Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 194 | 202 | theorem map_Hσ {D : Type*} [Category D] [Preadditive D] (G : C ⥤ D) [G.Additive]
(X : SimplicialObject C) (q n : ℕ) :
(Hσ q : K[((whiskering C D).obj G).obj X] ⟶ _).f n = G.map ((Hσ q : K[X] ⟶ _).f n) := by |
unfold Hσ
have eq := HomologicalComplex.congr_hom (map_nullHomotopicMap' G (@hσ' _ _ _ X q)) n
simp only [Functor.mapHomologicalComplex_map_f, ← map_hσ'] at eq
rw [eq]
let h := (Functor.congr_obj (map_alternatingFaceMapComplex G) X).symm
congr
|
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 84 | 93 | theorem lieIdeal_oper_eq_linear_span' :
(↑⁅I, N⁆ : Submodule R M) = Submodule.span R { m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m } := by |
rw [lieIdeal_oper_eq_linear_span]
congr
ext m
constructor
· rintro ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩
exact ⟨x, hx, n, hn, rfl⟩
· rintro ⟨x, hx, n, hn, rfl⟩
exact ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩
|
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
namespace NNRat
@[simp, norm_cast]
theorem cast_pow {K} [DivisionSemiring K] (q : ℚ≥0) (n : ℕ) :
NNRat.cast (q ^ n) = (NNRat.cast q :... | Mathlib/Data/Rat/Cast/Lemmas.lean | 69 | 75 | theorem cast_zpow_of_ne_zero {K} [DivisionSemiring K] (q : ℚ≥0) (z : ℤ) (hq : (q.num : K) ≠ 0) :
NNRat.cast (q ^ z) = (NNRat.cast q : K) ^ z := by |
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg
· simp
· simp_rw [zpow_neg, zpow_natCast, ← inv_pow, NNRat.cast_pow]
congr
rw [cast_inv_of_ne_zero hq]
|
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE
#align_import algebraic_geometry.morphisms.basic from "leanprover-community/mathlib"@"434e2fd21c1900747afc6d13d8be7f4eedba7218"
set_option lin... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 297 | 305 | theorem AffineTargetMorphismProperty.IsLocal.affine_target_iff {P : AffineTargetMorphismProperty}
(hP : P.IsLocal) {X Y : Scheme.{u}} (f : X ⟶ Y) [IsAffine Y] :
targetAffineLocally P f ↔ P f := by |
haveI : ∀ i, IsAffine (Scheme.OpenCover.obj (Scheme.openCoverOfIsIso (𝟙 Y)) i) := fun i => by
dsimp; infer_instance
rw [hP.affine_openCover_iff f (Scheme.openCoverOfIsIso (𝟙 Y))]
trans P (pullback.snd : pullback f (𝟙 _) ⟶ _)
· exact ⟨fun H => H PUnit.unit, fun H _ => H⟩
rw [← Category.comp_id pullback... |
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 448 | 456 | theorem chaar_self (K₀ : PositiveCompacts G) : chaar K₀ K₀.toCompacts = 1 := by |
let eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
have : Continuous eval := continuous_apply _
show chaar K₀ ∈ eval ⁻¹' {(1 : ℝ)}
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
unfold clPrehaar; rw [IsClosed.closure_subset_iff]
· rintro _ ⟨U, ⟨_, h2U, h3U⟩, rfl⟩; apply prehaar_self
r... |
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 334 | 337 | theorem image_prod_mk_subset_prod {f : α → β} {g : α → γ} {s : Set α} :
(fun x => (f x, g x)) '' s ⊆ (f '' s) ×ˢ (g '' s) := by |
rintro _ ⟨x, hx, rfl⟩
exact mk_mem_prod (mem_image_of_mem f hx) (mem_image_of_mem g hx)
|
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 729 | 733 | theorem TendstoLocallyUniformly.comp [TopologicalSpace γ] (h : TendstoLocallyUniformly F f p)
(g : γ → α) (cg : Continuous g) : TendstoLocallyUniformly (fun n => F n ∘ g) (f ∘ g) p := by |
rw [← tendstoLocallyUniformlyOn_univ] at h ⊢
rw [continuous_iff_continuousOn_univ] at cg
exact h.comp _ (mapsTo_univ _ _) cg
|
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 158 | 174 | theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
{t : L[[α]].Term β} {v : β → M} :
t.constantsToVars.realize (Sum.elim (fun a => ↑(L.con a)) v) = t.realize v := by |
induction' t with _ n f ts ih
· simp
· cases n
· cases f
· simp only [realize, ih, Nat.zero_eq, constantsOn, mk₂_Functions]
-- Porting note: below lemma does not work with simp for some reason
rw [withConstants_funMap_sum_inl]
· simp only [realize, constantsToVars, Sum.elim_inl, f... |
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 | 157 | 161 | theorem to_rightInverse (hf : AntilipschitzWith K f) {g : β → α} (hg : Function.RightInverse g f) :
LipschitzWith K g := by |
intro x y
have := hf (g x) (g y)
rwa [hg x, hg y] at this
|
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 236 | 237 | theorem comp_hom_eq_id (α : X ≅ Y) {f : Y ⟶ X} : f ≫ α.hom = 𝟙 Y ↔ f = α.inv := by |
rw [← eq_comp_inv, id_comp]
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 234 | 236 | theorem IsCompact.disjoint_nhdsSet_right {l : Filter X} (hs : IsCompact s) :
Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x) := by |
simpa only [disjoint_comm] using hs.disjoint_nhdsSet_left
|
import Mathlib.RingTheory.Trace
import Mathlib.FieldTheory.Finite.GaloisField
#align_import field_theory.finite.trace from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace FiniteField
| Mathlib/FieldTheory/Finite/Trace.lean | 25 | 32 | theorem trace_to_zmod_nondegenerate (F : Type*) [Field F] [Finite F]
[Algebra (ZMod (ringChar F)) F] {a : F} (ha : a ≠ 0) :
∃ b : F, Algebra.trace (ZMod (ringChar F)) F (a * b) ≠ 0 := by |
haveI : Fact (ringChar F).Prime := ⟨CharP.char_is_prime F _⟩
have htr := traceForm_nondegenerate (ZMod (ringChar F)) F a
