Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 113 | 115 | theorem iteratedDerivWithin_zero : iteratedDerivWithin 0 f s = f := by |
ext x
simp [iteratedDerivWithin]
| 2 |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.AlgebraicIndependent
#align_import field_theory.is_alg_closed.classification from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 96 | 100 | theorem isAlgClosure_of_transcendence_basis [IsAlgClosed K] (hv : IsTranscendenceBasis R v) :
IsAlgClosure (Algebra.adjoin R (Set.range v)) K :=
letI := RingHom.domain_nontrivial (algebraMap R K)
{ alg_closed := by | infer_instance
algebraic := hv.isAlgebraic }
| 2 |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 113 | 116 | theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :
a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) := by |
rw [← seqClosure_eq_closure]
rfl
| 2 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_theory.measure.stieltjes from "leanprover-community/mathlib"@"20d5763051978e9bc6428578ed070445df6a18b3"
noncomputable section
open scoped Classical
open Set Filter Function ENNReal NNReal T... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 71 | 73 | theorem rightLim_eq (f : StieltjesFunction) (x : ℝ) : Function.rightLim f x = f x := by |
rw [← f.mono.continuousWithinAt_Ioi_iff_rightLim_eq, continuousWithinAt_Ioi_iff_Ici]
exact f.right_continuous' x
| 2 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 90 | 93 | theorem associator_inv_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv := by |
dsimp
simp
| 2 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 108 | 110 | theorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n := by |
simp only [padicValNat, dite_eq_right_iff, PartENat.get_eq_iff_eq_coe, Nat.cast_zero,
multiplicity_eq_zero, and_imp, pos_iff_ne_zero, Ne, ← or_iff_not_imp_left]
| 2 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 149 | 151 | theorem mem_support_iff (x : X) (a : FreeAbelianGroup X) : x ∈ a.support ↔ coeff x a ≠ 0 := by |
rw [support, Finsupp.mem_support_iff]
exact Iff.rfl
| 2 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 56 | 59 | theorem integralNormalization_support {f : R[X]} :
(integralNormalization f).support ⊆ f.support := by |
intro
simp (config := { contextual := true }) [integralNormalization, coeff_monomial, mem_support_iff]
| 2 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 99 | 103 | theorem Path.cast_cast {u v u' v' u'' v'' : U} (p : Path u v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(p.cast hu hv).cast hu' hv' = p.cast (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
| 2 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 346 | 349 | theorem HasStrictDerivAt.finset_prod (hf : ∀ i ∈ u, HasStrictDerivAt (f i) (f' i) x) :
HasStrictDerivAt (∏ i ∈ u, f i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, f j x) • f' i) x := by |
simpa [ContinuousLinearMap.sum_apply, ContinuousLinearMap.smul_apply] using
(HasStrictFDerivAt.finset_prod (fun i hi ↦ (hf i hi).hasStrictFDerivAt)).hasStrictDerivAt
| 2 |
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.L... | Mathlib/CategoryTheory/Limits/Types.lean | 83 | 87 | theorem isLimitEquivSections_symm_apply {c : Cone F} (t : IsLimit c)
(x : F.sections) (j : J) :
c.π.app j ((isLimitEquivSections t).symm x) = (x : ∀ j, F.obj j) j := by |
conv_rhs => rw [← (isLimitEquivSections t).right_inv x]
rfl
| 2 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
namespace MeasureTheory
open Filter
open scoped ENNReal
variable {α E : Type*} {m m0 : MeasurableSpace α} {p : ℝ≥0∞} {q : ℝ} {μ :... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 48 | 51 | theorem essSup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
essSup f (μ.trim hm) = essSup f μ := by |
simp_rw [essSup]
exact limsup_trim hm hf
| 2 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
variable (K : Type*) [Field K] [NumberField K]
namespace NumberField
open scoped nonZeroDivisors
section Basis
open Module
