Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 774 | 784 | theorem tendsto_translation_number₀' :
Tendsto (fun n : ℕ => (f ^ (n + 1) : CircleDeg1Lift) 0 / ((n : ℝ) + 1)) atTop (𝓝 <| τ f) := by |
refine
tendsto_iff_dist_tendsto_zero.2 <|
squeeze_zero (fun _ => dist_nonneg) (fun n => ?_)
((tendsto_const_div_atTop_nhds_zero_nat 1).comp (tendsto_add_atTop_nat 1))
dsimp
have : (0 : ℝ) < n + 1 := n.cast_add_one_pos
rw [Real.dist_eq, div_sub' _ _ _ (ne_of_gt this), abs_div, ← Real.dist_eq, ... |
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Whiskering
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.Tactic.CategoryTheory.Slice
#align_import category_theory.equivalence from "leanprover-community/mathlib"@"9aba7801eeec... | Mathlib/CategoryTheory/Equivalence.lean | 166 | 170 | theorem counitInv_app_functor (e : C ≌ D) (X : C) :
e.counitInv.app (e.functor.obj X) = e.functor.map (e.unit.app X) := by |
symm
erw [← Iso.comp_hom_eq_id (e.counitIso.app _), functor_unit_comp]
rfl
|
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 110 | 114 | theorem UniformInducing.uniformContinuousOn_iff {f : α → β} {g : β → γ} {S : Set α}
(hg : UniformInducing g) :
UniformContinuousOn f S ↔ UniformContinuousOn (g ∘ f) S := by |
dsimp only [UniformContinuousOn, Tendsto]
rw [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, comp_def, comp_def]
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 125 | 126 | theorem leadingCoeff_hermite (n : ℕ) : (hermite n).leadingCoeff = 1 := by |
rw [← coeff_natDegree, natDegree_hermite, coeff_hermite_self]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 53 | 53 | theorem one_sub_K_pos' (hf : ContractingWith K f) : (0 : ℝ≥0∞) < 1 - K := by | simp [hf.1]
|
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 | 99 | 102 | theorem homogeneousSubmodule_eq_finsupp_supported [CommSemiring R] (n : ℕ) :
homogeneousSubmodule σ R n = Finsupp.supported _ R { d | degree d = n } := by |
simp_rw [← weightedDegree_one]
exact weightedHomogeneousSubmodule_eq_finsupp_supported R 1 n
|
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 | 104 | 104 | theorem two_smul : (2 : R) • x = x + x := by | rw [← one_add_one_eq_two, add_smul, one_smul]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Order.Monotone.Monovary
import Mathlib.Tactic.Abel
#align_impo... | Mathlib/Algebra/Order/Rearrangement.lean | 285 | 287 | theorem Monovary.sum_comp_perm_smul_lt_sum_smul_iff (hfg : Monovary f g) :
((∑ i, f (σ i) • g i) < ∑ i, f i • g i) ↔ ¬Monovary (f ∘ σ) g := by |
simp [(hfg.monovaryOn _).sum_comp_perm_smul_lt_sum_smul_iff fun _ _ ↦ mem_univ _]
|
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 55 | 58 | theorem rank_mul_rank (A : Type v) [AddCommGroup A]
[Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A] :
Module.rank F K * Module.rank K A = Module.rank F A := by |
convert lift_rank_mul_lift_rank F K A <;> rw [lift_id]
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 476 | 483 | theorem norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle y (y - x)) = ‖y‖ := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.norm_div_tan_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.l... |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Order.Partition.Equipartition
import Mathlib.SetTheory.Ordinal.Basic
#align_import combinatorics.simple_graph.regularity.uniform from "leanprover-community/mathlib"@"bf7ef0e83e5b... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 148 | 151 | theorem right_nonuniformWitnesses_subset (h : ¬G.IsUniform ε s t) :
(G.nonuniformWitnesses ε s t).2 ⊆ t := by |
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.1
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 62 | 63 | theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by |
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
open Set Function Filter MeasureTheory MeasureTheory.Measure
open ENNReal
variable {α : Type*} {m : MeasurableSpace α} (f : α → α) {s : Set α}
... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 74 | 76 | theorem prob_eq_zero_or_one [IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs : MeasurableSet s)
(hs' : f ⁻¹' s = s) : μ s = 0 ∨ μ s = 1 := by |
simpa [hs] using hf.measure_self_or_compl_eq_zero hs hs'
|
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 174 | 185 | theorem N₂Γ₂_inv_app_f_f (X : Karoubi (ChainComplex C ℕ)) (n : ℕ) :
(N₂Γ₂.inv.app X).f.f n =
X.p.f n ≫ ((Γ₀.splitting X.X).cofan _).inj (Splitting.IndexSet.id (op [n])) := by |
dsimp [N₂Γ₂]
simp only [whiskeringLeft_obj_preimage_app, NatTrans.comp_app, Functor.comp_map,
Karoubi.comp_f, N₂Γ₂ToKaroubiIso_inv_app, HomologicalComplex.comp_f,
N₁Γ₀_inv_app_f_f, toKaroubi_obj_X, Splitting.toKaroubiNondegComplexIsoN₁_hom_f_f,
Γ₀.obj_obj, PInfty_on_Γ₀_splitting_summand_eq_self, N₂_map... |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Dynamics.Minimal
