Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
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import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
noncomputable section
open scoped Classical
@[ext]
structure ComplexShape (ι : Type*) where
Rel : ι → ι → Prop
nex... | Mathlib/Algebra/Homology/ComplexShape.lean | 161 | 164 | theorem prev_eq' (c : ComplexShape ι) {i j : ι} (h : c.Rel i j) : c.prev j = i := by |
apply c.prev_eq _ h
rw [prev, dif_pos]
exact Exists.choose_spec (⟨i, h⟩ : ∃ k, c.Rel k j)
|
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.decomposition.unsigned_hahn from "leanprover-community/mathlib"@"0f1becb755b3d008b242c622e248a70556ad19e6"
open Set Filter
open scoped Classical
open Topology ENNReal
namespace MeasureTheory
variable {α : Type*} [MeasurableSpace α] {... | Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean | 37 | 176 | theorem hahn_decomposition [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
∃ s,
MeasurableSet s ∧
(∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t := by |
let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal
let c : Set ℝ := d '' { s | MeasurableSet s }
let γ : ℝ := sSup c
have hμ : ∀ s, μ s ≠ ∞ := measure_ne_top μ
have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν
have to_nnreal_μ : ∀ s, ((μ s).toNNReal : ℝ≥0∞) = μ s := fun s => ENNReal.coe_toNNReal ... |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 93 | 94 | theorem vars_X [Nontrivial R] : (X n : MvPolynomial σ R).vars = {n} := by |
rw [X, vars_monomial (one_ne_zero' R), Finsupp.support_single_ne_zero _ (one_ne_zero' ℕ)]
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 136 | 139 | theorem finite_setOf_isRoot {p : R[X]} (hp : p ≠ 0) : Set.Finite { x | IsRoot p x } := by |
classical
simpa only [← Finset.setOf_mem, Multiset.mem_toFinset, mem_roots hp]
using p.roots.toFinset.finite_toSet
|
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 312 | 315 | theorem LieSubmodule.trivial_lie_oper_zero [LieModule.IsTrivial L M] : ⁅I, N⁆ = ⊥ := by |
suffices ⁅I, N⁆ ≤ ⊥ from le_bot_iff.mp this
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
rintro m ⟨x, n, h⟩; rw [trivial_lie_zero] at h; simp [← h]
|
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 82 | 98 | theorem Sphere.eq_or_eq_secondInter_of_mem_mk'_span_singleton_iff_mem {s : Sphere P} {p : P}
(hp : p ∈ s) {v : V} {p' : P} (hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)) :
p' = p ∨ p' = s.secondInter p v ↔ p' ∈ s := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· rcases h with (h | h)
· rwa [h]
· rwa [h, Sphere.secondInter_mem]
· rw [AffineSubspace.mem_mk'_iff_vsub_mem, Submodule.mem_span_singleton] at hp'
rcases hp' with ⟨r, hr⟩
rw [eq_comm, ← eq_vadd_iff_vsub_eq] at hr
subst hr
by_cases hv : v = 0
· simp [... |
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 103 | 105 | theorem intervalIntegrable_iff_integrableOn_Icc_of_le {f : ℝ → E} {a b : ℝ} (hab : a ≤ b)
{μ : Measure ℝ} [NoAtoms μ] : IntervalIntegrable f μ a b ↔ IntegrableOn f (Icc a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hab, integrableOn_Icc_iff_integrableOn_Ioc]
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 130 | 135 | theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) :
FinMeasAdditive (c • μ) T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top]
simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff]
exact Or.inl hμs
|
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 154 | 156 | theorem comp_iff {r : α → Prop → Prop} : r ∘r (· ↔ ·) = r := by |
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, comp_eq]
|
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable (N : Type*) (G : Type*) {H : Type*} [Group N] [Group G] [Group H]
... | Mathlib/GroupTheory/SemidirectProduct.lean | 161 | 162 | theorem inl_aut_inv (g : G) (n : N) : (inl ((φ g)⁻¹ n) : N ⋊[φ] G) = inr g⁻¹ * inl n * inr g := by |
rw [← MonoidHom.map_inv, inl_aut, inv_inv]
|
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Powerset
#align_import data.fintype.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α]
open List
namespace Multiset
def lists : Multiset α → Finset (List α) := fun s =>
Quotient... | Mathlib/Data/Fintype/List.lean | 51 | 53 | theorem mem_lists_iff (s : Multiset α) (l : List α) : l ∈ lists s ↔ s = ⟦l⟧ := by |
induction s using Quotient.inductionOn
simpa using perm_comm
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 276 | 280 | theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n := by |
induction' n with n ih
· simp
· simp [ih, descPochhammer_succ_left, map_comp]
|