-- This is necessary to avoid several time... | Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 87 | 90 | theorem mem_span_basisOfFractionalIdeal {I : (FractionalIdeal (𝓞 K)⁰ K)ˣ} {x : K} :
x ∈ Submodule.span ℤ (Set.range (basisOfFractionalIdeal K I)) ↔ x ∈ (I : Set K) := by |
rw [basisOfFractionalIdeal, (fractionalIdealBasis K I.1).ofIsLocalizedModule_span ℚ ℤ⁰ _]
simp
| 2 |
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped ENNReal
namespace ProbabilityTheory
namespace kernel
variable {α β ι : Type*} {mα : MeasurableSpace α}... | Mathlib/Probability/Kernel/Composition.lean | 93 | 96 | theorem compProdFun_empty (κ : kernel α β) (η : kernel (α × β) γ) (a : α) :
compProdFun κ η a ∅ = 0 := by |
simp only [compProdFun, Set.mem_empty_iff_false, Set.setOf_false, measure_empty,
MeasureTheory.lintegral_const, zero_mul]
| 2 |
import Mathlib.CategoryTheory.Sites.Grothendieck
import Mathlib.CategoryTheory.Sites.Pretopology
import Mathlib.CategoryTheory.Limits.Lattice
import Mathlib.Topology.Sets.Opens
#align_import category_theory.sites.spaces from "leanprover-community/mathlib"@"b6fa3beb29f035598cf0434d919694c5e98091eb"
universe u
nam... | Mathlib/CategoryTheory/Sites/Spaces.lean | 92 | 95 | theorem pretopology_toGrothendieck :
Pretopology.toGrothendieck _ (Opens.pretopology T) = Opens.grothendieckTopology T := by |
rw [← pretopology_ofGrothendieck]
apply (Pretopology.gi (Opens T)).l_u_eq
| 2 |
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 | 33 | 35 | theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by |
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
| 2 |
import Mathlib.Data.Real.NNReal
import Mathlib.Tactic.GCongr.Core
#align_import analysis.normed.group.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
open Set
open NNReal
variable {ι R R' E F G : Type*}
structure AddGroupSeminorm (G : Type*) [AddGroup G] where
-- Port... | Mathlib/Analysis/Normed/Group/Seminorm.lean | 148 | 150 | theorem map_sub_le_max : f (x - y) ≤ max (f x) (f y) := by |
rw [sub_eq_add_neg, ← NonarchAddGroupSeminormClass.map_neg_eq_map' f y]
exact map_add_le_max _ _ _
| 2 |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
| Mathlib/Data/Complex/Determinant.lean | 24 | 26 | theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by |
rw [← LinearMap.det_toMatrix basisOneI, toMatrix_conjAe, Matrix.det_fin_two_of]
simp
| 2 |
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 | 326 | 329 | theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : ℕ) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) := by |
rw [descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, ← C_eq_natCast,
eval_C_mul, Nat.cast_comm, ← mul_sub]
| 2 |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 82 | 85 | theorem associator_hom_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).hom = (α_ X.V Y.V Z.V).hom := by |
dsimp
simp
| 2 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {𝕜 : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 131 | 134 | theorem removeZero_applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
(c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c := by |
ext v i
simp [applyComposition, zero_lt_one.trans_le (c.one_le_blocksFun i), removeZero_of_pos]
| 2 |
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 | 290 | 293 | theorem normAtPlace_apply_isComplex {w : InfinitePlace K} (hw : IsComplex w) (x : E K) :
normAtPlace w x = ‖x.2 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
dif_neg (not_isReal_iff_isComplex.mpr hw)]
| 2 |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 62 | 65 | theorem count_apply_finite' {s : Set α} (s_fin : s.Finite) (s_mble : MeasurableSet s) :
count s = s_fin.toFinset.card := by |
simp [←
@count_apply_finset' _ _ s_fin.toFinset (by simpa only [Finite.coe_toFinset] using s_mble)]
| 2 |
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 | 84 | 87 | theorem mdifferentiableAt_iff_differentiableAt :
MDifferentiableAt 𝓘(𝕜, E) 𝓘(𝕜, E') f x ↔ DifferentiableAt 𝕜 f x := by |
simp only [mdifferentiableAt_iff, differentiableWithinAt_univ, mfld_simps]
exact ⟨fun H => H.2, fun H => ⟨H.continuousAt, H⟩⟩
