Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
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import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 147 | 148 | theorem quotient_norm_sub_rev {S : AddSubgroup M} (x y : M ⧸ S) : ‖x - y‖ = ‖y - x‖ := by |
rw [← neg_sub, quotient_norm_neg]
| true |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 99 | 101 | theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by
contrapose! h |
contrapose! h
exact order_le _ h
| true |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 108 | 116 | theorem measurable_measure_mul_right (hs : MeasurableSet s) :
Measurable fun x => μ ((fun y => y * x) ⁻¹' s) := by
suffices |
suffices
Measurable fun y =>
μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s))
by convert this using 1; ext1 x; congr 1 with y : 1; simp
apply measurable_measure_prod_mk_right
apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs)
infer_instance... | true |
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {ι : Type*} {R : Type*} {S : Type*} {M : ι → Type*} {N : Type*}
n... | Mathlib/LinearAlgebra/DFinsupp.lean | 206 | 209 | theorem mapRange.linearMap_id :
(mapRange.linearMap fun i => (LinearMap.id : β₂ i →ₗ[R] _)) = LinearMap.id := by
ext |
ext
simp [linearMap]
| true |
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 53 | 56 | theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) := by
rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact, |
rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact,
CompleteLattice.isSupFiniteCompact_iff_all_elements_compact]
exact forall_congr' Opens.isCompactElement_iff
| true |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 98 | 100 | theorem kernelSubobject_arrow :
(kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow := by |
simp [kernelSubobjectIso]
| true |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 107 | 110 | theorem Monotone.isChain_range [LinearOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
IsChain (· ≤ ·) (range f) := by
rw [← image_univ] |
rw [← image_univ]
exact (isChain_of_trichotomous _).image (· ≤ ·) _ _ hf
| true |
import Mathlib.AlgebraicTopology.DoldKan.Normalized
#align_import algebraic_topology.dold_kan.homotopy_equivalence from "leanprover-community/mathlib"@"f951e201d416fb50cc7826171d80aa510ec20747"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
CategoryTheory.Preadditive Simplicial DoldKan
nonco... | Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean | 52 | 58 | theorem homotopyPToId_eventually_constant {q n : ℕ} (hqn : n < q) :
((homotopyPToId X (q + 1)).hom n (n + 1) : X _[n] ⟶ X _[n + 1]) =
(homotopyPToId X q).hom n (n + 1) := by
simp only [homotopyHσToZero, AlternatingFaceMapComplex.obj_X, Nat.add_eq, Homotopy.trans_hom, |
simp only [homotopyHσToZero, AlternatingFaceMapComplex.obj_X, Nat.add_eq, Homotopy.trans_hom,
Homotopy.ofEq_hom, Pi.zero_apply, Homotopy.add_hom, Homotopy.compLeft_hom, add_zero,
Homotopy.nullHomotopy'_hom, ComplexShape.down_Rel, hσ'_eq_zero hqn (c_mk (n + 1) n rfl),
dite_eq_ite, ite_self, comp_zero, zer... | true |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ι R M σ : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 145 | 147 | theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) :
i = j := by |
contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx]
| true |
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
#align_import topology.sheaves.skyscraper from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open TopologicalSpace TopCat CategoryTheory CategoryT... | Mathlib/Topology/Sheaves/Skyscraper.lean | 68 | 74 | theorem skyscraperPresheaf_eq_pushforward
[hd : ∀ U : Opens (TopCat.of PUnit.{u + 1}), Decidable (PUnit.unit ∈ U)] :
skyscraperPresheaf p₀ A =
ContinuousMap.const (TopCat.of PUnit) p₀ _*
skyscraperPresheaf (X := TopCat.of PUnit) PUnit.unit A := by
convert_to @skyscraperPresheaf X p₀ (fun U => hd... |
convert_to @skyscraperPresheaf X p₀ (fun U => hd <| (Opens.map <| ContinuousMap.const _ p₀).obj U)
C _ _ A = _ <;> congr
| true |
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a fu... | Mathlib/MeasureTheory/Group/LIntegral.lean | 46 | 49 | theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (x * g) ∂μ) = ∫⁻ x, f x ∂μ := by
convert (lintegral_map_equiv f <| MeasurableEquiv.mulRight g).symm using 1 |
convert (lintegral_map_equiv f <| MeasurableEquiv.mulRight g).symm using 1
simp [map_mul_right_eq_self μ g]
| true |
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 136 | 138 | theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 := by
refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p) |
refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)
split_ifs with h <;> simp only [h, zero_le', le_rfl]
| true |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 98 | 99 | theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l := by |
simpa [duplicate_cons_iff, hx.symm] using h
| true |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 94 | 97 | theorem Filter.Tendsto.atBot_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg