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 98 | 103 | theorem Nat.Prime.mod_four_ne_three_of_dvd_isSquare_neg_one {p n : ℕ} (hpp : p.Prime) (hp : p ∣ n)
(hs : IsSquare (-1 : ZMod n)) : p % 4 ≠ 3 := by |
obtain ⟨y, h⟩ := ZMod.isSquare_neg_one_of_dvd hp hs
rw [← sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h
haveI : Fact p.Prime := ⟨hpp⟩
exact ZMod.mod_four_ne_three_of_sq_eq_neg_sq' one_ne_zero h
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv
#align_import analysis.special_functions.trigonometric.arctan_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real... | Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | 82 | 85 | theorem hasStrictDerivAt_arctan (x : ℝ) : HasStrictDerivAt arctan (1 / (1 + x ^ 2)) x := by |
have A : cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne'
simpa [cos_sq_arctan] using
tanPartialHomeomorph.hasStrictDerivAt_symm trivial (by simpa) (hasStrictDerivAt_tan A)
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 205 | 207 | theorem isUnit_det_transpose (h : IsUnit A.det) : IsUnit Aᵀ.det := by |
rw [det_transpose]
exact h
|
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Hom
open Set Pointwise
theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) := by
lift m to Mˣ using hm
rw [Functio... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 72 | 84 | theorem preimage_smul_setₛₗ'
(hc : Function.Surjective (fun (m : M) ↦ c • m))
(hc' : Function.Injective (fun (n : N) ↦ σ c • n)) :
h ⁻¹' (σ c • t) = c • h ⁻¹' t := by |
apply le_antisymm
· intro m
obtain ⟨m', rfl⟩ := hc m
rintro ⟨n, hn, hn'⟩
refine ⟨m', ?_, rfl⟩
rw [map_smulₛₗ] at hn'
rw [mem_preimage, ← hc' hn']
exact hn
· exact smul_preimage_set_leₛₗ M N σ h c t
|
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 | 56 | 57 | theorem mem_orthogonal' (v : E) : v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪v, u⟫ = 0 := by |
simp_rw [mem_orthogonal, inner_eq_zero_symm]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 106 | 109 | theorem left_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f m x) :
IsPeriodicPt f n x := by |
rw [IsPeriodicPt, iterate_add] at hn
exact hn.left_of_comp hm
|
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {𝕜 : Type*} [RCLike 𝕜] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 85 | 93 | theorem ContinuousLinearMap.opNorm_bound_of_ball_bound {r : ℝ} (r_pos : 0 < r) (c : ℝ)
(f : E →L[𝕜] 𝕜) (h : ∀ z ∈ closedBall (0 : E) r, ‖f z‖ ≤ c) : ‖f‖ ≤ c / r := by |
apply ContinuousLinearMap.opNorm_le_bound
· apply div_nonneg _ r_pos.le
exact
(norm_nonneg _).trans
(h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]))
apply LinearMap.bound_of_ball_bound' r_pos
exact fun z hz => h z hz
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 97 | 97 | theorem mem_box : x ∈ box n d ↔ ∀ i, x i < d := by | simp only [box, Fintype.mem_piFinset, mem_range]
|
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
#align_import analysis.locally_convex.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {𝕜 E F ι : Type*}
open Topology
... | Mathlib/Analysis/LocallyConvex/WeakDual.lean | 73 | 76 | theorem toSeminorm_comp (f : F →ₗ[𝕜] 𝕜) (g : E →ₗ[𝕜] F) :
f.toSeminorm.comp g = (f.comp g).toSeminorm := by |
ext
simp only [Seminorm.comp_apply, toSeminorm_apply, coe_comp, Function.comp_apply]
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 98 | 100 | theorem intervalIntegrable_iff' [NoAtoms μ] :
IntervalIntegrable f μ a b ↔ IntegrableOn f (uIcc a b) μ := by |
rw [intervalIntegrable_iff, ← Icc_min_max, uIoc, integrableOn_Icc_iff_integrableOn_Ioc]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 129 | 136 | theorem IsCoprime.mul_dvd (H : IsCoprime x y) (H1 : x ∣ z) (H2 : y ∣ z) : x * y ∣ z := by |
obtain ⟨a, b, h⟩ := H
rw [← mul_one z, ← h, mul_add]
apply dvd_add
· rw [mul_comm z, mul_assoc]
exact (mul_dvd_mul_left _ H2).mul_left _
· rw [mul_comm b, ← mul_assoc]
exact (mul_dvd_mul_right H1 _).mul_right _
|
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 | 124 | 127 | theorem not_isField : ¬IsField (ringOfIntegers Fq F) := by |
simpa [← (IsIntegralClosure.isIntegral_algebra Fq[X] F).isField_iff_isField
(algebraMap_injective Fq F)] using
Polynomial.not_isField Fq
|
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
open LinearMap (BilinForm)
universe u v w
variable {R : Type*} {M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
variable {R₁ : Type*} {M₁ : Type*} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁]
va... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 119 | 130 | theorem linearIndependent_of_iIsOrtho {n : Type w} {B : BilinForm K V} {v : n → V}
(hv₁ : B.iIsOrtho v) (hv₂ : ∀ i, ¬B.IsOrtho (v i) (v i)) : LinearIndependent K v := by |
