Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | theorem ofReal_intCast (n : ℤ) : ((n : ℝ) : K) = n :=
map_intCast _ n
@[simp, rclike_simps] -- Porting note: removed `norm_cast`
| Mathlib/Analysis/RCLike/Basic.lean | 569 | 572 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... | variable [MulZeroOneClass M₀] {s t : Set ι} {i : ι}
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 58 | 59 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... |
@[to_additive]
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 120 | 121 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | Mathlib/Data/Nat/Factorial/Basic.lean | 463 | 465 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... |
theorem exists_clusterPt_of_compactSpace [CompactSpace X] (f : Filter X) [NeBot f] :
∃ x, ClusterPt x f := by
simpa using isCompact_univ (show f ≤ 𝓟 univ by simp)
nonrec theorem Ultrafilter.le_nhds_lim [CompactSpace X] (F : Ultrafilter X) : ↑F ≤ 𝓝 F.lim := by
rcases isCompact_univ.ultrafilter_le_nhds F (by ... | Mathlib/Topology/Compactness/Compact.lean | 676 | 682 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... |
end LinearOrderedField
/-!
#### Convex sets in an ordered space
| Mathlib/Analysis/Convex/Strict.lean | 375 | 380 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 301 | 309 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | · let a := toOrdinal (k + ∗y)
use toLeftMovesAdd (Sum.inl (toLeftMovesNim ⟨a, h⟩))
simp [a, grundyValue_nim_add_nim a y]
· let a := toOrdinal (k + ∗x)
use toLeftMovesAdd (Sum.inr (toLeftMovesNim ⟨a, h⟩))
simp [a, grundyValue_nim_add_nim x a, add_comm (∗x)]
termination_by (x, y)
theorem ... | Mathlib/SetTheory/Game/Nim.lean | 346 | 354 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | by_cases hq0 : q = 0
· simp [hq0]
· rw [mod_def, mod_def, leadingCoeff_map f, ← map_inv₀ f, ← map_C f, ← Polynomial.map_mul f,
map_modByMonic f (monic_mul_leadingCoeff_inv hq0)]
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 403 | 407 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.Set.Function
/-!
# Fixed points of a self-map
In this file we define
* the predicate `IsFixedPt f x := f x =... | Mathlib/Dynamics/FixedPoints/Basic.lean | 199 | 201 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | theorem Ici_bot : Ici (⊥ : α) = univ :=
isBot_bot.Ici_eq
| Mathlib/Order/Interval/Set/Basic.lean | 887 | 889 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | use s ^ natDegree (a * p) • s ^ (natDegree b + natDegree q - natDegree (b * q)) • b.scaleRoots r
simp only [s, smul_mul_assoc, ← mul_scaleRoots, smul_smul, Units.smul_def, mul_assoc,
← mul_pow, IsUnit.val_inv_mul, one_pow, mul_one, ← smul_add, one_smul, e, natDegree_one,
one_scaleRoots, ← add_scaleRoots_of_... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 251 | 255 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
@[nontriviality, simp]
lemma iIndepSets.of_subsingleton [Subsingleton ι] {m : ι → Set (Set Ω)} [IsProbabilityMeasure μ] :
iIndepSets m μ := Kernel.iIndepSets.of_subsingleton
| Mathlib/Probability/Independence/Basic.lean | 217 | 220 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.Set.Prod
/-!