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib.MeasureTheory.Measure.Regular
#align_import measure_theory.group.action from "leanprover-community/mathlib"@"f2ce6086713c78a7f8... | Mathlib/MeasureTheory/Group/Action.lean | 90 | 95 | theorem measurePreserving_smul : MeasurePreserving (c • ·) μ μ :=
{ measurable := measurable_const_smul c
map_eq := by |
ext1 s hs
rw [map_apply (measurable_const_smul c) hs]
exact SMulInvariantMeasure.measure_preimage_smul c hs }
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 171 | 172 | theorem degree_le_zero_iff_eq_one (hp : p.Monic) : p.degree ≤ 0 ↔ p = 1 := by |
rw [← hp.natDegree_eq_zero_iff_eq_one, natDegree_eq_zero_iff_degree_le_zero]
|
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 185 | 193 | theorem exists_dist_le_lt (hδ : 0 ≤ δ) (hε : 0 < ε) (h : dist x z < ε + δ) :
∃ y, dist x y ≤ δ ∧ dist y z < ε := by |
refine (exists_dist_eq x z (div_nonneg hε.le <| add_nonneg hε.le hδ)
(div_nonneg hδ <| add_nonneg hε.le hδ) <| by
rw [← add_div, div_self (add_pos_of_pos_of_nonneg hε hδ).ne']).imp
fun y hy => ?_
rw [hy.1, hy.2, div_mul_comm, div_mul_comm ε]
rw [← div_lt_one (add_pos_of_pos_of_nonneg hε hδ)] at h
... |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
open List
def Cycle (α : Type*) : Type _ :=
Quotient (IsRotated.setoid α)
#align cycle Cycle
namespace Cycle
variable {α : Type*}
--... | Mathlib/Data/List/Cycle.lean | 675 | 679 | theorem Nodup.nontrivial_iff {s : Cycle α} (h : Nodup s) : Nontrivial s ↔ ¬Subsingleton s := by |
rw [length_subsingleton_iff]
induction s using Quotient.inductionOn'
simp only [mk''_eq_coe, nodup_coe_iff] at h
simp [h, Nat.succ_le_iff]
|
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 561 | 567 | theorem sheafifyLift_unique {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : Presheaf.IsSheaf J Q)
(γ : J.sheafify P ⟶ Q) : J.toSheafify P ≫ γ = η → γ = sheafifyLift J η hQ := by |
intro h
apply plusLift_unique
apply plusLift_unique
rw [← Category.assoc, ← plusMap_toPlus]
exact h
|
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 158 | 162 | theorem iInter_setOf_subset {ι : Type*} (S : ι → L[[α]].Theory) :
⋂ i : ι, { p : T.CompleteType α | S i ⊆ p } =
{ p : T.CompleteType α | ⋃ i : ι, S i ⊆ p } := by |
ext
simp only [mem_iInter, mem_setOf_eq, iUnion_subset_iff]
|
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 283 | 290 | theorem hasProd_prod {f : γ → β → α} {a : γ → α} {s : Finset γ} :
(∀ i ∈ s, HasProd (f i) (a i)) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) := by |
classical
exact Finset.induction_on s (by simp only [hasProd_one, prod_empty, forall_true_iff]) <| by
-- Porting note: with some help, `simp` used to be able to close the goal
simp (config := { contextual := true }) only [mem_insert, forall_eq_or_imp, not_false_iff,
prod_insert, and_imp]
exact fu... |
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
#align_import measure_theory.measure.portmanteau from "leanprover-community/mathlib"@"fd5edc43dc4f... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 318 | 324 | theorem ProbabilityMeasure.limsup_measure_closed_le_of_tendsto {Ω ι : Type*} {L : Filter ι}
[MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] [HasOuterApproxClosed Ω]
{μ : ProbabilityMeasure Ω} {μs : ι → ProbabilityMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ))
{F : Set Ω} (F_closed : IsClosed F)... |
apply FiniteMeasure.limsup_measure_closed_le_of_tendsto
((ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds L).mp μs_lim) F_closed
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 620 | 624 | theorem isPreconnected_connectedComponentIn {x : α} {F : Set α} :
IsPreconnected (connectedComponentIn F x) := by |
rw [connectedComponentIn]; split_ifs
· exact inducing_subtype_val.isPreconnected_image.mpr isPreconnected_connectedComponent
· exact isPreconnected_empty
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 109 | 109 | theorem card_Iio : (Iio b).card = b := by | rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 224 | 225 | theorem forall_in_swap {p : α → β → Prop} : (∀ a ∈ s, ∀ (b), p a b) ↔ ∀ (b), ∀ a ∈ s, p a b := by |
tauto
|
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 | 116 | 118 | theorem comp_differentiable_iff {f : G → E} : Differentiable 𝕜 (iso ∘ f) ↔ Differentiable 𝕜 f := by |
rw [← differentiableOn_univ, ← differentiableOn_univ]
exact iso.comp_differentiableOn_iff
|
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 51 | 54 | theorem xgcdAux_rec {r s t r' s' t'} (h : 0 < r) :
xgcdAux r s t r' s' t' = xgcdAux (r' % r) (s' - r' / r * s) (t' - r' / r * t) r s t := by |
obtain ⟨r, rfl⟩ := Nat.exists_eq_succ_of_ne_zero h.ne'
simp [xgcdAux]
|
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 59 | 61 | theorem mk_add_mem_enumFrom_iff_get? {n i : ℕ} {x : α} {l : List α} :
(n + i, x) ∈ enumFrom n l ↔ l.get? i = x := by |
simp [mem_iff_get?]
