [← mul_left_cancel hy]
| 4 | 54.59815 | 2 | 2 | 3 | 2,365 |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 133 | 139 | theorem isComplement_singleton_right {g : G} : IsComplement S {g} ↔ S = univ := by |
refine
⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => h ▸ isComplement_univ_singleton⟩
obtain ⟨y, hy⟩ := h.2 (x * g)
conv_rhs at hy => rw [← show y.2.1 = g from y.2.2]
rw [← mul_right_cancel hy]
exact y.1.2
| 6 | 403.428793 | 2 | 2 | 3 | 2,365 |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asym... | Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 36 | 42 | theorem hasDerivAt_exp (x : ℂ) : HasDerivAt exp (exp x) x := by |
rw [hasDerivAt_iff_isLittleO_nhds_zero]
have : (1 : ℕ) < 2 := by norm_num
refine (IsBigO.of_bound ‖exp x‖ ?_).trans_isLittleO (isLittleO_pow_id this)
filter_upwards [Metric.ball_mem_nhds (0 : ℂ) zero_lt_one]
simp only [Metric.mem_ball, dist_zero_right, norm_pow]
exact fun z hz => exp_bound_sq x z hz.le
| 6 | 403.428793 | 2 | 2 | 2 | 2,366 |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asym... | Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 64 | 73 | theorem contDiff_exp : ∀ {n}, ContDiff 𝕜 n exp := by |
-- Porting note: added `@` due to `∀ {n}` weirdness above
refine @(contDiff_all_iff_nat.2 fun n => ?_)
have : ContDiff ℂ (↑n) exp := by
induction' n with n ihn
· exact contDiff_zero.2 continuous_exp
· rw [contDiff_succ_iff_deriv]
use differentiable_exp
rwa [deriv_exp]
exact this.restric... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,366 |
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import category_theory.preadditive.schur from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 114 | 125 | theorem finrank_endomorphism_eq_one {X : C} (isIso_iff_nonzero : ∀ f : X ⟶ X, IsIso f ↔ f ≠ 0)
[I : FiniteDimensional 𝕜 (X ⟶ X)] : finrank 𝕜 (X ⟶ X) = 1 := by |
have id_nonzero := (isIso_iff_nonzero (𝟙 X)).mp (by infer_instance)
refine finrank_eq_one (𝟙 X) id_nonzero ?_
intro f
have : Nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero
have : FiniteDimensional 𝕜 (End X) := I
obtain ⟨c, nu⟩ := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional 𝕜 (End.of f)... | 10 | 22,026.465795 | 2 | 2 | 1 | 2,367 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Tactic.Linarith
open Finset Set
variable {ι 𝕜 E : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E}
| Mathlib/Analysis/Convex/Radon.lean | 26 | 50 | theorem radon_partition (h : ¬ AffineIndependent 𝕜 f) :
∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty := by |
rw [affineIndependent_iff] at h
push_neg at h
obtain ⟨s, w, h_wsum, h_vsum, nonzero_w_index, h1, h2⟩ := h
let I : Finset ι := s.filter fun i ↦ 0 ≤ w i
let J : Finset ι := s.filter fun i ↦ w i < 0
let p : E := centerMass I w f -- point of intersection
have hJI : ∑ j ∈ J, w j + ∑ i ∈ I, w i = 0 := by
s... | 23 | 9,744,803,446.248903 | 2 | 2 | 1 | 2,368 |
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Galois
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.finite.galois_field from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb330... | Mathlib/FieldTheory/Finite/GaloisField.lean | 55 | 60 | theorem galois_poly_separable {K : Type*} [Field K] (p q : ℕ) [CharP K p] (h : p ∣ q) :
Separable (X ^ q - X : K[X]) := by |
use 1, X ^ q - X - 1
rw [← CharP.cast_eq_zero_iff K[X] p] at h
rw [derivative_sub, derivative_X_pow, derivative_X, C_eq_natCast, h]
ring
| 4 | 54.59815 | 2 | 2 | 2 | 2,369 |
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Galois
import Mathlib.FieldTheory.SplittingField.IsSplittingField
#align_import field_theory.finite.galois_field from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb330... | Mathlib/FieldTheory/Finite/GaloisField.lean | 96 | 143 | theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n := by |
set g_poly := (X ^ p ^ n - X : (ZMod p)[X])
have hp : 1 < p := h_prime.out.one_lt
have aux : g_poly ≠ 0 := FiniteField.X_pow_card_pow_sub_X_ne_zero _ h hp
-- Porting note: in the statment of `key`, replaced `g_poly` by its value otherwise the
-- proof fails
have key : Fintype.card (g_poly.rootSet (GaloisFi... | 47 | 258,131,288,619,006,750,000 | 2 | 2 | 2 | 2,369 |
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Basic
#align_import init.control.lawful from "leanprover-community/lean"@"9af482290ef68e8aaa5ead01aa7b09b7be7019fd"
set_option autoImplicit true
universe u v
#align is_lawful_functor LawfulFunctor
#align is_lawful_functor.map_const_eq LawfulFunctor.map_const
... | Mathlib/Init/Control/Lawful.lean | 213 | 219 | theorem run_map (f : α → β) [LawfulMonad m] : (f <$> x).run = Option.map f <$> x.run := by |
rw [← bind_pure_comp _ x.run]
change x.run >>= (fun
| some a => OptionT.run (pure (f a))
| none => pure none) = _
apply bind_congr
intro a; cases a <;> simp [Option.map, Option.bind]
| 6 | 403.428793 | 2 | 2 | 1 | 2,370 |
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.locally_convex.continuous_of_bounded from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open TopologicalSpace Bornology Filter Topology Pointwise
variable {𝕜 𝕜' E F : Type*}
var... | Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean | 96 | 166 | theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
(hf : ∀ s, IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (f '' s)) : ContinuousAt f 0 := by |
-- Assume that f is not continuous at 0
by_contra h
-- We use a decreasing balanced basis for 0 : E and a balanced basis for 0 : F
-- and reformulate non-continuity in terms of these bases
rcases (nhds_basis_balanced 𝕜 E).exists_antitone_subbasis with ⟨b, bE1, bE⟩
simp only [_root_.id] at bE
have bE' : ... | 69 | 925,378,172,558,778,900,000,000,000,000 | 2 | 2 | 1 | 2,371 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