| 2 |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.split_simplicial_object from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Categ... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 91 | 95 | theorem cofan_inj_comp_PInfty_eq_zero {X : SimplicialObject C} (s : SimplicialObject.Splitting X)
{n : ℕ} (A : SimplicialObject.Splitting.IndexSet (op [n])) (hA : ¬A.EqId) :
(s.cofan _).inj A ≫ PInfty.f n = 0 := by |
rw [SimplicialObject.Splitting.IndexSet.eqId_iff_mono] at hA
rw [SimplicialObject.Splitting.cofan_inj_eq, assoc, degeneracy_comp_PInfty X n A.e hA, comp_zero]
| 2 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 108 | 110 | theorem nhds_basis_closedBall {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (closedBall x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_closedBall.comap _
| 2 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 71 | 76 | theorem OrthonormalBasis.addHaar_eq_volume {ι F : Type*} [Fintype ι] [NormedAddCommGroup F]
[InnerProductSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F]
(b : OrthonormalBasis ι ℝ F) :
b.toBasis.addHaar = volume := by |
rw [Basis.addHaar_eq_iff]
exact b.volume_parallelepiped
| 2 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 68 | 71 | theorem compl_mem_generateMeasurableRec {s : Set (Set α)} {i j : ω₁} (h : j < i) {t : Set α}
(ht : t ∈ generateMeasurableRec s j) : tᶜ ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_right _ ⟨t, mem_iUnion.2 ⟨⟨j, h⟩, ht⟩, rfl⟩)
| 2 |
import Mathlib.CategoryTheory.NatIso
import Mathlib.Logic.Equiv.Defs
#align_import category_theory.functor.fully_faithful from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe v₁ v₂ v₃ u₁ u₂ u₃
namespac... | Mathlib/CategoryTheory/Functor/FullyFaithful.lean | 125 | 127 | theorem preimageIso_mapIso (f : X ≅ Y) : F.preimageIso (F.mapIso f) = f := by |
ext
simp
| 2 |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 107 | 109 | theorem exists_irreducible : ∃ ϖ : R, Irreducible ϖ := by |
simp_rw [irreducible_iff_uniformizer]
exact (IsPrincipalIdealRing.principal <| maximalIdeal R).principal
| 2 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 99 | 103 | theorem le_or_lt_of_mul_le_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ ≤ a₂ ∨ b₁ < b₂ := by |
contrapose!
exact fun h => mul_lt_mul_of_lt_of_le h.1 h.2
| 2 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 172 | 175 | theorem StrongEpi.iff_of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) : StrongEpi f ↔ StrongEpi g := by |
constructor <;> intro
exacts [StrongEpi.of_arrow_iso e, StrongEpi.of_arrow_iso e.symm]
| 2 |
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 | 90 | 93 | theorem ascPochhammer_eval₂ (f : S →+* T) (n : ℕ) (t : T) :
(ascPochhammer T n).eval t = (ascPochhammer S n).eval₂ f t := by |
rw [← ascPochhammer_map f]
exact eval_map f t
| 2 |
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 | 78 | 81 | theorem summable_sigma_of_nonneg {β : α → Type*} {f : (Σ x, β x) → ℝ} (hf : ∀ x, 0 ≤ f x) :
Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ := by |
lift f to (Σx, β x) → ℝ≥0 using hf
exact mod_cast NNReal.summable_sigma
| 2 |
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 | 44 | 47 | theorem ι_mul_ι_mul_invOf_ι (a b : M) [Invertible (ι Q a)] [Invertible (Q a)] :
ι Q a * ι Q b * ⅟ (ι Q a) = ι Q ((⅟ (Q a) * QuadraticForm.polar Q a b) • a - b) := by |
rw [invOf_ι, map_smul, mul_smul_comm, ι_mul_ι_mul_ι, ← map_smul, smul_sub, smul_smul, smul_smul,
invOf_mul_self, one_smul]
| 2 |
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 | 83 | 86 | theorem nonempty_monotone_mem {s : Set α} (hs : s.Nonempty) :
Nonempty { u // Monotone u ∧ ∀ i : ℕ, u i ∈ s } := by |
obtain ⟨x, hx⟩ := hs
exact ⟨⟨fun _ => x, fun i j _ => le_rfl, fun _ => hx⟩⟩
| 2 |
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 | 144 | 146 | theorem fourier_eval_zero (n : ℤ) : fourier n (0 : AddCircle T) = 1 := by |
rw [← QuotientAddGroup.mk_zero, fourier_coe_apply, Complex.ofReal_zero, mul_zero,
zero_div, Complex.exp_zero]
| 2 |