classical
rw [linearIndependent_iff']
intro s w hs i hi
have : B (s.sum fun i : n => w i • v i) (v i) = 0 := by rw [hs, zero_left]
have hsum : (s.sum fun j : n => w j * B (v j) (v i)) = w i * B (v i) (v i) := by
apply Finset.sum_eq_single_of_mem i hi
intro j _ hij
rw [iIsOrtho_def.1... |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace List
@[simp]
theorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l := by
-- Porting ... | Mathlib/Data/Bool/Count.lean | 105 | 117 | theorem two_mul_count_bool_eq_ite (hl : Chain' (· ≠ ·) l) (b : Bool) :
2 * count b l =
if Even (length l) then length l else
if Option.some b == l.head? then length l + 1 else length l - 1 := by |
by_cases h2 : Even (length l)
· rw [if_pos h2, hl.two_mul_count_bool_of_even h2]
· cases' l with x l
· exact (h2 even_zero).elim
simp only [if_neg h2, count_cons, mul_add, head?, Option.mem_some_iff, @eq_comm _ x]
rw [length_cons, Nat.even_add_one, not_not] at h2
replace hl : l.Chain' (· ≠ ·) := ... |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 106 | 107 | theorem hamming_zero_eq_dist {x y : ∀ i, β i} : 0 = hammingDist x y ↔ x = y := by |
rw [eq_comm, hammingDist_eq_zero]
|
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
noncomputable section
open scoped Classical
open scoped RealInnerProductSpace
namespace Affine
namespace Simplex
open Finset AffineSubspac... | Mathlib/Geometry/Euclidean/MongePoint.lean | 169 | 182 | theorem mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub {n : ℕ} {i₁ i₂ : Fin (n + 3)}
(h : i₁ ≠ i₂) :
mongePointVSubFaceCentroidWeightsWithCircumcenter i₁ i₂ =
mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i₁, i₂}ᶜ := by |
ext i
cases' i with i
· rw [Pi.sub_apply, mongePointWeightsWithCircumcenter, centroidWeightsWithCircumcenter,
mongePointVSubFaceCentroidWeightsWithCircumcenter]
have hu : card ({i₁, i₂}ᶜ : Finset (Fin (n + 3))) = n + 1 := by
simp [card_compl, Fintype.card_fin, h]
rw [hu]
by_cases hi : i =... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
import Mathlib.CategoryTheory.Elementwise
#align_import analysis.normed.group.SemiNormedGroup from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean | 121 | 129 | theorem iso_isometry_of_normNoninc {V W : SemiNormedGroupCat} (i : V ≅ W) (h1 : i.hom.NormNoninc)
(h2 : i.inv.NormNoninc) : Isometry i.hom := by |
apply AddMonoidHomClass.isometry_of_norm
intro v
apply le_antisymm (h1 v)
calc
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
‖v‖ = ‖i.inv (i.hom v)‖ := by erw [Iso.hom_inv_id_apply]
_ ≤ ‖i.hom v‖ := h2 _
|
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 47 | 48 | theorem id_tensor_rightUnitor_inv (X Y : C) : 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ _).inv ≫ (α_ _ _ _).hom := by |
coherence
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 66 | 67 | theorem interedges_empty_left (t : Finset β) : interedges r ∅ t = ∅ := by |
rw [interedges, Finset.empty_product, filter_empty]
|
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.RingTheory.Polynomial.Pochhammer
section Multichoose
open Function Polynomial
class BinomialRing (R : Type*) [AddCommMonoid R] [Pow R ℕ] where
nsmul_right_injective (n : ℕ) (h : n ≠ 0) : Injective (n • · : R →... | Mathlib/RingTheory/Binomial.lean | 90 | 97 | theorem ascPochhammer_smeval_cast (R : Type*) [Semiring R] {S : Type*} [NonAssocSemiring S]
[Pow S ℕ] [Module R S] [IsScalarTower R S S] [NatPowAssoc S]
(x : S) (n : ℕ) : (ascPochhammer R n).smeval x = (ascPochhammer ℕ n).smeval x := by |
induction' n with n hn
· simp only [Nat.zero_eq, ascPochhammer_zero, smeval_one, one_smul]
· simp only [ascPochhammer_succ_right, mul_add, smeval_add, smeval_mul_X, ← Nat.cast_comm]
simp only [← C_eq_natCast, smeval_C_mul, hn, ← nsmul_eq_smul_cast R n]
exact rfl
|
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset
universe u v
namespace HallMarriageTheorem
variable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}
s... | Mathlib/Combinatorics/Hall/Finite.lean | 136 | 158 | theorem hall_cond_of_compl {ι : Type u} {t : ι → Finset α} {s : Finset ι}
(hus : s.card = (s.biUnion t).card) (ht : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card)
(s' : Finset (sᶜ : Set ι)) : s'.card ≤ (s'.biUnion fun x' => t x' \ s.biUnion t).card := by |
haveI := Classical.decEq ι
have disj : Disjoint s (s'.image fun z => z.1) := by
simp only [disjoint_left, not_exists, mem_image, exists_prop, SetCoe.exists, exists_and_right,
exists_eq_right, Subtype.coe_mk]
intro x hx hc _
exact absurd hx hc