simp_rw [Algebra.traceForm_apply] at htr
by_contra! hf
exact ha (htr hf)
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 54 | 61 | theorem oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arcsin (‖y‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arcsin_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.left_ne_zero_of_oangle_eq_pi_div_... |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 213 | 219 | theorem TendstoUniformlyOn.congr {F' : ι → α → β} (hf : TendstoUniformlyOn F f p s)
(hff' : ∀ᶠ n in p, Set.EqOn (F n) (F' n) s) : TendstoUniformlyOn F' f p s := by |
rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at hf ⊢
refine hf.congr ?_
rw [eventually_iff] at hff' ⊢
simp only [Set.EqOn] at hff'
simp only [mem_prod_principal, hff', mem_setOf_eq]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 135 | 143 | theorem le_fib_self {n : ℕ} (five_le_n : 5 ≤ n) : n ≤ fib n := by |
induction' five_le_n with n five_le_n IH
·-- 5 ≤ fib 5
rfl
· -- n + 1 ≤ fib (n + 1) for 5 ≤ n
rw [succ_le_iff]
calc
n ≤ fib n := IH
_ < fib (n + 1) := fib_lt_fib_succ (le_trans (by decide) five_le_n)
|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 203 | 219 | theorem of_diff_of_diff_eq_zero {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
(h' : v (B \ A) = 0) : v (A \ B) + v B = v A := by |
symm
calc
v A = v (A \ B ∪ A ∩ B) := by simp only [Set.diff_union_inter]
_ = v (A \ B) + v (A ∩ B) := by
rw [of_union]
· rw [disjoint_comm]
exact Set.disjoint_of_subset_left A.inter_subset_right disjoint_sdiff_self_right
· exact hA.diff hB
· exact hA.inter hB
_ = v (A \ ... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 416 | 416 | theorem coeff_zero_mul_X (φ : R⟦X⟧) : coeff R 0 (φ * X) = 0 := by | simp
|
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 | 167 | 177 | theorem basisSets_smul_left (x : 𝕜) (U : Set E) (hU : U ∈ p.basisSets) :
∃ V ∈ p.addGroupFilterBasis.sets, V ⊆ (fun y : E => x • y) ⁻¹' U := by |
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
rw [hU]
by_cases h : x ≠ 0
· rw [(s.sup p).smul_ball_preimage 0 r x h, smul_zero]
use (s.sup p).ball 0 (r / ‖x‖)
exact ⟨p.basisSets_mem s (div_pos hr (norm_pos_iff.mpr h)), Subset.rfl⟩
refine ⟨(s.sup p).ball 0 r, p.basisSets_mem s hr, ?_⟩
simp only [... |
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 121 | 124 | theorem integral_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) (hts : t ⊆ s) :
∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ - ∫ x in t, f x ∂μ := by |
rw [eq_sub_iff_add_eq, ← integral_union, diff_union_of_subset hts]
exacts [disjoint_sdiff_self_left, ht, hfs.mono_set diff_subset, hfs.mono_set hts]
|
import Mathlib.Topology.UniformSpace.AbsoluteValue
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.UniformSpace.Completion
#align_import topology.uniform_space.compare_reals from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
open Set... | Mathlib/Topology/UniformSpace/CompareReals.lean | 60 | 65 | theorem Rat.uniformSpace_eq :
(AbsoluteValue.abs : AbsoluteValue ℚ ℚ).uniformSpace = PseudoMetricSpace.toUniformSpace := by |
ext s
rw [(AbsoluteValue.hasBasis_uniformity _).mem_iff, Metric.uniformity_basis_dist_rat.mem_iff]
simp only [Rat.dist_eq, AbsoluteValue.abs_apply, ← Rat.cast_sub, ← Rat.cast_abs, Rat.cast_lt,
abs_sub_comm]
|
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import... | Mathlib/Analysis/Calculus/MeanValue.lean | 728 | 738 | theorem exists_ratio_hasDerivAt_eq_ratio_slope :
∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c := by |
let h x := (g b - g a) * f x - (f b - f a) * g x
have hI : h a = h b := by simp only [h]; ring
let h' x := (g b - g a) * f' x - (f b - f a) * g' x
have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx =>
((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))
have hhc : Continu... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 402 | 402 | theorem cos_coe_pi : cos (π : Angle) = -1 := by | rw [cos_coe, Real.cos_pi]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 90 | 91 | theorem map_add_nat [AddMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (x : G) (n : ℕ) : f (x + n) = f x + n := by | simp
|
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 102 | 127 | theorem integral_comp_abs {f : ℝ → ℝ} :
∫ x, f |x| = 2 * ∫ x in Ioi (0:ℝ), f x := by |
have eq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x := by
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [abs_eq_self.mpr (le_of_lt (by exact hx))]
by_cases hf : IntegrableOn (fun x => f |x|) (Ioi 0)
· have int_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) := by
rw [← Measure... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 87 | 89 | theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by |
ext x
simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range]
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 473 | 474 | theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ := by |
simp only [compl_iInter, compl_compl]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Ring.Divisibility.Basic
#align_import ring_theory.prime from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
section CommRing