#align_import measure_theory.measure.complex from "leanprover-community/mathlib"@"17b3357baa47f48697ca9c243e300eb8cdd16a15"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
variable {α β : Type*} {m : MeasurableSpace α}
namespace Measur... | Mathlib/MeasureTheory/Measure/Complex.lean | 116 | 122 | theorem absolutelyContinuous_ennreal_iff (c : ComplexMeasure α) (μ : VectorMeasure α ℝ≥0∞) :
c ≪ᵥ μ ↔ ComplexMeasure.re c ≪ᵥ μ ∧ ComplexMeasure.im c ≪ᵥ μ := by |
constructor <;> intro h
· constructor <;> · intro i hi; simp [h hi]
· intro i hi
rw [← Complex.re_add_im (c i), (_ : (c i).re = 0), (_ : (c i).im = 0)]
exacts [by simp, h.2 hi, h.1 hi]
| 5 | 148.413159 | 2 | 2 | 1 | 2,372 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.inner_product_space.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
noncomputable section
open RCLike
open scoped ComplexConjugate Classical
variable ... | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 140 | 147 | theorem adjointAux_inner (y : T.adjointDomain) (x : T.domain) :
⟪adjointAux hT y, x⟫ = ⟪(y : F), T x⟫ := by |
simp only [adjointAux, LinearMap.coe_mk, InnerProductSpace.toDual_symm_apply,
adjointDomainMkCLMExtend_apply]
-- Porting note(https://github.com/leanprover-community/mathlib4/issues/5026):
-- mathlib3 was finished here
simp only [AddHom.coe_mk, InnerProductSpace.toDual_symm_apply]
rw [adjointDomainMkCLME... | 6 | 403.428793 | 2 | 2 | 2 | 2,373 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.inner_product_space.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
noncomputable section
open RCLike
open scoped ComplexConjugate Classical
variable ... | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 171 | 178 | theorem mem_adjoint_domain_of_exists (y : F) (h : ∃ w : E, ∀ x : T.domain, ⟪w, x⟫ = ⟪y, T x⟫) :
y ∈ T†.domain := by |
cases' h with w hw
rw [T.mem_adjoint_domain_iff]
-- Porting note: was `by continuity`
have : Continuous ((innerSL 𝕜 w).comp T.domain.subtypeL) := ContinuousLinearMap.continuous _
convert this using 1
exact funext fun x => (hw x).symm
| 6 | 403.428793 | 2 | 2 | 2 | 2,373 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
universe u... | Mathlib/Analysis/MeanInequalitiesPow.lean | 72 | 86 | theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
(∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by |
rcases s.eq_empty_or_nonempty with (rfl | hs)
· simp_rw [Finset.sum_empty, zero_pow n.succ_ne_zero, zero_div]; rfl
· have hs0 : 0 < (s.card : ℝ) := Nat.cast_pos.2 hs.card_pos
suffices (∑ x ∈ s, f x / s.card) ^ (n + 1) ≤ ∑ x ∈ s, f x ^ (n + 1) / s.card by
rwa [← Finset.sum_div, ← Finset.sum_div, div_pow... | 13 | 442,413.392009 | 2 | 2 | 2 | 2,374 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
universe u... | Mathlib/Analysis/MeanInequalitiesPow.lean | 101 | 110 | theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
(hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) := by |
have : 0 < p := by positivity
rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one]
· exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp
all_goals
apply_rules [sum_nonneg, rpow_nonneg]
intro i hi
apply_rules [mul_nonneg, rpow_nonneg, hw i hi, hz i hi]
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,374 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 44 | 68 | theorem norm_coe_mul (x : ℝ) (t : ℝ) :
‖(↑(t * x) : AddCircle (t * p))‖ = |t| * ‖(x : AddCircle p)‖ := by |
have aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b) := fun {a b c} h => by
simp only [mem_zmultiples_iff] at h ⊢
obtain ⟨n, rfl⟩ := h
exact ⟨n, (mul_smul_comm n c b).symm⟩
rcases eq_or_ne t 0 with (rfl | ht); · simp
have ht' : |t| ≠ 0 := (not_congr abs_eq_zero).mpr ht
simp only ... | 23 | 9,744,803,446.248903 | 2 | 2 | 5 | 2,375 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 71 | 75 | theorem norm_neg_period (x : ℝ) : ‖(x : AddCircle (-p))‖ = ‖(x : AddCircle p)‖ := by |
suffices ‖(↑(-1 * x) : AddCircle (-1 * p))‖ = ‖(x : AddCircle p)‖ by
rw [← this, neg_one_mul]
simp
simp only [norm_coe_mul, abs_neg, abs_one, one_mul]
| 4 | 54.59815 | 2 | 2 | 5 | 2,375 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 79 | 83 | theorem norm_eq_of_zero {x : ℝ} : ‖(x : AddCircle (0 : ℝ))‖ = |x| := by |
suffices { y : ℝ | (y : AddCircle (0 : ℝ)) = (x : AddCircle (0 : ℝ)) } = {x} by
rw [quotient_norm_eq, this, image_singleton, Real.norm_eq_abs, csInf_singleton]
ext y
simp [QuotientAddGroup.eq_iff_sub_mem, mem_zmultiples_iff, sub_eq_zero]
| 4 | 54.59815 | 2 | 2 | 5 | 2,375 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 86 | 117 | theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p| := by |
suffices ∀ x : ℝ, ‖(x : AddCircle (1 : ℝ))‖ = |x - round x| by
rcases eq_or_ne p 0 with (rfl | hp)
· simp
have hx := norm_coe_mul p x p⁻¹
rw [abs_inv, eq_inv_mul_iff_mul_eq₀ ((not_congr abs_eq_zero).mpr hp)] at hx
rw [← hx, inv_mul_cancel hp, this, ← abs_mul, mul_sub, mul_inv_cancel_left₀ hp, mul... | 31 | 29,048,849,665,247.426 | 2 | 2 | 5 | 2,375 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 120 | 124 | theorem norm_eq' (hp : 0 < p) {x : ℝ} : ‖(x : AddCircle p)‖ = p * |p⁻¹ * x - round (p⁻¹ * x)| := by |
conv_rhs =>
congr
rw [← abs_eq_self.mpr hp.le]
rw [← abs_mul, mul_sub, mul_inv_cancel_left₀ hp.ne.symm, norm_eq, mul_comm p]
| 4 | 54.59815 | 2 | 2 | 5 | 2,375 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
noncomputable section
open scoped ENNReal MeasureTheory Topology
open Set MeasureTheory Filter Measure
namespace MeasureTheory
se... | Mathlib/MeasureTheory/Integral/Layercake.lean | 73 | 82 | theorem countable_meas_le_ne_meas_lt (g : α → R) :
{t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by |
-- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
-- jumps down on the right of `t`. This jump set is countable for any function.