have : s'.card = (s ∪ s'.image fun z => z.1).card - ... |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 124 | 127 | theorem restrict_extend_compl_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f)ᶜ.restrict (extend f g g') = g' ∘ Subtype.val := by |
classical
exact restrict_dite_compl _ _
|
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 | 104 | 110 | theorem IsEquipartition.card_small_parts_eq_mod (hP : P.IsEquipartition) :
(P.parts.filter fun p ↦ p.card = s.card / P.parts.card).card =
P.parts.card - s.card % P.parts.card := by |
conv_rhs =>
arg 1
rw [← filter_card_add_filter_neg_card_eq_card (p := fun p ↦ p.card = s.card / P.parts.card + 1)]
rw [hP.card_large_parts_eq_mod, add_tsub_cancel_left, hP.filter_ne_average_add_one_eq_average]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
#align_import data.real.pi.bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
-- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
open scoped Real
namespace Real
| Mathlib/Data/Real/Pi/Bounds.lean | 28 | 37 | theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
(2 : ℝ) ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) < π := by |
have : √(2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]
focus
apply sin_lt
apply div_pos pi_pos
all_goals apply pow_pos; norm_num
apply lt_of_le_of_lt (le_of_eq _) this
rw [pow_succ' _ (n + 1), ← mul_assoc, div_mul_cancel₀, ... |
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 96 | 99 | theorem mdifferentiableOn_iff_differentiableOn :
MDifferentiableOn 𝓘(𝕜, E) 𝓘(𝕜, E') f s ↔ DifferentiableOn 𝕜 f s := by |
simp only [MDifferentiableOn, DifferentiableOn,
mdifferentiableWithinAt_iff_differentiableWithinAt]
|
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 133 | 135 | theorem AddCommMonoidWithOne.toAddMonoidWithOne_injective :
Function.Injective (@AddCommMonoidWithOne.toAddMonoidWithOne R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
|
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 92 | 94 | theorem tendsto_nhds_coe_iff {α : Type*} {l : Filter α} {x : ℝ≥0} {f : ℝ≥0∞ → α} :
Tendsto f (𝓝 ↑x) l ↔ Tendsto (f ∘ (↑) : ℝ≥0 → α) (𝓝 x) l := by |
rw [nhds_coe, tendsto_map'_iff]
|
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 66 | 67 | theorem separable_of_subsingleton [Subsingleton R] (f : R[X]) : f.Separable := by |
simp [Separable, IsCoprime, eq_iff_true_of_subsingleton]
|
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 Union
#align list.nil_union List.nil_unio... | Mathlib/Data/List/Lattice.lean | 109 | 110 | theorem forall_mem_union : (∀ x ∈ l₁ ∪ l₂, p x) ↔ (∀ x ∈ l₁, p x) ∧ ∀ x ∈ l₂, p x := by |
simp only [mem_union_iff, or_imp, forall_and]
|
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 42 | 43 | theorem leftUnitor_tensor_inv' (X Y : C) :
(λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom := by | coherence
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
nam... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 56 | 64 | theorem dart_fst_fiber [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by |
ext d
simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
constructor
· rintro rfl
exact ⟨_, d.adj, by ext <;> rfl⟩
· rintro ⟨e, he, rfl⟩
rfl
|
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 | 133 | 134 | theorem exists_duplicate_iff_not_nodup : (∃ x : α, x ∈+ l) ↔ ¬Nodup l := by |
simp [nodup_iff_forall_not_duplicate]
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 35 | 44 | theorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : α) : SurjOn f (Ico a b) (Ico (f a) (f b)) := by |
obtain hab | hab := lt_or_le a b
· intro p hp
rcases eq_left_or_mem_Ioo_of_mem_Ico hp with (rfl | hp')
· exact mem_image_of_mem f (left_mem_Ico.mpr hab)
· have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
exact image_subset f Ioo_subset_Ico_self this
· rw [Ico_eq_empty (h_mono hab... |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antid... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 84 | 86 | theorem trunc_succ (f : R⟦X⟧) (n : ℕ) :
trunc n.succ f = trunc n f + Polynomial.monomial n (coeff R n f) := by |
rw [trunc, Ico_zero_eq_range, sum_range_succ, trunc, Ico_zero_eq_range]
|
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 97 | 98 | theorem Convex.combo_self {a b : R} (h : a + b = 1) (x : M) : a • x + b • x = x := by |
rw [← add_smul, h, one_smul]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 119 | 121 | theorem abs_im_pow_eventuallyLE_exp_re (hl : IsExpCmpFilter l) (n : ℕ) :
(fun z : ℂ => |z.im| ^ n) ≤ᶠ[l] fun z => Real.exp z.re := by |
simpa using (hl.isLittleO_im_pow_exp_re n).bound zero_lt_one
|