|
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 120 | 122 | theorem weightedTotalDegree_zero (w : σ → M) :
weightedTotalDegree w (0 : MvPolynomial σ R) = ⊥ := by |
simp only [weightedTotalDegree, support_zero, Finset.sup_empty]
|
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 98 | 99 | theorem dependent_iff_not_independent : Dependent f ↔ ¬Independent f := by |
rw [dependent_iff, independent_iff]
|
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
def HomRel (C) [Quiver C] :=
∀ ⦃X Y : C⦄, (X ⟶ Y) → (X ⟶ Y) → Prop
#align hom_rel HomRel
-- Porting Note: `deriving I... | Mathlib/CategoryTheory/Quotient.lean | 223 | 227 | theorem lift_map_functor_map {X Y : C} (f : X ⟶ Y) :
(lift r F H).map ((functor r).map f) = F.map f := by |
rw [← NatIso.naturality_1 (lift.isLift r F H)]
dsimp [lift, functor]
simp
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 111 | 116 | theorem degree_hermite (n : ℕ) : (hermite n).degree = n := by |
rw [degree_eq_of_le_of_coeff_ne_zero]
· simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
rintro m hnm
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n]
|
import Mathlib.Analysis.Complex.Isometry
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.complex.conformal from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
noncomputable section
open Complex Continuous... | Mathlib/Analysis/Complex/Conformal.lean | 78 | 91 | theorem IsConformalMap.is_complex_or_conj_linear (h : IsConformalMap g) :
(∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g) ∨
∃ map : ℂ →L[ℂ] ℂ, map.restrictScalars ℝ = g ∘L ↑conjCLE := by |
rcases h with ⟨c, -, li, rfl⟩
obtain ⟨li, rfl⟩ : ∃ li' : ℂ ≃ₗᵢ[ℝ] ℂ, li'.toLinearIsometry = li :=
⟨li.toLinearIsometryEquiv rfl, by ext1; rfl⟩
rcases linear_isometry_complex li with ⟨a, rfl | rfl⟩
-- let rot := c • (a : ℂ) • ContinuousLinearMap.id ℂ ℂ,
· refine Or.inl ⟨c • (a : ℂ) • ContinuousLinearMap.i... |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
namespace Submodule
open Pointwise
variable {R M M' F G : Type*} [CommRing R] [AddCommGroup M] [Module R M]
variable {N N₁ N₂ P P₁ P₂ : Submodule R M}
def colon (N P : Submodule R M) : Ideal R :=
annihilator (P.map N.mkQ)
#align ... | Mathlib/RingTheory/Ideal/Colon.lean | 81 | 84 | theorem annihilator_quotient {N : Submodule R M} :
Module.annihilator R (M ⧸ N) = N.colon ⊤ := by |
simp_rw [SetLike.ext_iff, Module.mem_annihilator, colon, mem_annihilator, map_top,
LinearMap.range_eq_top.mpr (mkQ_surjective N), mem_top, forall_true_left, forall_const]
|
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Types
#align_import category_theory.limits.filtered from "leanprover-community/mathlib"@"e4ee4e30418efcb8cf304ba76ad653aeec04ba6e"
universe w' w v u
noncomputable section
open Categor... | Mathlib/CategoryTheory/Limits/Filtered.lean | 40 | 48 | theorem IsFiltered.iff_nonempty_limit : IsFiltered C ↔
∀ {J : Type v} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
∃ (X : C), Nonempty (limit (F.op ⋙ yoneda.obj X)) := by |
rw [IsFiltered.iff_cocone_nonempty.{v}]
refine ⟨fun h J _ _ F => ?_, fun h J _ _ F => ?_⟩
· obtain ⟨c⟩ := h F
exact ⟨c.pt, ⟨(limitCompYonedaIsoCocone F c.pt).inv c.ι⟩⟩
· obtain ⟨pt, ⟨ι⟩⟩ := h F
exact ⟨⟨pt, (limitCompYonedaIsoCocone F pt).hom ι⟩⟩
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 93 | 95 | theorem card_option [Finite α] : Nat.card (Option α) = Nat.card α + 1 := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_option]
|
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 140 | 143 | theorem map₂_right_comm {f : δ → γ → ε} {g : α → β → δ} {f' : α → γ → δ'} {g' : δ' → β → ε}
(h_right_comm : ∀ a b c, f (g a b) c = g' (f' a c) b) :
map₂ f (map₂ g a b) c = map₂ g' (map₂ f' a c) b := by |
cases a <;> cases b <;> cases c <;> simp [h_right_comm]
|
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Algebra.Module.Submodule.LinearMap
open Function Pointwise Set
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*} {M₃ : Type*}
namespace Submodule
section AddCommMonoid
variable [Semiring R] [... | Mathlib/Algebra/Module/Submodule/Map.lean | 121 | 123 | theorem map_add_le (f g : M →ₛₗ[σ₁₂] M₂) : map (f + g) p ≤ map f p ⊔ map g p := by |
rintro x ⟨m, hm, rfl⟩
exact add_mem_sup (mem_map_of_mem hm) (mem_map_of_mem hm)
|
import Mathlib.Data.List.Nodup
#align_import data.list.dedup from "leanprover-community/mathlib"@"d9e96a3e3e0894e93e10aff5244f4c96655bac1c"
universe u
namespace List
variable {α : Type u} [DecidableEq α]
@[simp]
theorem dedup_nil : dedup [] = ([] : List α) :=
rfl
#align list.dedup_nil List.dedup_nil
theorem... | Mathlib/Data/List/Dedup.lean | 109 | 114 | theorem dedup_eq_nil (l : List α) : l.dedup = [] ↔ l = [] := by |
induction' l with a l hl
· exact Iff.rfl
· by_cases h : a ∈ l
· simp only [List.dedup_cons_of_mem h, hl, List.ne_nil_of_mem h]
· simp only [List.dedup_cons_of_not_mem h, List.cons_ne_nil]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 338 | 346 | theorem intCast_dvd (z : ℤ) (a : ℤ√d) : ↑z ∣ a ↔ z ∣ a.re ∧ z ∣ a.im := by |
constructor
· rintro ⟨x, rfl⟩
simp only [add_zero, intCast_re, zero_mul, mul_im, dvd_mul_right, and_self_iff,
mul_re, mul_zero, intCast_im]
· rintro ⟨⟨r, hr⟩, ⟨i, hi⟩⟩
use ⟨r, i⟩
rw [smul_val, Zsqrtd.ext_iff]
exact ⟨hr, hi⟩
|
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.ShrinkingLemma
#align_import topology.metric_space.shrinking_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