let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
apply (countable_image_gt_image_Ioi F).mono
intro t ht
have : μ {a | t < g a} < μ ... | 8 | 2,980.957987 | 2 | 2 | 2 | 2,376 |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
noncomputable section
open scoped ENNReal MeasureTheory Topology
open Set MeasureTheory Filter Measure
namespace MeasureTheory
se... | Mathlib/MeasureTheory/Integral/Layercake.lean | 105 | 183 | theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
(μ : Measure α) [SigmaFinite μ]
(f_nn : 0 ≤ f) (f_mble : Measurable f)
(g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
(g_nn : ∀ t > 0, 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ... |
have g_intble' : ∀ t : ℝ, 0 ≤ t → IntervalIntegrable g volume 0 t := by
intro t ht
cases' eq_or_lt_of_le ht with h h
· simp [← h]
· exact g_intble t h
have integrand_eq : ∀ ω,
ENNReal.ofReal (∫ t in (0)..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t) := by
intro ω
have g_ae_nn ... | 72 | 18,586,717,452,841,279,000,000,000,000,000 | 2 | 2 | 2 | 2,376 |
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 114 | 119 | theorem Coloring.card_colorClasses_le [Fintype α] [Fintype C.colorClasses] :
Fintype.card C.colorClasses ≤ Fintype.card α := by |
simp [colorClasses]
-- Porting note: brute force instance declaration `[Fintype (Setoid.classes (Setoid.ker C))]`
haveI : Fintype (Setoid.classes (Setoid.ker C)) := by assumption
convert Setoid.card_classes_ker_le C
| 4 | 54.59815 | 2 | 2 | 2 | 2,377 |
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 151 | 155 | theorem isEmpty_of_colorable_zero (h : G.Colorable 0) : IsEmpty V := by |
constructor
intro v
obtain ⟨i, hi⟩ := h.some v
exact Nat.not_lt_zero _ hi
| 4 | 54.59815 | 2 | 2 | 2 | 2,377 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.Preserves
universe v u w
namespace CategoryTheory
open Limits
variable {C : Type u} [Category.{v} C]
variable [FinitaryPreExten... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 52 | 58 | theorem isSheafFor_extensive_of_preservesFiniteProducts {X : C} (S : Presieve X) [S.Extensive]
(F : Cᵒᵖ ⥤ Type w) [PreservesFiniteProducts F] : S.IsSheafFor F := by |
obtain ⟨α, _, Z, π, rfl, ⟨hc⟩⟩ := Extensive.arrows_nonempty_isColimit (R := S)
have : (ofArrows Z (Cofan.mk X π).inj).hasPullbacks :=
(inferInstance : (ofArrows Z π).hasPullbacks)
cases nonempty_fintype α
exact isSheafFor_of_preservesProduct _ _ hc
| 5 | 148.413159 | 2 | 2 | 4 | 2,378 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.Preserves
universe v u w
namespace CategoryTheory
open Limits
variable {C : Type u} [Category.{v} C]
variable [FinitaryPreExten... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 64 | 70 | theorem extensiveTopology.isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (extensiveTopology C)
(yoneda.obj W) := by |
erw [isSheaf_coverage]
intro X R ⟨Y, α, Z, π, hR, hi⟩
have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩
exact isSheafFor_extensive_of_preservesFiniteProducts _ _
| 5 | 148.413159 | 2 | 2 | 4 | 2,378 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.Preserves
universe v u w
namespace CategoryTheory
open Limits
variable {C : Type u} [Category.{v} C]
variable [FinitaryPreExten... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 80 | 110 | theorem Presieve.isSheaf_iff_preservesFiniteProducts [FinitaryExtensive C] (F : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (extensiveTopology C) F ↔
Nonempty (PreservesFiniteProducts F) := by |
refine ⟨fun hF ↦ ⟨⟨fun α _ ↦ ⟨fun {K} ↦ ?_⟩⟩⟩, fun hF ↦ ?_⟩
· erw [Presieve.isSheaf_coverage] at hF
let Z : α → C := fun i ↦ unop (K.obj ⟨i⟩)
have : (Presieve.ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).hasPullbacks :=
(inferInstance : (Presieve.ofArrows Z (Sigma.ι Z)).hasPullbacks)
have : ∀ (i ... | 28 | 1,446,257,064,291.475 | 2 | 2 | 4 | 2,378 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.Preserves
universe v u w
namespace CategoryTheory
open Limits
variable {C : Type u} [Category.{v} C]
variable [FinitaryPreExten... | Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean | 115 | 132 | theorem Presheaf.isSheaf_iff_preservesFiniteProducts {D : Type*} [Category D]
[FinitaryExtensive C] (F : Cᵒᵖ ⥤ D) :
IsSheaf (extensiveTopology C) F ↔ Nonempty (PreservesFiniteProducts F) := by |
constructor
· intro h
rw [IsSheaf] at h
refine ⟨⟨fun J _ ↦ ⟨fun {K} ↦ ⟨fun {c} hc ↦ ?_⟩⟩⟩⟩
apply coyonedaJointlyReflectsLimits
intro ⟨E⟩
specialize h E
rw [Presieve.isSheaf_iff_preservesFiniteProducts] at h
have : PreservesLimit K (F.comp (coyoneda.obj ⟨E⟩)) := (h.some.preserves J).pres... | 15 | 3,269,017.372472 | 2 | 2 | 4 | 2,378 |
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.Analysis.RCLike.Lemmas
#align_import analysis.normed_space.hahn_banach.extension from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
univers... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 44 | 59 | theorem exists_extension_norm_eq (p : Subspace ℝ E) (f : p →L[ℝ] ℝ) :
∃ g : E →L[ℝ] ℝ, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ := by |
rcases exists_extension_of_le_sublinear ⟨p, f⟩ (fun x => ‖f‖ * ‖x‖)
(fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])
(fun x y => by -- Porting note: placeholder filled here
rw [← left_distrib]
exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 2,379 |
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.Analysis.RCLike.Lemmas
#align_import analysis.normed_space.hahn_banach.extension from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
univers... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 151 | 163 | theorem coord_norm' {x : E} (h : x ≠ 0) : ‖(‖x‖ : 𝕜) • coord 𝕜 x h‖ = 1 := by |
#adaptation_note
/--
`set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119
Alternatively, we can add:
```
let X : SeminormedAddCommGroup (↥(span 𝕜 {x}) →L[𝕜] 𝕜) := inferInstance
have : BoundedSMul 𝕜 (↥(span 𝕜 {x}) →L[𝕜] 𝕜) := @NormedSpace.boundedSMul 𝕜 _... | 12 | 162,754.791419 | 2 | 2 | 2 | 2,379 |
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import probability.martingale.convergence from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Filter Me... | Mathlib/Probability/Martingale/Convergence.lean | 110 | 127 | theorem not_frequently_of_upcrossings_lt_top (hab : a < b) (hω : upcrossings a b f ω ≠ ∞) :
¬((∃ᶠ n in atTop, f n ω < a) ∧ ∃ᶠ n in atTop, b < f n ω) := by |
rw [← lt_top_iff_ne_top, upcrossings_lt_top_iff] at hω
replace hω : ∃ k, ∀ N, upcrossingsBefore a b f N ω < k := by
obtain ⟨k, hk⟩ := hω
exact ⟨k + 1, fun N => lt_of_le_of_lt (hk N) k.lt_succ_self⟩
rintro ⟨h₁, h₂⟩
rw [frequently_atTop] at h₁ h₂
refine Classical.not_not.2 hω ?_
push_neg
intro k
... | 16 | 8,886,110.520508 | 2 | 2 | 3 | 2,380 |
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import probability.martingale.convergence from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Filter Me... | Mathlib/Probability/Martingale/Convergence.lean | 141 | 152 | theorem tendsto_of_uncrossing_lt_top (hf₁ : liminf (fun n => (‖f n ω‖₊ : ℝ≥0∞)) atTop < ∞)
(hf₂ : ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞) :
∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by |
by_cases h : IsBoundedUnder (· ≤ ·) atTop fun n => |f n ω|
· rw [isBoundedUnder_le_abs] at h
refine tendsto_of_no_upcrossings Rat.denseRange_cast ?_ h.1 h.2
intro a ha b hb hab
obtain ⟨⟨a, rfl⟩, ⟨b, rfl⟩⟩ := ha, hb
exact not_frequently_of_upcrossings_lt_top hab (hf₂ a b (Rat.cast_lt.1 hab)).ne
· ... | 9 | 8,103.083928 | 2 | 2 | 3 | 2,380 |
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import probability.martingale.convergence from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Filter Me... | Mathlib/Probability/Martingale/Convergence.lean | 156 | 183 | theorem Submartingale.upcrossings_ae_lt_top' [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, snorm (f n) 1 μ ≤ R) (hab : a < b) : ∀ᵐ ω ∂μ, upcrossings a b f ω < ∞ := by |
refine ae_lt_top (hf.adapted.measurable_upcrossings hab) ?_
have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b
rw [mul_comm, ← ENNReal.le_div_iff_mul_le] at this
· refine (lt_of_le_of_lt this (ENNReal.div_lt_top ?_ ?_)).ne
· have hR' : ∀ n, ∫⁻ ω, ‖f n ω - a‖₊ ∂μ ≤ R + ‖a‖₊ * μ Set.univ := by
... | 26 | 195,729,609,428.83878 | 2 | 2 | 3 | 2,380 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Nilpotent
import Mathlib.Order.Radical
def frattini (G : Type*) [Group G] : Subgroup G :=
Order.radical (Subgroup G)
variable {G H : Type*} [Group G] [Group H] {φ : G →* H} (hφ : Function.Surjective φ)
lemma... | Mathlib/GroupTheory/Frattini.lean | 59 | 74 | theorem frattini_nilpotent [Finite G] : Group.IsNilpotent (frattini G) := by |
-- We use the characterisation of nilpotency in terms of all Sylow subgroups being normal.