variable {α : Ty... | Mathlib/RingTheory/Prime.lean | 70 | 73 | theorem Prime.abs [LinearOrder α] {p : α} (hp : Prime p) : Prime (abs p) := by |
obtain h | h := abs_choice p <;> rw [h]
· exact hp
· exact hp.neg
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 47 | 48 | theorem constantCoeff_invUnitsSub (u : Rˣ) : constantCoeff R (invUnitsSub u) = 1 /ₚ u := by |
rw [← coeff_zero_eq_constantCoeff_apply, coeff_invUnitsSub, zero_add, pow_one]
|
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 60 | 64 | theorem hasseDeriv_apply :
hasseDeriv k f = f.sum fun i r => monomial (i - k) (↑(i.choose k) * r) := by |
dsimp [hasseDeriv]
congr; ext; congr
apply nsmul_eq_mul
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 207 | 209 | theorem mem_top_objs (c : C) : c ∈ (⊤ : Subgroupoid C).objs := by |
dsimp [Top.top, objs]
simp only [univ_nonempty]
|
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.Topology.Algebra.ValuedField
#align_import number_theory.function_field from "leanprover-community/mathlib"@"70fd9563... | Mathlib/NumberTheory/FunctionField.lean | 179 | 195 | theorem InftyValuation.map_add_le_max' (x y : RatFunc Fq) :
inftyValuationDef Fq (x + y) ≤ max (inftyValuationDef Fq x) (inftyValuationDef Fq y) := by |
by_cases hx : x = 0
· rw [hx, zero_add]
conv_rhs => rw [inftyValuationDef, if_pos (Eq.refl _)]
rw [max_eq_right (WithZero.zero_le (inftyValuationDef Fq y))]
· by_cases hy : y = 0
· rw [hy, add_zero]
conv_rhs => rw [max_comm, inftyValuationDef, if_pos (Eq.refl _)]
rw [max_eq_right (WithZer... |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 496 | 516 | theorem oangle_add_right_smul_rotation_pi_div_two {x : V} (h : x ≠ 0) (r : ℝ) :
o.oangle x (x + r • o.rotation (π / 2 : ℝ) x) = Real.arctan r := by |
rcases lt_trichotomy r 0 with (hr | rfl | hr)
· have ha : o.oangle x (r • o.rotation (π / 2 : ℝ) x) = -(π / 2 : ℝ) := by
rw [o.oangle_smul_right_of_neg _ _ hr, o.oangle_neg_right h, o.oangle_rotation_self_right h, ←
sub_eq_zero, add_comm, sub_neg_eq_add, ← Real.Angle.coe_add, ← Real.Angle.coe_add,
... |
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Integral.DominatedConve... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,347 | 1,356 | theorem integral_deriv_mul_eq_sub_of_hasDeriv_right {u v u' v' : ℝ → A}
(hu : ContinuousOn u [[a, b]])
(hv : ContinuousOn v [[a, b]])
(huu' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt u (u' x) (Ioi x) x)
(hvv' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt v (v' x) (Ioi x) x)
(hu' : Interva... |
apply integral_eq_sub_of_hasDeriv_right (hu.mul hv) fun x hx ↦ (huu' x hx).mul (hvv' x hx)
exact (hu'.mul_continuousOn hv).add (hv'.continuousOn_mul hu)
|
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Algebra.Ring.Subring.Basic
#align_import algebraic_geometry.struct... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 108 | 118 | theorem IsFraction.eq_mk' {U : Opens (PrimeSpectrum.Top R)} {f : ∀ x : U, Localizations R x}
(hf : IsFraction f) :
∃ r s : R,
∀ x : U,
∃ hs : s ∉ x.1.asIdeal,
f x =
IsLocalization.mk' (Localization.AtPrime _) r
(⟨s, hs⟩ : (x : PrimeSpectrum.Top R).asIdeal.primeC... |
rcases hf with ⟨r, s, h⟩
refine ⟨r, s, fun x => ⟨(h x).1, (IsLocalization.mk'_eq_iff_eq_mul.mpr ?_).symm⟩⟩
exact (h x).2.symm
|
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 400 | 403 | theorem ι_mul_ι (r₁ r₂) : ι (0 : QuadraticForm R R) r₁ * ι (0 : QuadraticForm R R) r₂ = 0 := by |
rw [← mul_one r₁, ← mul_one r₂, ← smul_eq_mul R, ← smul_eq_mul R, LinearMap.map_smul,
LinearMap.map_smul, smul_mul_smul, ι_sq_scalar, QuadraticForm.zero_apply, RingHom.map_zero,
smul_zero]
|
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 101 | 107 | theorem embeddingPiTangent_ker_mfderiv (x : M) (hx : x ∈ s) :
LinearMap.ker (mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) = ⊥ := by |
apply bot_unique
rw [← (mdifferentiable_chart I (f.c (f.ind x hx))).ker_mfderiv_eq_bot
(f.mem_chartAt_ind_source x hx),
← comp_embeddingPiTangent_mfderiv]
exact LinearMap.ker_le_ker_comp _ _
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 239 | 247 | theorem ae_eq_condexp_of_forall_setIntegral_eq (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
{f g : α → F'} (hf : Integrable f μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hg_eq : ∀ s : Set α, MeasurableSet[m] s → μ s < ∞ → ∫ x in s, g x ∂μ = ∫ x in s, f x ∂μ)
(hgm : AEStrongly... |
refine ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condexp.integrableOn) (fun s hs hμs => ?_) hgm
(StronglyMeasurable.aeStronglyMeasurable' stronglyMeasurable_condexp)
rw [hg_eq s hs hμs, setIntegral_condexp hm hf hs]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 143 | 143 | theorem evariance_zero : evariance 0 μ = 0 := by | simp [evariance]
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 206 | 211 | theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SigmaFinite ν]
[TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν := by |
filter_upwards [ae_ae_of_ae_prod hf.ae_eq_mk] with x hx
exact
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Basic
#align_import algebra.covariant_and_contravariant from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
-- TODO: convert `ExistsMulOfLE`, `ExistsAddOfLE`?