unive... | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | 62 | 69 | theorem exists_subset_iUnion_ball_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i) (hs : IsClosed s)
(uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i ∈ Ioo 0 (r i) := by |
rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with
⟨v, hsv, hvc, hcv⟩
have := fun i => exists_pos_lt_subset_ball (hr i) (hvc i) (hcv i)
choose r' hlt hsub using this
exact ⟨r', hsv.trans <| iUnion_mono hsub, hlt⟩
|
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Limits.Preserves.Limits
#align_import category_theory.limits.functor_category from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
open CategoryTheory CategoryTheory.Category CategoryTheory.Functor
-- morphism ... | Mathlib/CategoryTheory/Limits/FunctorCategory.lean | 243 | 248 | theorem limit_obj_ext {H : J ⥤ K ⥤ C} [HasLimitsOfShape J C] {k : K} {W : C}
{f g : W ⟶ (limit H).obj k}
(w : ∀ j, f ≫ (Limits.limit.π H j).app k = g ≫ (Limits.limit.π H j).app k) : f = g := by |
apply (cancel_mono (limitObjIsoLimitCompEvaluation H k).hom).1
ext j
simpa using w j
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 505 | 510 | theorem galoisConnection_map_comap (hφ : Function.Injective φ.obj) :
GaloisConnection (map φ hφ) (comap φ) := by |
rintro S T; simp_rw [le_iff]; constructor
· exact fun h c d f fS => h (Map.Arrows.im f fS)
· rintro h _ _ g ⟨a, gφS⟩
exact h gφS
|
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Pi
#align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Finset
open Multiset
section Pi
variable {α : Type*}
def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=... | Mathlib/Data/Finset/Pi.lean | 74 | 83 | theorem Pi.cons_injective {a : α} {b : δ a} {s : Finset α} (hs : a ∉ s) :
Function.Injective (Pi.cons s a b) := fun e₁ e₂ eq =>
@Multiset.Pi.cons_injective α _ δ a b s.1 hs _ _ <|
funext fun e =>
funext fun h =>
have :
Pi.cons s a b e₁ e (by simpa only [Multiset.mem_cons, mem_insert] u... |
rw [eq]
this
|
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 213 | 225 | theorem Fintype.all_card_le_filter_rel_iff_exists_injective {α : Type u} {β : Type v} [Fintype β]
(r : α → β → Prop) [∀ a, DecidablePred (r a)] :
(∀ A : Finset α, A.card ≤ (univ.filter fun b : β => ∃ a ∈ A, r a b).card) ↔
∃ f : α → β, Function.Injective f ∧ ∀ x, r x (f x) := by |
haveI := Classical.decEq β
let r' a := univ.filter fun b => r a b
have h : ∀ A : Finset α, (univ.filter fun b : β => ∃ a ∈ A, r a b) = A.biUnion r' := by
intro A
ext b
simp [r']
have h' : ∀ (f : α → β) (x), r x (f x) ↔ f x ∈ r' x := by simp [r']
simp_rw [h, h']
apply Finset.all_card_le_biUnion_... |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 421 | 423 | theorem digits_lt_base {b m d : ℕ} (hb : 1 < b) (hd : d ∈ digits b m) : d < b := by |
rcases b with (_ | _ | b) <;> try simp_all
exact digits_lt_base' hd
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 862 | 863 | theorem sdiff_mem_sdiff [SDiff α] (a b : Part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma \ mb ∈ a \ b := by | simp [sdiff_def]; aesop
|
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 | 273 | 274 | theorem clog_one_right (b : ℕ) : clog b (1 : R) = 0 := by |
rw [clog_of_one_le_right _ le_rfl, Nat.ceil_one, Nat.clog_one_right, Int.ofNat_zero]
|
import Mathlib.Data.Finsupp.Defs
#align_import data.list.to_finsupp from "leanprover-community/mathlib"@"06a655b5fcfbda03502f9158bbf6c0f1400886f9"
namespace List
variable {M : Type*} [Zero M] (l : List M) [DecidablePred (getD l · 0 ≠ 0)] (n : ℕ)
def toFinsupp : ℕ →₀ M where
toFun i := getD l i 0
support := ... | Mathlib/Data/List/ToFinsupp.lean | 92 | 94 | theorem toFinsupp_singleton (x : M) [DecidablePred (getD [x] · 0 ≠ 0)] :
toFinsupp [x] = Finsupp.single 0 x := by |
ext ⟨_ | i⟩ <;> simp [Finsupp.single_apply, (Nat.zero_lt_succ _).ne]
|
import Mathlib.CategoryTheory.Sites.Coherent.CoherentSheaves
namespace CategoryTheory
variable {C : Type*} [Category C] [Precoherent C] {X : C}
theorem coherentTopology.mem_sieves_of_hasEffectiveEpiFamily (S : Sieve X) :
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
EffectiveEpiFam... | Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean | 44 | 68 | theorem EffectiveEpiFamily.transitive_of_finite {α : Type} [Finite α] {Y : α → C}
(π : (a : α) → (Y a ⟶ X)) (h : EffectiveEpiFamily Y π) {β : α → Type} [∀ (a: α), Finite (β a)]
{Y_n : (a : α) → β a → C} (π_n : (a : α) → (b : β a) → (Y_n a b ⟶ Y a))
(H : ∀ a, EffectiveEpiFamily (Y_n a) (π_n a)) :
Effecti... |
rw [← Sieve.effectiveEpimorphic_family]
suffices h₂ : (Sieve.generate (Presieve.ofArrows (fun (⟨a, b⟩ : Σ _, β _) => Y_n a b)
(fun ⟨a,b⟩ => π_n a b ≫ π a))) ∈ GrothendieckTopology.sieves (coherentTopology C) X by
change Nonempty _
rw [← Sieve.forallYonedaIsSheaf_iff_colimit]
exact fun W => cohe... |
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import number_theory.sum_four_squares from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
open Finset Polynomial FiniteField Equiv
| Mathlib/NumberTheory/SumFourSquares.lean | 28 | 31 | theorem euler_four_squares {R : Type*} [CommRing R] (a b c d x y z w : R) :
(a * x - b * y - c * z - d * w) ^ 2 + (a * y + b * x + c * w - d * z) ^ 2 +
(a * z - b * w + c * x + d * y) ^ 2 + (a * w + b * z - c * y + d * x) ^ 2 =
(a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2) := by | ring