simpa [(· ∘ ·)] using tendsto_neg_atTop_atBot.comp this
| true |
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
namespace Nat
variable {ι : Type*}
lemma modEq_list_prod_iff {a b} {l : List ℕ} (co : l.Pairwise Coprime) :
a ≡ b [MOD l.prod] ↔ ∀ i, a ≡ b [MOD l.get i] := by
induction' l with m l ih
· si... | Mathlib/Data/Nat/ChineseRemainder.lean | 93 | 105 | theorem chineseRemainderOfList_modEq_unique (l : List ι)
(co : l.Pairwise (Coprime on s)) {z} (hz : ∀ i ∈ l, z ≡ a i [MOD s i]) :
z ≡ chineseRemainderOfList a s l co [MOD (l.map s).prod] := by
induction' l with i l ih |
induction' l with i l ih
· simp [modEq_one]
· simp only [List.map_cons, List.prod_cons, chineseRemainderOfList]
have : Coprime (s i) (l.map s).prod := by
simp only [coprime_list_prod_right_iff, List.mem_map, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
intro j hj
exact (L... | true |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 54 | 59 | theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by
ext x y; simp only [AddMonoidHom.id_comp] |
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
| true |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 50 | 52 | theorem fold_cons (h : a ∉ s) : (cons a s h).fold op b f = f a * s.fold op b f := by
dsimp only [fold] |
dsimp only [fold]
rw [cons_val, Multiset.map_cons, fold_cons_left]
| true |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 168 | 173 | theorem weightedHomogeneousSubmodule_eq_finsupp_supported (w : σ → M) (m : M) :
weightedHomogeneousSubmodule R w m = Finsupp.supported R R { d | weightedDegree w d = m } := by
ext x |
ext x
rw [mem_supported, Set.subset_def]
simp only [Finsupp.mem_support_iff, mem_coe]
rfl
| true |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 331 | 337 | theorem value_at (a : ℤ) {R : Type*} [CommSemiring R] (χ : R →* ℤ)
(hp : ∀ (p : ℕ) (pp : p.Prime), p ≠ 2 → @legendreSym p ⟨pp⟩ a = χ p) {b : ℕ} (hb : Odd b) :
J(a | b) = χ b := by
conv_rhs => rw [← prod_factors hb.pos.ne', cast_list_prod, map_list_prod χ] |
conv_rhs => rw [← prod_factors hb.pos.ne', cast_list_prod, map_list_prod χ]
rw [jacobiSym, List.map_map, ← List.pmap_eq_map Nat.Prime _ _ fun _ => prime_of_mem_factors]
congr 1; apply List.pmap_congr
exact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <| dvd_of_mem_factors h)
| true |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 538 | 541 | theorem subalgebra_top_rank_eq_submodule_top_rank :
Module.rank F (⊤ : Subalgebra F E) = Module.rank F (⊤ : Submodule F E) := by
rw [← Algebra.top_toSubmodule] |
rw [← Algebra.top_toSubmodule]
rfl
| true |
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace Multiset
open List
instance functor : Functor Multiset... | Mathlib/Data/Multiset/Functor.lean | 137 | 143 | theorem naturality {G H : Type _ → Type _} [Applicative G] [Applicative H] [CommApplicative G]
[CommApplicative H] (eta : ApplicativeTransformation G H) {α β : Type _} (f : α → G β)
(x : Multiset α) : eta (traverse f x) = traverse (@eta _ ∘ f) x := by
refine Quotient.inductionOn x ?_ |
refine Quotient.inductionOn x ?_
intro
simp only [quot_mk_to_coe, traverse, lift_coe, Function.comp_apply,
ApplicativeTransformation.preserves_map, LawfulTraversable.naturality]
| true |
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 117 | 120 | theorem lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) :
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) = snorm' f q μ ^ q := by
rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one] |
rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one]
exact (ne_of_lt hq0_lt).symm
| true |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 53 | 62 | theorem mem_splitCenterBox {s : Set ι} {y : ι → ℝ} :
y ∈ I.splitCenterBox s ↔ y ∈ I ∧ ∀ i, (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s := by
simp only [splitCenterBox, mem_def, ← forall_and] |
simp only [splitCenterBox, mem_def, ← forall_and]
refine forall_congr' fun i ↦ ?_
dsimp only [Set.piecewise]
split_ifs with hs <;> simp only [hs, iff_true_iff, iff_false_iff, not_lt]
exacts [⟨fun H ↦ ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩,
fun H ↦ ⟨H.2, H.1.2⟩⟩,
⟨fun H... | true |
import Mathlib.CategoryTheory.Preadditive.ProjectiveResolution
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Tactic.SuppressCompilation
suppress_compilation
noncomputable section
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
open Category Limits Projective
set_... | Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean | 99 | 102 | theorem lift_commutes {Y Z : C} (f : Y ⟶ Z) (P : ProjectiveResolution Y)
(Q : ProjectiveResolution Z) : lift f P Q ≫ Q.π = P.π ≫ (ChainComplex.single₀ C).map f := by
ext |
ext
simp [lift, liftFZero, liftFOne]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 401 | 407 | theorem nmem_nonZeroDivisors_iff {P : R[X]} : P ∉ R[X]⁰ ↔ ∃ a : R, a ≠ 0 ∧ a • P = 0 := by
refine ⟨fun hP ↦ ?_, fun ⟨a, ha, h⟩ h1 ↦ ha <| C_eq_zero.1 <| (h1 _) <| smul_eq_C_mul a ▸ h⟩ |
refine ⟨fun hP ↦ ?_, fun ⟨a, ha, h⟩ h1 ↦ ha <| C_eq_zero.1 <| (h1 _) <| smul_eq_C_mul a ▸ h⟩