have q := (isNilpotent_of_finite_tfae (G := frattini G)).out 0 3
rw [q]; clear q
-- Consider each prime `p` and Sylow `p`-subgroup `P` of `frattini G`.
intro p p_prime P
-- The Frattini argument shows that the normal... | 15 | 3,269,017.372472 | 2 | 2 | 1 | 2,381 |
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Algebra.ContinuedFractions.TerminatedStable
import Mathlib.Algebra.ContinuedFractions.ContinuantsRecurrence
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import algebra.continued_... | Mathlib/Algebra/ContinuedFractions/Computation/CorrectnessTerminating.lean | 104 | 212 | theorem compExactValue_correctness_of_stream_eq_some :
∀ {ifp_n : IntFractPair K}, IntFractPair.stream v n = some ifp_n →
v = compExactValue ((of v).continuantsAux n) ((of v).continuantsAux <| n + 1) ifp_n.fr := by |
let g := of v
induction' n with n IH
· intro ifp_zero stream_zero_eq
-- Nat.zero
have : IntFractPair.of v = ifp_zero := by
have : IntFractPair.stream v 0 = some (IntFractPair.of v) := rfl
simpa only [Nat.zero_eq, this, Option.some.injEq] using stream_zero_eq
cases this
cases' Decidabl... | 106 | 10,844,638,552,900,231,000,000,000,000,000,000,000,000,000,000 | 2 | 2 | 1 | 2,382 |
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
universe u v w
structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domai... | Mathlib/LinearAlgebra/LinearPMap.lean | 64 | 70 | theorem ext {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g := by |
rcases f with ⟨f_dom, f⟩
rcases g with ⟨g_dom, g⟩
obtain rfl : f_dom = g_dom := h
obtain rfl : f = g := LinearMap.ext fun x => h' rfl
rfl
| 5 | 148.413159 | 2 | 2 | 2 | 2,383 |
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
universe u v w
structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domai... | Mathlib/LinearAlgebra/LinearPMap.lean | 151 | 157 | theorem mkSpanSingleton'_apply (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (c : R) (h) :
mkSpanSingleton' x y H ⟨c • x, h⟩ = c • y := by |
dsimp [mkSpanSingleton']
rw [← sub_eq_zero, ← sub_smul]
apply H
simp only [sub_smul, one_smul, sub_eq_zero]
apply Classical.choose_spec (mem_span_singleton.1 h)
| 5 | 148.413159 | 2 | 2 | 2 | 2,383 |
import Mathlib.NumberTheory.Cyclotomic.Embeddings
import Mathlib.NumberTheory.Cyclotomic.Rat
import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
open NumberField Units InfinitePlace nonZeroDivisors Polynomial
namespace IsCyclotomicExtension.Rat.Three
variable {K : Type*} [Field K] [NumberField K] [IsC... | Mathlib/NumberTheory/Cyclotomic/Three.lean | 41 | 68 | theorem Units.mem : u ∈ [1, -1, η, -η, η ^ 2, -η ^ 2] := by |
have hrank : rank K = 0 := by
dsimp only [rank]
rw [card_eq_nrRealPlaces_add_nrComplexPlaces, nrRealPlaces_eq_zero (n := 3) K (by decide),
zero_add, nrComplexPlaces_eq_totient_div_two (n := 3)]
rfl
obtain ⟨⟨x, e⟩, hxu, -⟩ := exist_unique_eq_mul_prod _ u
replace hxu : u = x := by
rw [← mul_o... | 27 | 532,048,240,601.79865 | 2 | 2 | 2 | 2,384 |
import Mathlib.NumberTheory.Cyclotomic.Embeddings
import Mathlib.NumberTheory.Cyclotomic.Rat
import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
open NumberField Units InfinitePlace nonZeroDivisors Polynomial
namespace IsCyclotomicExtension.Rat.Three
variable {K : Type*} [Field K] [NumberField K] [IsC... | Mathlib/NumberTheory/Cyclotomic/Three.lean | 85 | 111 | theorem eq_one_or_neg_one_of_unit_of_congruent (hcong : ∃ n : ℤ, λ ^ 2 ∣ (u - n : 𝓞 K)) :
u = 1 ∨ u = -1 := by |
replace hcong : ∃ n : ℤ, (3 : 𝓞 K) ∣ (↑u - n : 𝓞 K) := by
obtain ⟨n, x, hx⟩ := hcong
exact ⟨n, -η * x, by rw [← mul_assoc, mul_neg, ← neg_mul, ← lambda_sq, hx]⟩
have hζ := IsCyclotomicExtension.zeta_spec 3 ℚ K
have := Units.mem hζ u
fin_cases this
· left; rfl
· right; rfl
all_goals exfalso
· ... | 25 | 72,004,899,337.38586 | 2 | 2 | 2 | 2,384 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Int
#align_import data.int.associated from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
| Mathlib/Data/Int/Associated.lean | 21 | 30 | theorem Int.natAbs_eq_iff_associated {a b : ℤ} : a.natAbs = b.natAbs ↔ Associated a b := by |
refine Int.natAbs_eq_natAbs_iff.trans ?_
constructor
· rintro (rfl | rfl)
· rfl
· exact ⟨-1, by simp⟩
· rintro ⟨u, rfl⟩
obtain rfl | rfl := Int.units_eq_one_or u
· exact Or.inl (by simp)
· exact Or.inr (by simp)
| 9 | 8,103.083928 | 2 | 2 | 1 | 2,385 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
| Mathlib/Algebra/Ring/Center.lean | 24 | 37 | theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by | rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul]
mid_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, mul_zero, zero_... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,386 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) :... | Mathlib/Algebra/Ring/Center.lean | 46 | 67 | theorem intCast_mem_center [NonAssocRing M] (n : ℤ) : (n : M) ∈ Set.center M where
comm _ := by | rw [Int.commute_cast]
left_assoc _ _ := match n with
| (n : ℕ) => by rw [Int.cast_natCast, (natCast_mem_center _ n).left_assoc _ _]
| Int.negSucc n => by
rw [Int.cast_negSucc, Nat.cast_add, Nat.cast_one, neg_add_rev, add_mul, add_mul, add_mul,
neg_mul, one_mul, neg_mul 1, one_mul, ← neg_mul, ad... | 21 | 1,318,815,734.483215 | 2 | 2 | 4 | 2,386 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) :... | Mathlib/Algebra/Ring/Center.lean | 72 | 77 | theorem add_mem_center [Distrib M] {a b : M} (ha : a ∈ Set.center M) (hb : b ∈ Set.center M) :
a + b ∈ Set.center M where
comm _ := by | rw [add_mul, mul_add, ha.comm, hb.comm]
left_assoc _ _ := by rw [add_mul, ha.left_assoc, hb.left_assoc, ← add_mul, ← add_mul]
mid_assoc _ _ := by rw [mul_add, add_mul, ha.mid_assoc, hb.mid_assoc, ← mul_add, ← add_mul]
right_assoc _ _ := by rw [mul_add, ha.right_assoc, hb.right_assoc, ← mul_add, ← mul_add]
| 4 | 54.59815 | 2 | 2 | 4 | 2,386 |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) :... | Mathlib/Algebra/Ring/Center.lean | 81 | 86 | theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by | rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, neg_mul]
right_assoc _ _ := by rw [mul_neg, ha.right_assoc, mul_neg, mul_neg]
| 4 | 54.59815 | 2 | 2 | 4 | 2,386 |
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.EffectiveEpi.Comp
import Mathlib.CategoryTheory.EffectiveEpi.Extensive
namespace CategoryTheory
open Limits GrothendieckTopology Sieve
variable (C : Type*) [Category C]
instance [Precoherent C] [HasFiniteCoproducts C] : Preregular C ... | Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean | 57 | 94 | theorem extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive C] :
((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck =
(coherentTopology C) := by |
ext B S
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· induction h with
| of Y T hT =>
apply Coverage.saturate.of
simp only [Coverage.sup_covering, Set.mem_union] at hT
exact Or.elim hT
(fun ⟨α, x, X, π, ⟨h, _⟩⟩ ↦ ⟨α, x, X, π, ⟨h, inferInstance⟩⟩)
(fun ⟨Z, f, ⟨h, _⟩⟩ ↦ ⟨Unit, inferInstan... | 35 | 1,586,013,452,313,430.8 | 2 | 2 | 1 | 2,387 |
import Mathlib.NumberTheory.NumberField.ClassNumber
import Mathlib.NumberTheory.Cyclotomic.Rat
import Mathlib.NumberTheory.Cyclotomic.Embeddings
universe u
namespace IsCyclotomicExtension.Rat
