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 |
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/-
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.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | alias mem_tangentCone_of_openSegment_subset := mem_tangentConeAt_of_openSegment_subset
/-- If a subset of a real vector space contains a segment, then the direction of this
segment belongs to the tangent cone at its endpoints. -/
theorem mem_tangentConeAt_of_segment_subset {s : Set G} {x y : G} (h : segment ℝ x y ⊆ s)... | Mathlib/Analysis/Calculus/TangentCone.lean | 255 | 259 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | lt_of_le_of_lt (pos_of_mem_properDivisors h) (mem_properDivisors.1 h).2
lemma one_lt_div_of_mem_properDivisors {m n : ℕ} (h : m ∈ n.properDivisors) :
| Mathlib/NumberTheory/Divisors.lean | 275 | 277 |
/-
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.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.RingHom.Locally
/-!
# Properties of morp... | (Q : outParam (∀ {R S : Type u} [CommRing R] [CommRing S], (R →+* S) → Prop)) : Prop where
isLocal_ringHomProperty : RingHom.PropertyIsLocal Q
eq_affineLocally' : P = affineLocally Q
namespace HasRingHomProperty
variable (P : MorphismProperty Scheme.{u}) {Q} [HasRingHomProperty P Q]
variable {X Y Z : Scheme.{... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 249 | 267 |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.LinearAlgebra.Aff... |
theorem lineMap_le_left_iff_le (h : 0 < r) : lineMap a b r ≤ a ↔ b ≤ a :=
left_le_lineMap_iff_le (E := Eᵒᵈ) h
| Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 127 | 130 |
/-
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 ... | (f₂ : β →ₛ γ) {y : β} (h : ¬∃ x, g x = y) : (f₁.extend g hg f₂) y = f₂ y :=
Function.extend_apply' _ _ _ h
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 317 | 318 |
/-
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.Algebra.RestrictScalars
import Mathlib.Algebra.Lie.TensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.TensorPro... | · simp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]
· intro a l
simp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]
· intro z₁ z₂ h₁ h₂
suffices bracket' R A L L z₁ z₂ + bracket' R A L L z₂ z₁ = 0 by
rw [LinearMap.map_add, LinearMap.map_add, LinearMap.... | Mathlib/Algebra/Lie/BaseChange.lean | 63 | 83 |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Amelia Livingston, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ImageToKernel... | (imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.assoc, image.fac, w, zero_comp])).unop ≫
(kernelSubobjectIso _ ≪≫ kernelUnopUnop _).inv := by
ext
dsimp only [imageUnopUnop]
simp only [Iso.t... | Mathlib/Algebra/Homology/Opposite.lean | 53 | 63 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | @[simp]
theorem pointsWithCircumcenter_eq_circumcenter {n : ℕ} (s : Simplex ℝ P n) :
s.pointsWithCircumcenter circumcenterIndex = s.circumcenter :=
rfl
/-- The weights for a single vertex of a simplex, in terms of
`pointsWithCircumcenter`. -/
def pointWeightsWithCircumcenter {n : ℕ} (i : Fin (n + 1)) : PointsWit... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 461 | 474 |
/-
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.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | Finite.measurableSet hs
| Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 315 | 316 |
/-
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
-/
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Data.FunLike.Fintype
import Mathlib.Logic.Embedding.Set
/-!
# Maps between graphs
This file ... | abbrev refl : G ↪g G :=
RelEmbedding.refl _
/-- An embedding of graphs gives rise to a homomorphism of graphs. -/
| Mathlib/Combinatorics/SimpleGraph/Maps.lean | 327 | 330 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... |
protected theorem orthogonalProjection_eq_sum {U : Submodule 𝕜 E} [CompleteSpace U]
(b : OrthonormalBasis ι 𝕜 U) (x : E) :
U.orthogonalProjection x = ∑ i, ⟪(b i : E), x⟫ • b i := by
simpa only [b.repr_apply_apply, inner_orthogonalProjection_eq_of_mem_left] using
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 461 | 465 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... |
theorem deriv_eq_derivFamily (f : Ordinal → Ordinal) : deriv f = derivFamily fun _ : Unit => f :=
rfl
@[simp]
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 307 | 311 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Images of pairs of submodules under bilinear maps
This file provides `Subm... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 160 | 163 | |
/-
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.LocalizerMorphism
import Mathlib.CategoryTheory.HomCongr
/-!