by_contra! h
obtain ⟨Q, hQ⟩ := _root_.nmem_nonZeroDivisors_iff.1 hP
refine hQ.2 (eq_zero_of_mul_eq_zero_of_smul P (fun a ha ↦ ?_) Q (mul_comm P _ ▸ hQ.1))
contrapose! ha
exact h a ha
| true |
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α β M E : Type*}
open Set Filter
@[to_additive]
| Mathlib/Order/Filter/IndicatorFunction.lean | 63 | 66 | theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
(hs : Monotone s) (f : α → β) (a : α) :
(fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a := by |
classical exact hs.piecewise_eventually_eq_iUnion f 1 a
| true |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 75 | 76 | theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by |
rw [nhdsWithin, principal_univ, inf_top_eq]
| true |
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] |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_iInter_mem hic
| true |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 52 | 54 | theorem lift_le_continuum {c : Cardinal.{u}} : lift.{v} c ≤ 𝔠 ↔ c ≤ 𝔠 := by
-- Porting note: added explicit universes |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_le]
| true |
import Mathlib.CategoryTheory.Adjunction.Whiskering
import Mathlib.CategoryTheory.Sites.PreservesSheafification
#align_import category_theory.sites.adjunction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open GrothendieckTopology CategoryTheory Limits Op... | Mathlib/CategoryTheory/Sites/Adjunction.lean | 148 | 160 | theorem adjunctionToTypes_counit_app_val {G : Type max v u ⥤ D} (adj : G ⊣ forget D)
(X : Sheaf J D) :
((adjunctionToTypes J adj).counit.app X).val =
sheafifyLift J ((Functor.associator _ _ _).hom ≫ (adj.whiskerRight _).counit.app _) X.2 := by
apply sheafifyLift_unique |
apply sheafifyLift_unique
dsimp only [adjunctionToTypes, Adjunction.comp, NatTrans.comp_app,
instCategorySheaf_comp_val, instCategorySheaf_id_val]
rw [adjunction_counit_app_val]
erw [Category.id_comp, sheafifyMap_sheafifyLift, toSheafify_sheafifyLift]
ext
dsimp [sheafEquivSheafOfTypes, Equivalence.symm... | true |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
#align_import data.nat.parity from "leanpr... | Mathlib/Algebra/Order/Ring/Abs.lean | 192 | 193 | theorem abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b := by |
cases' abs_choice a with h h <;> simp only [h, neg_dvd]
| true |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 63 | 73 | theorem IsPrimePow.exists_ord_compl_eq_one {n : ℕ} (h : IsPrimePow n) :
∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by
rcases eq_or_ne n 0 with (rfl | hn0); · cases not_isPrimePow_zero h |
rcases eq_or_ne n 0 with (rfl | hn0); · cases not_isPrimePow_zero h
rcases isPrimePow_iff_factorization_eq_single.mp h with ⟨p, k, hk0, h1⟩
rcases em' p.Prime with (pp | pp)
· refine absurd ?_ hk0.ne'
simp [← Nat.factorization_eq_zero_of_non_prime n pp, h1]
refine ⟨p, pp, ?_⟩
refine Nat.eq_of_factoriza... | true |
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
#align_import group_theory.noncomm_pi_coprod from "leanprover-community/mathlib"@"6f9f36364eae3f42368b04858fd66d6d9ae730d8"
... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 125 | 137 | theorem noncommPiCoprod_mulSingle (i : ι) (y : N i) :
noncommPiCoprod ϕ hcomm (Pi.mulSingle i y) = ϕ i y := by
change Finset.univ.noncommProd (fun j => ϕ j (Pi.mulSingle i y j)) (fun _ _ _ _ h => hcomm h _ _) |
change Finset.univ.noncommProd (fun j => ϕ j (Pi.mulSingle i y j)) (fun _ _ _ _ h => hcomm h _ _)
= ϕ i y
rw [← Finset.insert_erase (Finset.mem_univ i)]
rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ (Finset.not_mem_erase i _)]
rw [Pi.mulSingle_eq_same]
rw [Finset.noncommProd_eq_pow_card]
· rw [one_p... | true |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 113 | 124 | theorem sinZeta_two_mul_nat_add_one' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) / (2 * k + 1) / Gammaℂ (2 * k + 1) *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by
rw [sinZeta_two_mul_nat_add_one hk hx] |
rw [sinZeta_two_mul_nat_add_one hk hx]
congr 1
have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by
rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)),
Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul,
← Nat.cast_add_one, ← Nat.cast_m... | true |
import Mathlib.Data.Fin.VecNotation
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.SetTheory.Cardinal.Basic
#align_import group_theory.solvable from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298... | Mathlib/GroupTheory/Solvable.lean | 56 | 59 | theorem derivedSeries_normal (n : ℕ) : (derivedSeries G n).Normal := by
induction' n with n ih |
induction' n with n ih
· exact (⊤ : Subgroup G).normal_of_characteristic
· exact @Subgroup.commutator_normal G _ (derivedSeries G n) (derivedSeries G n) ih ih
| true |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import data.nat.choose.bounds from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
open Nat
variable {α : Type*} [LinearOrderedSemif... | Mathlib/Data/Nat/Choose/Bounds.lean | 32 | 37 | theorem choose_le_pow (r n : ℕ) : (n.choose r : α) ≤ (n ^ r : α) / r ! := by