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 74 | 87 | theorem isCofiltered_costructuredArrow_of_isCofiltered_of_exists [IsCofilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) (h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d),
∃ (c' : C) (t : c' ⟶ c), F.map t ≫ s = F.map t ≫ s') (d : D) :
IsCofiltered (CostructuredArrow F d) := by |
suffices IsFiltered (CostructuredArrow F d)ᵒᵖ from isCofiltered_of_isFiltered_op _
suffices IsFiltered (StructuredArrow (op d) F.op) from
IsFiltered.of_equivalence (costructuredArrowOpEquivalence _ _).symm
apply isFiltered_structuredArrow_of_isFiltered_of_exists
· intro d
obtain ⟨c, ⟨t⟩⟩ := h₁ d.unop
... |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 97 | 99 | theorem toPMF_dirac [Countable α] [h : MeasurableSingletonClass α] :
(Measure.dirac a).toPMF = pure a := by |
rw [toPMF_eq_iff_toMeasure_eq, toMeasure_pure]
|
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 102 | 104 | theorem IsEquivalent.refl : u ~[l] u := by |
rw [IsEquivalent, sub_self]
exact isLittleO_zero _ _
|
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.clifford_algebra.grading from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
namespace CliffordAlgebra
variable {R M : Type*} [Co... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 40 | 42 | theorem range_ι_le_evenOdd_one : LinearMap.range (ι Q) ≤ evenOdd Q 1 := by |
refine le_trans ?_ (le_iSup _ ⟨1, Nat.cast_one⟩)
exact (pow_one _).ge
|
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 | 166 | 171 | theorem TendstoUniformlyOnFilter.tendsto_at (h : TendstoUniformlyOnFilter F f p p')
(hx : 𝓟 {x} ≤ p') : Tendsto (fun n => F n x) p <| 𝓝 (f x) := by |
refine Uniform.tendsto_nhds_right.mpr fun u hu => mem_map.mpr ?_
filter_upwards [(h u hu).curry]
intro i h
simpa using h.filter_mono hx
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 200 | 210 | theorem balancedSz_up {l r₁ r₂ : ℕ} (h₁ : r₁ ≤ r₂) (h₂ : l + r₂ ≤ 1 ∨ r₂ ≤ delta * l)
(H : BalancedSz l r₁) : BalancedSz l r₂ := by |
refine or_iff_not_imp_left.2 fun h => ?_
refine ⟨?_, h₂.resolve_left h⟩
cases H with
| inl H =>
cases r₂
· cases h (le_trans (Nat.add_le_add_left (Nat.zero_le _) _) H)
· exact le_trans (le_trans (Nat.le_add_right _ _) H) (Nat.le_add_left 1 _)
| inr H =>
exact le_trans H.1 (Nat.mul_le_mul_left... |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 68 | 70 | theorem sInf_inv (s : Set α) : sInf s⁻¹ = (sSup s)⁻¹ := by |
rw [← image_inv, sInf_image]
exact ((OrderIso.inv α).map_sSup _).symm
|
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 | 121 | 127 | theorem basisSets_add (U) (hU : U ∈ p.basisSets) :
∃ V ∈ p.basisSets, V + V ⊆ U := by |
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
use (s.sup p).ball 0 (r / 2)
refine ⟨p.basisSets_mem s (div_pos hr zero_lt_two), ?_⟩
refine Set.Subset.trans (ball_add_ball_subset (s.sup p) (r / 2) (r / 2) 0 0) ?_
rw [hU, add_zero, add_halves']
|
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 90 | 93 | theorem IsMatching.support_eq_verts {M : Subgraph G} (h : M.IsMatching) : M.support = M.verts := by |
refine M.support_subset_verts.antisymm fun v hv => ?_
obtain ⟨w, hvw, -⟩ := h hv
exact ⟨_, hvw⟩
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 108 | 124 | theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (log b r + 1) := by |
rcases le_or_lt r 0 with hr | hr
· rw [log_of_right_le_zero _ hr, zero_add, zpow_one]
exact hr.trans_lt (zero_lt_one.trans_le <| mod_cast hb.le)
rcases le_or_lt 1 r with hr1 | hr1
· rw [log_of_one_le_right _ hr1]
rw [Int.ofNat_add_one_out, zpow_natCast, ← Nat.cast_pow]
apply Nat.lt_of_floor_lt
... |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Finite
#align_import order.conditionally_complete_lattice.finset from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
open Set
variable {ι α β γ : Type*}
namespace Finset
section ConditionallyCompleteLat... | Mathlib/Order/ConditionallyCompleteLattice/Finset.lean | 85 | 86 | theorem sup'_id_eq_csSup (s : Finset α) (hs) : s.sup' hs id = sSup s := by |
rw [sup'_eq_csSup_image s hs, Set.image_id]
|
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 | 64 | 67 | theorem antilipschitzWith_iff_le_mul_dist :
AntilipschitzWith K f ↔ ∀ x y, dist x y ≤ K * dist (f x) (f y) := by |
simp only [antilipschitzWith_iff_le_mul_nndist, dist_nndist]
norm_cast
|
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 224... | Mathlib/Data/Complex/ExponentialBounds.lean | 59 | 71 | theorem log_two_near_10 : |log 2 - 287209 / 414355| ≤ 1 / 10 ^ 10 := by |
suffices |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34) by
norm_num1 at *
assumption
have t : |(2⁻¹ : ℝ)| = 2⁻¹ := by rw [abs_of_pos]; norm_num
have z := Real.abs_log_sub_add_sum_range_le (show |(2⁻¹ : ℝ)| < 1 by rw [t]; norm_num) 34