# Bijections between morphisms in two localized categories
Given two ... | lemma homEquiv_trans (f : L₁.obj X ⟶ L₁.obj Y) :
homEquiv W L₂ L₃ (homEquiv W L₁ L₂ f) = homEquiv W L₁ L₃ f := by
dsimp only [homEquiv_apply]
apply LocalizerMorphism.homMap_homMap
| Mathlib/CategoryTheory/Localization/HomEquiv.lean | 142 | 145 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... |
/-- The integral `PowerBasis` of `𝓞 K` given by `ζ - 1`, where `K` is a `p ^ k` cyclotomic
| Mathlib/NumberTheory/Cyclotomic/Rat.lean | 252 | 254 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | mapAccumr₂_eq_map₂ (fun _ => true) rfl (fun _ _ _ _ => rfl) (fun a b s s' _ _ => h a b s s')
/-- If `f` takes a pair of states, but always returns the same value for both elements of the
pair, then we can simplify to just a single element of state.
-/
@[simp]
theorem mapAccumr_redundant_pair (f : α → (σ × σ) → (σ ×... | Mathlib/Data/Vector/MapLemmas.lean | 307 | 314 |
/-
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.Hom.Basic
/-!
# Unbounded lattice homomorphisms
This file defines unbounded lattice homomorphisms. _Bounded_ lattice homomorphisms are defined in
`... |
@[simp]
| Mathlib/Order/Hom/Lattice.lean | 316 | 317 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... | _ = a + b * c + a * (b * c) := by simp only [mul_self, add_self, add_zero]
theorem le_sup_inf (a b c : α) : (a ⊔ b) ⊓ (a ⊔ c) ⊔ (a ⊔ b ⊓ c) = a ⊔ b ⊓ c := by
| Mathlib/Algebra/Ring/BooleanRing.lean | 197 | 199 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | reflect_C_mul_X_pow, revAt_le (hkm.trans hmn).le, revAt_le hmn.le, revAt_le le_rfl, add_mul,
add_mul, mul_assoc, mul_assoc, mul_assoc, ← pow_add, ← pow_add, ← pow_add,
Nat.sub_add_cancel (hkm.trans hmn).le, Nat.sub_self, zero_add, add_comm, add_comm (C u * X ^ n),
| Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 105 | 107 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Complement
import Mathlib.GroupTheory.Sylow
/-!
# The Transfer Homomorphism
In this file we construct the transfer homomorphism.
## Ma... |
variable (H) in
/-- The transfer transversal as a function. Given a `⟨g⟩`-orbit `q₀, g • q₀, ..., g ^ (m - 1) • q₀`
in `G ⧸ H`, an element `g ^ k • q₀` is mapped to `g ^ k • g₀` for a fixed choice of
representative `g₀` of `q₀`. -/
| Mathlib/GroupTheory/Transfer.lean | 79 | 83 |
/-
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.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc (attr := simp)]
lemma inrX_d (i j : ι) :
inrX φ i ≫ d φ i j = G.d i j ≫ inrX φ j := by
by_cases hij : c.Rel i j
· by_cases hj : c.Rel j (c.next j)
· apply ext_to_X _ _ _ hj
· simp [d_fstX φ _ _ _ hij]
· simp [d_sndX φ _ _ hij]
· apply ext_to_X' _ _ hj
simp [d_sndX φ _ _ hij]
... | Mathlib/Algebra/Homology/HomotopyCofiber.lean | 203 | 213 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | by_cases hf : Integrable f μ
· rw [integral_sub hf (integrable_const _), integral_average, sub_self]
refine integral_undef fun h => hf ?_
convert h.add (integrable_const (⨍ a, f a ∂μ))
exact (sub_add_cancel _ _).symm
| Mathlib/MeasureTheory/Integral/Average.lean | 430 | 435 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... |
theorem mul_inv_lt_of_lt_mul₀ (h : a < b * c) : a * c⁻¹ < b := by
contrapose! h
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 158 | 160 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.PathCategory.Basic
/-!