-- TODO: relationship with `Con/AddC... | Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 170 | 176 | theorem Group.covariant_swap_iff_contravariant_swap [Group N] :
Covariant N N (swap (· * ·)) r ↔ Contravariant N N (swap (· * ·)) r := by |
refine ⟨fun h a b c bc ↦ ?_, fun h a b c bc ↦ ?_⟩
· rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a]
exact h a⁻¹ bc
· rw [← mul_inv_cancel_right b a, ← mul_inv_cancel_right c a] at bc
exact h a⁻¹ bc
|
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 | 1,101 | 1,107 | theorem ContinuousOn.preimage_interior_subset_interior_preimage {f : α → β} {s : Set α} {t : Set β}
(hf : ContinuousOn f s) (hs : IsOpen s) : s ∩ f ⁻¹' interior t ⊆ s ∩ interior (f ⁻¹' t) :=
calc
s ∩ f ⁻¹' interior t ⊆ interior (s ∩ f ⁻¹' t) :=
interior_maximal (inter_subset_inter (Subset.refl _) (preim... | rw [interior_inter, hs.interior_eq]
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 197 | 199 | theorem setIntegral_indicator (ht : MeasurableSet t) :
∫ x in s, t.indicator f x ∂μ = ∫ x in s ∩ t, f x ∂μ := by |
rw [integral_indicator ht, Measure.restrict_restrict ht, Set.inter_comm]
|
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f76"
universe u
section
open Decidable
variable {α : Type u} [LinearOrder α]
theorem min_def (a b : α) : min a b = if a ≤ b then a else b := by
rw [LinearOrder.min_def a]... | Mathlib/Init/Order/LinearOrder.lean | 97 | 97 | theorem min_self (a : α) : min a a = a := by | simp [min_def]
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 72 | 73 | theorem mem_iInter₂ {x : γ} {s : ∀ i, κ i → Set γ} : (x ∈ ⋂ (i) (j), s i j) ↔ ∀ i j, x ∈ s i j := by |
simp_rw [mem_iInter]
|
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.GDelta
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.SpecificLimits.Basic
#align_import topology.urysohns_lemma from "lea... | Mathlib/Topology/UrysohnsLemma.lean | 251 | 252 | theorem lim_of_nmem_U (c : CU P) (x : X) (h : x ∉ c.U) : c.lim x = 1 := by |
simp only [CU.lim, approx_of_nmem_U c _ h, ciSup_const]
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 236 | 240 | theorem MDifferentiableWithinAt.hasMFDerivWithinAt (h : MDifferentiableWithinAt I I' f s x) :
HasMFDerivWithinAt I I' f s x (mfderivWithin I I' f s x) := by |
refine ⟨h.1, ?_⟩
simp only [mfderivWithin, h, if_pos, mfld_simps]
exact DifferentiableWithinAt.hasFDerivWithinAt h.2
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 1,387 | 1,388 | theorem sep_eq_self_iff_mem_true : { x ∈ s | p x } = s ↔ ∀ x ∈ s, p x := by |
simp_rw [ext_iff, mem_sep_iff, and_iff_left_iff_imp]
|
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {α G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 210 | 225 | theorem mul_normal (H N : Subgroup G) [hN : N.Normal] : (↑(H ⊔ N) : Set G) = H * N := by |
rw [sup_eq_closure_mul]
refine Set.Subset.antisymm (fun x hx => ?_) subset_closure
induction hx using closure_induction'' with
| one => exact ⟨1, one_mem _, 1, one_mem _, mul_one 1⟩
| mem _ hx => exact hx
| inv_mem x hx =>
obtain ⟨x, hx, y, hy, rfl⟩ := hx
simpa only [mul_inv_rev, mul_assoc, inv_inv... |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 356 | 373 | theorem partiallyWellOrderedOn_iff_finite_antichains [IsSymm α r] :
s.PartiallyWellOrderedOn r ↔ ∀ t, t ⊆ s → IsAntichain r t → t.Finite := by |
refine ⟨fun h t ht hrt => hrt.finite_of_partiallyWellOrderedOn (h.mono ht), ?_⟩
rintro hs f hf
by_contra! H
refine infinite_range_of_injective (fun m n hmn => ?_) (hs _ (range_subset_iff.2 hf) ?_)
· obtain h | h | h := lt_trichotomy m n
· refine (H _ _ h ?_).elim
rw [hmn]
exact refl _
· e... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 293 | 294 | theorem liftHom_root (h : IsAdjoinRoot S f) : h.liftHom x hx' h.root = x := by |
rw [← lift_algebraMap_apply, lift_root]
|
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 404 | 409 | theorem nullHomotopicMap_f_of_not_rel_left {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₀ : ∀ l : ι, ¬c.Rel k₀ l) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₀ = hom k₀ k₁ ≫ D.d k₁ k₀ := by |
dsimp only [nullHomotopicMap]
rw [prevD_eq hom r₁₀, dNext, AddMonoidHom.mk'_apply, C.shape, zero_comp, zero_add]
exact hk₀ _
|
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Tactic.Lift
import Mathlib.Tactic.Monotonicity.Attr
open Function
variable {β G M : Type*}
section Monoid
variable [Monoid M]
section Preorder
variable [Preorder M]
section Left
variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
@[to_additive (... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 88 | 92 | theorem pow_lt_pow_right' [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ} (ha : 1 < a)
(h : n < m) : a ^ n < a ^ m := by |
rcases Nat.le.dest h with ⟨k, rfl⟩; clear h
rw [pow_add, pow_succ, mul_assoc, ← pow_succ']
exact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)
|
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 347 | 350 | theorem mk_smul_mk (r : R) (m : M) (s t : S) :
Localization.mk r s • mk m t = mk (r • m) (s * t) := by |
rw [Localization.mk_eq_mk']
exact mk'_smul_mk ..