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous
import Mathlib.Algebra.Polynomial.Roots
#align_i... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 116 | 119 | theorem isHomogeneous_monomial {d : σ →₀ ℕ} (r : R) {n : ℕ} (hn : degree d = n) :
IsHomogeneous (monomial d r) n := by |
simp_rw [← weightedDegree_one] at hn
exact isWeightedHomogeneous_monomial 1 d r hn
| 2 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 43 | 46 | theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i := by |
rw [PiToModule.fromMatrix_apply, Fintype.total_apply, Matrix.mulVec_single]
simp_rw [mul_one]
| 2 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousFunction.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.homotopy.basic from "leanprover-community/mathlib"@"11c53f174270aa43140c0b26dabce5fc4a253e80"
noncomputable section
universe u v ... | Mathlib/Topology/Homotopy/Basic.lean | 166 | 169 | theorem extend_apply_of_le_zero (F : Homotopy f₀ f₁) {t : ℝ} (ht : t ≤ 0) (x : X) :
F.extend t x = f₀ x := by |
rw [← F.apply_zero]
exact ContinuousMap.congr_fun (Set.IccExtend_of_le_left (zero_le_one' ℝ) F.curry ht) x
| 2 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 183 | 186 | theorem not_dvd_of_natDegree_lt {p q : R[X]} (h0 : q ≠ 0) (hl : q.natDegree < p.natDegree) :
¬p ∣ q := by |
by_contra hcontra
exact h0 (eq_zero_of_dvd_of_natDegree_lt hcontra hl)
| 2 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 112 | 115 | theorem integerNormalization_aeval_eq_zero [Algebra R R'] [Algebra S R'] [IsScalarTower R S R']
(p : S[X]) {x : R'} (hx : aeval x p = 0) : aeval x (integerNormalization M p) = 0 := by |
rw [aeval_def, IsScalarTower.algebraMap_eq R S R',
integerNormalization_eval₂_eq_zero _ (algebraMap _ _) _ hx]
| 2 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 110 | 112 | theorem lcm_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).lcm = GCDMonoid.lcm s₁.lcm s₂.lcm := by |
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add]
simp
| 2 |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 77 | 81 | theorem succ_nth_stream_eq_none_iff :
IntFractPair.stream v (n + 1) = none ↔
IntFractPair.stream v n = none ∨ ∃ ifp, IntFractPair.stream v n = some ifp ∧ ifp.fr = 0 := by |
rw [IntFractPair.stream]
cases IntFractPair.stream v n <;> simp [imp_false]
| 2 |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 86 | 90 | theorem continuous_circleTransformDeriv {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
(hf : ContinuousOn f (sphere z R)) (hw : w ∈ ball z R) :
Continuous (circleTransformDeriv R z w f) := by |
rw [circleTransformDeriv_eq]
exact (continuous_circleMap_inv hw).smul (continuous_circleTransform hR hf hw)
| 2 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 74 | 77 | theorem integerNormalization_coeff (p : S[X]) (i : ℕ) :
(integerNormalization M p).coeff i = coeffIntegerNormalization M p i := by |
simp (config := { contextual := true }) [integerNormalization, coeff_monomial,
coeffIntegerNormalization_of_not_mem_support]
| 2 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 173 | 176 | theorem Matrix.toLinearMapRight'_one :
Matrix.toLinearMapRight' (1 : Matrix m m R) = LinearMap.id := by |
ext
simp [LinearMap.one_apply, stdBasis_apply]
| 2 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Analysis.Convex.Segment
import Mathlib.Tactic.GCongr
#align_import analysis.convex.star from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
open Convex Pointwise
variable {𝕜 E F : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Star.lean | 136 | 139 | theorem starConvex_sUnion {S : Set (Set E)} (hS : ∀ s ∈ S, StarConvex 𝕜 x s) :
StarConvex 𝕜 x (⋃₀ S) := by |
rw [sUnion_eq_iUnion]
exact starConvex_iUnion fun s => hS _ s.2
| 2 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 75 | 77 | theorem P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] := by |
rw [Q]
abel
| 2 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 188 | 195 | theorem ext_iff' {f g : α →₀ M} : f = g ↔ f.support = g.support ∧ ∀ x ∈ f.support, f x = g x :=
⟨fun h => h ▸ ⟨rfl, fun _ _ => rfl⟩, fun ⟨h₁, h₂⟩ =>