# Free groupoid on a quiver
This file defines the free groupoid on a q... |
section Functoriality
variable {V' : Type u'} [Quiver.{v' + 1} V'] {V'' : Type u''} [Quiver.{v'' + 1} V'']
| Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 168 | 171 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | · rintro _ ⟨_, c', rfl⟩ -
exact ⟨c', rfl⟩
· exact h.apply _
@[simp]
theorem wellFoundedOn_image {s : Set β} : (f '' s).WellFoundedOn r ↔ s.WellFoundedOn (r on f) := by
rw [image_eq_range]; exact wellFoundedOn_range
| Mathlib/Order/WellFoundedSet.lean | 102 | 109 |
/-
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.Data.Set.Finite.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Grade
import Mathlib.Order.Nat
/-!
# Finsets and multisets form a graded order
This... | lemma covBy_iff : s ⋖ t ↔ ∃ a, a ::ₘ s = t :=
⟨CovBy.exists_multiset_cons, by rintro ⟨a, rfl⟩; exact covBy_cons _ _⟩
| Mathlib/Data/Finset/Grade.lean | 37 | 38 |
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | wittStructureRat p Φ n = C (1 / (p : ℚ) ^ n) * (wittStructureRat p Φ n * C ((p : ℚ) ^ n)) := ?_
_ = _ := by rw [wittStructureRat_rec_aux]
rw [mul_left_comm, ← C_mul, div_mul_cancel₀, C_1, mul_one]
exact pow_ne_zero _ (Nat.cast_ne_zero.2 hp.1.ne_zero)
/-- `wittStructureInt Φ` is a family of polynomials `ℕ →... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 180 | 189 |
/-
Copyright (c) 2017 Johannes Hölzl. 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.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,475 | 2,475 | |
/-
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.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... |
@[continuity, fun_prop]
| Mathlib/Analysis/Complex/Basic.lean | 145 | 146 |
/-
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... |
@[to_additive]
| Mathlib/Topology/Algebra/Group/Basic.lean | 640 | 641 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | p.totalDegree = 0 ↔ IsHomogeneous p 0 := by
rw [← weightedTotalDegree_one,
← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsHomogeneous]
| Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 124 | 127 |
/-
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.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
/-!
# Uniformly locally do... | simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,
isClosed_closedBall, true_and, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
measurableSet_closedBall]
/- The measure is doubling on scales smaller than `R`. Therefore, we treat differently small
... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 71 | 109 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... |
@[simp]
theorem lift_symm_apply (F : MonoidAlgebra k G →ₐ[k] A) (x : G) :
(lift k G A).symm F x = F (single x 1) :=
rfl
@[simp]
theorem lift_single (F : G →* A) (a b) : lift k G A F (single a b) = b • F a := by
rw [lift_def, liftNC_single, Algebra.smul_def, AddMonoidHom.coe_coe]
| Mathlib/Algebra/MonoidAlgebra/Basic.lean | 202 | 210 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | nonrec theorem IsPWO.exists_monotone_subseq (h : s.IsPWO) {f : ℕ → α} (hf : ∀ n, f n ∈ s) :
∃ g : ℕ ↪o ℕ, Monotone (f ∘ g) :=
h.exists_monotone_subseq hf
theorem isPWO_iff_exists_monotone_subseq :
s.IsPWO ↔ ∀ f : ℕ → α, (∀ n, f n ∈ s) → ∃ g : ℕ ↪o ℕ, Monotone (f ∘ g) :=
partiallyWellOrderedOn_iff_exists_mo... | Mathlib/Order/WellFoundedSet.lean | 399 | 405 |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | @[deprecated (since := "2025-03-10")]
alias UniformContinuous.prod_mk := UniformContinuous.prodMk
| Mathlib/Topology/UniformSpace/Basic.lean | 741 | 742 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | refine ⟨fun h => ⟨toIocDiv hp a c - toIocDiv hp a b, ?_⟩, fun h => ?_⟩
· conv_lhs => rw [← toIocMod_add_toIocDiv_zsmul hp a b, ← toIocMod_add_toIocDiv_zsmul hp a c]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 480 | 481 |
/-
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... | @[ext]
theorem hom_ext {g₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A}
| Mathlib/Algebra/Lie/UniversalEnveloping.lean | 148 | 149 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
theorem encard_le_encard_iff_encard_diff_le_encard_diff (h : (s ∩ t).Finite) :
| Mathlib/Data/Set/Card.lean | 192 | 193 |
/-
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.Finset.Max
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Multiset.Sort
import Mathlib.Order.RelIso.Set
/-!