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 1,043 | 1,048 | theorem _root_.ContinuousOn.angle_sign_comp {α : Type*} [TopologicalSpace α] {f : α → Angle}
{s : Set α} (hf : ContinuousOn f s) (hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ π) :
ContinuousOn (sign ∘ f) s := by |
refine (ContinuousAt.continuousOn fun θ hθ => ?_).comp hf (Set.mapsTo_image f s)
obtain ⟨z, hz, rfl⟩ := hθ
exact continuousAt_sign (hs _ hz).1 (hs _ hz).2
|
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 220 | 224 | theorem Complex.exp_eq_exp_ℂ : Complex.exp = NormedSpace.exp ℂ := by |
refine funext fun x => ?_
rw [Complex.exp, exp_eq_tsum_div]
have : CauSeq.IsComplete ℂ norm := Complex.instIsComplete
exact tendsto_nhds_unique x.exp'.tendsto_limit (expSeries_div_summable ℝ x).hasSum.tendsto_sum_nat
|
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 93 | 99 | theorem braiding_tensor_left (X Y Z : C) :
(β_ (X ⊗ Y) Z).hom =
(α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
(β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom := by |
apply (cancel_epi (α_ X Y Z).inv).1
apply (cancel_mono (α_ Z X Y).inv).1
simp [hexagon_reverse]
|
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 142 | 143 | theorem cast_lt_one : (n : α) < 1 ↔ n = 0 := by |
rw [← cast_one, cast_lt, Nat.lt_succ_iff, ← bot_eq_zero, le_bot_iff]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 463 | 471 | theorem IsPreconnected.prod [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ×ˢ t) := by |
apply isPreconnected_of_forall_pair
rintro ⟨a₁, b₁⟩ ⟨ha₁, hb₁⟩ ⟨a₂, b₂⟩ ⟨ha₂, hb₂⟩
refine ⟨Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s, ?_, .inl ⟨b₁, hb₁, rfl⟩, .inr ⟨a₂, ha₂, rfl⟩, ?_⟩
· rintro _ (⟨y, hy, rfl⟩ | ⟨x, hx, rfl⟩)
exacts [⟨ha₁, hy⟩, ⟨hx, hb₂⟩]
· exact (ht.image _ (Continuous.Prod.mk _).continuous... |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 190 | 203 | theorem closure_inverse_graph (hf : LinearMap.ker f.toFun = ⊥) (hf' : f.IsClosable)
(hcf : LinearMap.ker f.closure.toFun = ⊥) :
f.closure.inverse.graph = f.inverse.graph.topologicalClosure := by |
rw [inverse_graph hf, inverse_graph hcf, ← hf'.graph_closure_eq_closure_graph]
apply SetLike.ext'
simp only [Submodule.topologicalClosure_coe, Submodule.map_coe, LinearEquiv.prodComm_apply]
apply (image_closure_subset_closure_image continuous_swap).antisymm
have h1 := Set.image_equiv_eq_preimage_symm f.graph... |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fin
import Mathlib.Tactic.NormNum.Ineq
#align_import group_theory.perm.sign from "leanprover-community/math... | Mathlib/GroupTheory/Perm/Sign.lean | 602 | 604 | theorem sign_subtypeCongr {p : α → Prop} [DecidablePred p] (ep : Perm { a // p a })
(en : Perm { a // ¬p a }) : sign (ep.subtypeCongr en) = sign ep * sign en := by |
simp [subtypeCongr]
|
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 366 | 368 | theorem norm_eq_norm (x : K) :
mixedEmbedding.norm (mixedEmbedding K x) = |Algebra.norm ℚ x| := by |
simp_rw [mixedEmbedding.norm_apply, normAtPlace_apply, prod_eq_abs_norm]
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 129 | 132 | theorem nat_card : Nat.card (DihedralGroup n) = 2 * n := by |
cases n
· rw [Nat.card_eq_zero_of_infinite]
· rw [Nat.card_eq_fintype_card, card]
|
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.complex.removable_singularity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric S... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 46 | 57 | theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s : Set ℂ} {c : ℂ}
(hs : s ∈ 𝓝 c) :
DifferentiableOn ℂ f (s \ {c}) ∧ ContinuousAt f c ↔ DifferentiableOn ℂ f s := by |
refine ⟨?_, fun hd => ⟨hd.mono diff_subset, (hd.differentiableAt hs).continuousAt⟩⟩
rintro ⟨hd, hc⟩ x hx
rcases eq_or_ne x c with (rfl | hne)
· refine (analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
?_ hc).differentiableAt.differentiableWithinAt
refine eventually_nhdsWithin_iff.2 ((ev... |
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {α : Type u} {β : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 214 | 214 | theorem span_one : span (1 : Set α) = ⊤ := by | rw [← Set.singleton_one, span_singleton_one]
|
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 60 | 62 | theorem count_eq_card_fintype (n : ℕ) : count p n = Fintype.card { k : ℕ // k < n ∧ p k } := by |
rw [count_eq_card_filter_range, ← Fintype.card_ofFinset, ← CountSet.fintype]
rfl
|
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"... | Mathlib/RingTheory/WittVector/Frobenius.lean | 293 | 306 | theorem coeff_frobenius_charP (x : 𝕎 R) (n : ℕ) : coeff (frobenius x) n = x.coeff n ^ p := by |
rw [coeff_frobenius]
letI : Algebra (ZMod p) R := ZMod.algebra _ _