rw [t] at z
norm_num1 at z
rw [one_div ... |
import Mathlib.Topology.Category.TopCat.Adjunctions
#align_import topology.category.Top.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
open CategoryTheory
open TopCat
namespace TopCat
theorem epi_iff_surjective {X Y : TopCat.{u}} (f : X ⟶ Y) : Epi f ↔ Functi... | Mathlib/Topology/Category/TopCat/EpiMono.lean | 38 | 45 | theorem mono_iff_injective {X Y : TopCat.{u}} (f : X ⟶ Y) : Mono f ↔ Function.Injective f := by |
suffices Mono f ↔ Mono ((forget TopCat).map f) by
rw [this, CategoryTheory.mono_iff_injective]
rfl
constructor
· intro
infer_instance
· apply Functor.mono_of_mono_map
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {x y : ℝ}
| Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 32 | 38 | theorem log_mul_self_monotoneOn : MonotoneOn (fun x : ℝ => log x * x) { x | 1 ≤ x } := by |
-- TODO: can be strengthened to exp (-1) ≤ x
simp only [MonotoneOn, mem_setOf_eq]
intro x hex y hey hxy
have y_pos : 0 < y := lt_of_lt_of_le zero_lt_one hey
gcongr
rwa [le_log_iff_exp_le y_pos, Real.exp_zero]
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 140 | 141 | theorem idRel_subset {s : Set (α × α)} : idRel ⊆ s ↔ ∀ a, (a, a) ∈ s := by |
simp [subset_def]
|
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α R : Type*}
open Filter Set Function
@[simp]
theorem Nat.comap_cast_atTop [S... | Mathlib/Order/Filter/Archimedean.lean | 69 | 71 | theorem tendsto_intCast_atTop_iff [StrictOrderedRing R] [Archimedean R] {f : α → ℤ}
{l : Filter α} : Tendsto (fun n => (f n : R)) l atTop ↔ Tendsto f l atTop := by |
rw [← @Int.comap_cast_atTop R, tendsto_comap_iff]; rfl
|
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ... | Mathlib/RingTheory/Int/Basic.lean | 121 | 123 | theorem Int.exists_prime_and_dvd {n : ℤ} (hn : n.natAbs ≠ 1) : ∃ p, Prime p ∧ p ∣ n := by |
obtain ⟨p, pp, pd⟩ := Nat.exists_prime_and_dvd hn
exact ⟨p, Nat.prime_iff_prime_int.mp pp, Int.natCast_dvd.mpr pd⟩
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 90 | 92 | theorem hasFDerivWithinAt_iff_hasGradientWithinAt {frechet : F →L[𝕜] 𝕜} {s : Set F} :
HasFDerivWithinAt f frechet s x ↔ HasGradientWithinAt f ((toDual 𝕜 F).symm frechet) s x := by |
rw [hasGradientWithinAt_iff_hasFDerivWithinAt, (toDual 𝕜 F).apply_symm_apply frechet]
|
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 38 | 39 | theorem list_succ (n) : list (n+1) = 0 :: (list n).map Fin.succ := by |
apply List.ext_get; simp; intro i; cases i <;> simp
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
#align_import data.sym.card from "leanprover-community/mathlib"@"0bd2ea37bcba5769e14866170f251c9bc64e35d7"
open Finset Fintype Function Sum Nat
variable {α β : Type*}
... | Mathlib/Data/Sym/Card.lean | 120 | 122 | theorem card_sym_eq_choose {α : Type*} [Fintype α] (k : ℕ) [Fintype (Sym α k)] :
card (Sym α k) = (card α + k - 1).choose k := by |
rw [card_sym_eq_multichoose, Nat.multichoose_eq]
|
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Order Set TopologicalSpace Filter
variable {α : Type*} [TopologicalSp... | Mathlib/Topology/Instances/Discrete.lean | 80 | 108 | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by |
refine (eq_bot_of_singletons_open fun a => ?_).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_inter_Iic, Icc_self a]
by_cases ha_top : IsTop a
· rw [ha_top.Iic_eq, inter_comm, inter_univ] at h_singleton_eq_inter
by_cases ha_bot : IsBot a
· rw [ha_bot.Ici_eq] at h_singlet... |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 36 | 36 | theorem log_im (x : ℂ) : x.log.im = x.arg := by | simp [log]
|
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 | 77 | 85 | theorem IsClosable.leIsClosable {f g : E →ₗ.[R] F} (hf : f.IsClosable) (hfg : g ≤ f) :
g.IsClosable := by |
cases' hf with f' hf
have : g.graph.topologicalClosure ≤ f'.graph := by
rw [← hf]
exact Submodule.topologicalClosure_mono (le_graph_of_le hfg)
use g.graph.topologicalClosure.toLinearPMap
rw [Submodule.toLinearPMap_graph_eq]
exact fun _ hx hx' => f'.graph_fst_eq_zero_snd (this hx) hx'
|
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 | 34 | 37 | theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (g * x) ∂μ) = ∫⁻ x, f x ∂μ := by |
convert (lintegral_map_equiv f <| MeasurableEquiv.mulLeft g).symm
simp [map_mul_left_eq_self μ g]
|
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 119 | 122 | theorem mulSupport_disjoint_iff {f : α → M} {s : Set α} :
Disjoint (mulSupport f) s ↔ EqOn f 1 s := by |
simp_rw [← subset_compl_iff_disjoint_right, mulSupport_subset_iff', not_mem_compl_iff, EqOn,
Pi.one_apply]
|
import Mathlib.AlgebraicGeometry.OpenImmersion