open NumberField Polynomial InfinitePlace Nat Real cyclotomic
variable (K : Type u) [Field K] [NumberField K]
| Mathlib/NumberTheory/Cyclotomic/PID.lean | 30 | 41 | theorem three_pid [IsCyclotomicExtension {3} ℚ K] : IsPrincipalIdealRing (𝓞 K) := by |
apply RingOfIntegers.isPrincipalIdealRing_of_abs_discr_lt
rw [absdiscr_prime 3 K, IsCyclotomicExtension.finrank (n := 3) K
(irreducible_rat (by norm_num)), nrComplexPlaces_eq_totient_div_two 3, totient_prime
PNat.prime_three]
simp only [Int.reduceNeg, PNat.val_ofNat, succ_sub_succ_eq_sub, tsub_zero, ze... | 11 | 59,874.141715 | 2 | 2 | 2 | 2,388 |
import Mathlib.NumberTheory.NumberField.ClassNumber
import Mathlib.NumberTheory.Cyclotomic.Rat
import Mathlib.NumberTheory.Cyclotomic.Embeddings
universe u
namespace IsCyclotomicExtension.Rat
open NumberField Polynomial InfinitePlace Nat Real cyclotomic
variable (K : Type u) [Field K] [NumberField K]
theorem ... | Mathlib/NumberTheory/Cyclotomic/PID.lean | 44 | 55 | theorem five_pid [IsCyclotomicExtension {5} ℚ K] : IsPrincipalIdealRing (𝓞 K) := by |
apply RingOfIntegers.isPrincipalIdealRing_of_abs_discr_lt
rw [absdiscr_prime 5 K, IsCyclotomicExtension.finrank (n := 5) K
(irreducible_rat (by norm_num)), nrComplexPlaces_eq_totient_div_two 5, totient_prime
PNat.prime_five]
simp only [Int.reduceNeg, PNat.val_ofNat, succ_sub_succ_eq_sub, tsub_zero, red... | 11 | 59,874.141715 | 2 | 2 | 2 | 2,388 |
import Mathlib.Geometry.Euclidean.Sphere.Power
import Mathlib.Geometry.Euclidean.Triangle
#align_import geometry.euclidean.sphere.ptolemy from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open Real
open scoped EuclideanGeometry RealInnerProductSpace Real
namespace EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Sphere/Ptolemy.lean | 53 | 70 | theorem mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {a b c d p : P}
(h : Cospherical ({a, b, c, d} : Set P)) (hapc : ∠ a p c = π) (hbpd : ∠ b p d = π) :
dist a b * dist c d + dist b c * dist d a = dist a c * dist b d := by |
have h' : Cospherical ({a, c, b, d} : Set P) := by rwa [Set.insert_comm c b {d}]
have hmul := mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi h' hapc hbpd
have hbp := left_dist_ne_zero_of_angle_eq_pi hbpd
have h₁ : dist c d = dist c p / dist b p * dist a b := by
rw [dist_mul_of_eq_angle_of_dist_mul b p ... | 15 | 3,269,017.372472 | 2 | 2 | 1 | 2,389 |
import Mathlib.MeasureTheory.Constructions.Cylinders
import Mathlib.MeasureTheory.Measure.Typeclasses
open Set
namespace MeasureTheory
variable {ι : Type*} {α : ι → Type*} [∀ i, MeasurableSpace (α i)]
{P : ∀ J : Finset ι, Measure (∀ j : J, α j)}
def IsProjectiveMeasureFamily (P : ∀ J : Finset ι, Measure (∀ j ... | Mathlib/MeasureTheory/Constructions/Projective.lean | 143 | 150 | theorem unique [∀ i, IsFiniteMeasure (P i)]
(hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
μ = ν := by |
haveI : IsFiniteMeasure μ := hμ.isFiniteMeasure
refine ext_of_generate_finite (measurableCylinders α) generateFrom_measurableCylinders.symm
isPiSystem_measurableCylinders (fun s hs ↦ ?_) (hμ.measure_univ_unique hν)
obtain ⟨I, S, hS, rfl⟩ := (mem_measurableCylinders _).mp hs
rw [hμ.measure_cylinder _ hS, hν... | 5 | 148.413159 | 2 | 2 | 1 | 2,390 |
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.preadditive.hom_orthogonal from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb... | Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean | 130 | 143 | theorem matrixDecomposition_id (o : HomOrthogonal s) {α : Type} [Finite α] {f : α → ι} (i : ι) :
o.matrixDecomposition (𝟙 (⨁ fun a => s (f a))) i = 1 := by |
ext ⟨b, ⟨⟩⟩ ⟨a, j_property⟩
simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property
simp only [Category.comp_id, Category.id_comp, Category.assoc, End.one_def, eqToHom_refl,
Matrix.one_apply, HomOrthogonal.matrixDecomposition_apply, biproduct.components]
split_ifs with h
· cases h
simp
· ... | 12 | 162,754.791419 | 2 | 2 | 2 | 2,391 |
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.LinearAlgebra.Matrix.InvariantBasisNumber
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.preadditive.hom_orthogonal from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb... | Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean | 146 | 166 | theorem matrixDecomposition_comp (o : HomOrthogonal s) {α β γ : Type} [Finite α] [Fintype β]
[Finite γ] {f : α → ι} {g : β → ι} {h : γ → ι} (z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b))
(w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)) (i : ι) :
o.matrixDecomposition (z ≫ w) i = o.matrixDecomposition w i * ... |
ext ⟨c, ⟨⟩⟩ ⟨a, j_property⟩
simp only [Set.mem_preimage, Set.mem_singleton_iff] at j_property
simp only [Matrix.mul_apply, Limits.biproduct.components,
HomOrthogonal.matrixDecomposition_apply, Category.comp_id, Category.id_comp, Category.assoc,
End.mul_def, eqToHom_refl, eqToHom_trans_assoc, Finset.sum_c... | 17 | 24,154,952.753575 | 2 | 2 | 2 | 2,391 |
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.NormedSpace.Complemented
#align_import analysis.calculus.implicit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/Analysis/Calculus/Implicit.lean | 201 | 214 | theorem implicitFunction_hasStrictFDerivAt (g'inv : G →L[𝕜] E)
(hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G)
(hg'invf : φ.leftDeriv.comp g'inv = 0) :
HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt) := by |
have := φ.hasStrictFDerivAt.to_localInverse
simp only [prodFun] at this
convert this.comp (φ.rightFun φ.pt) ((hasStrictFDerivAt_const _ _).prod (hasStrictFDerivAt_id _))
-- Porting note: added parentheses to help `simp`
simp only [ContinuousLinearMap.ext_iff, (ContinuousLinearMap.comp_apply)] at hg'inv hg'in... | 10 | 22,026.465795 | 2 | 2 | 1 | 2,392 |
import Mathlib.Algebra.Polynomial.DenomsClearable
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Data.Real.Irrational
import Mathlib.Topology.Algebra.Polynomial
#align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"04e80bb7e851095... | Mathlib/NumberTheory/Liouville/Basic.lean | 95 | 120 | theorem exists_one_le_pow_mul_dist {Z N R : Type*} [PseudoMetricSpace R] {d : N → ℝ}
{j : Z → N → R} {f : R → R} {α : R} {ε M : ℝ}
-- denominators are positive
(d0 : ∀ a : N, 1 ≤ d a)
(e0 : 0 < ε)
-- function is Lipschitz at α
(B : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M)... |
-- A useful inequality to keep at hand
have me0 : 0 < max (1 / ε) M := lt_max_iff.mpr (Or.inl (one_div_pos.mpr e0))
-- The maximum between `1 / ε` and `M` works
refine ⟨max (1 / ε) M, me0, fun z a => ?_⟩
-- First, let's deal with the easy case in which we are far away from `α`
by_cases dm1 : 1 ≤ dist α (j ... | 16 | 8,886,110.520508 | 2 | 2 | 2 | 2,393 |
import Mathlib.Algebra.Polynomial.DenomsClearable
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Data.Real.Irrational
import Mathlib.Topology.Algebra.Polynomial
#align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"04e80bb7e851095... | Mathlib/NumberTheory/Liouville/Basic.lean | 123 | 173 | theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : ℤ[X]} (f0 : f ≠ 0)
(fa : eval α (map (algebraMap ℤ ℝ) f) = 0) :
∃ A : ℝ, 0 < A ∧ ∀ a : ℤ, ∀ b : ℕ,
(1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|α - a / (b + 1)| * A) := by |
-- `fR` is `f` viewed as a polynomial with `ℝ` coefficients.