-- outline of the calculation, proofs follow below
calc
aeval (fun k => x.coeff k) (frobeniusPoly p n) =
aeval (fun k => x.coeff k)
(MvPolynomial.map (Int.castRingHom (ZMod p)) (frobeniusPoly p n)) := ?_
_ = aeval (fun... |
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 542 | 548 | theorem setIntegral_indicatorConstLp [CompleteSpace E]
{p : ℝ≥0∞} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμt : μ t ≠ ∞) (e : E) :
∫ x in s, indicatorConstLp p ht hμt e x ∂μ = (μ (t ∩ s)).toReal • e :=
calc
∫ x in s, indicatorConstLp p ht hμt e x ∂μ = ∫ x in s, t.indicator (fun _ => e) x ∂μ := by |
rw [setIntegral_congr_ae hs (indicatorConstLp_coeFn.mono fun x hx _ => hx)]
_ = (μ (t ∩ s)).toReal • e := by rw [integral_indicator_const _ ht, Measure.restrict_apply ht]
|
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 | 117 | 119 | theorem Filter.Tendsto.neg_mul_atBot {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atBot) : Tendsto (fun x => f x * g x) l atTop := by |
simpa only [mul_comm] using hg.atBot_mul_neg hC hf
|
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 76 | 84 | theorem mapAccumr₂_mapAccumr_right (f₁ : α → γ → σ₁ → σ₁ × ζ) (f₂ : β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ xs (mapAccumr f₂ ys s₂).snd s₁)
= let m := (mapAccumr₂ (fun x y s =>
let r₂ := f₂ y s.snd
let r₁ := f₁ x r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂))
(m.... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 436 | 443 | theorem norm_div_cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.cos (o.oangle y (y - x)) = ‖y - x‖ := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.r... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 129 | 129 | theorem bot_symmDiff : ⊥ ∆ a = a := by | rw [symmDiff_comm, symmDiff_bot]
|
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Topology
namespa... | Mathlib/Probability/Process/Adapted.lean | 188 | 198 | theorem progMeasurable_of_tendsto' {γ} [MeasurableSpace ι] [PseudoMetrizableSpace β]
(fltr : Filter γ) [fltr.NeBot] [fltr.IsCountablyGenerated] {U : γ → ι → Ω → β}
(h : ∀ l, ProgMeasurable f (U l)) (h_tendsto : Tendsto U fltr (𝓝 u)) : ProgMeasurable f u := by |
intro i
apply @stronglyMeasurable_of_tendsto (Set.Iic i × Ω) β γ
(MeasurableSpace.prod _ (f i)) _ _ fltr _ _ _ _ fun l => h l i
rw [tendsto_pi_nhds] at h_tendsto ⊢
intro x
specialize h_tendsto x.fst
rw [tendsto_nhds] at h_tendsto ⊢
exact fun s hs h_mem => h_tendsto {g | g x.snd ∈ s} (hs.preimage (con... |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 517 | 520 | theorem restrict_sum_of_countable [Countable ι] (μ : ι → Measure α) (s : Set α) :
(sum μ).restrict s = sum fun i => (μ i).restrict s := by |
ext t ht
simp_rw [sum_apply _ ht, restrict_apply ht, sum_apply_of_countable]
|
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.quotient_operations from "leanprover-community/mathlib"@"b88d81c84530450a8989e918608e5960f015e6c8"
universe u v w
namespace Ideal
open Function RingHom
var... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 336 | 340 | theorem snd_comp_quotientMulEquivQuotientProd (I J : Ideal R) (coprime : IsCoprime I J) :
(RingHom.snd _ _).comp
(quotientMulEquivQuotientProd I J coprime : R ⧸ I * J →+* (R ⧸ I) × R ⧸ J) =
Ideal.Quotient.factor (I * J) J mul_le_left := by |
apply Quotient.ringHom_ext; ext; rfl
|
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 520 | 532 | theorem comp_le_of_antitoneOn (f : α → E) {s : Set α} {t : Set β} (φ : β → α) (hφ : AntitoneOn φ t)
(φst : MapsTo φ t s) : eVariationOn (f ∘ φ) t ≤ eVariationOn f s := by |
refine iSup_le ?_
rintro ⟨n, u, hu, ut⟩
rw [← Finset.sum_range_reflect]
refine (Finset.sum_congr rfl fun x hx => ?_).trans_le <| le_iSup_of_le
⟨n, fun i => φ (u <| n - i), fun x y xy => hφ (ut _) (ut _) (hu <| Nat.sub_le_sub_left xy n),
fun i => φst (ut _)⟩
le_rfl
rw [Finset.mem_range] at hx
... |
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 326 | 332 | theorem Filter.HasBasis.cauchySeq_iff {γ} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop}
{s : γ → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → (u m, u n) ∈ s i := by |
rw [cauchySeq_iff_tendsto, ← prod_atTop_atTop_eq]
refine (atTop_basis.prod_self.tendsto_iff h).trans ?_
simp only [exists_prop, true_and_iff, MapsTo, preimage, subset_def, Prod.forall, mem_prod_eq,
mem_setOf_eq, mem_Ici, and_imp, Prod.map, ge_iff_le, @forall_swap (_ ≤ _) β]
|
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import ring_theory.witt_vector.structure_polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open MvPolynomial Set
open Finset (range)
o... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 301 | 307 | theorem wittStructureInt_prop (Φ : MvPolynomial idx ℤ) (n) :
bind₁ (wittStructureInt p Φ) (wittPolynomial p ℤ n) =