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fin
import Mathlib.Tactic.NormNum.Ineq
#align_import group_theory.perm.sign from "leanprover-community/math... | Mathlib/GroupTheory/Perm/Sign.lean | 223 | 243 | theorem signAux_mul {n : ℕ} (f g : Perm (Fin n)) : signAux (f * g) = signAux f * signAux g := by |
rw [← signAux_inv g]
unfold signAux
rw [← prod_mul_distrib]
refine prod_nbij (signBijAux g) signBijAux_mem signBijAux_injOn signBijAux_surj ?_
rintro ⟨a, b⟩ hab
dsimp only [signBijAux]
rw [mul_apply, mul_apply]
rw [mem_finPairsLT] at hab
by_cases h : g b < g a
· rw [dif_pos h]
simp only [not_le... |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 347 | 364 | theorem lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow {p q : ℝ}
(hpq : p.IsConjExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
(hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) :
(∫⁻ a, f a * g a ^ (p - 1) ∂μ) ≤
(∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ p ∂μ) ^ (1 / q) := by |
refine le_trans (ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf (hg.pow_const _)) ?_
by_cases hf_zero_rpow : (∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) = 0
· rw [hf_zero_rpow, zero_mul]
exact zero_le _
have hf_top_rpow : (∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
by_contra h
refine hf_top ?_
have hp_not_neg... |
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 | 222 | 225 | theorem updateRow_subsingleton [Subsingleton m] (A : Matrix m n R) (i : m) (b : n → R) :
A.updateRow i b = (row b).submatrix (Function.const m ()) id := by |
ext x y
simp [updateColumn_apply, Subsingleton.elim i x]
|
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_limits_deriv from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Filter
open scoped uniformity Filter Topology
section L... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 307 | 393 | theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
(hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
(hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasFDerivAt g (g' x) x := by |
-- The proof strategy follows several steps:
-- 1. The quantifiers in the definition of the derivative are
-- `∀ ε > 0, ∃δ > 0, ∀y ∈ B_δ(x)`. We will introduce a quantifier in the middle:
-- `∀ ε > 0, ∃N, ∀n ≥ N, ∃δ > 0, ∀y ∈ B_δ(x)` which will allow us to introduce the `f(') n`
-- 2. The order... |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 459 | 462 | theorem toReal_eq_toReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toReal = b.toReal ↔ a = b := by |
lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
simp only [coe_inj, NNReal.coe_inj, coe_toReal]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 157 | 164 | theorem IsPreconnected.sUnion_directed {S : Set (Set α)} (K : DirectedOn (· ⊆ ·) S)
(H : ∀ s ∈ S, IsPreconnected s) : IsPreconnected (⋃₀ S) := by |
rintro u v hu hv Huv ⟨a, ⟨s, hsS, has⟩, hau⟩ ⟨b, ⟨t, htS, hbt⟩, hbv⟩
obtain ⟨r, hrS, hsr, htr⟩ : ∃ r ∈ S, s ⊆ r ∧ t ⊆ r := K s hsS t htS
have Hnuv : (r ∩ (u ∩ v)).Nonempty :=
H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv) ⟨a, hsr has, hau⟩ ⟨b, htr hbt, hbv⟩
have Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v... |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 253 | 254 | theorem rpow_neg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ := by |
simp only [rpow_def_of_nonneg hx]; split_ifs <;> simp_all [exp_neg]
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 707 | 714 | theorem Tape.write_move_right_n {Γ} [Inhabited Γ] (f : Γ → Γ) (L R : ListBlank Γ) (n : ℕ) :
((Tape.move Dir.right)^[n] (Tape.mk' L R)).write (f (R.nth n)) =
(Tape.move Dir.right)^[n] (Tape.mk' L (R.modifyNth f n)) := by |
induction' n with n IH generalizing L R
· simp only [ListBlank.nth_zero, ListBlank.modifyNth, iterate_zero_apply, Nat.zero_eq]
rw [← Tape.write_mk', ListBlank.cons_head_tail]
simp only [ListBlank.head_cons, ListBlank.nth_succ, ListBlank.modifyNth, Tape.move_right_mk',
ListBlank.tail_cons, iterate_succ_ap... |
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.Ideal.Maps
#align_import data.polynomial.div from "leanprover-community/mathlib"@"e1e7190efdcefc925cb36f257a8362ef22944204"
noncomputable section
open Polynomial
... | Mathlib/Algebra/Polynomial/Div.lean | 166 | 173 | theorem natDegree_modByMonic_lt (p : R[X]) {q : R[X]} (hmq : Monic q) (hq : q ≠ 1) :
natDegree (p %ₘ q) < q.natDegree := by |
by_cases hpq : p %ₘ q = 0
· rw [hpq, natDegree_zero, Nat.pos_iff_ne_zero]
contrapose! hq
exact eq_one_of_monic_natDegree_zero hmq hq
· haveI := Nontrivial.of_polynomial_ne hpq
exact natDegree_lt_natDegree hpq (degree_modByMonic_lt p hmq)
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 425 | 446 | theorem ContinuousLinearEquiv.iteratedFDerivWithin_comp_right (g : G ≃L[𝕜] E) (f : E → F)
(hs : UniqueDiffOn 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) :
iteratedFDerivWithin 𝕜 i (f ∘ g) (g ⁻¹' s) x =
(iteratedFDerivWithin 𝕜 i f s (g x)).compContinuousLinearMap fun _ => g := by |
induction' i with i IH generalizing x
· ext1
simp only [Nat.zero_eq, iteratedFDerivWithin_zero_apply, comp_apply,
ContinuousMultilinearMap.compContinuousLinearMap_apply]
· ext1 m
simp only [ContinuousMultilinearMap.compContinuousLinearMap_apply,
ContinuousLinearEquiv.coe_coe, iteratedFDerivWit... |