rw [le_div_iff'] |
rw [le_div_iff']
· norm_cast
rw [← Nat.descFactorial_eq_factorial_mul_choose]
exact n.descFactorial_le_pow r
exact mod_cast r.factorial_pos
| true |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 49 | 49 | theorem gal_X_isSolvable : IsSolvable (X : F[X]).Gal := by | infer_instance
| true |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 179 | 181 | theorem dropWhile_idempotent : dropWhile p (dropWhile p l) = dropWhile p l := by
simp only [dropWhile_eq_self_iff] |
simp only [dropWhile_eq_self_iff]
exact fun h => dropWhile_nthLe_zero_not p l h
| true |
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
open scoped Classical
open Polynomial Set Function minpoly
namespace... | Mathlib/FieldTheory/Minpoly/Field.lean | 86 | 90 | theorem dvd_map_of_isScalarTower (A K : Type*) {R : Type*} [CommRing A] [Field K] [CommRing R]
[Algebra A K] [Algebra A R] [Algebra K R] [IsScalarTower A K R] (x : R) :
minpoly K x ∣ (minpoly A x).map (algebraMap A K) := by
refine minpoly.dvd K x ?_ |
refine minpoly.dvd K x ?_
rw [aeval_map_algebraMap, minpoly.aeval]
| true |
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def ImpartialAux : PGame → Prop
| G => (G ≈ -G) ∧ (∀ i... | Mathlib/SetTheory/Game/Impartial.lean | 35 | 38 | theorem impartialAux_def {G : PGame} :
G.ImpartialAux ↔
(G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j) := by |
rw [ImpartialAux]
| true |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section Group
variable [Group α] (e : α) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 142 | 145 | theorem mulETransformLeft.fst_mul_snd_subset :
(mulETransformLeft e x).1 * (mulETransformLeft e x).2 ⊆ x.1 * x.2 := by
refine inter_mul_union_subset_union.trans (union_subset Subset.rfl ?_) |
refine inter_mul_union_subset_union.trans (union_subset Subset.rfl ?_)
rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]
| true |
import Mathlib.Topology.ContinuousFunction.ZeroAtInfty
open Topology Filter
variable {E F 𝓕 : Type*}
variable [SeminormedAddGroup E] [SeminormedAddCommGroup F]
variable [FunLike 𝓕 E F] [ZeroAtInftyContinuousMapClass 𝓕 E F]
| Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean | 24 | 34 | theorem ZeroAtInftyContinuousMapClass.norm_le (f : 𝓕) (ε : ℝ) (hε : 0 < ε) :
∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε := by
have h := zero_at_infty f |
have h := zero_at_infty f
rw [tendsto_zero_iff_norm_tendsto_zero, tendsto_def] at h
specialize h (Metric.ball 0 ε) (Metric.ball_mem_nhds 0 hε)
rcases Metric.closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩
use r
intro x hr'
suffices x ∈ (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε by aesop
apply hr
a... | true |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 83 | 85 | theorem fold_op_distrib {f g : α → β} {b₁ b₂ : β} :
(s.fold op (b₁ * b₂) fun x => f x * g x) = s.fold op b₁ f * s.fold op b₂ g := by |
simp only [fold, fold_distrib]
| true |
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Sites.CoverLifting
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.sites.dense_subsite from "leanprover-community/mathlib"@"1d650c2e131f500f3c17f33b4d19d2ea15987f2c"
universe w v u
namespace CategoryTheory... | Mathlib/CategoryTheory/Sites/DenseSubsite.lean | 133 | 141 | theorem functorPullback_pushforward_covering [Full G] {X : C}
(T : K (G.obj X)) : (T.val.functorPullback G).functorPushforward G ∈ K (G.obj X) := by
refine K.superset_covering ?_ (K.bind_covering T.property |
refine K.superset_covering ?_ (K.bind_covering T.property
fun Y f _ => G.is_cover_of_isCoverDense K Y)
rintro Y _ ⟨Z, _, f, hf, ⟨W, g, f', ⟨rfl⟩⟩, rfl⟩
use W; use G.preimage (f' ≫ f); use g
constructor
· simpa using T.val.downward_closed hf f'
· simp
| true |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 56 | 66 | theorem borel_eq_generateFrom_Ioi_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ioi (a : ℝ)}) := by
rw [borel_eq_generateFrom_Ioi] |
rw [borel_eq_generateFrom_Ioi]
refine le_antisymm
(generateFrom_le ?_)
(generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
rintro _ ⟨a, rfl⟩
have : IsGLB (range ((↑) : ℚ → ℝ) ∩ Ioi a) a := by
simp [isGLB_iff_le_iff, mem_lowerBounds, ← le_iff_forall_lt_rat_imp_l... | true |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 83 | 85 | theorem one_lt_card_iff_nontrivial [Finite α] : 1 < Nat.card α ↔ Nontrivial α := by
haveI := Fintype.ofFinite α |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.one_lt_card_iff_nontrivial]
| true |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
section IsLocalizedModule
universe u v
variable {R : Type*} [CommSemiring R] (S : Submonoid R)
variabl... | Mathlib/Algebra/Module/LocalizedModule.lean | 574 | 588 | theorem IsLocalizedModule.of_linearEquiv (e : M' ≃ₗ[R] M'') [hf : IsLocalizedModule S f] :
IsLocalizedModule S (e ∘ₗ f : M →ₗ[R] M'') where
map_units s := by
rw [show algebraMap R (Module.End R M'') s = e ∘ₗ (algebraMap R (Module.End R M') s) ∘ₗ e.symm |
rw [show algebraMap R (Module.End R M'') s = e ∘ₗ (algebraMap R (Module.End R M') s) ∘ₗ e.symm