# Construct a sorted list f... | theorem sort_nodup (s : Finset α) : (sort r s).Nodup :=
(by rw [sort_eq]; exact s.2 : @Multiset.Nodup α (sort r s))
| Mathlib/Data/Finset/Sort.lean | 49 | 50 |
/-
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.Real
/-!
# Power function... | rw [prod_insert hi, prod_insert hi, ih h2f, ← mul_rpow_of_ne_top <| hf i <| mem_insert_self ..]
apply prod_ne_top h2f
theorem prod_rpow_of_nonneg {ι} {s : Finset ι} {f : ι → ℝ≥0∞} {r : ℝ} (hr : 0 ≤ r) :
∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r := by
classical
induction s using Finset.induction with
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 685 | 691 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 493 | 499 | |
/-
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... | (LinearMap.mk₂ 𝕜 compContinuousMultilinearMap (fun _ _ _ => rfl) (fun _ _ _ => rfl)
(fun f g₁ g₂ => by ext1; apply f.map_add)
(fun c f g => by ext1; simp))
1
fun f g => by rw [one_mul]; exact f.norm_compContinuousMultilinearMap_le g
variable {𝕜 G G'}
/-- `ContinuousLinearMap.compContinuousMu... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 852 | 860 |
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Yaël Dillies, Jineon Baek
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebr... | /-- Statement of Fermat's Last Theorem: `a ^ n + b ^ n = c ^ n` has no nontrivial natural solution
when `n ≥ 3`.
| Mathlib/NumberTheory/FLT/Basic.lean | 54 | 56 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | @[to_additive]
theorem prod_eq_prod_coe [CommMonoid α] (m : Multiset α) : m.prod = ∏ x : m, (x : α) := by
congr
| Mathlib/Data/Multiset/Fintype.lean | 213 | 215 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib... | (hP : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [HasPullback π π], Function.Bijective
(MapToEqualizer P π (pullback.fst π π) (pullback.snd π π)
pullback.condition)) : EqualizerCondition P := by
intro X B π _ c hc
have : HasPullback π π := ⟨c, hc⟩
specialize hP X B π
rw [Types.type_equalizer_iff_uniqu... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 102 | 120 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | -- This matches the precedence of the infix `~` for `List.Perm`, and of other relation infixes
infixr:50 " ~r " => IsRotated
variable {l l'}
@[refl]
theorem IsRotated.refl (l : List α) : l ~r l :=
⟨0, by simp⟩
| Mathlib/Data/List/Rotate.lean | 361 | 369 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | rw [apply_inv_self]; rwa [pow_succ', mul_apply] at h)
⟨i + 1, by
rw [add_comm, zpow_add, mul_apply, hi, zpow_one, mul_apply, apply_inv_self,
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 369 | 371 |
/-
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 =... | protected theorem equiv_symm (h : IsFixedPt e x) : IsFixedPt e.symm x :=
| Mathlib/Dynamics/FixedPoints/Basic.lean | 94 | 94 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | exact G.ne_of_adj he
instance decidableMemEdgeSet [DecidableRel G.Adj] : DecidablePred (· ∈ G.edgeSet) :=
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 526 | 528 |
/-
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, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Round
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
/-... |
section NormNum
open Mathlib.Meta.NormNum Qq
theorem isNat_intFloor {R} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorRing R]
(r : R) (m : ℕ) :