# Finite products of types
This file defines the product of types over a list. For `l : List ι` ... | @[simp]
protected def elim' (h : ∀ i, i ∈ l) (v : TProd α l) (i : ι) : α i :=
v.elim (h i)
theorem mk_elim (hnd : l.Nodup) (h : ∀ i, i ∈ l) (v : TProd α l) : TProd.mk l (v.elim' h) = v :=
TProd.ext hnd fun i hi => by simp [elim_mk]
/-- Pi-types are equivalent to iterated products. -/
def piEquivTProd (hnd : l.Nod... | Mathlib/Data/Prod/TProd.lean | 115 | 123 |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... | le_sup_left := AEEqFun.le_sup_left
le_sup_right := AEEqFun.le_sup_right
sup_le := AEEqFun.sup_le
inf := min
inf_le_left := AEEqFun.inf_le_left
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 554 | 558 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... |
lemma IsBasis.map {X : Set α} (hIX : M.IsBasis I X) {f : α → β} (hf) :
(M.map f hf).IsBasis (f '' I) (f '' X) := by
refine (hIX.indep.map f hf).isBasis_of_forall_insert (image_subset _ hIX.subset) ?_
rintro _ ⟨⟨e,he,rfl⟩, he'⟩
have hss := insert_subset (hIX.subset_ground he) hIX.indep.subset_ground
rw [← n... | Mathlib/Data/Matroid/Map.lean | 408 | 418 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | obtain ⟨x, hx⟩ := hcard
obtain (rfl : f = x) := hx.subset ⟨hfB, hf⟩
simp_rw [← h, ← singleton_union, ← hx, sdiff_sdiff_right_self, inf_eq_inter, inter_comm B,
diff_union_inter]
exact hB'
| Mathlib/Data/Matroid/Basic.lean | 652 | 657 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | exact h.memLp_snd hf
theorem integrable_iff [NormedAddCommGroup γ] [BorelSpace γ] (h : IdentDistrib f g μ ν) :
Integrable f μ ↔ Integrable g ν :=
⟨fun hf => h.integrable_snd hf, fun hg => h.symm.integrable_snd hg⟩
| Mathlib/Probability/IdentDistrib.lean | 224 | 228 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | @[simp]
lemma symm_add : φ.symm.add = φ.add := by
dsimp [add, symm]
congr 1
apply add_comm
| Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 62 | 66 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
/-!
# Topological star (sub)algebras
A topological... | eq_induced := rfl
injective := Subtype.coe_injective
isClosed_range := by simpa using isClosed R x
@[deprecated (since := "2024-11-05")]
alias _root_.elementalStarAlgebra.isClosedEmbedding_coe := isClosedEmbedding_coe
@[elab_as_elim]
theorem induction_on {x y : A}
(hy : y ∈ elemental R x) {P : (u : A) → u ∈ ... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 234 | 243 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Data.Complex.ExponentialBounds
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.SpecialF... | lemma strictAnti_eulerMascheroniSeq' : StrictAnti eulerMascheroniSeq' := by
refine strictAnti_nat_of_succ_lt (fun n ↦ ?_)
rcases Nat.eq_zero_or_pos n with rfl | hn
· simp [eulerMascheroniSeq']
simp_rw [eulerMascheroniSeq', eq_false_intro hn.ne', reduceCtorEq, if_false]
rw [← sub_pos, sub_sub_sub_comm,
har... | Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | 83 | 95 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Scott Carnahan
-/
import Mathlib.LinearAlgebra.PerfectPairing.Basic
import Mathlib.LinearAlgebra.Reflection
/-!
# Root data and root systems
This file con... | Module.reflection (P.root_coroot_two i)
@[simp]
lemma coroot_reflection_perm (j : ι) :
P.coroot (P.reflection_perm i j) = (P.coreflection i) (P.coroot j) :=
(P.reflection_perm_coroot i j).symm
theorem mapsTo_coreflection_coroot :
| Mathlib/LinearAlgebra/RootSystem/Defs.lean | 297 | 304 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | rw [ENNReal.coe_toNNReal]
exact measure_ne_top _ _
lemma tendsto_eLpNorm_one_densityProcess_limitProcess (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
(a : α) {s : Set β} (hs : MeasurableSet s) :
Tendsto (fun n ↦ eLpNorm ((fun x ↦ densityProcess κ ν n a x s)
- (countableFiltration γ).limitProcess (fun n x ↦ d... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 392 | 398 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | @[simp]
lemma corootForm_rootSystem_eq_killing :
(rootSystem H).CorootForm = (killingForm K L).restrict H := by
rw [restrict_killingForm_eq_sum, RootPairing.CorootForm, ← Finset.sum_coe_sort (s := H.root)]
rfl
@[simp] lemma rootSystem_toPerfectPairing_apply (f x) : (rootSystem H).toPerfectPairing f x = f x :=
... | Mathlib/Algebra/Lie/Weights/RootSystem.lean | 398 | 406 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | fun a x ↦ ⨅ q : { q' : ℚ // x < q' }, f a q
lemma IsMeasurableRatCDF.stieltjesFunctionAux_def' (f : α → ℚ → ℝ) (a : α) :
IsMeasurableRatCDF.stieltjesFunctionAux f a
= fun (t : ℝ) ↦ ⨅ r : { r' : ℚ // t < r' }, f a r := by
ext t; exact IsMeasurableRatCDF.stieltjesFunctionAux_def f a t
lemma IsMeasurableRa... | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 274 | 284 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | · rw [← mul_sum]
| Mathlib/Data/Complex/Exponential.lean | 422 | 422 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | have ha : a ∉ Ico (a + 1) b := by simp
rw [← prod_insert ha, Nat.Ico_insert_succ_left hab]
@[to_additive]
| Mathlib/Algebra/BigOperators/Intervals.lean | 118 | 121 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Basic
/-!