by ext; simp, Module.End_isUnit_iff, LinearMap.coe_comp, LinearMap.coe_comp,
LinearEquiv.coe_coe, LinearEquiv.coe_coe, EquivLike.comp_bijective, EquivLike.bijective_comp]
exact (Module.End_isUnit_iff _).m... | true |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 86 | 86 | theorem lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b := by | rw [mul_comm, lt_div_iff hc]
| true |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Conj
universe w v u
namespace CategoryTheory.Limits.Types
variable (C : Type u) [Category.{v} C]
def constPUnitFunctor : C ⥤ Type w := (Functor.const C).o... | Mathlib/CategoryTheory/Limits/IsConnected.lean | 87 | 93 | theorem zigzag_of_eqvGen_quot_rel (F : C ⥤ Type w) (c d : Σ j, F.obj j)
(h : EqvGen (Quot.Rel F) c d) : Zigzag c.1 d.1 := by
induction h with |
induction h with
| rel _ _ h => exact Zigzag.of_hom <| Exists.choose h
| refl _ => exact Zigzag.refl _
| symm _ _ _ ih => exact zigzag_symmetric ih
| trans _ _ _ _ _ ih₁ ih₂ => exact ih₁.trans ih₂
| true |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
| Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 57 | 64 | theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.InjOn fun x => (x ⊔ u) \ v := by
rintro a ha b hb hab |
rintro a ha b hb hab
have h : ((a ⊔ u) \ v) \ u ⊔ v = ((b ⊔ u) \ v) \ u ⊔ v := by
dsimp at hab
rw [hab]
rwa [sdiff_sdiff_comm, ha.1.symm.sup_sdiff_cancel_right, sdiff_sdiff_comm,
hb.1.symm.sup_sdiff_cancel_right, sdiff_sup_cancel ha.2, sdiff_sup_cancel hb.2] at h
| true |
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.SuccPred
#align_import data.int.succ_pred from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Order
namespace Int
-- so that Lean reads `Int.succ` through `SuccOrder.succ`
@[instance] abbrev instSuccOrder : Su... | Mathlib/Data/Int/SuccPred.lean | 79 | 79 | theorem sub_one_covBy (z : ℤ) : z - 1 ⋖ z := by | rw [Int.covBy_iff_succ_eq, sub_add_cancel]
| true |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 99 | 105 | theorem t_inv (i j : D.J) : D.t i j ≫ D.t j i = 𝟙 _ := by
have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd ≫ inv pullback.fst := by simp |
have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd ≫ inv pullback.fst := by simp
have := D.cocycle i j i
rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this
simp only [Category.assoc, IsIso.inv_hom_id_assoc] at this
rw [← IsIso.eq_inv_comp, ← Category.assoc, IsIso.comp_inv_e... | true |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 161 | 162 | theorem cylinder_empty (s : Finset ι) : cylinder s (∅ : Set (∀ i : s, α i)) = ∅ := by |
rw [cylinder, preimage_empty]
| true |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 166 | 168 | theorem expComparison_iso_of_frobeniusMorphism_iso (h : L ⊣ F) (A : C)
[i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) := by |
rw [← frobeniusMorphism_mate F h]; infer_instance
| true |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 73 | 75 | theorem Gamma_one_top : Gamma 1 = ⊤ := by
ext |
ext
simp [eq_iff_true_of_subsingleton]
| true |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
variable {R : Type*} [CommRing R] (M : Submonoid R) ... | Mathlib/RingTheory/Localization/Submodule.lean | 125 | 133 | theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
(coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal := by
constructor <;> rintro ⟨⟨x, hx⟩⟩ |
constructor <;> rintro ⟨⟨x, hx⟩⟩
· have x_mem : x ∈ coeSubmodule S I := hx.symm ▸ Submodule.mem_span_singleton_self x
obtain ⟨x, _, rfl⟩ := (mem_coeSubmodule _ _).mp x_mem
refine ⟨⟨x, coeSubmodule_injective S h ?_⟩⟩
rw [Ideal.submodule_span_eq, hx, coeSubmodule_span_singleton]
· refine ⟨⟨algebraMap R... | true |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 188 | 196 | theorem comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor |
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars ... | true |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 233 | 235 | theorem compl_mem_coprodᵢ {s : Set (∀ i, α i)} :
sᶜ ∈ Filter.coprodᵢ f ↔ ∀ i, (eval i '' s)ᶜ ∈ f i := by |
simp only [Filter.coprodᵢ, mem_iSup, compl_mem_comap]
| true |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.integral.interval_average from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open MeasureTheory Set TopologicalSpace
open scoped Interval
variable {E : Ty... | Mathlib/MeasureTheory/Integral/IntervalAverage.lean | 39 | 40 | theorem interval_average_symm (f : ℝ → E) (a b : ℝ) : (⨍ x in a..b, f x) = ⨍ x in b..a, f x := by |
rw [setAverage_eq, setAverage_eq, uIoc_comm]
| true |
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 94 | 113 | theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
(p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure := by
-- `p` itself is in the closure of polynomials, by the Weierstrass theorem, |
-- `p` itself is in the closure of polynomials, by the Weierstrass theorem,
have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure :=
continuousMap_mem_polynomialFunctions_closure _ _ p
-- and so there are polynomials arbitrarily close.