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
noncomputable section
open TopologicalSpace CategoryTheory Opposite
open CategoryTheory.Limits
namespace AlgebraicGeometry
universe v v₁ v₂... | Mathlib/AlgebraicGeometry/Restrict.lean | 131 | 135 | theorem Scheme.restrictFunctor_map_base {U V : Opens X} (i : U ⟶ V) :
(X.restrictFunctor.map i).1.1.base = (Opens.toTopCat _).map i := by |
ext a; refine Subtype.ext ?_ -- Porting note: `ext` did not pick up `Subtype.ext`
exact (congr_arg (fun f : X.restrict U.openEmbedding ⟶ X => f.1.base a)
(X.restrictFunctor_map_ofRestrict i))
|
import Mathlib.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
section genericPoint
def IsGenericPoint (x : α) (S : Set α) : Prop :=
closure ({x} : Set α)... | Mathlib/Topology/Sober.lean | 53 | 54 | theorem isGenericPoint_iff_specializes : IsGenericPoint x S ↔ ∀ y, x ⤳ y ↔ y ∈ S := by |
simp only [specializes_iff_mem_closure, IsGenericPoint, Set.ext_iff]
|
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 41 | 70 | theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) {r : ℝ}
(hr : r < 1) : ∃ x₀ : E, x₀ ∉ F ∧ ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖ := by |
classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1... |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 85 | 87 | theorem opRingEquiv_symm_C_mul_X_pow (r : Rᵐᵒᵖ) (n : ℕ) :
(opRingEquiv R).symm (C r * X ^ n : Rᵐᵒᵖ[X]) = op (C (unop r) * X ^ n) := by |
rw [C_mul_X_pow_eq_monomial, opRingEquiv_symm_monomial, C_mul_X_pow_eq_monomial]
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 120 | 124 | theorem addHaarMeasure_eq_volume_pi (ι : Type*) [Fintype ι] :
addHaarMeasure (piIcc01 ι) = volume := by |
convert (addHaarMeasure_unique volume (piIcc01 ι)).symm
simp only [piIcc01, volume_pi_pi fun _ => Icc (0 : ℝ) 1, PositiveCompacts.coe_mk,
Compacts.coe_mk, Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero]
|
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 70 | 75 | theorem convexBodyLT_neg_mem (x : E K) (hx : x ∈ (convexBodyLT K f)) :
-x ∈ (convexBodyLT K f) := by |
simp only [Set.mem_prod, Prod.fst_neg, Set.mem_pi, Set.mem_univ, Pi.neg_apply,
mem_ball_zero_iff, norm_neg, Real.norm_eq_abs, forall_true_left, Subtype.forall,
Prod.snd_neg, Complex.norm_eq_abs] at hx ⊢
exact hx
|
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 83 | 84 | theorem centerAndRescale_center : a.centerAndRescale.c (last N) = 0 := by |
simp [SatelliteConfig.centerAndRescale]
|
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 91 | 95 | theorem Functor.final_of_exists_of_isFiltered [IsFilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) (h₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),
∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t) : Functor.Final F := by |
suffices ∀ d, IsFiltered (StructuredArrow d F) from final_of_isFiltered_structuredArrow F
exact isFiltered_structuredArrow_of_isFiltered_of_exists F h₁ h₂
|
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 45 | 48 | theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by |
unfold stdBasisMatrix
ext
simp
|
import Mathlib.Data.PNat.Defs
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Basic
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.Order.Hom.Basic
#align_import data.pnat.basic from "leanprover-community/mathlib"@"172bf2812857f5e56938cc148b7a5... | Mathlib/Data/PNat/Basic.lean | 33 | 34 | theorem one_add_natPred (n : ℕ+) : 1 + n.natPred = n := by |
rw [natPred, add_tsub_cancel_iff_le.mpr <| show 1 ≤ (n : ℕ) from n.2]
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 221 | 224 | theorem card_sigmaLift :
(sigmaLift f a b).card = dite (a.1 = b.1) (fun h => (f (h ▸ a.2) b.2).card) fun _ => 0 := by |
simp_rw [sigmaLift]
split_ifs with h <;> simp [h]
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 108 | 110 | theorem intervalIntegrable_iff_integrableOn_Ico_of_le [NoAtoms μ] (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ico]
|
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 76 | 79 | theorem reesAlgebra.monomial_mem {I : Ideal R} {i : ℕ} {r : R} :
monomial i r ∈ reesAlgebra I ↔ r ∈ I ^ i := by |
simp (config := { contextual := true }) [mem_reesAlgebra_iff_support, coeff_monomial, ←
imp_iff_not_or]
|
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 102 | 102 | theorem zero : mapFun f (0 : 𝕎 R) = 0 := by | map_fun_tac
|
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 | 70 | 72 | theorem cauchy_map_iff {l : Filter β} {f : β → α} :
Cauchy (l.map f) ↔ NeBot l ∧ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) := by |
rw [Cauchy, map_neBot_iff, prod_map_map_eq, Tendsto]