bind₁ (fun i => rename (Prod.mk i) (W_ ℤ n)) Φ := by |
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
have := wittStructureRat_prop p (map (Int.castRingHom ℚ) Φ) n
simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,
AlgHom.coe_toRingHom, map_wittStructureInt]
|
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 | 1,160 | 1,162 | theorem Inducing.continuousWithinAt_iff {f : α → β} {g : β → γ} (hg : Inducing g) {s : Set α}
{x : α} : ContinuousWithinAt f s x ↔ ContinuousWithinAt (g ∘ f) s x := by |
simp_rw [ContinuousWithinAt, Inducing.tendsto_nhds_iff hg]; rfl
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 808 | 812 | theorem toReal_add {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :
toReal (x + y) = toReal x + toReal y := by |
lift x to ℝ using ⟨hx, h'x⟩
lift y to ℝ using ⟨hy, h'y⟩
rfl
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 127 | 131 | theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by |
rw [orthogonal_eq_inter K]
have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E))
convert isClosed_iInter this
simp only [iInf_coe]
|
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 38 | 42 | theorem isEquipartition_iff_card_parts_eq_average :
P.IsEquipartition ↔
∀ a : Finset α,
a ∈ P.parts → a.card = s.card / P.parts.card ∨ a.card = s.card / P.parts.card + 1 := by |
simp_rw [IsEquipartition, Finset.equitableOn_iff, P.sum_card_parts]
|
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 352 | 365 | theorem kreplace_self {a : α} {b : β a} {l : List (Sigma β)} (nd : NodupKeys l)
(h : Sigma.mk a b ∈ l) : kreplace a b l = l := by |
refine (lookmap_congr ?_).trans (lookmap_id' (Option.guard fun (s : Sigma β) => a = s.1) ?_ _)
· rintro ⟨a', b'⟩ h'
dsimp [Option.guard]
split_ifs
· subst a'
simp [nd.eq_of_mk_mem h h']
· rfl
· rintro ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
dsimp [Option.guard]
split_ifs
· simp
· rintro ⟨⟩
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 346 | 348 | theorem ContinuousLinearEquiv.comp_contDiffAt_iff (e : F ≃L[𝕜] G) :
ContDiffAt 𝕜 n (e ∘ f) x ↔ ContDiffAt 𝕜 n f x := by |
simp only [← contDiffWithinAt_univ, e.comp_contDiffWithinAt_iff]
|
import Mathlib.Topology.Category.LightProfinite.Basic
import Mathlib.Topology.Category.Profinite.Limits
namespace LightProfinite
universe u w
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
section Pullbacks
variable {X Y B : LightProfinite.{u}} (f : X ⟶ B) (g ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 128 | 131 | theorem pullback_snd_eq :
LightProfinite.pullback.snd f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.snd := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
|
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import ring_theory.roots_of_unity.complex from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Complex
open Polynomial Real
open scoped Nat Real
theorem isPrimitiveRoot_e... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 53 | 55 | theorem isPrimitiveRoot_exp (n : ℕ) (h0 : n ≠ 0) : IsPrimitiveRoot (exp (2 * π * I / n)) n := by |
simpa only [Nat.cast_one, one_div] using
isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left
|
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 | 165 | 166 | theorem pi_inf_principal_univ_pi_neBot :
NeBot (pi f ⊓ 𝓟 (Set.pi univ s)) ↔ ∀ i, NeBot (f i ⊓ 𝓟 (s i)) := by | simp [neBot_iff]
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 335 | 344 | theorem ae_nonneg_of_forall_setIntegral_nonneg_of_sigmaFinite [SigmaFinite μ] {f : α → ℝ}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by |
apply ae_of_forall_measure_lt_top_ae_restrict
intro t t_meas t_lt_top
apply ae_nonneg_restrict_of_forall_setIntegral_nonneg_inter (hf_int_finite t t_meas t_lt_top)
intro s s_meas _
exact
hf_zero _ (s_meas.inter t_meas)
(lt_of_le_of_lt (measure_mono (Set.inter_subset_right)) t_lt_top)
|
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 254 | 256 | theorem reverse_revzip (l : List α) : reverse l.revzip = revzip l.reverse := by |
rw [← zip_unzip (revzip l).reverse]
simp [unzip_eq_map, revzip, map_reverse, map_fst_zip, map_snd_zip]
|
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import algebra.quaternion from "leanprover-community/mathlib"@"cf7a7252c19... | Mathlib/Algebra/Quaternion.lean | 670 | 670 | theorem smul_coe : x • (y : ℍ[R,c₁,c₂]) = ↑(x * y) := by | rw [coe_mul, coe_mul_eq_smul]
|
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 110 | 114 | theorem t'_fst_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.fst ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
simp only [t', Category.assoc, pullbackSymmetry_hom_comp_fst_assoc,
pullbackRightPullbackFstIso_inv_snd_fst_assoc, pullback.lift_fst_assoc, t_fst_fst,