have frequently_mem_polynomials := mem_c... | true |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 71 | 73 | theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_left' bc a⁻¹
| true |
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 107 | 114 | theorem natDegree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
(cyclotomic' n R).natDegree = Nat.totient n := by
rw [cyclotomic'] |
rw [cyclotomic']
rw [natDegree_prod (primitiveRoots n R) fun z : R => X - C z]
· simp only [IsPrimitiveRoot.card_primitiveRoots h, mul_one, natDegree_X_sub_C, Nat.cast_id,
Finset.sum_const, nsmul_eq_mul]
intro z _
exact X_sub_C_ne_zero z
| true |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 138 | 139 | theorem toIocMod_sub_self (a b : α) : toIocMod hp a b - b = -toIocDiv hp a b • p := by |
rw [toIocMod, sub_sub_cancel_left, neg_smul]
| true |
set_option autoImplicit true
namespace Array
@[simp]
theorem extract_eq_nil_of_start_eq_end {a : Array α} :
a.extract i i = #[] := by
refine extract_empty_of_stop_le_start a ?h
exact Nat.le_refl i
theorem extract_append_left {a b : Array α} {i j : Nat} (h : j ≤ a.size) :
(a ++ b).extract i j = a.extrac... | Mathlib/Data/Array/ExtractLemmas.lean | 40 | 42 | theorem extract_eq_of_size_le_end {a : Array α} (h : a.size ≤ l) :
a.extract p l = a.extract p a.size := by |
simp only [extract, Nat.min_eq_right h, Nat.sub_eq, mkEmpty_eq, Nat.min_self]
| true |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 161 | 168 | theorem areaForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
[hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
(Orientation.map (Fin 2) φ.toLinearEquiv o).areaForm x y =
o.areaForm (φ.symm x) (φ.symm y) := by
have : φ.symm ∘ ![x, y] = ![φ.symm x, φ.symm y] := by |
have : φ.symm ∘ ![x, y] = ![φ.symm x, φ.symm y] := by
ext i
fin_cases i <;> rfl
simp [areaForm_to_volumeForm, volumeForm_map, this]
| true |
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
variable {α : Type*}
namespace Order
open Function Set OrderDual
section LT
variable [LT α]
def IsSuccLimit (a : α) : Pr... | Mathlib/Order/SuccPred/Limit.lean | 46 | 47 | theorem not_isSuccLimit_iff_exists_covBy (a : α) : ¬IsSuccLimit a ↔ ∃ b, b ⋖ a := by |
simp [IsSuccLimit]
| true |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 56 | 58 | theorem coprime_fintype_prod_right_iff [Fintype ι] {x : ℕ} {s : ι → ℕ} :
Coprime x (∏ i, s i) ↔ ∀ i, Coprime x (s i) := by |
simp [coprime_prod_right_iff]
| true |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 135 | 144 | theorem of_le (N : Submonoid R) (h₁ : M ≤ N) (h₂ : ∀ r ∈ N, IsUnit (algebraMap R S r)) :
IsLocalization N S where
map_units' r := h₂ r r.2
surj' s :=
have ⟨⟨x, y, hy⟩, H⟩ := IsLocalization.surj M s
⟨⟨x, y, h₁ hy⟩, H⟩
exists_of_eq {x y} := by
rw [IsLocalization.eq_iff_exists M] |
rw [IsLocalization.eq_iff_exists M]
rintro ⟨c, hc⟩
exact ⟨⟨c, h₁ c.2⟩, hc⟩
| true |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 64 | 65 | theorem withDensityᵥ_zero : μ.withDensityᵥ (0 : α → E) = 0 := by |
ext1 s hs; erw [withDensityᵥ_apply (integrable_zero α E μ) hs]; simp
| true |
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import model_theory.graph from "leanprover-community/mathlib"@"e56b8fea84d60fe434632b9d3b829ee685fb0c8f"
set_option linter.uppercaseLean3 false
universe u v w w'
namespace FirstOrder
namespace Language
open FirstOr... | Mathlib/ModelTheory/Graph.lean | 107 | 110 | theorem _root_.SimpleGraph.simpleGraphOfStructure (G : SimpleGraph V) :
@simpleGraphOfStructure V G.structure _ = G := by
ext |
ext
rfl
| true |
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 143 | 153 | theorem basisSets_smul_right (v : E) (U : Set E) (hU : U ∈ p.basisSets) :
∀ᶠ x : 𝕜 in 𝓝 0, x • v ∈ U := by
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩ |
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
rw [hU, Filter.eventually_iff]
simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul]
by_cases h : 0 < (s.sup p) v
· simp_rw [(lt_div_iff h).symm]
rw [← _root_.ball_zero_eq]
exact Metric.ball_mem_nhds 0 (div_pos hr h)
simp_rw [le_antisymm (not_lt.mp h) (... | true |
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] |
rw [f.decomp]
apply f.contLinear.contDiff.add
exact contDiff_const
| true |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_theory.measure.stieltjes from "leanprover-community/mathlib"@"20d5763051978e9bc6428578ed070445df6a18b3"
noncomputable section
open scoped Classical
open Set Filter Function ENNReal NNReal T... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 76 | 80 | theorem iInf_Ioi_eq (f : StieltjesFunction) (x : ℝ) : ⨅ r : Ioi x, f r = f x := by
suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.rightLim_eq] |
suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.rightLim_eq]
rw [f.mono.rightLim_eq_sInf, sInf_image']
rw [← neBot_iff]
infer_instance
