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) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | Mathlib/Order/Basic.lean | 1,532 | 1,534 | |
/-
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.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... |
/-- The preimage of a star-convex set under an affine map is star-convex. -/
theorem StarConvex.affine_preimage (f : E →ᵃ[𝕜] F) {s : Set F} (hs : StarConvex 𝕜 (f x) s) :
StarConvex 𝕜 x (f ⁻¹' s) := by
intro y hy a b ha hb hab
rw [mem_preimage, Convex.combo_affine_apply hab]
| Mathlib/Analysis/Convex/Star.lean | 333 | 338 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | theorem basisSets_mem (i : Finset ι) {r : ℝ} (hr : 0 < r) : (i.sup p).ball 0 r ∈ p.basisSets :=
(basisSets_iff _).mpr ⟨i, _, hr, rfl⟩
| Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 81 | 83 |
/-
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.Limits.Final
import Mathlib.CategoryTheory.Functor.TwoSquare
/-!
# Guitart exact squares
Given four functors `T`, `L`, `R` and `B`, a 2-square `... | /-- Given `w : TwoSquare T L R B` and a morphism `g : R.obj X₂ ⟶ B.obj X₃`, this is
the obvious functor `w.StructuredArrowRightwards g ⥤ w.CostructuredArrowDownwards g`. -/
@[simps]
def functor : w.StructuredArrowRightwards g ⥤ w.CostructuredArrowDownwards g where
obj f := CostructuredArrow.mk (Y := StructuredArrow.m... | Mathlib/CategoryTheory/GuitartExact/Basic.lean | 141 | 148 |
/-
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.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analy... |
theorem differentiableAt_euclidean :
DifferentiableAt 𝕜 f y ↔ ∀ i, DifferentiableAt 𝕜 (fun x => f x i) y :=
differentiableAt_piLp _
| Mathlib/Analysis/InnerProductSpace/Calculus.lean | 281 | 285 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Jacobson.Ideal
import Mathlib.RingTheory.Ideal.Quo... | map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- Criterion for checking that an adic cauchy sequence is mapped to zero in the adic completion. -/
| Mathlib/RingTheory/AdicCompletion/Basic.lean | 460 | 463 |
/-
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.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | simp_rw [← ofFinsupp_add, coeff]
exact Finsupp.add_apply _ _ _
@[simp]
theorem coeff_smul [SMulZeroClass S R] (r : S) (p : R[X]) (n : ℕ) :
| Mathlib/Algebra/Polynomial/Coeff.lean | 40 | 44 |
/-
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.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,234 | 3,235 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | congr_arg₂ (· ⊗ₜ[R] ·) ha.eq hb.eq
| Mathlib/RingTheory/TensorProduct/Basic.lean | 296 | 297 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
/-!
# Wide pullbacks
We define the category `WidePullbackShape`, (re... |
variable {X : C} (f : B ⟶ X) (fs : ∀ j : J, objs j ⟶ X) (w : ∀ j, arrows j ≫ fs j = f)
| Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 364 | 365 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | lemma Equation.baseChange (h : (W'.baseChange A).toAffine.Equation x y) :
(W'.baseChange B).toAffine.Equation (f x) (f y) := by
convert Equation.map f.toRingHom h using 1
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 833 | 835 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option α`
In this file we define
* `Option.t... | eraseNone (s.image some) = s := by simpa only [map_eq_image] using eraseNone_map_some s
@[simp]
theorem coe_eraseNone (s : Finset (Option α)) : (eraseNone s : Set α) = some ⁻¹' s :=
| Mathlib/Data/Finset/Option.lean | 105 | 108 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... |
@[simp]
theorem toMeasure_apply_fintype [Fintype α] : p.toMeasure s = ∑ x, s.indicator p x :=
(p.toMeasure_apply_eq_toOuterMeasure_apply s s.toFinite.measurableSet).trans
| Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 279 | 282 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | exact ConcreteCategory.hom_ext _ _ <| toSpec_base_apply_eq 𝒜
lemma mk_mem_toSpec_base_apply {f} (x : Proj| pbo f)
(z : NumDenSameDeg 𝒜 (.powers f)) :
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 676 | 679 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... | by_cases ha' : a ^ (Fintype.card F / 2) = 1
· simp only [ha', if_true, Int.cast_one]
· have ha'' := Or.resolve_left (FiniteField.pow_dichotomy hF ha) ha'
simp only [ha'', Int.cast_ite, Int.cast_one, Int.cast_neg, ite_eq_right_iff]
exact Eq.symm
| Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 190 | 194 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | · simp [hf.emultiplicity_eq_multiplicity]
· simp [hf, emultiplicity_eq_top.2]