IsNat r m → IsNat ⌊r⌋ m := by rintro ⟨⟨⟩⟩; exact ⟨by simp⟩
theorem isInt_intFloor {R} [Ring R] [LinearOrder R] [IsStrictOrderedRing R] [FloorRing R]
(r... | Mathlib/Data/Rat/Floor.lean | 106 | 125 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | rw [← mul_apply, Commute.self_pow f, mul_apply, apply_eq_iff_eq]
@[simp]
theorem apply_zpow_apply_eq_iff (f : Perm α) (n : ℤ) {x : α} :
| Mathlib/GroupTheory/Perm/Support.lean | 264 | 267 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,460 | 1,463 | |
/-
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 =... | · simp
theorem antidiagonal_eq_map_powerset [DecidableEq α] (s : Multiset α) :
s.antidiagonal = s.powerset.map fun t ↦ (s - t, t) := by
induction' s using Multiset.induction_on with a s hs
· simp only [antidiagonal_zero, powerset_zero, Multiset.zero_sub, map_singleton]
· simp_rw [antidiagonal_cons, power... | Mathlib/Data/Multiset/Antidiagonal.lean | 76 | 87 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... |
lemma primeFactors_mono (hmn : m ∣ n) (hn : n ≠ 0) : primeFactors m ⊆ primeFactors n := by
| Mathlib/Data/Nat/PrimeFin.lean | 40 | 41 |
/-
Copyright (c) 2023 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.MeasureComp
import Mathlib.Probability.Kernel.CondDistrib
import Mathlib.Probability.ConditionalProbability
/-!
# Kernel as... | theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_id [NormedAddCommGroup F] (hm : m ≤ mΩ)
(hf : Integrable f μ) : Integrable (fun x : Ω × Ω => f x.2)
(@Measure.map Ω (Ω × Ω) mΩ (m.prod mΩ) (fun ω => (id ω, id ω)) μ) := by
rw [← integrable_comp_snd_map_prodMk_iff (measurable_id'' hm)] at hf
simp_rw... | Mathlib/Probability/Kernel/Condexp.lean | 52 | 57 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn, Sabbir Rahman
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algeb... | let iα ← synthInstanceQ q(IsStrictOrderedRing $α)
orElse (← catchNone (ofNonneg q(le_of_lt $pa) sα oα iα))
(ofNonzero q(ne_of_gt $pa) q(inferInstance))
| .nonnegative pa =>
| Mathlib/Algebra/Order/Field/Power.lean | 166 | 169 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.Data.Fin.FlagRange
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Dual.Basis
import Mathlib.RingTheory.SimpleR... | Mathlib/LinearAlgebra/Basis/Flag.lean | 32 | 32 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | section LE
variable [LE β] {l : Filter α}
| Mathlib/Order/Filter/Basic.lean | 1,037 | 1,039 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... |
namespace Iso
variable {E : Type u₃} [Category.{v₃} E] {F : C ⥤ E} {G : C ⥤ D} {H : D ⥤ E}
| Mathlib/CategoryTheory/Equivalence.lean | 636 | 640 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Algebra.Module.LinearMap.Rat
import Mathlib.Tactic.Module
/... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 269 | 276 | |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Eric Wieser
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.T... | Mathlib/Algebra/Group/Pi/Basic.lean | 510 | 514 | |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Group.Indicator
imp... | agree on inverses of natural numbers in `R` and `S`. -/
theorem inv_natCast_smul_eq {E : Type*} (R S : Type*) [AddCommMonoid E] [DivisionSemiring R]
[DivisionSemiring S] [Module R E] [Module S E] (n : ℕ) (x : E) :
(n⁻¹ : R) • x = (n⁻¹ : S) • x :=
map_inv_natCast_smul (AddMonoidHom.id E) R S n x
| Mathlib/Algebra/Module/Basic.lean | 64 | 69 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
/-!