| true |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 46 | 46 | theorem zero_of_nonpos {x : ℝ} (hx : x ≤ 0) : expNegInvGlue x = 0 := by | simp [expNegInvGlue, hx]
| true |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 91 | 95 | theorem iteratedFDerivWithin_eq_equiv_comp :
iteratedFDerivWithin 𝕜 n f s =
ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDerivWithin n f s := by
rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm, |
rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,
Function.id_comp]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 124 | 126 | theorem natDegree_mul (hp : p ≠ 0) (hq : q ≠ 0) : (p*q).natDegree = p.natDegree + q.natDegree := by
rw [← Nat.cast_inj (R := WithBot ℕ), ← degree_eq_natDegree (mul_ne_zero hp hq), |
rw [← Nat.cast_inj (R := WithBot ℕ), ← degree_eq_natDegree (mul_ne_zero hp hq),
Nat.cast_add, ← degree_eq_natDegree hp, ← degree_eq_natDegree hq, degree_mul]
| true |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_import ring_theory.chain_of_divisors from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {M : Type*} [CancelCommMonoidWithZero... | Mathlib/RingTheory/ChainOfDivisors.lean | 224 | 231 | theorem factor_orderIso_map_one_eq_bot {m : Associates M} {n : Associates N}
(d : { l : Associates M // l ≤ m } ≃o { l : Associates N // l ≤ n }) :
(d ⟨1, one_dvd m⟩ : Associates N) = 1 := by
letI : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot bot_le |
letI : OrderBot { l : Associates M // l ≤ m } := Subtype.orderBot bot_le
letI : OrderBot { l : Associates N // l ≤ n } := Subtype.orderBot bot_le
simp only [← Associates.bot_eq_one, Subtype.mk_bot, bot_le, Subtype.coe_eq_bot_iff]
letI : BotHomClass ({ l // l ≤ m } ≃o { l // l ≤ n }) _ _ := OrderIsoClass.toBotH... | true |
import Mathlib.CategoryTheory.Equivalence
#align_import algebraic_topology.dold_kan.compatibility from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category
namespace AlgebraicTopology
namespace DoldKan
namespace Compatibility
variable {A A' B B'... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 103 | 105 | theorem equivalence₁UnitIso_eq : (equivalence₁ hF).unitIso = equivalence₁UnitIso hF := by
ext X |
ext X
simp [equivalence₁]
| true |
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u → Type u) [Functor F] where
P : PFunctor.{u}
abs : ∀ {α}, P α → F α
repr : ∀ {α}, F α → P α
abs_repr : ∀ {α} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 377 | 379 | theorem corecF_eq {α : Type _} (g : α → F α) (x : α) :
PFunctor.M.dest (corecF g x) = q.P.map (corecF g) (repr (g x)) := by |
rw [corecF, PFunctor.M.dest_corec]
| true |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 65 | 68 | theorem intDegree_polynomial {p : K[X]} :
intDegree (algebraMap K[X] (RatFunc K) p) = natDegree p := by
rw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one, |
rw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one,
Int.ofNat_zero, sub_zero]
| true |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 111 | 114 | theorem isLittleO_pow_pow_of_abs_lt_left {r₁ r₂ : ℝ} (h : |r₁| < |r₂|) :
(fun n : ℕ ↦ r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n := by
refine (IsLittleO.of_norm_left ?_).of_norm_right |
refine (IsLittleO.of_norm_left ?_).of_norm_right
exact (isLittleO_pow_pow_of_lt_left (abs_nonneg r₁) h).congr (pow_abs r₁) (pow_abs r₂)
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 100 | 112 | theorem rpow_def_of_neg {x : ℝ} (hx : x < 0) (y : ℝ) : x ^ y = exp (log x * y) * cos (y * π) := by
rw [rpow_def, Complex.cpow_def, if_neg] |
rw [rpow_def, Complex.cpow_def, if_neg]
· have : Complex.log x * y = ↑(log (-x) * y) + ↑(y * π) * Complex.I := by
simp only [Complex.log, abs_of_neg hx, Complex.arg_ofReal_of_neg hx, Complex.abs_ofReal,
Complex.ofReal_mul]
ring
rw [this, Complex.exp_add_mul_I, ← Complex.ofReal_exp, ← Comple... | true |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 211 | 214 | theorem cons_bagInter_of_neg (l₁ : List α) (h : a ∉ l₂) :
(a :: l₁).bagInter l₂ = l₁.bagInter l₂ := by
cases l₂; · simp only [bagInter_nil] |
cases l₂; · simp only [bagInter_nil]
simp only [erase_of_not_mem h, List.bagInter, if_neg (mt mem_of_elem_eq_true h)]
| true |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 68 | 69 | theorem gcd_mul_left_add_left (m n k : ℕ) : gcd (n * k + m) n = gcd m n := by |
rw [gcd_comm, gcd_mul_left_add_right, gcd_comm]
| true |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 135 | 149 | theorem isBigO_iff'' {g : α → E'''} :
f =O[l] g ↔ ∃ c > 0, ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ := by
refine ⟨fun h => ?mp, fun h => ?mpr⟩ |
refine ⟨fun h => ?mp, fun h => ?mpr⟩
case mp =>
rw [isBigO_iff'] at h
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
filter_upwards [hc] with x hx
rwa [inv_mul_le_iff (by positivity)]
case mpr =>
rw [isBigO_iff']
obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