ext fun a => by
classical
exact if h : a ∈ f.support then h₂ a h else by
have hf : f a = 0 := not_mem_support_iff.1 h
have hg : g a = 0 :... | rwa [h₁, not_mem_support_iff] at h
rw [hf, hg]⟩
| 2 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 101 | 103 | theorem aleph_one_le_continuum : aleph 1 ≤ 𝔠 := by |
rw [← succ_aleph0]
exact Order.succ_le_of_lt aleph0_lt_continuum
| 2 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 129 | 131 | theorem prod_univ_three [CommMonoid β] (f : Fin 3 → β) : ∏ i, f i = f 0 * f 1 * f 2 := by |
rw [prod_univ_castSucc, prod_univ_two]
rfl
| 2 |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 717 | 719 | theorem of_d (j : α) : (of X d sq).d (j + 1) j = d j := by |
dsimp [of]
rw [if_pos rfl, Category.id_comp]
| 2 |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 82 | 85 | theorem map_prodComm_prod :
map ((RingEquiv.prodComm : R × S ≃+* S × R) : R × S →+* S × R) (prod I J) = prod J I := by |
refine Trans.trans (ideal_prod_eq _) ?_
simp [map_map]
| 2 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Bisim
variable {xs : Vector α n}
theorem mapAccumr_bisim {f₁ : α → σ₁ → σ₁ × β} {f₂ : α → σ₂ → σ₂ × β} {s₁ : σ₁} {s₂ : σ₂}
(R : σ₁ → σ₂ → Prop) (h₀ : R s₁ s₂)
(hR : ∀ {... | Mathlib/Data/Vector/MapLemmas.lean | 205 | 211 | theorem mapAccumr₂_bisim_tail {ys : Vector β n} {f₁ : α → β → σ₁ → σ₁ × γ}
{f₂ : α → β → σ₂ → σ₂ × γ} {s₁ : σ₁} {s₂ : σ₂}
(h : ∃ R : σ₁ → σ₂ → Prop, R s₁ s₂ ∧
∀ {s q} a b, R s q → R (f₁ a b s).1 (f₂ a b q).1 ∧ (f₁ a b s).2 = (f₂ a b q).2) :
(mapAccumr₂ f₁ xs ys s₁).2 = (mapAccumr₂ f₂ xs ys s₂).2 := by |
rcases h with ⟨R, h₀, hR⟩
exact (mapAccumr₂_bisim R h₀ hR).2
| 2 |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 189 | 192 | theorem mk_one' (p : K[X]) :
RatFunc.mk p 1 = ofFractionRing (algebraMap K[X] (FractionRing K[X]) p) := by |
-- Porting note: had to hint `M := K[X]⁰` below
rw [← IsLocalization.mk'_one (M := K[X]⁰) (FractionRing K[X]) p, ← mk_coe_def, Submonoid.coe_one]
| 2 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 70 | 72 | theorem IsUpperSet.mul_right (hs : IsUpperSet s) : IsUpperSet (s * t) := by |
rw [mul_comm]
exact hs.mul_left
| 2 |
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 | 137 | 139 | theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ := by |
rw [← image_inv]
exact ((OrderIso.inv α).map_csSup' hs₀ hs₁).symm
| 2 |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 199 | 201 | theorem apply_mono (μ : ProbabilityMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ := by |
rw [← coeFn_comp_toFiniteMeasure_eq_coeFn]
exact MeasureTheory.FiniteMeasure.apply_mono _ h
| 2 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 66 | 68 | theorem gauge_def' : gauge s x = sInf {r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s} := by |
congrm sInf {r | ?_}
exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
| 2 |
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 | 137 | 139 | theorem inter_prod : (s₁ ∩ s₂) ×ˢ t = s₁ ×ˢ t ∩ s₂ ×ˢ t := by |
ext ⟨x, y⟩
simp only [← and_and_right, mem_inter_iff, mem_prod]
| 2 |
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Matrix
namespace SlashInvariantForm
theorem vAdd_width_periodic (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f ... | Mathlib/NumberTheory/ModularForms/Identities.lean | 34 | 37 | theorem T_zpow_width_invariant (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f (((ModularGroup.T ^ (N * n))) • z) = f z := by |
rw [modular_T_zpow_smul z (N * n)]
simpa only [Int.cast_mul, Int.cast_natCast] using vAdd_width_periodic N k n f z
| 2 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 333 | 337 | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']
rfl
| 2 |
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Algebra.Algebra.Pi
#align_import order.filter.zero_and_bounded_at_filter from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Filt... | Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | 84 | 87 | theorem ZeroAtFilter.boundedAtFilter [NormedAddCommGroup β] {l : Filter α} {f : α → β}