# Teichmüller lifts
This file defines `WittVector.teichmuller`, a monoid hom `R →* 𝕎 R`, which embeds `r : R` as the... | map_teichmullerFun _ _ _
| Mathlib/RingTheory/WittVector/Teichmuller.lean | 116 | 117 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 924 | 927 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... |
theorem range_vecMulLinear (M : Matrix m n R) :
LinearMap.range M.vecMulLinear = span R (range M.row) := by
letI := Classical.decEq m
simp_rw [range_eq_map, ← iSup_range_single, Submodule.map_iSup, range_eq_map, ←
Ideal.span_singleton_one, Ideal.span, Submodule.map_span, image_image, image_singleton,
M... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 91 | 99 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | end Real
section Continuity
open Real
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 423 | 427 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Function.Injective (liftAlgHom φ hφ') :=
liftMonoidWithZeroHom_injective _ hφ
@[simp]
theorem liftAlgHom_apply_div' (p q : K[X]) :
liftAlgHom φ hφ (algebraMap _ _ p) / liftAlgHom φ hφ (algebraMap _ _ q) = φ p / φ q :=
liftMonoidWithZeroHom_apply_div' _ hφ _ _
theorem liftAlgHom_apply_div (p q : K[X]) :
... | Mathlib/FieldTheory/RatFunc/Basic.lean | 636 | 646 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y / x‖ < ε } := by
simp_rw [← ball_eq']
| Mathlib/Analysis/Normed/Group/Basic.lean | 662 | 663 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angle between complex numbers
This file rel... | Mathlib/Analysis/Complex/Angle.lean | 47 | 47 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 577 | 580 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
/-!
# Casts of rational numbers in... | @[simp, norm_cast]
| Mathlib/Data/Rat/Cast/Order.lean | 59 | 59 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | -- Porting note: is there a better way compared to the old `congr_arg coe h`?
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsCoprime.prod_right_iff.mp (hp i hi)
exact Finset.mem_sdiff.mpr ⟨hj, fun f ↦ h <| Subtype.ext (Finset.mem_singleton.mp f).symm⟩
| Mathlib/RingTheory/Coprime/Lemmas.lean | 178 | 181 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Data.Nat.Log
/-!