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 478 | 502 | theorem uniformContinuous_uniformly_extend [CompleteSpace γ] : UniformContinuous ψ := fun d hd =>
let ⟨s, hs, hs_comp⟩ := comp3_mem_uniformity hd
have h_pnt : ∀ {a m}, m ∈ 𝓝 a → ∃ c ∈ f '' (e ⁻¹' m), (c, ψ a) ∈ s ∧ (ψ a, c) ∈ s :=
fun {a m} hm =>
have nb : NeBot (map f (comap e (𝓝 a))) :=
((h_e.dens... |
rwa [← h_e.comap_uniformity] at this
let ⟨t, ht, ts⟩ := this
show (Prod.map ψ ψ) ⁻¹' d ∈ 𝓤 α from
mem_of_superset (interior_mem_uniformity ht) fun ⟨x₁, x₂⟩ hx_t => by
have : interior t ∈ 𝓝 (x₁, x₂) := isOpen_interior.mem_nhds hx_t
let ⟨m₁, hm₁, m₂, hm₂, (hm : m₁ ×ˢ m₂ ⊆ interior t)⟩ := mem_nh... |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 229 | 232 | theorem oangle_eq_zero_or_eq_pi_iff_collinear {p₁ p₂ p₃ : P} :
∡ p₁ p₂ p₃ = 0 ∨ ∡ p₁ p₂ p₃ = π ↔ Collinear ℝ ({p₁, p₂, p₃} : Set P) := by |
rw [← not_iff_not, not_or, oangle_ne_zero_and_ne_pi_iff_affineIndependent,
affineIndependent_iff_not_collinear_set]
|
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureTheory.Integral.Bochner
#align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"e0736bb5b48bdadbca19dbd857e12bee38ccf... | Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 78 | 134 | theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
(s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}
(hε : ε ≠ 0) :
∃ f : α → E,
Continuous f ∧
snorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
... |
obtain ⟨η, η_pos, hη⟩ :
∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ ε :=
exists_snorm_indicator_le hp c hε
have ηpos : (0 : ℝ≥0∞) < η := ENNReal.coe_lt_coe.2 η_pos
obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α), V ⊇ s ∧ IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η :=
s_cl... |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 156 | 165 | theorem matPolyEquiv_symm_map_eval (M : (Matrix n n R)[X]) (r : R) :
(matPolyEquiv.symm M).map (eval r) = M.eval (scalar n r) := by |
suffices ((aeval r).mapMatrix.comp matPolyEquiv.symm.toAlgHom : (Matrix n n R)[X] →ₐ[R] _) =
(eval₂AlgHom' (AlgHom.id R _) (scalar n r)
fun x => (scalar_commute _ (Commute.all _) _).symm) from
DFunLike.congr_fun this M
ext : 1
· ext M : 1
simp [Function.comp]
· simp [smul_eq_diagonal_mul]... |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 126 | 135 | theorem card_edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t - 1 := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_adj ha, card_sdiff (by simp [ha]),
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_... |
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 441 | 446 | theorem isCompact_closure_singleton : IsCompact (closure {x}) := by |
refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi
|
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 52 | 54 | theorem mem_parallelepiped_iff (v : ι → E) (x : E) :
x ∈ parallelepiped v ↔ ∃ t ∈ Icc (0 : ι → ℝ) 1, x = ∑ i, t i • v i := by |
simp [parallelepiped, eq_comm]
|
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 183 | 186 | theorem toNat_add (hc : c < ℵ₀) (hd : d < ℵ₀) : toNat (c + d) = toNat c + toNat d := by |
lift c to ℕ using hc
lift d to ℕ using hd
norm_cast
|
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 | 155 | 159 | theorem ιSummand_comp_d_comp_πSummand_eq_zero (j k : ℕ) (A : IndexSet (op [j])) (hA : ¬A.EqId) :
(s.cofan _).inj A ≫ K[X].d j k ≫ s.πSummand (IndexSet.id (op [k])) = 0 := by |
rw [A.eqId_iff_mono] at hA
rw [← assoc, ← s.comp_PInfty_eq_zero_iff, assoc, ← PInfty.comm j k, s.cofan_inj_eq, assoc,
degeneracy_comp_PInfty_assoc X j A.e hA, zero_comp, comp_zero]
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 987 | 1,012 | theorem ContDiffWithinAt.hasFDerivWithinAt_nhds {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ}
{x₀ : E} (hf : ContDiffWithinAt 𝕜 (n + 1) (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : ContDiffWithinAt 𝕜 n g s x₀) (hgt : t ∈ 𝓝[g '' s] g x₀) :
∃ v ∈ 𝓝[insert x₀ s] x₀, v ⊆ insert x₀ s ∧ ∃ f' : E → F →L[... |
have hst : insert x₀ s ×ˢ t ∈ 𝓝[(fun x => (x, g x)) '' s] (x₀, g x₀) := by
refine nhdsWithin_mono _ ?_ (nhdsWithin_prod self_mem_nhdsWithin hgt)
simp_rw [image_subset_iff, mk_preimage_prod, preimage_id', subset_inter_iff, subset_insert,
true_and_iff, subset_preimage_image]
obtain ⟨v, hv, hvs, f', hv... |
import Mathlib.Logic.Function.CompTypeclasses
import Mathlib.Algebra.Group.Hom.Defs
section MonoidHomCompTriple
namespace MonoidHom
class CompTriple {M N P : Type*} [Monoid M] [Monoid N] [Monoid P]
(φ : M →* N) (ψ : N →* P) (χ : outParam (M →* P)) : Prop where
comp_eq : ψ.comp φ = χ
attribute [simp] C... | Mathlib/Algebra/Group/Hom/CompTypeclasses.lean | 98 | 106 | theorem comp_assoc {Q : Type*} [Monoid Q]
{φ₁ : M →* N} {φ₂ : N →* P} {φ₁₂ : M →* P}
(κ : CompTriple φ₁ φ₂ φ₁₂)
{φ₃ : P →* Q} {φ₂₃ : N →* Q} (κ' : CompTriple φ₂ φ₃ φ₂₃)
{φ₁₂₃ : M →* Q} :
CompTriple φ₁ φ₂₃ φ₁₂₃ ↔ CompTriple φ₁₂ φ₃ φ₁₂₃ := by |
constructor <;>
· rintro ⟨h⟩
exact ⟨by simp only [← κ.comp_eq, ← h, ← κ'.comp_eq, MonoidHom.comp_assoc]⟩
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 560 | 567 | theorem X_pow_dvd_iff {n : ℕ} {φ : R⟦X⟧} :
(X : R⟦X⟧) ^ n ∣ φ ↔ ∀ m, m < n → coeff R m φ = 0 := by |
convert@MvPowerSeries.X_pow_dvd_iff Unit R _ () n φ
constructor <;> intro h m hm
· rw [Finsupp.unique_single m]
convert h _ hm
· apply h
simpa only [Finsupp.single_eq_same] using hm
|
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 | 451 | 453 | theorem toNonUnitalNonAssocRing_injective :