# Convex hull
This file defines the convex hull of a set `s` in a module. `con... | Mathlib/Analysis/Convex/Hull.lean | 214 | 217 | |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Constructi... | rw [Polynomial.map_map]; rfl⟩
-- The map `K ↦ g` is injective
have hinj : Function.Injective g := fun K K' heq ↦ by
rw [Subtype.mk.injEq] at heq
apply_fun fun f : E[X] ↦ adjoin F (f.coeffs : Set E) at heq
simpa only [adjoin_minpoly_coeff_of_exists_primitive_element F hprim] using heq
-- Therefor... | Mathlib/FieldTheory/PrimitiveElement.lean | 318 | 340 |
/-
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... | isBasis_self_iff_indep.mpr h
@[simp] theorem isBasis_empty_iff (M : Matroid α) : M.IsBasis I ∅ ↔ I = ∅ :=
⟨fun h ↦ subset_empty_iff.mp h.subset, fun h ↦ by (rw [h]; exact M.empty_indep.isBasis_self)⟩
| Mathlib/Data/Matroid/Basic.lean | 914 | 917 |
/-
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... | MeasurableSet[(hτ.min hπ).measurableSpace] (s ∩ {ω | τ ω ≤ π ω}) := by
simp_rw [IsStoppingTime.measurableSet] at hs ⊢
intro i
have : s ∩ {ω | τ ω ≤ π ω} ∩ {ω | min (τ ω) (π ω) ≤ i} =
s ∩ {ω | τ ω ≤ i} ∩ {ω | min (τ ω) (π ω) ≤ i} ∩
{ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i} := by
ext1 ω
... | Mathlib/Probability/Process/Stopping.lean | 557 | 569 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | theorem one_lt_encard_iff_nontrivial : 1 < s.encard ↔ s.Nontrivial := by
rw [← not_iff_not, not_lt, Set.not_nontrivial_iff, ← encard_le_one_iff_subsingleton]
theorem one_lt_encard_iff : 1 < s.encard ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b := by
rw [← not_iff_not, not_exists, not_lt, encard_le_one_iff]; aesop
theorem exist... | Mathlib/Data/Set/Card.lean | 324 | 336 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... | toKaroubi (ChainComplex C ℕ) ⋙ (Γ₂ ⋙ N₂) := (Functor.associator _ _ _).symm
_ ≅ (Γ₀ ⋙ toKaroubi (SimplicialObject C)) ⋙ N₂ :=
isoWhiskerRight ((functorExtension₂CompWhiskeringLeftToKaroubiIso _ _).app Γ₀) N₂
_ ≅ Γ₀ ⋙ toKaroubi (SimplicialObject C) ⋙ N₂ := Functor.associator _ _ _
| Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 113 | 116 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.Algebra.Module.LocalizedModule.IsLocalization
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.RingTheory.Localization.FractionRing
import ... | choose! a fg hfg using IsLocalization.exist_integer_multiples S (t.disjSum t) (Sum.elim g₁ g₂)
simp_rw [Sum.forall, Finset.inl_mem_disjSum, Sum.elim_inl, Finset.inr_mem_disjSum, Sum.elim_inr,
Subtype.forall'] at hfg
apply_fun ((a : R) • ·) at eq
simp_rw [← t.sum_coe_sort, Finset.smul_sum, ← smul_assoc, ← hf... | Mathlib/RingTheory/Localization/Module.lean | 56 | 71 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | variable {f g : α → β} {s : Finset α} {t : Finset β} {a : α} {b c : β}
@[simp]
| Mathlib/Data/Finset/Image.lean | 279 | 281 |
/-
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, Mitchell Lee, Johan Commelin
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.Module.Linear... | lemma reflection_symm (h : f x = 2) :
(reflection h).symm = reflection h :=
rfl
lemma invOn_reflection_of_mapsTo {Φ : Set M} (h : f x = 2) :
InvOn (reflection h) (reflection h) Φ Φ :=
⟨fun x _ ↦ involutive_reflection h x, fun x _ ↦ involutive_reflection h x⟩
lemma bijOn_reflection_of_mapsTo {Φ : Set M} (h... | Mathlib/LinearAlgebra/Reflection.lean | 117 | 151 |
/-
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.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | theorem image₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : image₂ f s t = image₂ g t s :=
(image₂_swap _ _ _).trans <| by simp_rw [h_comm]
| Mathlib/Data/Finset/NAry.lean | 313 | 314 |
/-
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.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
@[simp]