(hf : ZeroAtFilter l f) : BoundedAtFilter l f := by |
rw [ZeroAtFilter, ← Asymptotics.isLittleO_const_iff (one_ne_zero' ℝ)] at hf
exact hf.isBigO
| 2 |
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Algebra.Ring.Basic
import Mathlib.Algebra.Field.MinimalAxioms
variable {K : Type*}
namespace FirstOrder
namespace Field
open Language Ring Structure BoundedFormula
inductive FieldAxiom : Type
| addAssoc : Field... | Mathlib/ModelTheory/Algebra/Field/Basic.lean | 81 | 86 | theorem FieldAxiom.realize_toSentence_iff_toProp {K : Type*}
[Add K] [Mul K] [Neg K] [Zero K] [One K] [CompatibleRing K]
(ax : FieldAxiom) :
(K ⊨ (ax.toSentence : Sentence Language.ring)) ↔ ax.toProp K := by |
cases ax <;>
simp [Sentence.Realize, Formula.Realize, Fin.snoc]
| 2 |
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 210 | 213 | theorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :
eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow := by |
induction h
simp
| 2 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 144 | 147 | theorem mapRange_id (h : ∀ i, id (0 : β₁ i) = 0 := fun i => rfl) (g : Π₀ i : ι, β₁ i) :
mapRange (fun i => (id : β₁ i → β₁ i)) h g = g := by |
ext
rfl
| 2 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v ≠ 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 137 | 139 | theorem submodule_eq (v : ℙ K V) : v.submodule = K ∙ v.rep := by |
conv_lhs => rw [← v.mk_rep]
rfl
| 2 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 61 | 64 | theorem isIntegrallyClosed_eq_field_fractions' [IsDomain S] [Algebra K S] [IsScalarTower R K S]
{s : S} (hs : IsIntegral R s) : minpoly K s = (minpoly R s).map (algebraMap R K) := by |
let L := FractionRing S
rw [← isIntegrallyClosed_eq_field_fractions K L hs, algebraMap_eq (IsFractionRing.injective S L)]
| 2 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 104 | 107 | theorem comp_differentiableAt_iff {f : G → E} {x : G} :
DifferentiableAt 𝕜 (iso ∘ f) x ↔ DifferentiableAt 𝕜 f x := by |
rw [← differentiableWithinAt_univ, ← differentiableWithinAt_univ,
iso.comp_differentiableWithinAt_iff]
| 2 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 200 | 202 | theorem transAssocReparamAux_mem_I (t : I) : transAssocReparamAux t ∈ I := by |
unfold transAssocReparamAux
split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]
| 2 |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 57 | 60 | theorem const_bind_eq_comp_const (κ : kernel α β) (μ : Measure α) :
const α (μ.bind κ) = κ ∘ₖ const α μ := by |
ext a s hs
simp_rw [comp_apply' _ _ _ hs, const_apply, Measure.bind_apply hs (kernel.measurable _)]
| 2 |
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section OrderTopology
variable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]
[OrderTopology β]
theorem IsLUB.fr... | Mathlib/Topology/Order/IsLUB.lean | 77 | 80 | theorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (hsf : a ∈ closure s) :
IsLUB s a := by |
rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
exact isLUB_of_mem_nhds hsa (mem_principal_self s)
| 2 |
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
noncomputable section
universe w v₁ v₂ u₁ u₂
open Cate... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 104 | 108 | theorem PreservesEqualizer.iso_inv_ι :
(PreservesEqualizer.iso G f g).inv ≫ G.map (equalizer.ι f g) =
equalizer.ι (G.map f) (G.map g) := by |
rw [← Iso.cancel_iso_hom_left (PreservesEqualizer.iso G f g), ← Category.assoc, Iso.hom_inv_id]
simp
| 2 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {α β : Type*} {n : ℕ}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 61 | 64 | theorem card_filter_univ_succ' (p : Fin (n + 1) → Prop) [DecidablePred p] :
(univ.filter p).card = ite (p 0) 1 0 + (univ.filter (p ∘ Fin.succ)).card := by |
rw [Fin.univ_succ, filter_cons, card_disjUnion, filter_map, card_map]
split_ifs <;> simp
| 2 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 56 | 59 | theorem norm_sub_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two (x y : V) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2 := by |