set fR : ℝ[X] := map (algebraMap ℤ ℝ) f
-- `fR` is non-zero, since `f` is non-zero.
obtain fR0 : fR ≠ 0 := fun fR0 =>
(map_injective (algebraMap ℤ ℝ) fun _ _ A => Int.cast_inj.mp A).ne f0
(fR0.trans (Polynomial.map_zero _).symm)
-- reform... | 47 | 258,131,288,619,006,750,000 | 2 | 2 | 2 | 2,393 |
import Mathlib.Control.Traversable.Equiv
import Mathlib.Control.Traversable.Instances
import Batteries.Data.LazyList
import Mathlib.Lean.Thunk
#align_import data.lazy_list.basic from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace LazyList
open Function
def listE... | Mathlib/Data/LazyList/Basic.lean | 143 | 147 | theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by |
induction' xs using LazyList.rec with _ _ _ _ ih
· simp only [Thunk.pure, append, Thunk.get]
· simpa only [append, cons.injEq, true_and]
· ext; apply ih
| 4 | 54.59815 | 2 | 2 | 3 | 2,394 |
import Mathlib.Control.Traversable.Equiv
import Mathlib.Control.Traversable.Instances
import Batteries.Data.LazyList
import Mathlib.Lean.Thunk
#align_import data.lazy_list.basic from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace LazyList
open Function
def listE... | Mathlib/Data/LazyList/Basic.lean | 150 | 155 | theorem append_assoc {α} (xs ys zs : LazyList α) :
(xs.append ys).append zs = xs.append (ys.append zs) := by |
induction' xs using LazyList.rec with _ _ _ _ ih
· simp only [append, Thunk.get]
· simpa only [append, cons.injEq, true_and]
· ext; apply ih
| 4 | 54.59815 | 2 | 2 | 3 | 2,394 |
import Mathlib.Control.Traversable.Equiv
import Mathlib.Control.Traversable.Instances
import Batteries.Data.LazyList
import Mathlib.Lean.Thunk
#align_import data.lazy_list.basic from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace LazyList
open Function
def listE... | Mathlib/Data/LazyList/Basic.lean | 159 | 168 | theorem append_bind {α β} (xs : LazyList α) (ys : Thunk (LazyList α)) (f : α → LazyList β) :
(xs.append ys).bind f = (xs.bind f).append (ys.get.bind f) := by |
match xs with
| LazyList.nil =>
simp only [append, Thunk.get, LazyList.bind]
| LazyList.cons x xs =>
simp only [append, Thunk.get, LazyList.bind]
have := append_bind xs.get ys f
simp only [Thunk.get] at this
rw [this, append_assoc]
| 8 | 2,980.957987 | 2 | 2 | 3 | 2,394 |
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Calculus.Deriv.Basic
variable {𝕜 ι : Type*} [DecidableEq ι] [Fintype ι] [NontriviallyNormedField 𝕜]
| Mathlib/Analysis/Calculus/Deriv/Pi.lean | 15 | 22 | theorem hasDerivAt_update (x : ι → 𝕜) (i : ι) (y : 𝕜) :
HasDerivAt (Function.update x i) (Pi.single i (1 : 𝕜)) y := by |
convert (hasFDerivAt_update x y).hasDerivAt
ext z j
rw [Pi.single, Function.update_apply]
split_ifs with h
· simp [h]
· simp [Pi.single_eq_of_ne h]
| 6 | 403.428793 | 2 | 2 | 1 | 2,395 |
import Mathlib.Topology.MetricSpace.PseudoMetric
open Filter
open scoped Uniformity Topology
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (hB : ∀ n, 0 < B n)
(H : ∀ u : ℕ → α, (∀ N n m... | Mathlib/Topology/MetricSpace/Cauchy.lean | 72 | 91 | theorem Metric.uniformCauchySeqOn_iff {γ : Type*} {F : β → γ → α} {s : Set γ} :
UniformCauchySeqOn F atTop s ↔ ∀ ε > (0 : ℝ),
∃ N : β, ∀ m ≥ N, ∀ n ≥ N, ∀ x ∈ s, dist (F m x) (F n x) < ε := by |
constructor
· intro h ε hε
let u := { a : α × α | dist a.fst a.snd < ε }
have hu : u ∈ 𝓤 α := Metric.mem_uniformity_dist.mpr ⟨ε, hε, by simp [u]⟩
rw [← @Filter.eventually_atTop_prod_self' _ _ _ fun m =>
∀ x ∈ s, dist (F m.fst x) (F m.snd x) < ε]
specialize h u hu
rw [prod_atTop_atTop_eq]... | 17 | 24,154,952.753575 | 2 | 2 | 2 | 2,396 |
import Mathlib.Topology.MetricSpace.PseudoMetric
open Filter
open scoped Uniformity Topology
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (hB : ∀ n, 0 < B n)
(H : ∀ u : ℕ → α, (∀ N n m... | Mathlib/Topology/MetricSpace/Cauchy.lean | 113 | 123 | theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R := by |
rcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩
rsuffices ⟨R, R0, H⟩ : ∃ R > 0, ∀ n, dist (u n) (u N) < R
· exact ⟨_, add_pos R0 R0, fun m n =>
lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩
let R := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)
refine ⟨↑R +... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,396 |
import Mathlib.FieldTheory.Adjoin
open Polynomial
namespace IntermediateField
variable (F E K : Type*) [Field F] [Field E] [Field K] [Algebra F E] [Algebra F K] {S : Set E}
structure Lifts where
carrier : IntermediateField F E
emb : carrier →ₐ[F] K
#align intermediate_field.lifts IntermediateField.Lif... | Mathlib/FieldTheory/Extension.lean | 57 | 70 | theorem Lifts.exists_upper_bound (c : Set (Lifts F E K)) (hc : IsChain (· ≤ ·) c) :
∃ ub, ∀ a ∈ c, a ≤ ub := by |
let t (i : ↑(insert ⊥ c)) := i.val.carrier
let t' (i) := (t i).toSubalgebra
have hc := hc.insert fun _ _ _ ↦ .inl bot_le
have dir : Directed (· ≤ ·) t := hc.directedOn.directed_val.mono_comp _ fun _ _ h ↦ h.1
refine ⟨⟨iSup t, (Subalgebra.iSupLift t' dir (fun i ↦ i.val.emb) (fun i j h ↦ ?_) _ rfl).comp
... | 12 | 162,754.791419 | 2 | 2 | 1 | 2,397 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 92 | 102 | theorem condexpIndL1Fin_add (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x y : G) :
condexpIndL1Fin hm hs hμs (x + y) =
condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm hs hμs y := by |
ext1
refine (Memℒp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_add _ _).symm
refine EventuallyEq.trans ?_
(EventuallyEq.add (Memℒp.coeFn_toLp q).symm (Memℒp.coeFn_toLp q).symm)
rw [condexpIndSMul_add]
refine (Lp.coeFn_add _ _).trans (eventually_of_forall fun a => ?_)
rfl
| 8 | 2,980.957987 | 2 | 2 | 4 | 2,398 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 105 | 113 | theorem condexpIndL1Fin_smul (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : ℝ) (x : G) :
condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x := by |
ext1