# Integer logarithms in a field with respect to a natural base
This file defines two `ℤ`-value... | · rw [log_of_one_le_right _ hr1]
rw [zpow_natCast, ← Nat.cast_pow, ← Nat.le_floor_iff hr.le]
exact Nat.pow_log_le_self b (Nat.floor_pos.mpr hr1).ne'
· rw [log_of_right_le_one _ hr1, zpow_neg, zpow_natCast, ← Nat.cast_pow]
| Mathlib/Data/Int/Log.lean | 93 | 96 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... |
/-- Sending `Fin (n+1)` to `Fin n` by subtracting one from anything above `p`
| Mathlib/Data/Fin/Basic.lean | 1,277 | 1,278 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probability measures
This file i... | lemma map_prod_map {α' : Type*} [MeasurableSpace α'] {β' : Type*} [MeasurableSpace β']
{f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) :
(μ.map f).prod (ν.map g) = (μ.prod ν).map (Prod.map f g) := by
apply Subtype.ext
simp only [val_eq_toMeasure, toMeasure_prod, toMeasure_map]
rw... | Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 73 | 78 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | (hπ : IsStoppingTime f π) (s : Set Ω) :
MeasurableSet[hτ.measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) ↔
MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
constructor <;> intro h
· have : s ∩ {ω | τ ω ≤ π ω} = s ∩ {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ π ω} := by
rw [Set.inter_assoc, Set.int... | Mathlib/Probability/Process/Stopping.lean | 586 | 598 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | Mathlib/RingTheory/WittVector/Basic.lean | 105 | 105 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,858 | 1,862 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 833 | 834 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | sup_inf_le_assoc_of_le _ _ := by
simp only [← toSubmodule_le_toSubmodule, sup_toSubmodule, inf_toSubmodule]
exact IsModularLattice.sup_inf_le_assoc_of_le _
| Mathlib/Algebra/Lie/Submodule.lean | 565 | 567 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | exact (Nat.findGreatest_spec (P := fun b => f n = b) (hg n) rfl).symm
-- We show that division is primitive recursive by showing that the graph is
| Mathlib/Computability/Primrec.lean | 678 | 680 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.Lebesgu... | simpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)
apply le_trans _ (f.measure_closedBall_le_integral μ)
rw [div_le_iff₀ (pow_pos K_pos _), addHaar_real_closedBall' _ _ f.rIn_pos.le,
addHaar_real_closedBall' _ _ f.rOut_pos.le, mul_assoc, mul_comm _ (K ^ _), ← mul_assoc,
... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 125 | 133 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Subalgebras over Comm... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,072 | 1,072 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAl... |
theorem cardinalMk_le_finrank [Module.Finite R M]
{ι : Type w} {b : ι → M} (h : LinearIndependent R b) : #ι ≤ finrank R M := by
rw [← lift_le.{max v w}]
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 173 | 176 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Module.Defs
/-!
# Ordered algebras
This file proves properties of algebras where both rings are ordered... | lemma algebraMap_pos (ha : 0 < a) : 0 < algebraMap α β a := by
simpa using algebraMap_strictMono β ha
| Mathlib/Algebra/Order/Module/Algebra.lean | 62 | 63 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | mul_assoc _ _ _ := rfl
one_mul _ := ext fun _ => rfl
| Mathlib/Topology/MetricSpace/Isometry.lean | 472 | 473 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,401 | 2,403 | |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... | @[simp] theorem quot_sub (a b : PGame) : ⟦a - b⟧ = (⟦a⟧ : Game) - ⟦b⟧ := rfl
@[simp]
| Mathlib/SetTheory/Game/Basic.lean | 225 | 227 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | LeftHomologyData.leftHomologyIso, RightHomologyData.rightHomologyIso,
homologyMap, homologyMap']
simp only [assoc, rightHomologyι_naturality', rightHomologyι_naturality'_assoc,
leftHomologyπ_naturality'_assoc, HomologyData.comm_assoc, p_opcyclesMap'_assoc,
id_τ₂, p_opcyclesMap', id_comp, cyclesMap'_i_... | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 997 | 1,010 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 999 | 1,002 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... | rwa [← ((hs _ hq).dvd_iff_eq hp.ne_one).1 hpq]
lemma primeFactors_div_gcd (hm : Squarefree m) (hn : n ≠ 0) :
primeFactors (m / m.gcd n) = primeFactors m \ primeFactors n := by
ext p
| Mathlib/Data/Nat/Squarefree.lean | 373 | 377 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,174 | 1,186 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finite.Card
import Mathlib.Data.Set.Finite.Range
/-!