rw [norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero]
exact inner_eq_zero_iff_angle_eq_pi_div_two x y
| 2 |
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
#align_import topology.homotopy.path from "leanprover-community/mathlib"@"bb9d1c5085e0b7ea619806a68c5021927cecb2a6"
universe u v
variable {X : Type u} {Y : Type v} [TopologicalSpace X] [Top... | Mathlib/Topology/Homotopy/Path.lean | 83 | 85 | theorem eval_zero (F : Homotopy p₀ p₁) : F.eval 0 = p₀ := by |
ext t
simp [eval]
| 2 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 62 | 65 | theorem prod_range_mul {f : ℕ → M} {k : ℕ} (h : HasProd (fun n ↦ f (n + k)) m) :
HasProd f ((∏ i ∈ range k, f i) * m) := by |
refine ((range k).hasProd f).mul_compl ?_
rwa [← (notMemRangeEquiv k).symm.hasProd_iff]
| 2 |
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
#align_import number_theory.fermat_psp from "leanprover-community/mathlib"@"c0439b4877c24a117bfdd9e32faf62eee9b115eb"
namespace Nat
def ProbablePrime (n b : ℕ) : Prop :=
n ∣ b ^ (n - 1) - 1
#align fermat_psp.probable_prime Nat.Probabl... | Mathlib/NumberTheory/FermatPsp.lean | 120 | 122 | theorem coprime_of_fermatPsp {n b : ℕ} (h : FermatPsp n b) (h₁ : 1 ≤ b) : Nat.Coprime n b := by |
rcases h with ⟨hp, _, hn₂⟩
exact coprime_of_probablePrime hp (by omega) h₁
| 2 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 104 | 106 | theorem mul_self_num (q : ℚ) : (q * q).num = q.num * q.num := by |
rw [mul_num, Int.natAbs_mul, Nat.Coprime.gcd_eq_one, Int.ofNat_one, Int.ediv_one]
exact (q.reduced.mul_right q.reduced).mul (q.reduced.mul_right q.reduced)
| 2 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Set.UnionLift
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [... | Mathlib/Algebra/Algebra/Subalgebra/Directed.lean | 90 | 93 | theorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :
iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by |
dsimp [iSupLift, inclusion]
rw [Set.iUnionLift_mk]
| 2 |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 112 | 114 | theorem conjTranspose_row [Star α] (v : m → α) : (row v)ᴴ = col (star v) := by |
ext
rfl
| 2 |
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 81 | 83 | theorem lift_succ (a : Ordinal.{v}) : lift.{u} (succ a) = succ (lift.{u} a) := by |
rw [← add_one_eq_succ, lift_add, lift_one]
rfl
| 2 |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 168 | 171 | theorem not_irreducible_expand (R p) [CommSemiring R] [Fact p.Prime] [CharP R p] [PerfectRing R p]
(f : R[X]) : ¬ Irreducible (expand R p f) := by |
rw [polynomial_expand_eq]
exact not_irreducible_pow (Fact.out : p.Prime).ne_one
| 2 |
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 | 284 | 287 | theorem descPochhammer_eval_cast (n : ℕ) (k : ℤ) :
(((descPochhammer ℤ n).eval k : ℤ) : R) = ((descPochhammer R n).eval k : R) := by |
rw [← descPochhammer_map (algebraMap ℤ R), eval_map, ← eq_intCast (algebraMap ℤ R)]
simp only [algebraMap_int_eq, eq_intCast, eval₂_at_intCast, Nat.cast_id, eq_natCast, Int.cast_id]
| 2 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 353 | 356 | theorem contDiffAt_euclidean {n : ℕ∞} :
ContDiffAt 𝕜 n f y ↔ ∀ i, ContDiffAt 𝕜 n (fun x => f x i) y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffAt_iff, contDiffAt_pi]
rfl
| 2 |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 221 | 224 | theorem reprW_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.reprX) :
(F.reprW.app X).hom f = F.map f.op F.reprx := by |
simp only [yoneda_obj_obj, Iso.app_hom, op_unop, reprx, ← FunctorToTypes.naturality,
yoneda_obj_map, unop_op, Quiver.Hom.unop_op, Category.comp_id]
| 2 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 275 | 278 | theorem mem_rootsOfUnity_prime_pow_mul_iff' (p k : ℕ) (m : ℕ+) [ExpChar R p]
{ζ : Rˣ} : ζ ^ (p ^ k * ↑m) = 1 ↔ ζ ∈ rootsOfUnity m R := by |
rw [← PNat.mk_coe p (expChar_pos R p), ← PNat.pow_coe, ← PNat.mul_coe, ← mem_rootsOfUnity,
mem_rootsOfUnity_prime_pow_mul_iff]
| 2 |
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 97 | 100 | theorem IsTensorProduct.lift_eq (h : IsTensorProduct f) (f' : M₁ →ₗ[R] M₂ →ₗ[R] M') (x₁ : M₁)