refine (Memℒp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_smul _ _).symm
rw [condexpIndSMul_smul hs hμs c x]
refine (Lp.coeFn_smul _ _).trans ?_
refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun y hy => ?_
simp only [Pi.smul_apply, hy]
| 7 | 1,096.633158 | 2 | 2 | 4 | 2,398 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 116 | 125 | theorem condexpIndL1Fin_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s)
(hμs : μ s ≠ ∞) (c : 𝕜) (x : F) :
condexpIndL1Fin hm hs hμs (c • x) = c • condexpIndL1Fin hm hs hμs x := by |
ext1
refine (Memℒp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_smul _ _).symm
rw [condexpIndSMul_smul' hs hμs c x]
refine (Lp.coeFn_smul _ _).trans ?_
refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x).mono fun y hy => ?_
simp only [Pi.smul_apply, hy]
| 7 | 1,096.633158 | 2 | 2 | 4 | 2,398 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 128 | 143 | theorem norm_condexpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
‖condexpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by |
have : 0 ≤ ∫ a : α, ‖condexpIndL1Fin hm hs hμs x a‖ ∂μ := by positivity
rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul, ←
ENNReal.toReal_ofReal this,
ENNReal.toReal_le_toReal ENNReal.ofReal_ne_top (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
ofReal_integr... | 14 | 1,202,604.284165 | 2 | 2 | 4 | 2,398 |
import Mathlib.Topology.Perfect
import Mathlib.Topology.MetricSpace.Polish
import Mathlib.Topology.MetricSpace.CantorScheme
#align_import topology.perfect from "leanprover-community/mathlib"@"3905fa80e62c0898131285baab35559fbc4e5cda"
open Set Filter
section CantorInjMetric
open Function ENNReal
variable {α : T... | Mathlib/Topology/MetricSpace/Perfect.lean | 62 | 73 | theorem Perfect.small_diam_splitting (ε_pos : 0 < ε) :
∃ C₀ C₁ : Set α, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧
(Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁ := by |
rcases hC.splitting hnonempty with ⟨D₀, D₁, ⟨perf0, non0, sub0⟩, ⟨perf1, non1, sub1⟩, hdisj⟩
cases' non0 with x₀ hx₀
cases' non1 with x₁ hx₁
rcases perf0.small_diam_aux ε_pos hx₀ with ⟨perf0', non0', sub0', diam0⟩
rcases perf1.small_diam_aux ε_pos hx₁ with ⟨perf1', non1', sub1', diam1⟩
refine
⟨closure ... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,399 |
import Mathlib.Topology.Perfect
import Mathlib.Topology.MetricSpace.Polish
import Mathlib.Topology.MetricSpace.CantorScheme
#align_import topology.perfect from "leanprover-community/mathlib"@"3905fa80e62c0898131285baab35559fbc4e5cda"
open Set Filter
section CantorInjMetric
open Function ENNReal
variable {α : T... | Mathlib/Topology/MetricSpace/Perfect.lean | 80 | 129 | theorem Perfect.exists_nat_bool_injection [CompleteSpace α] :
∃ f : (ℕ → Bool) → α, range f ⊆ C ∧ Continuous f ∧ Injective f := by |
obtain ⟨u, -, upos', hu⟩ := exists_seq_strictAnti_tendsto' (zero_lt_one' ℝ≥0∞)
have upos := fun n => (upos' n).1
let P := Subtype fun E : Set α => Perfect E ∧ E.Nonempty
choose C0 C1 h0 h1 hdisj using
fun {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε) =>
hC.small_diam_spl... | 48 | 701,673,591,209,763,100,000 | 2 | 2 | 2 | 2,399 |
import Mathlib.Data.Set.Basic
#align_import order.well_founded from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104e7bbb471381592"
variable {α β γ : Type*}
namespace WellFounded
variable {r r' : α → α → Prop}
#align well_founded_relation.r WellFoundedRelation.rel
protected theorem isAsymm (h : Well... | Mathlib/Order/WellFounded.lean | 82 | 89 | theorem wellFounded_iff_has_min {r : α → α → Prop} :
WellFounded r ↔ ∀ s : Set α, s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬r x m := by |
refine ⟨fun h => h.has_min, fun h => ⟨fun x => ?_⟩⟩
by_contra hx
obtain ⟨m, hm, hm'⟩ := h {x | ¬Acc r x} ⟨x, hx⟩
refine hm ⟨_, fun y hy => ?_⟩
by_contra hy'
exact hm' y hy' hy
| 6 | 403.428793 | 2 | 2 | 1 | 2,400 |
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.hofer from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Topology
open Filter Finset
local notation "d" => dist
#noalign pos_div_pow_pos
| Mathlib/Analysis/Hofer.lean | 33 | 104 | theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by |
by_contra H
have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
intro x' k
rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
positivity
-- Now let's specialize to `ε/2^k`
replace H : ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ... | 69 | 925,378,172,558,778,900,000,000,000,000 | 2 | 2 | 1 | 2,401 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 50 | 64 | theorem norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) :
‖cderiv r f z‖ ≤ M / r := by |
have hM : 0 ≤ M := by
obtain ⟨w, hw⟩ : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
exact (norm_nonneg _).trans (hf w hw)
have h1 : ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2 := by
intro w hw
simp only [mem_sphere_iff_norm, norm_eq_abs] at hw
simp only [norm_smul, i... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,402 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 67 | 76 | theorem cderiv_sub (hr : 0 < r) (hf : ContinuousOn f (sphere z r))
(hg : ContinuousOn g (sphere z r)) : cderiv r (f - g) z = cderiv r f z - cderiv r g z := by |
have h1 : ContinuousOn (fun w : ℂ => ((w - z) ^ 2)⁻¹) (sphere z r) := by
refine ((continuous_id'.sub continuous_const).pow 2).continuousOn.inv₀ fun w hw h => hr.ne ?_
rwa [mem_sphere_iff_norm, sq_eq_zero_iff.mp h, norm_zero] at hw
simp_rw [cderiv, ← smul_sub]
congr 1
simpa only [Pi.sub_apply, smul_sub]... | 8 | 2,980.957987 | 2 | 2 | 4 | 2,402 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 79 | 86 | theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r := by |
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
ex... | 6 | 403.428793 | 2 | 2 | 4 | 2,402 |