# ... |
@[to_additive]
| Mathlib/Algebra/Group/Subgroup/Finite.lean | 171 | 172 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... |
/-- Given a subgraph of a subgraph of `G`, construct a subgraph of `G`. -/
protected abbrev coeSubgraph {G' : G.Subgraph} : G'.coe.Subgraph → G.Subgraph :=
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 975 | 977 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | · simp
· simp; ring
end bernsteinPolynomial
| Mathlib/RingTheory/Polynomial/Bernstein.lean | 384 | 410 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.RingTheory.Finiteness.Nakayama
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheor... | Mathlib/RingTheory/Filtration.lean | 493 | 502 | |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | theorem coe_image_Icc_eq : ((↑) : 𝕜 → AddCircle p) '' Icc a (a + p) = univ :=
eq_top_mono (image_subset _ Ico_subset_Icc_self) <| coe_image_Ico_eq _ _
end LinearOrderedAddCommGroup
| Mathlib/Topology/Instances/AddCircle.lean | 304 | 308 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
/-!
# Tri... | rw [← top_le_iff, ← Submodule.eq_top_of_disjoint ER ES h_dim_add.ge h_disjoint]
apply sup_le hER hES
end Module.End
namespace Submodule
variable {p : Submodule K V} {f : Module.End K V}
theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈ p) (k : ℕ∞) :
p ⊓ ⨆ μ, f.genEigenspace μ k ... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 132 | 192 |
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
/-!
# Adjunction between `Γ` and ... | def toΓSpecBase : X.toTopCat ⟶ Spec.topObj (Γ.obj (op X)) :=
TopCat.ofHom
{ toFun := X.toΓSpecFun
continuous_toFun := X.toΓSpec_continuous }
| Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 91 | 95 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | · calc
∫ x in Ι a b, |x - a| ^ n = ∫ x in b..a, |x - a| ^ n := by
rw [uIoc_of_ge hab.le, ← integral_of_le hab.le]
_ = ∫ x in b - a..0, (-x) ^ n := by
simp only [integral_comp_sub_right fun x => |x| ^ n, sub_self]
refine integral_congr fun x hx => congr_arg₂ Pow.pow (abs_of_nonpos <... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 417 | 437 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,752 | 1,754 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... |
lemma divInt_one_one : 1 /. 1 = 1 := by rw [divInt_one, intCast_one]
| Mathlib/Data/Rat/Defs.lean | 250 | 251 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,381 | 1,395 | |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Ring.Subsemiring.MulOpposite
import Mathlib.Algebra.Ring.Subring.Basic
/-!
# Subring of opposite rings
For every ring `R`, we construct an equivalence betwe... |
theorem unop_iSup (S : ι → Subring Rᵐᵒᵖ) : (iSup S).unop = ⨆ i, (S i).unop :=
opEquiv.symm.map_iSup _
| Mathlib/Algebra/Ring/Subring/MulOpposite.lean | 126 | 129 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | @[simp, mfld_simps]
theorem restr_source (s : Set α) : (e.restr s).source = e.source ∩ s :=
rfl
| Mathlib/Logic/Equiv/PartialEquiv.lean | 486 | 488 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
/-!
# Universal enveloping algebra
Given a commu... | Mathlib/Algebra/Lie/UniversalEnveloping.lean | 164 | 170 | |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 1,262 | 1,264 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.List.NatAntidiagonal
import Mathlib.Data.Multiset.MapFold
/-!
# Antidiagonals in ℕ × ℕ as multisets
This file defines the antidiagonals of ℕ × ℕ... | antidiagonal (n + 2) =
(0, n + 2) ::ₘ (n + 2, 0) ::ₘ (antidiagonal n).map (Prod.map Nat.succ Nat.succ) := by
rw [antidiagonal_succ, antidiagonal_succ', map_cons, map_map, Prod.map_apply]
rfl
| Mathlib/Data/Multiset/NatAntidiagonal.lean | 64 | 67 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | refine iSup_mono fun y ↦ LieSubmodule.map_le_iff_le_comap.mpr fun m hm ↦ ?_
simp only [mem_posFittingCompOf] at hm
simp only [LieSubmodule.mem_comap, mem_posFittingCompOf]
intro k
obtain ⟨n, hn⟩ := hm k
use f n
rw [LieModule.toEnd_pow_apply_map, hn]
lemma map_genWeightSpace_le :
(genWeightSpace M χ).... | Mathlib/Algebra/Lie/Weights/Basic.lean | 505 | 519 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | theorem le_cof_iff_blsub {b : Ordinal} {a : Cardinal} :
a ≤ cof b ↔ ∀ {o} (f : ∀ a < o, Ordinal), blsub.{u, u} o f = b → a ≤ o.card :=
le_cof_iff_lsub.trans
⟨fun H o f hf => by simpa using H _ hf, fun H ι f hf => by
rcases Cardinal.ord_eq ι with ⟨r, hr, hι'⟩
rw [← @blsub_eq_lsub' ι r hr] at hf
... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 317 | 324 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
/-!