pullbackRightPullbackFstIso_hom_fst_assoc]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 608 | 612 | theorem index_center_le_pow [Finite (commutatorSet G)] [Group.FG G] :
(center G).index ≤ Nat.card (commutatorSet G) ^ Group.rank G := by |
obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
rw [← hS1, ← Fintype.card_coe, ← Nat.card_eq_fintype_card, ← Finset.coe_sort_coe, ← Nat.card_fun]
exact Finite.card_le_of_embedding (quotientCenterEmbedding hS2)
|
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Filter
variable {ι α β : Type*}
class Bornology (α : Type*) where
cobounded' : Filter α
le_cofinite' : cobounded' ≤ cofinite
#align borno... | Mathlib/Topology/Bornology/Basic.lean | 143 | 144 | theorem isBounded_compl_iff : IsBounded sᶜ ↔ IsCobounded s := by |
rw [isBounded_def, isCobounded_def, compl_compl]
|
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 127 | 129 | theorem fourier_coe_apply' {n : ℤ} {x : ℝ} :
toCircle (n • (x : AddCircle T) :) = Complex.exp (2 * π * Complex.I * n * x / T) := by |
rw [← fourier_apply]; exact fourier_coe_apply
|
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11361a98ae4acd77f5c1837686f6f0102cdc25"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n : ℕ} [Division... | Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean | 63 | 72 | theorem denominators_recurrence {gp : Pair K} {ppredB predB : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_denom_eq : g.denominators n = ppredB)
(succ_nth_denom_eq : g.denominators (n + 1) = predB) :
g.denominators (n + 2) = gp.b * predB + gp.a * ppredB := by |
obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.b = ppredB :=
exists_conts_b_of_denom nth_denom_eq
obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.b = predB :=
exists_conts_b_of_denom succ_nth_denom_eq
rw [denom_eq_co... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 394 | 394 | theorem arccos_neg_one : arccos (-1) = π := by | simp [arccos, add_halves]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
#align_import analysis.calculus.deriv.slope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open Topology Filter TopologicalSpace
open Filter Set
secti... | Mathlib/Analysis/Calculus/Deriv/Slope.lean | 66 | 69 | theorem hasDerivWithinAt_iff_tendsto_slope :
HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s \ {x}] x) (𝓝 f') := by |
simp only [HasDerivWithinAt, nhdsWithin, diff_eq, ← inf_assoc, inf_principal.symm]
exact hasDerivAtFilter_iff_tendsto_slope
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 347 | 354 | theorem integral_cexp_neg_mul_sq_norm_add
(hb : 0 < b.re) (c : ℂ) (w : V) :
∫ v : V, cexp (- b * ‖v‖^2 + c * ⟪w, v⟫) =
(π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (c ^ 2 * ‖w‖^2 / (4 * b)) := by |
let e := (stdOrthonormalBasis ℝ V).repr.symm
rw [← e.measurePreserving.integral_comp e.toHomeomorph.measurableEmbedding]
convert integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
hb c (e.symm w) <;> simp [LinearIsometryEquiv.inner_map_eq_flip]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 388 | 398 | theorem IsPreconnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β}
(hs : IsPreconnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f)
(hsf : s ⊆ range f) : IsPreconnected (f ⁻¹' s) :=
isPreconnected_closed_iff.2 fun u v hu hv hsuv hsu hsv => by
replace hsf : f '' (f ⁻¹' s)... |
refine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_
· simpa only [hsf, image_union] using image_subset f hsuv
· simpa only [image_preimage_inter] using hsu.image f
· simpa only [image_preimage_inter] using hsv.image f
· exact ⟨a, has, hau, hinj.mem_set_image... |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 588 | 596 | theorem isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i)) (U : Set X) :
IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by |
constructor
· exact fun ⟨h₁, h₂⟩ ↦ eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open _ hb U h₁ h₂
· rintro ⟨s, hs, rfl⟩
constructor
· exact hs.isCompact_biUnion fun i _ => hb' i
· exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _)
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.Tactic.Spread
#align_import data.pi.algebra from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
-- We enforce to only import `Algebra.Group.Defs` and ... | Mathlib/Algebra/Group/Pi/Basic.lean | 404 | 410 | theorem apply_mulSingle₂ (f' : ∀ i, f i → g i → h i) (hf' : ∀ i, f' i 1 1 = 1) (i : I)
(x : f i) (y : g i) (j : I) :
f' j (mulSingle i x j) (mulSingle i y j) = mulSingle i (f' i x y) j := by |
by_cases h : j = i
· subst h
simp only [mulSingle_eq_same]
· simp only [mulSingle_eq_of_ne h, hf']
|
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