Function.Injective (@toNonUnitalNonAssocRing R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
|
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α β γ : Type*}
namespace Prod.Lex
@[inherit_doc] notation:35 α " ×ₗ " β:34 => Lex (Prod α β)
instance decidableEq (α β : Type*) [DecidableEq α] [DecidableEq β] ... | Mathlib/Data/Prod/Lex.lean | 105 | 109 | theorem monotone_fst [Preorder α] [LE β] (t c : α ×ₗ β) (h : t ≤ c) :
(ofLex t).1 ≤ (ofLex c).1 := by |
cases (Prod.Lex.le_iff t c).mp h with
| inl h' => exact h'.le
| inr h' => exact h'.1.le
|
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 117 | 117 | theorem symm_symm_eq {X Y : C} (α : X ≅ Y) : α.symm.symm = α := by | cases α; rfl
|
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 | 90 | 92 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
|
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 88 | 109 | theorem select_add_select_not : ∀ x : 𝕎 R, select P x + select (fun i => ¬P i) x = x := by |
-- Porting note: TC search was insufficient to find this instance, even though all required
-- instances exist. See zulip: [https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/WittVector.20saga/near/370073526]
have : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x :=
... |
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Tactic.ClearExclamation
import Mathlib.Data.Nat.Cast.Basic
set_option autoImplicit true
namespace Mathlib
open Lean hidi... | Mathlib/Tactic/NormNum/Basic.lean | 125 | 126 | theorem isintCast {R} [Ring R] (n m : ℤ) :
IsInt n m → IsInt (n : R) m := by | rintro ⟨⟨⟩⟩; exact ⟨rfl⟩
|
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import field_theory.mv_polynomial from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open scoped Classical
... | Mathlib/FieldTheory/MvPolynomial.lean | 34 | 40 | theorem quotient_mk_comp_C_injective (I : Ideal (MvPolynomial σ K)) (hI : I ≠ ⊤) :
Function.Injective ((Ideal.Quotient.mk I).comp MvPolynomial.C) := by |
refine (injective_iff_map_eq_zero _).2 fun x hx => ?_
rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem] at hx
refine _root_.by_contradiction fun hx0 => absurd (I.eq_top_iff_one.2 ?_) hI
have := I.mul_mem_left (MvPolynomial.C x⁻¹) hx
rwa [← MvPolynomial.C.map_mul, inv_mul_cancel hx0, MvPolynomial.C_1] a... |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 53 | 56 | theorem FiniteDimensional.rank_linearMap :
Module.rank S (M →ₗ[R] N) = lift.{w} (Module.rank R M) * lift.{v} (Module.rank S N) := by |
rw [(linearMapEquivFun R S M N).rank_eq, rank_fun_eq_lift_mul,
← finrank_eq_card_chooseBasisIndex, ← finrank_eq_rank R, lift_natCast]
|
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 158 | 163 | theorem Filter.HasBasis.uniformEmbedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
UniformEmbedding f ↔ Injective f ∧
(∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x... |
rw [uniformEmbedding_iff, and_comm, h.uniformInducing_iff h']
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 640 | 653 | theorem findIdx_cons (p : α → Bool) (b : α) (l : List α) :
(b :: l).findIdx p = bif p b then 0 else (l.findIdx p) + 1 := by |
cases H : p b with
| true => simp [H, findIdx, findIdx.go]
| false => simp [H, findIdx, findIdx.go, findIdx_go_succ]
where
findIdx_go_succ (p : α → Bool) (l : List α) (n : Nat) :
List.findIdx.go p l (n + 1) = (findIdx.go p l n) + 1 := by
cases l with
| nil => unfold findIdx.go; exact Nat.succ_eq_... |
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
universe u v
namespace MvQPF
open TypeVec
open MvFunctor (LiftP LiftR)
open MvFunctor
var... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 245 | 258 | theorem Fix.ind_aux (a : q.P.A) (f' : q.P.drop.B a ⟹ α) (f : q.P.last.B a → q.P.W α) :
Fix.mk (abs ⟨a, q.P.appendContents f' fun x => ⟦f x⟧⟩) = ⟦q.P.wMk a f' f⟧ := by |
have : Fix.mk (abs ⟨a, q.P.appendContents f' fun x => ⟦f x⟧⟩) = ⟦wrepr (q.P.wMk a f' f)⟧ := by
apply Quot.sound; apply wEquiv.abs'
rw [MvPFunctor.wDest'_wMk', abs_map, abs_repr, ← abs_map, MvPFunctor.map_eq]
conv =>
rhs
rw [wrepr_wMk, q.P.wDest'_wMk', abs_repr, MvPFunctor.map_eq]
congr 2;... |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 197 | 198 | theorem preimage_add_const_Ico : (fun x => x + a) ⁻¹' Ico b c = Ico (b - a) (c - a) := by |
simp [← Ici_inter_Iio]
|
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 431 | 436 | theorem chromaticNumber_top_eq_top_of_infinite (V : Type*) [Infinite V] :
(⊤ : SimpleGraph V).chromaticNumber = ⊤ := by |
by_contra hc
rw [← Ne, chromaticNumber_ne_top_iff_exists] at hc
obtain ⟨n, ⟨hn⟩⟩ := hc
exact not_injective_infinite_finite _ hn.injective_of_top_hom
|
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f76"
universe u
section
open Decidable
variable {α : Type u} [LinearOrder α]
theorem min_def (a b : α) : min a b = if a ≤ b then a else b := by
rw [LinearOrder.min_def a]... | Mathlib/Init/Order/LinearOrder.lean | 130 | 130 | theorem max_self (a : α) : max a a = a := by | simp [max_def]
|
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
universe v u
namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where... | Mathlib/CategoryTheory/Monoidal/End.lean | 235 | 238 | theorem ε_inv_app_obj (n : M) (X : C) :
F.εIso.inv.app ((F.obj n).obj X) = (F.μ n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X := by |