| Mathlib/RingTheory/Multiplicity.lean | 137 | 139 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 665 | 671 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | ext n
rw [coeff_contract hp, coeff_mul, coeff_mul, ← sum_subset
(s₁ := (antidiagonal n).image fun x ↦ (x.1 * p, x.2 * p)), sum_image]
· simp_rw [coeff_expand_mul hp.bot_lt, coeff_contract hp]
· intro x hx y hy eq; simpa only [Prod.ext_iff, Nat.mul_right_cancel_iff hp.bot_lt] using eq
· simp_rw [subset_iff... | Mathlib/Algebra/Polynomial/Expand.lean | 212 | 221 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | (huv : Integrable (u' * v + u * v'))
(h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
∫ (x : ℝ), u' x * v x + u x * v' x = b' - a' :=
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,234 | 1,236 |
/-
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.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | theorem natDegree_le_iff_coeff_eq_zero : p.natDegree ≤ n ↔ ∀ N : ℕ, n < N → p.coeff N = 0 := by
simp_rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero, Nat.cast_lt]
theorem natDegree_add_le_iff_left {n : ℕ} (p q : R[X]) (qn : q.natDegree ≤ n) :
(p + q).natDegree ≤ n ↔ p.natDegree ≤ n := by
refine ⟨fun h... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 76 | 81 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 934 | 941 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Preserving (co)kernels
Constructions to relate the ... | cokernel.map (G.map f) (G.map g) (G.map p) (G.map q)
(by rw [← G.map_comp, hpq, G.map_comp]) =
G.map (cokernel.map f g p q hpq) ≫ (PreservesCokernel.iso G _).hom := by
rw [← Iso.comp_inv_eq, Category.assoc, ← Iso.eq_inv_comp, PreservesCokernel.iso_inv,
cokernel_map_comp_cokernelComparison,... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean | 279 | 287 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheo... | term (f ⍟ g) s = fun n ↦
∑' (b : (fun p : ℕ × ℕ ↦ p.1 * p.2) ⁻¹' {n}), term f s b.val.1 * term g s b.val.2 := by
ext n
rcases eq_or_ne n 0 with rfl | hn
· -- show that both sides vanish when `n = 0`; this is the hardest part of the proof!
refine (term_zero ..).trans ?_
-- the right hand sum is o... | Mathlib/NumberTheory/LSeries/Convolution.lean | 111 | 134 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Ackermann function
In this file, we def... | theorem ack_ack_lt_ack_max_add_two (m n k : ℕ) : ack m (ack n k) < ack (max m n + 2) k :=
calc
ack m (ack n k) ≤ ack (max m n) (ack n k) := ack_mono_left _ (le_max_left _ _)
_ < ack (max m n) (ack (max m n + 1) k) :=
ack_strictMono_right _ <| ack_strictMono_left k <| lt_succ_of_le <| le_max_right m n
... | Mathlib/Computability/Ackermann.lean | 251 | 256 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
import Mathlib.Order.Directed
/-!
# Monotone functions on intervals
This file sho... | lemma MonotoneOn.mapsTo_Iic (h : MonotoneOn f (Iic b)) : MapsTo f (Iic b) (Iic (f b)) :=
fun _ _ ↦ by aesop
| Mathlib/Order/Interval/Set/Image.lean | 27 | 28 |
/-
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 | 839 | 841 | |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | .of_isLittleOTVS <| by
simpa only using hf.isLittleOTVS.comp_tendsto (tendsto_id.prodMk_nhds tendsto_const_nhds)
protected theorem HasStrictFDerivAt.differentiableAt (hf : HasStrictFDerivAt f f' x) :
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 456 | 459 |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Real.Sqrt
import Mathlib.Data.Real... | 4 * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) := by
-- polyrith suggests:
linear_combination
(2 * B₀ * B₁ + 2) * A₀_inv + (B₀ ^ 2 - 2 * B₀ * B₁ + B₁ ^ 2) * A₁_inv +
(A₀ ^ 2 + 2 * A₀ * A₁ + 1) * B₀_inv +
(A₀ ^ 2 - 2 * A₀ * A₁ + 1) * B₁_inv
/-- Given a CHSH tuple (A₀, A₁, B₀, B₁) in a *comm... | Mathlib/Algebra/Star/CHSH.lean | 105 | 113 |
/-
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... | ‖b i‖ₑ = 1 := b.orthonormal.enorm_eq_one i
@[simp]
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 394 | 396 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | theorem oangle_add_cyc3_neg_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle (-x) y + o.oangle (-y) z + o.oangle (-z) x = π := by
rw [o.oangle_neg_left hx hy, o.oangle_neg_left hy hz, o.oangle_neg_left hz hx,
show o.oangle x y + π + (o.oangle y z + π) + (o.oangle z x + π) =
o.oangle x ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 478 | 491 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... | theorem isHermitian_one [DecidableEq n] : (1 : Matrix n n α).IsHermitian :=
conjTranspose_one
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 210 | 211 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,349 | 1,350 | |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... | refine forall_congr' fun x => ?_
rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
theorem mem_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) (f : A) :
f ∈ vanishingIdeal t ↔ ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal := by
rw [← SetLike.mem_coe, coe_vanish... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 99 | 106 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | repeat' apply prod_primeFactors_dvd
simp only [prod_primeFactors_gcd_mul_prod_primeFactors_mul]
rw [eq_comm, mul_comm, ← mul_assoc, ← Nat.mul_div_assoc]
exact mul_dvd_mul (prod_primeFactors_dvd a) (prod_primeFactors_dvd b)
theorem totient_super_multiplicative (a b : ℕ) : φ a * φ b ≤ φ (a * b) := by
let d := ... | Mathlib/Data/Nat/Totient.lean | 314 | 327 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | theorem sin_toReal (θ : Angle) : Real.sin θ.toReal = sin θ := by
conv_rhs => rw [← coe_toReal θ, sin_coe]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 598 | 600 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
/-!
# Lemmas which are consequences of monoidal coher... | end CategoryTheory.MonoidalCategory
| Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 72 | 75 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | ‖f y - f x‖ ≤ C * ‖y - x‖ :=
hs.norm_image_sub_le_of_norm_hasDerivWithin_le (fun x hx => (hf x hx).hasDerivWithinAt) bound xs
ys
| Mathlib/Analysis/Calculus/MeanValue.lean | 708 | 710 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... |
/-- We say a solution to `a ^ 4 + b ^ 4 = c ^ 2` is minimal if there is no other solution with
a smaller `c` (in absolute value). -/
def Minimal (a b c : ℤ) : Prop :=
Fermat42 a b c ∧ ∀ a1 b1 c1 : ℤ, Fermat42 a1 b1 c1 → Int.natAbs c ≤ Int.natAbs c1
| Mathlib/NumberTheory/FLT/Four.lean | 58 | 62 |
/-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analysis.PSeries
import Mat... | lemma summand_bound_of_mem_verticalStrip {k : ℝ} (hk : 0 ≤ k) (x : Fin 2 → ℤ)
{A B : ℝ} (hB : 0 < B) (hz : z ∈ verticalStrip A B) :
‖x 0 * (z : ℂ) + x 1‖ ^ (-k) ≤ r ⟨⟨A, B⟩, hB⟩ ^ (-k) * ‖x‖ ^ (-k) := by
refine (summand_bound z hk x).trans (mul_le_mul_of_nonneg_right ?_ (by positivity))
exact Real.rpow_le_r... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean | 137 | 142 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | _root_.by_contradiction fun hne ↦
not_lt_of_ge (by rw [padicNormE.add_eq_max_of_ne hne]; apply le_max_right) h
theorem eq_of_norm_add_lt_left {z1 z2 : ℚ_[p]} (h : ‖z1 + z2‖ < ‖z1‖) : ‖z1‖ = ‖z2‖ :=
_root_.by_contradiction fun hne ↦
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 869 | 873 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | HasFiniteIntegral f (μ.restrict s) ↔ HasFiniteIntegral f (μ.restrict t) := by
dsimp only [HasFiniteIntegral]
rw [hs.setLIntegral_eq ht]
intro g x; rw [hf]
@[to_additive]
protected theorem integrableOn_iff (hs : IsFundamentalDomain G s μ) (ht : IsFundamentalDomain G t μ)
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 355 | 361 |
/-
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.... | Mathlib/Data/Fin/Basic.lean | 1,555 | 1,560 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.Kronecker
import Mathlib.RingTheory.TensorProduct.Basic
... | (Algebra.lsmul R R (Matrix n n A)).toLinearMap.compl₂ (Algebra.linearMap R A).mapMatrix
@[simp]
| Mathlib/RingTheory/MatrixAlgebra.lean | 99 | 101 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | intro _ _ _
ext <;> congr
end NonUnitalRing
| Mathlib/Algebra/Ring/Ext.lean | 203 | 207 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,223 | 1,236 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 607 | 612 | |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | /-- The Schnirelmann density is unaffected by removing `0`. -/
lemma schnirelmannDensity_diff_singleton_zero [DecidablePred (· ∈ A \ {0})] :
schnirelmannDensity (A \ {0}) = schnirelmannDensity A :=
schnirelmannDensity_congr' (by aesop)
| Mathlib/Combinatorics/Schnirelmann.lean | 163 | 166 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | S ⟶ T
↗ ↗
X ⟶ Z
```
-/
theorem pullback_map_isOpenEmbedding {W X Y Z S T : TopCat.{u}} (f₁ : W ⟶ S)
(f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z} (H₁ : IsOpenEmbedding i₁)
(H₂ : IsOpenEmbedding i₂) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁)
(eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) : IsOp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 262 | 271 |
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Order.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
import Batteries.WF
/-!