| Mathlib/Order/Heyting/Basic.lean | 940 | 941 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Semirings and ... | Mathlib/Algebra/Ring/Semiconj.lean | 89 | 91 | |
/-
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.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | refine ⟨u, ?_, ?_, ?_⟩
· intro n
have : u n ∈ T' := by rw [hu]; exact mem_range_self n
rcases mem_insert_iff.1 this with h|h
· rw [h]
exact isOpen_empty
· exact T_open _ h
· intro x hx
obtain ⟨v, hv, h'v⟩ : ∃ v, v ∈ T ∧ x ∈ v := by simpa only [mem_iUnion, exists_prop] using sT hx
hav... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 105 | 129 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Aaron Anderson
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Fi... | instance instSMulPosStrictMono [SMulPosStrictMono α β] : SMulPosStrictMono α (ι →₀ β) :=
SMulPosStrictMono.lift _ coe_le_coe coe_smul coe_zero
-- `PosSMulReflectLT α (ι →₀ β)` already follows from the other instances
| Mathlib/Data/Finsupp/Order.lean | 201 | 204 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | mem_eraseNone
@[simp]
theorem iUnion_ofWithBot (boxes : Finset (WithBot (Box ι)))
| Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 370 | 373 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,073 | 1,076 | |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | theorem dichotomy (b : Bool) : b = false ∨ b = true := by cases b <;> simp
| Mathlib/Data/Bool/Basic.lean | 102 | 102 |
/-
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.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Additive Haar measure constructed from a basis
Given a ba... | convert parallelepiped_comp_equiv b e.symm
ext i
simp only [OrthonormalBasis.coe_reindex]
rw [← B]
let F : ℝ → Fin 1 → ℝ := fun t => fun _i => t
have A : Icc (0 : Fin 1 → ℝ) 1 = F '' Icc (0 : ℝ) 1 := by
apply Subset.antisymm
· intro x hx
refine ⟨x 0, ⟨hx.1 0, hx.2 0⟩, ?_⟩
ext j
... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 98 | 125 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | (↑(f.sum g) : R) = f.sum fun a b => ↑(g a b) :=
Nat.cast_sum _ _
| Mathlib/Data/Finsupp/Basic.lean | 352 | 354 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | IsZeroOrMarkovKernel (const α μβ) := by
rcases eq_zero_or_isProbabilityMeasure μβ with rfl | h
· simp only [const_zero]
infer_instance
· infer_instance
| Mathlib/Probability/Kernel/Basic.lean | 206 | 211 |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | let ⟨_, K0, HK⟩ := abv_pos_of_not_limZero hf
let ⟨_, δ0, Hδ⟩ := rat_inv_continuous_lemma abv ε0 K0
let ⟨i, H⟩ := exists_forall_ge_and HK (f.cauchy₃ δ0)
⟨i, fun _ ij =>
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 534 | 537 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... | calc
[[x, z]] ⊆ [[x, y]] ∪ [[y, z]] := uIcc_subset_uIcc_union_uIcc
| Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 73 | 74 |
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike... | /-- The `ratio` is equal to the distance ratio for any two points
with nonzero finite distance; `dist` version -/
theorem ratio_unique_of_dist_ne_zero {α β} {F : Type*} [PseudoMetricSpace α] [PseudoMetricSpace β]
[FunLike F α β] [DilationClass F α β] {f : F} {x y : α} {r : ℝ≥0} (hxy : dist x y ≠ 0)
| Mathlib/Topology/MetricSpace/Dilation.lean | 183 | 186 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,206 | 1,210 | |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 677 | 678 | |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Partition.Equipartition
/-!
# Turán's theorem
In this file we prove Turán's theorem, the first importan... | exact hz.injOn hv hw (by rwa [← h.not_adj_iff_part_eq])
rw [Finset.card_eq_of_equiv hz.equiv] at ncf
exact absurd (h.1.mono (Nat.succ_le_of_lt l)) ncf
/-- There are `min n r` parts in a graph on `n` vertices satisfying `G.IsTuranMaximal r`.