rw [← cancel_mono (F.ε.app ((F.obj n).obj X)), ε_inv_hom_app]
simp
|
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 48 | 52 | theorem isQuadratic_χ₄ : χ₄.IsQuadratic := by |
intro a
-- Porting note (#11043): was `decide!`
fin_cases a
all_goals decide
|
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.continuants_recurrence from "leanprover-community/mathlib"@"5f11361a98ae4acd77f5c1837686f6f0102cdc25"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n : ℕ} [Division... | Mathlib/Algebra/ContinuedFractions/ContinuantsRecurrence.lean | 50 | 59 | theorem numerators_recurrence {gp : Pair K} {ppredA predA : K}
(succ_nth_s_eq : g.s.get? (n + 1) = some gp) (nth_num_eq : g.numerators n = ppredA)
(succ_nth_num_eq : g.numerators (n + 1) = predA) :
g.numerators (n + 2) = gp.b * predA + gp.a * ppredA := by |
obtain ⟨ppredConts, nth_conts_eq, ⟨rfl⟩⟩ : ∃ conts, g.continuants n = conts ∧ conts.a = ppredA :=
exists_conts_a_of_num nth_num_eq
obtain ⟨predConts, succ_nth_conts_eq, ⟨rfl⟩⟩ :
∃ conts, g.continuants (n + 1) = conts ∧ conts.a = predA :=
exists_conts_a_of_num succ_nth_num_eq
rw [num_eq_conts_a, con... |
import Mathlib.Data.Finset.Pointwise
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.DFinsupp.Order
import Mathlib.Order.Interval.Finset.Basic
#align_import data.dfinsupp.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open DFinsupp Finset
open Pointwise
vari... | Mathlib/Data/DFinsupp/Interval.lean | 48 | 58 | theorem mem_dfinsupp_iff : f ∈ s.dfinsupp t ↔ f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i := by |
refine mem_map.trans ⟨?_, ?_⟩
· rintro ⟨f, hf, rfl⟩
rw [Function.Embedding.coeFn_mk] -- Porting note: added to avoid heartbeat timeout
refine ⟨support_mk_subset, fun i hi => ?_⟩
convert mem_pi.1 hf i hi
exact mk_of_mem hi
· refine fun h => ⟨fun i _ => f i, mem_pi.2 h.2, ?_⟩
ext i
dsimp
... |
import Mathlib.MeasureTheory.Measure.AEDisjoint
import Mathlib.MeasureTheory.Constructions.EventuallyMeasurable
#align_import measure_theory.measure.null_measurable from "leanprover-community/mathlib"@"e4edb23029fff178210b9945dcb77d293f001e1c"
open Filter Set Encodable
variable {ι α β γ : Type*}
namespace Measu... | Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 221 | 227 | theorem exists_measurable_superset_ae_eq (h : NullMeasurableSet s μ) :
∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s := by |
rcases h with ⟨t, htm, hst⟩
refine ⟨t ∪ toMeasurable μ (s \ t), ?_, htm.union (measurableSet_toMeasurable _ _), ?_⟩
· exact diff_subset_iff.1 (subset_toMeasurable _ _)
· have : toMeasurable μ (s \ t) =ᵐ[μ] (∅ : Set α) := by simp [ae_le_set.1 hst.le]
simpa only [union_empty] using hst.symm.union this
|
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 91 | 102 | theorem integrableOn_rpow_mul_exp_neg_mul_rpow {p s b : ℝ} (hs : -1 < s) (hp : 1 ≤ p) (hb : 0 < b) :
IntegrableOn (fun x : ℝ => x ^ s * exp (- b * x ^ p)) (Ioi 0) := by |
have hib : 0 < b ^ (-p⁻¹) := rpow_pos_of_pos hb _
suffices IntegrableOn (fun x ↦ (b ^ (-p⁻¹)) ^ s * (x ^ s * exp (-x ^ p))) (Ioi 0) by
rw [show 0 = b ^ (-p⁻¹) * 0 by rw [mul_zero], ← integrableOn_Ioi_comp_mul_left_iff _ _ hib]
refine this.congr_fun (fun _ hx => ?_) measurableSet_Ioi
rw [← mul_assoc, mu... |
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 132 | 148 | theorem isJacobson_of_isIntegral [Algebra R S] [Algebra.IsIntegral R S] (hR : IsJacobson R) :
IsJacobson S := by |
rw [isJacobson_iff_prime_eq]
intro P hP
by_cases hP_top : comap (algebraMap R S) P = ⊤
· simp [comap_eq_top_iff.1 hP_top]
· haveI : Nontrivial (R ⧸ comap (algebraMap R S) P) := Quotient.nontrivial hP_top
rw [jacobson_eq_iff_jacobson_quotient_eq_bot]
refine eq_bot_of_comap_eq_bot (R := R ⧸ comap (alge... |
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 | 1,458 | 1,461 | theorem bsup_eq_sup' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → Ordinal.{max u v}) :
bsup.{_, v} _ (bfamilyOfFamily' r f) = sup.{_, v} f := by |
simp (config := { unfoldPartialApp := true }) only [← sup_eq_bsup' r, enum_typein,
familyOfBFamily', bfamilyOfFamily']
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 56 | 58 | theorem Ici_mul_Ici_subset' (a b : α) : Ici a * Ici b ⊆ Ici (a * b) := by |
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_le_mul' hya hzb
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 168 | 168 | theorem arsinh_nonpos_iff : arsinh x ≤ 0 ↔ x ≤ 0 := by | rw [← sinh_le_sinh, sinh_zero, sinh_arsinh]
|
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 76 | 78 | theorem inv : t⁻¹ = t := by |
rcases ht with ⟨w, i, rfl⟩
simp [mul_assoc]
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 169 | 172 | theorem simpleFunc_bot {α} (f : @SimpleFunc α ⊥ β) [Nonempty β] : ∃ c, ∀ x, f x = c := by |
have hf_meas := @SimpleFunc.measurableSet_fiber α _ ⊥ f
simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas
exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc ↦ congr_fun hc
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 217 | 221 | theorem IndepFun.exp_mul {X Y : Ω → ℝ} (h_indep : IndepFun X Y μ) (s t : ℝ) :
IndepFun (fun ω => exp (s * X ω)) (fun ω => exp (t * Y ω)) μ := by |
have h_meas : ∀ t, Measurable fun x => exp (t * x) := fun t => (measurable_id'.const_mul t).exp
change IndepFun ((fun x => exp (s * x)) ∘ X) ((fun x => exp (t * x)) ∘ Y) μ
exact IndepFun.comp h_indep (h_meas s) (h_meas t)
|
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