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 95 | 110 | theorem _root_.TendstoUniformlyOn.cderiv (hF : TendstoUniformlyOn F f φ (cthickening δ K))
(hδ : 0 < δ) (hFn : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K)) :
TendstoUniformlyOn (cderiv δ ∘ F) (cderiv δ f) φ K := by |
rcases φ.eq_or_neBot with rfl | hne
· simp only [TendstoUniformlyOn, eventually_bot, imp_true_iff]
have e1 : ContinuousOn f (cthickening δ K) := TendstoUniformlyOn.continuousOn hF hFn
rw [tendstoUniformlyOn_iff] at hF ⊢
rintro ε hε
filter_upwards [hF (ε * δ) (mul_pos hε hδ), hFn] with n h h' z hz
simp_rw... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,402 |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheory.Abelian.Exact
#align_import algebra.category.Module.abelian from "leanprover-community/mathlib"@"09f981f72d43749f1fa072deade828d9c1e185bb"
open... | Mathlib/Algebra/Category/ModuleCat/Abelian.lean | 123 | 127 | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by |
rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>
⟨range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm⟩⟩
| 4 | 54.59815 | 2 | 2 | 1 | 2,403 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Linarith.Frontend
#align_import algebra.quadratic_discriminant from "leanprover-community/mathlib"@"e085d1df33274f4b32f611f483aae678ba0b42df"
open Filter
section Ring
variable {R : ... | Mathlib/Algebra/QuadraticDiscriminant.lean | 63 | 70 | theorem quadratic_eq_zero_iff_discrim_eq_sq [NeZero (2 : R)] [NoZeroDivisors R]
(ha : a ≠ 0) (x : R) :
a * x * x + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2 := by |
refine ⟨discrim_eq_sq_of_quadratic_eq_zero, fun h ↦ ?_⟩
rw [discrim] at h
have ha : 2 * 2 * a ≠ 0 := mul_ne_zero (mul_ne_zero (NeZero.ne _) (NeZero.ne _)) ha
apply mul_left_cancel₀ ha
linear_combination -h
| 5 | 148.413159 | 2 | 2 | 1 | 2,404 |
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
namespace PosNum
def minFacAux (n : PosNum) : ℕ → PosNum → PosNum
| 0, _ => n
| fuel + 1, k =>
if n < k.bit1... | Mathlib/Data/Num/Prime.lean | 44 | 54 | theorem minFacAux_to_nat {fuel : ℕ} {n k : PosNum} (h : Nat.sqrt n < fuel + k.bit1) :
(minFacAux n fuel k : ℕ) = Nat.minFacAux n k.bit1 := by |
induction' fuel with fuel ih generalizing k <;> rw [minFacAux, Nat.minFacAux]
· rw [Nat.zero_add, Nat.sqrt_lt] at h
simp only [h, ite_true]
simp_rw [← mul_to_nat]
simp only [cast_lt, dvd_to_nat]
split_ifs <;> try rfl
rw [ih] <;> [congr; convert Nat.lt_succ_of_lt h using 1] <;>
simp only [_root_.bit... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,405 |
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.Ring
#align_import data.num.prime from "leanprover-community/mathlib"@"58581d0fe523063f5651df0619be2bf65012a94a"
namespace PosNum
def minFacAux (n : PosNum) : ℕ → PosNum → PosNum
| 0, _ => n
| fuel + 1, k =>
if n < k.bit1... | Mathlib/Data/Num/Prime.lean | 65 | 83 | theorem minFac_to_nat (n : PosNum) : (minFac n : ℕ) = Nat.minFac n := by |
cases' n with n
· rfl
· rw [minFac, Nat.minFac_eq, if_neg]
swap
· simp
rw [minFacAux_to_nat]
· rfl
simp only [cast_one, cast_bit1]
unfold _root_.bit1 _root_.bit0
rw [Nat.sqrt_lt]
calc
(n : ℕ) + (n : ℕ) + 1 ≤ (n : ℕ) + (n : ℕ) + (n : ℕ) := by simp
_ = (n : ℕ) * (1 + 1 +... | 18 | 65,659,969.137331 | 2 | 2 | 2 | 2,405 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
#align_import dynamics.ergodic.conservative from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open scoped Classi... | Mathlib/Dynamics/Ergodic/Conservative.lean | 83 | 106 | theorem frequently_measure_inter_ne_zero (hf : Conservative f μ) (hs : MeasurableSet s)
(h0 : μ s ≠ 0) : ∃ᶠ m in atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0 := by |
by_contra H
simp only [not_frequently, eventually_atTop, Ne, Classical.not_not] at H
rcases H with ⟨N, hN⟩
induction' N with N ihN
· apply h0
simpa using hN 0 le_rfl
rw [imp_false] at ihN
push_neg at ihN
rcases ihN with ⟨n, hn, hμn⟩
set T := s ∩ ⋃ n ≥ N + 1, f^[n] ⁻¹' s
have hT : MeasurableSet ... | 22 | 3,584,912,846.131591 | 2 | 2 | 3 | 2,406 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
#align_import dynamics.ergodic.conservative from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open scoped Classi... | Mathlib/Dynamics/Ergodic/Conservative.lean | 121 | 130 | theorem measure_mem_forall_ge_image_not_mem_eq_zero (hf : Conservative f μ) (hs : MeasurableSet s)
(n : ℕ) : μ ({ x ∈ s | ∀ m ≥ n, f^[m] x ∉ s }) = 0 := by |
by_contra H
have : MeasurableSet (s ∩ { x | ∀ m ≥ n, f^[m] x ∉ s }) := by
simp only [setOf_forall, ← compl_setOf]
exact
hs.inter (MeasurableSet.biInter (to_countable _) fun m _ => hf.measurable.iterate m hs.compl)
rcases (hf.exists_gt_measure_inter_ne_zero this H) n with ⟨m, hmn, hm⟩
rcases nonem... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,406 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
#align_import dynamics.ergodic.conservative from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open scoped Classi... | Mathlib/Dynamics/Ergodic/Conservative.lean | 135 | 140 | theorem ae_mem_imp_frequently_image_mem (hf : Conservative f μ) (hs : MeasurableSet s) :
∀ᵐ x ∂μ, x ∈ s → ∃ᶠ n in atTop, f^[n] x ∈ s := by |
simp only [frequently_atTop, @forall_swap (_ ∈ s), ae_all_iff]
intro n
filter_upwards [measure_zero_iff_ae_nmem.1 (hf.measure_mem_forall_ge_image_not_mem_eq_zero hs n)]
simp
| 4 | 54.59815 | 2 | 2 | 3 | 2,406 |
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory
open CategoryTheory.Limits
open Opposite
universe w v u
variable {C : Type u} [Ca... | Mathlib/CategoryTheory/Sites/Plus.lean | 71 | 77 | theorem diagramNatTrans_id (X : C) (P : Cᵒᵖ ⥤ D) :
J.diagramNatTrans (𝟙 P) X = 𝟙 (J.diagram P X) := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
simp only [limit.lift_π, Multifork.ofι_pt, Multifork.ofι_π_app, Category.id_comp]
erw [Category.comp_id]
| 5 | 148.413159 | 2 | 2 | 3 | 2,407 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.