# Continu... | exact continuous_inv.continuousAt.comp (continuousAt_rpow_const_of_pos z_pos)
theorem tendsto_const_mul_rpow_nhds_zero_of_pos {c : ℝ≥0∞} (hc : c ≠ ∞) {y : ℝ} (hy : 0 < y) :
Tendsto (fun x : ℝ≥0∞ => c * x ^ y) (𝓝 0) (𝓝 0) := by
convert ENNReal.Tendsto.const_mul (ENNReal.continuous_rpow_const.tendsto 0) _
... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 475 | 483 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | apply (le_div_iff_of_nonzero hI).mpr _
intro y hy x hx
rw [mul_comm]
exact mul_mem_mul hy hx
theorem mul_div_self_cancel_iff {I : FractionalIdeal R₁⁰ K} : I * (1 / I) = 1 ↔ ∃ J, I * J = 1 :=
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 455 | 460 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,421 | 1,429 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Andrew Yang, Yuyang Zhao
-/
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# The... | theorem support_integralNormalization_subset :
(integralNormalization p).support ⊆ p.support := by
intro
simp +contextual [sum_def, integralNormalization, coeff_monomial, mem_support_iff]
| Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 56 | 59 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 801 | 802 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | Mathlib/Data/TypeVec.lean | 797 | 800 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection.Basic
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own ... |
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) :
| Mathlib/Order/Heyting/Regular.lean | 70 | 71 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | ```
f g h
X ───> Y ───> Z ───> X⟦1⟧
```
the composition `g ≫ h = 0`. -/
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 126 | 130 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | refine (mdifferentiableAt_id.mfderiv_prod mdifferentiableAt_const).trans ?_
rw [mfderiv_id, mfderiv_const, ContinuousLinearMap.inl]
| Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 495 | 497 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | /-- The lattice formed by the image of the logarithmic embedding. -/
noncomputable def _root_.NumberField.Units.unitLattice :
Submodule ℤ (logSpace K) :=
Submodule.map (logEmbedding K).toIntLinearMap ⊤
open scoped Classical in
theorem unitLattice_inter_ball_finite (r : ℝ) :
((unitLattice K : Set (logSpace K)... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 160 | 179 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | (hf' : ∀ x ∈ f.source, HasFDerivAt f (f' x) x) (g : E → F) :
∫ x in f.target, g x ∂μ = ∫ x in f.source, |(f' x).det| • g (f x) ∂μ := by
have : f '' f.source = f.target := PartialEquiv.image_source_eq_target f.toPartialEquiv
rw [← this]
apply integral_image_eq_integral_abs_det_fderiv_smul μ f.open_source.m... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,207 | 1,215 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Topology.Neighborhoods
/-!