|
import Mathlib.Data.DFinsupp.Interval
import Mathlib.Data.DFinsupp.Multiset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.multiset.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset DFinsupp Function
open Pointwise
variable {α : Type*}
namespace Mu... | Mathlib/Data/Multiset/Interval.lean | 62 | 64 | theorem card_Ico :
(Finset.Ico s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (t.count i + 1 - s.count i) - 1 := by |
rw [Finset.card_Ico_eq_card_Icc_sub_one, card_Icc]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Dynamics.FixedPoints.Basic
#align_import data.set.pointwise.iterate from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Pointwise
open Set Function
@[to_additive
"Let `n : ℤ`... | Mathlib/Data/Set/Pointwise/Iterate.lean | 31 | 42 | theorem smul_eq_self_of_preimage_zpow_eq_self {G : Type*} [CommGroup G] {n : ℤ} {s : Set G}
(hs : (fun x => x ^ n) ⁻¹' s = s) {g : G} {j : ℕ} (hg : g ^ n ^ j = 1) : g • s = s := by |
suffices ∀ {g' : G} (_ : g' ^ n ^ j = 1), g' • s ⊆ s by
refine le_antisymm (this hg) ?_
conv_lhs => rw [← smul_inv_smul g s]
replace hg : g⁻¹ ^ n ^ j = 1 := by rw [inv_zpow, hg, inv_one]
simpa only [le_eq_subset, set_smul_subset_set_smul_iff] using this hg
rw [(IsFixedPt.preimage_iterate hs j : (zp... |
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 | 102 | 103 | theorem nodupKeys_cons {s : Sigma β} {l : List (Sigma β)} :
NodupKeys (s :: l) ↔ s.1 ∉ l.keys ∧ NodupKeys l := by | simp [keys, NodupKeys]
|
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
#align_import topology.is_locally_homeomorph from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
open Topology
variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : Y →... | Mathlib/Topology/IsLocalHomeomorph.lean | 66 | 77 | theorem mk (h : ∀ x ∈ s, ∃ e : PartialHomeomorph X Y, x ∈ e.source ∧ Set.EqOn f e e.source) :
IsLocalHomeomorphOn f s := by |
intro x hx
obtain ⟨e, hx, he⟩ := h x hx
exact
⟨{ e with
toFun := f
map_source' := fun _x hx ↦ by rw [he hx]; exact e.map_source' hx
left_inv' := fun _x hx ↦ by rw [he hx]; exact e.left_inv' hx
right_inv' := fun _y hy ↦ by rw [he (e.map_target' hy)]; exact e.right_inv' hy
... |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 97 | 99 | theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by |
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 67 | 75 | theorem antidiagonal_succ (n : ℕ) :
antidiagonal (n + 1) =
cons (0, n + 1)
((antidiagonal n).map
(Embedding.prodMap ⟨Nat.succ, Nat.succ_injective⟩ (Embedding.refl _)))
(by simp) := by |
apply eq_of_veq
rw [cons_val, map_val]
apply Multiset.Nat.antidiagonal_succ
|
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
namespace Sigma
variable {ι : Type*} {E : ι → Type... | Mathlib/Topology/MetricSpace/Gluing.lean | 352 | 355 | theorem one_le_dist_of_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
1 ≤ dist (⟨i, x⟩ : Σk, E k) ⟨j, y⟩ := by |
rw [Sigma.dist_ne h x y]
linarith [@dist_nonneg _ _ x (Nonempty.some ⟨x⟩), @dist_nonneg _ _ (Nonempty.some ⟨y⟩) y]
|
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import data.pi.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Fintype
variable {ι : Type*} {α : ι → Type*} [Fintype ι] [DecidableEq ι] [∀ i, DecidableEq (α i)]
namespac... | Mathlib/Data/Pi/Interval.lean | 69 | 70 | theorem card_Iio : (Iio b).card = (∏ i, (Iic (b i)).card) - 1 := by |
rw [card_Iio_eq_card_Iic_sub_one, card_Iic]
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
| Mathlib/Data/Nat/Cast/Field.lean | 29 | 33 | theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_dvd : n ∣ m) (hn : (n : α) ≠ 0) :
((m / n : ℕ) : α) = m / n := by |
rcases n_dvd with ⟨k, rfl⟩
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Nat.mul_div_cancel_left _ this.bot_lt, mul_comm n, cast_mul, mul_div_cancel_right₀ _ hn]
|
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 51 | 54 | theorem invOf_ι_mul_ι_mul_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
⅟ (ι Q a) * ι Q b * ι Q a = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by |
rw [invOf_ι, map_smul, smul_mul_assoc, smul_mul_assoc, ι_mul_ι_mul_ι, ← map_smul, smul_sub,
smul_smul, smul_smul, invOf_mul_self, one_smul]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Instances.ENNReal
#align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Filte... | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | 67 | 70 | theorem not_summable_iff_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by |
lift f to ℕ → ℝ≥0 using hf
exact mod_cast NNReal.not_summable_iff_tendsto_nat_atTop
|
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