(x₂ : M₂) : h.lift f' (f x₁ x₂) = f' x₁ x₂ := by |
delta IsTensorProduct.lift
simp
| 2 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 145 | 147 | theorem zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [Epi f] (h : f ≫ g = 0) : g = 0 := by |
rw [← comp_zero, cancel_epi] at h
exact h
| 2 |
import Aesop.Nanos
import Aesop.Util.UnionFind
import Aesop.Util.UnorderedArraySet
import Batteries.Data.String
import Batteries.Lean.Expr
import Batteries.Lean.Meta.DiscrTree
import Batteries.Lean.PersistentHashSet
import Lean.Meta.Tactic.TryThis
open Lean
open Lean.Meta Lean.Elab.Tactic
namespace Aesop.Array
| .lake/packages/aesop/Aesop/Util/Basic.lean | 21 | 24 | theorem size_modify (a : Array α) (i : Nat) (f : α → α) :
(a.modify i f).size = a.size := by |
simp only [Array.modify, Id.run, Array.modifyM]
split <;> simp
| 2 |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 100 | 102 | theorem eqToIso_refl {n : ℕ} (h : n = n) : X.eqToIso h = Iso.refl _ := by |
ext
simp [eqToIso]
| 2 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.Tactic.IntervalCases
#align_import group_theory.p_gr... | Mathlib/GroupTheory/PGroup.lean | 84 | 87 | theorem of_surjective {H : Type*} [Group H] (ϕ : G →* H) (hϕ : Function.Surjective ϕ) :
IsPGroup p H := by |
refine fun h => Exists.elim (hϕ h) fun g hg => Exists.imp (fun k hk => ?_) (hG g)
rw [← hg, ← ϕ.map_pow, hk, ϕ.map_one]
| 2 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 90 | 93 | theorem lintegral_mono' {m : MeasurableSpace α} ⦃μ ν : Measure α⦄ (hμν : μ ≤ ν) ⦃f g : α → ℝ≥0∞⦄
(hfg : f ≤ g) : ∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂ν := by |
rw [lintegral, lintegral]
exact iSup_mono fun φ => iSup_mono' fun hφ => ⟨le_trans hφ hfg, lintegral_mono (le_refl φ) hμν⟩
| 2 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 111 | 115 | theorem HasDerivWithinAt.complexToReal_fderiv' {f : ℂ → E} {s : Set ℂ} {x : ℂ} {f' : E}
(h : HasDerivWithinAt f f' s x) :
HasFDerivWithinAt f (reCLM.smulRight f' + I • imCLM.smulRight f') s x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using
h.hasFDerivWithinAt.restrictScalars ℝ
| 2 |
import Mathlib.Data.List.Basic
#align_import data.list.count from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists Set.range
assert_not_exists GroupWithZero
assert_not_exists Ring
open Nat
variable {α : Type*} {l : List α}
namespace List
section CountP
variable (p q ... | Mathlib/Data/List/Count.lean | 54 | 57 | theorem length_filter_lt_length_iff_exists (l) :
length (filter p l) < length l ↔ ∃ x ∈ l, ¬p x := by |
simpa [length_eq_countP_add_countP p l, countP_eq_length_filter] using
countP_pos (fun x => ¬p x) (l := l)
| 2 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 50 | 52 | theorem natAbs_lt_iff_sq_lt {a b : ℤ} : a.natAbs < b.natAbs ↔ a ^ 2 < b ^ 2 := by |
rw [sq, sq]
exact natAbs_lt_iff_mul_self_lt
| 2 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 104 | 106 | theorem disjoint_inv_left_iff : Disjoint f⁻¹ g ↔ Disjoint f g := by |
refine ⟨fun h => ?_, Disjoint.inv_left⟩
convert h.inv_left
| 2 |
import Mathlib.NumberTheory.Zsqrtd.GaussianInt
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.zsqrtd.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open scoped Comple... | Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | 86 | 93 | theorem prime_of_nat_prime_of_mod_four_eq_three (p : ℕ) [hp : Fact p.Prime] (hp3 : p % 4 = 3) :
Prime (p : ℤ[i]) :=
irreducible_iff_prime.1 <|
by_contradiction fun hpi =>
let ⟨a, b, hab⟩ := sq_add_sq_of_nat_prime_of_not_irreducible p hpi
have : ∀ a b : ZMod 4, a ^ 2 + b ^ 2 ≠ (p : ZMod 4) := by |
erw [← ZMod.natCast_mod p 4, hp3]; decide
this a b (hab ▸ by simp)
| 2 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 157 | 160 | theorem linearMapAt_symmₗ (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : F) : e.linearMapAt R b (e.symmₗ R b y) = y := by |
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).right_inv y
| 2 |
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