# Neighborhoods of a set
In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all ne... | lemma Filter.HasBasis.nhdsSet_interior {ι : Sort*} {p : ι → Prop} {s : ι → Set X} {t : Set X}
(h : (𝓝ˢ t).HasBasis p s) : (𝓝ˢ t).HasBasis p (interior <| s ·) :=
| Mathlib/Topology/NhdsSet.lean | 78 | 79 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | HasSum (fun n ↦ (n + k).choose k * r ^ n) (1 / (1 - r) ^ (k + 1)) := by
convert hasSum_choose_mul_geometric_of_norm_lt_one' k hr
simp
| Mathlib/Analysis/SpecificLimits/Normed.lean | 426 | 428 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | e.preimageHomeomorph_apply subset_rfl ⟨p, e.mem_source.mp p.2⟩
/-- Auxiliary definition to avoid looping in `dsimp`
| Mathlib/Topology/FiberBundle/Trivialization.lean | 439 | 441 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,440 | 1,441 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | Mathlib/Order/SupClosed.lean | 44 | 44 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCofiber
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Algebra.Homology.QuasiIso
import Mathlib.CategoryTheory.Localiz... | lemma quasiIso_eq_quasiIso_map_quotient :
quasiIso C c = (HomologicalComplex.quasiIso C c).map (quotient C c) := by
ext ⟨K⟩ ⟨L⟩ f
obtain ⟨f, rfl⟩ := (HomotopyCategory.quotient C c).map_surjective f
constructor
· intro hf
rw [quotient_map_mem_quasiIso_iff] at hf
exact MorphismProperty.map_mem_map _ _... | Mathlib/Algebra/Homology/Localization.lean | 126 | 136 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | ContinuousMultilinearMap 𝕜 E₁ G ≃L[𝕜] ContinuousMultilinearMap 𝕜 E G :=
{ compContinuousLinearMapL fun i => (f i : E i →L[𝕜] E₁ i) with
invFun := compContinuousLinearMapL fun i => ((f i).symm : E₁ i →L[𝕜] E i)
continuous_toFun := (compContinuousLinearMapL fun i => (f i : E i →L[𝕜] E₁ i)).continuous
... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 1,152 | 1,164 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | @[deprecated (since := "2025-01-02")]
alias ContinuousOn.fin_insertNth := ContinuousOn.finInsertNth
end Fin
theorem Set.LeftInvOn.map_nhdsWithin_eq {f : α → β} {g : β → α} {x : β} {s : Set β}
(h : LeftInvOn f g s) (hx : f (g x) = x) (hf : ContinuousWithinAt f (g '' s) (g x))
| Mathlib/Topology/ContinuousOn.lean | 1,266 | 1,272 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | degree (derivative p) = (natDegree p - 1 : ℕ) := by
apply le_antisymm
· rw [derivative_apply]
| Mathlib/Algebra/Polynomial/Derivative.lean | 307 | 309 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.Basic
/-!
# Functors which commute with shifts
Let `C` and `D` be two categories equipped with shifts by an additive monoid `A`. In this ... | shiftFunctor C (a + b) ⋙ F ≅ F ⋙ shiftFunctor D (a + b) :=
CommShift.isoAdd' rfl e₁ e₂
@[simp]
lemma isoAdd_hom_app {a b : A}
(e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a)
(e₂ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b) (X : C) :
(CommShift.isoAdd e₁ e₂).hom.app X =
| Mathlib/CategoryTheory/Shift/CommShift.lean | 81 | 88 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 527 | 531 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.LocalExtr.Basic
/-!
# Darboux's theorem
In ... | Mathlib/Analysis/Calculus/Darboux.lean | 148 | 154 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Powerset
/-!
# The antidiagonal on a multiset.
The antidiagonal of a multiset `s` consists of all pairs `(t₁, t₂)`
such that `t₁ + t₂ =... | List.mem_map, mem_powersetAux]
obtain ⟨x₁, x₂⟩ := x
exact ⟨x₁, le_add_right _ _, by rw [add_tsub_cancel_left x₁ x₂]⟩
@[simp]
theorem antidiagonal_map_fst (s : Multiset α) : (antidiagonal s).map Prod.fst = powerset s :=
Quotient.inductionOn s fun l ↦ by simp [powersetAux']
@[simp]
theorem antidiagonal_... | Mathlib/Data/Multiset/Antidiagonal.lean | 48 | 57 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.Algebra.Homology.TotalComplex
/-!
# Behaviour of the total complex with respect to shifts
There are two ... |
variable {K L} in
@[reassoc]
lemma totalShift₂Iso_hom_naturality [L.HasTotal (up ℤ)] :
total.map ((shiftFunctor₂ C y).map f) (up ℤ) ≫ (L.totalShift₂Iso y).hom =
(K.totalShift₂Iso y).hom ≫ (total.map f (up ℤ))⟦y⟧' := by
ext n i₁ i₂ h
dsimp at h ⊢
rw [ιTotal_map_assoc, L.ι_totalShift₂Iso_hom_f y i₁ i₂ n ... | Mathlib/Algebra/Homology/TotalComplexShift.lean | 331 | 342 |
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