`min` handles the `n < r` case, when `G` is complete but still `r + 1`... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 236 | 253 |
/-
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.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | variable [DecidableEq α]
mutual
instance Equiv.decidable : ∀ l₁ l₂ : Lists α, Decidable (l₁ ~ l₂)
| ⟨false, l₁⟩, ⟨false, l₂⟩ =>
decidable_of_iff' (l₁ = l₂) <| by
cases l₁
apply equiv_atom.trans
simp only [atom]
constructor <;> (rintro ⟨rfl⟩; rfl)
| ⟨false, l₁⟩, ⟨true, l₂... | Mathlib/SetTheory/Lists.lean | 313 | 349 |
/-
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 _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} :=
eq_singleton_iff_unique_mem.2 ⟨left_mem_Ici, fun _ => h.eq_of_ge⟩
| Mathlib/Order/Interval/Set/Basic.lean | 815 | 816 |
/-
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.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... | theorem linearProjOfIsCompl_apply_left {F : Type*} [AddCommGroup F] [Module R F]
(i : F →ₗ[R] E) (hi : Function.Injective i)
(h : IsCompl (LinearMap.range i) q) (x : F) :
linearProjOfIsCompl q i hi h (i x) = x := by
let ix : LinearMap.range i := ⟨i x, mem_range_self i x⟩
| Mathlib/LinearAlgebra/Projection.lean | 211 | 215 |
/-
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.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | simp only [add_quot, f.map_add x y]
map_one' := by simp only [← one_quot, f.map_one]
map_mul' := by
rintro ⟨⟨x⟩⟩ ⟨⟨y⟩⟩
| Mathlib/Algebra/RingQuot.lean | 425 | 428 |
/-
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... | (fun _ _ => (isometry_mul_right _).lipschitz.lipschitzOnWith) fun _ _ =>
(isometry_mul_left _).lipschitz.lipschitzOnWith).trans_eq <|
| Mathlib/Analysis/Normed/Group/Pointwise.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... | constructor
· intro ⟨h₁, h₃⟩
have := T.epi₂ hT h₃
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 356 | 358 |
/-
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.Algebra.Order.Group.Multiset
/-!
# Disjoint sum of multisets
This file defines the disjoint sum of two multisets as `Multiset (α ⊕ β)`. Beware not to con... | Mathlib/Data/Multiset/Sum.lean | 104 | 109 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... |
end MonoidalPreadditive
| Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 60 | 61 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | (h₂ : ∀ i ∈ s, f.IsBoundedUnder (· ≤ ·) (F i) := by exact fun _ _ ↦ by isBoundedDefault) :
limsup (fun a ↦ sup s (fun i ↦ F i a)) f = sup s (fun i ↦ limsup (F i) f) := by
rcases eq_or_neBot f with (rfl | _)
· simp [limsup_eq, csInf_univ]
| Mathlib/Order/LiminfLimsup.lean | 1,173 | 1,176 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
/-!
#... | Mathlib/GroupTheory/Perm/Basic.lean | 591 | 592 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | obtain ⟨b', h'⟩ := Option.isSome_iff_exists.mp (dlookup_isSome.mpr (mem_keys_of_mem h))
cases nd.eq_of_mk_mem h (of_mem_dlookup h')
exact h'
theorem map_dlookup_eq_find (a : α) :
∀ l : List (Sigma β), (dlookup a l).map (Sigma.mk a) = find? (fun s => a = s.1) l
| [] => rfl
| Mathlib/Data/List/Sigma.lean | 188 | 194 |
/-
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.Analysis.Convex.Deriv
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.NumberTheory.Harmo... | lemma hasDerivAt_Gammaℝ_one : HasDerivAt Gammaℝ (-(γ + log (4 * π)) / 2) 1 := by
let f (s : ℂ) : ℂ := π ^ (-s / 2)
let g (s : ℂ) : ℂ := Gamma (s / 2)
have aux : (π : ℂ) ^ (1 / 2 : ℂ) = ↑√π := by
rw [Real.sqrt_eq_rpow, ofReal_cpow Real.pi_pos.le, ofReal_div, ofReal_one, ofReal_ofNat]
have aux2 : (√π : ℂ) ≠ 0... | Mathlib/NumberTheory/Harmonic/GammaDeriv.lean | 206 | 229 |
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