refine ⟨c⁻¹, ⟨by posi... | true |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 113 | 116 | theorem discr_of_matrix_vecMul (b : ι → B) (P : Matrix ι ι A) :
discr A (b ᵥ* P.map (algebraMap A B)) = P.det ^ 2 * discr A b := by
rw [discr_def, traceMatrix_of_matrix_vecMul, det_mul, det_mul, det_transpose, mul_comm, ← |
rw [discr_def, traceMatrix_of_matrix_vecMul, det_mul, det_mul, det_transpose, mul_comm, ←
mul_assoc, discr_def, pow_two]
| true |
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 108 | 112 | theorem pairwise_append_comm {R : α → α → Prop} (s : ∀ {x y}, R x y → R y x) {l₁ l₂ : List α} :
Pairwise R (l₁ ++ l₂) ↔ Pairwise R (l₂ ++ l₁) := by
have (l₁ l₂ : List α) (H : ∀ x : α, x ∈ l₁ → ∀ y : α, y ∈ l₂ → R x y) |
have (l₁ l₂ : List α) (H : ∀ x : α, x ∈ l₁ → ∀ y : α, y ∈ l₂ → R x y)
(x : α) (xm : x ∈ l₂) (y : α) (ym : y ∈ l₁) : R x y := s (H y ym x xm)
simp only [pairwise_append, and_left_comm]; rw [Iff.intro (this l₁ l₂) (this l₂ l₁)]
| true |
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.antilipschitz from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 77 | 79 | theorem mul_le_nndist (hf : AntilipschitzWith K f) (x y : α) :
K⁻¹ * nndist x y ≤ nndist (f x) (f y) := by |
simpa only [div_eq_inv_mul] using NNReal.div_le_of_le_mul' (hf.le_mul_nndist x y)
| true |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 98 | 100 | theorem leftUnitor_hom_hom {X : Action V G} : Hom.hom (λ_ X).hom = (λ_ X.V).hom := by
dsimp |
dsimp
simp
| true |
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open FirstOrder
namespace FirstOrder
namespace Language
open Structure
variable (L : Language) (M : Type*) (N : T... | Mathlib/ModelTheory/ElementaryMaps.lean | 98 | 100 | theorem map_formula (f : M ↪ₑ[L] N) {α : Type*} (φ : L.Formula α) (x : α → M) :
φ.Realize (f ∘ x) ↔ φ.Realize x := by |
rw [Formula.Realize, Formula.Realize, ← f.map_boundedFormula, Unique.eq_default (f ∘ default)]
| true |
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 149 | 149 | theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by | simp only [mem_Ioc, and_true_iff, le_rfl]
| true |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 165 | 166 | theorem Left.inv_lt_one_iff : a⁻¹ < 1 ↔ 1 < a := by |
rw [← mul_lt_mul_iff_left a, mul_inv_self, mul_one]
| true |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 98 | 101 | theorem map_scaleRoots (p : R[X]) (x : R) (f : R →+* S) (h : f p.leadingCoeff ≠ 0) :
(p.scaleRoots x).map f = (p.map f).scaleRoots (f x) := by
ext |
ext
simp [Polynomial.natDegree_map_of_leadingCoeff_ne_zero _ h]
| true |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 320 | 326 | theorem toDual_total_right (f : ι →₀ R) (i : ι) :
b.toDual (b i) (Finsupp.total ι M R b f) = f i := by
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum] |
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
| true |
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 483 | 486 | theorem sheafifyMap_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) :
J.sheafifyMap (η ≫ γ) = J.sheafifyMap η ≫ J.sheafifyMap γ := by
dsimp [sheafifyMap, sheafify] |
dsimp [sheafifyMap, sheafify]
simp
| true |
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 |
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]
| true |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 181 | 185 | theorem mk_eq_localization_mk (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
RatFunc.mk p q =
ofFractionRing (Localization.mk p ⟨q, mem_nonZeroDivisors_iff_ne_zero.mpr hq⟩) := by
-- Porting note: the original proof, did not need to pass `hq` |
-- Porting note: the original proof, did not need to pass `hq`
rw [mk_def_of_ne _ hq, Localization.mk_eq_mk']
| true |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 152 | 155 | theorem single_mul_single [Fintype n] [DecidableEq k] [DecidableEq m] [DecidableEq n] [Semiring α]
(a : m) (b : n) (c : k) :
((single a b).toMatrix : Matrix _ _ α) * (single b c).toMatrix = (single a c).toMatrix := by |
rw [← toMatrix_trans, single_trans_single]
| true |
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 64 | 66 | theorem pi_eq_zero (f : (i : ι) → M₂ →ₗ[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 := by
simp only [LinearMap.ext_iff, pi_apply, funext_iff]; |
simp only [LinearMap.ext_iff, pi_apply, funext_iff];
exact ⟨fun h a b => h b a, fun h a b => h b a⟩
| true |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 142 | 149 | theorem dual_ordConnectedSection (s : Set α) :
ordConnectedSection (ofDual ⁻¹' s) = ofDual ⁻¹' ordConnectedSection s := by
simp only [ordConnectedSection] |
simp only [ordConnectedSection]
simp (config := { unfoldPartialApp := true }) only [ordConnectedProj]
ext x
simp only [mem_range, Subtype.exists, mem_preimage, OrderDual.exists, dual_ordConnectedComponent,
ofDual_toDual]
tauto
| true |
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