# Unbundled... | Mathlib/Order/RelClasses.lean | 608 | 608 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.Shapes.Prod... | { pt := iInf F.obj
π := { app := fun _ => homOfLE (CompleteLattice.sInf_le _ _ (Set.mem_range_self _)) } }
isLimit :=
{ lift := fun s =>
homOfLE (CompleteLattice.le_sInf _ _ (by rintro _ ⟨j, rfl⟩; exact (s.π.app j).le)) }
/-- The colimit cocone over any functor into a complete lattice.
-/
| Mathlib/CategoryTheory/Limits/Lattice.lean | 168 | 175 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Equiv
/-!
# Partially defined linear operators on Hilbert spaces
We will develop... | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 251 | 261 | |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... | instance : OrderHomClass (α →𝒄 β) α β where
map_rel f _ _ h := f.mono h
instance : PartialOrder (α →𝒄 β) :=
| Mathlib/Order/OmegaCompletePartialOrder.lean | 541 | 544 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | that `L e p q` is close to `L e p r` (where `r` is large enough), as both approximate `f` at
scale `2 ^(- p)`. And `L e p r` is close to `L e' p' r` as both approximate `f` at scale
`2 ^ (- r)`. And `L e' p' r` is close to `L e' p' q'` as both approximate `f` at scale
`2 ^ (- p')`. -/
have M :... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 562 | 574 |
/-
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.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Continuity of series of functions
We show that series ... | refine tendstoUniformlyOn_iff.2 fun ε εpos => ?_
have := (tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ εpos
simp only [not_forall, Classical.not_imp, not_le, gt_iff_lt,
eventually_atTop, ge_iff_le, Finset.le_eq_subset] at *
| Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 53 | 56 |
/-
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... | @[simp] theorem encard_eq_top_iff : s.encard = ⊤ ↔ s.Infinite := by
rw [← not_iff_not, ← Ne, ← lt_top_iff_ne_top, encard_lt_top_iff, not_infinite]
| Mathlib/Data/Set/Card.lean | 140 | 141 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,409 | 1,412 | |
/-
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... | (hε : ∀ i, 0 < ε i) (hc : ∀ i, 1 < ‖c i‖)
(hf : ∀ m : ∀ i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
(m : ∀ i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ :=
bound_of_shell_of_norm_map_coord_zero f
(fun h ↦ by rw [map_coord_zero f _ (norm_eq_zero.1 h), norm_zero]) hε hc hf m
en... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 1,336 | 1,342 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | lemma exists_isNilRegular_of_finrank_le_card (h : finrank R M ≤ #R) :
∃ x : L, IsNilRegular φ x := by
let b := chooseBasis R L
let bₘ := chooseBasis R M
let n := Fintype.card (ChooseBasisIndex R M)
have aux :
((polyCharpoly φ b).coeff (nilRank φ)).IsHomogeneous (n - nilRank φ) :=
polyCharpoly_coeff_... | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 538 | 554 |
/-
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... | maxGenEigenspace_eq_top x := IsTriangularizable.maxGenEigenspace_eq_top (x : L)
instance [IsTriangularizable R L M] : IsTriangularizable R (LieModule.toEnd R L M).range M where
maxGenEigenspace_eq_top := by
rintro ⟨-, x, rfl⟩
exact IsTriangularizable.maxGenEigenspace_eq_top x
| Mathlib/Algebra/Lie/Weights/Basic.lean | 722 | 727 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | constructor
· rw [← hy, ← abs_repr y, h, ← abs_map]
rfl
intro i
apply (f i).property
rintro ⟨a, f, h₀, h₁⟩
use abs ⟨a, fun i => ⟨f i, h₁ i⟩⟩
rw [← abs_map, h₀]; rfl
theorem liftp_iff' {α : Type u} (p : α → Prop) (x : F α) :
Liftp p x ↔ ∃ u : q.P α, abs u = x ∧ ∀ i, p (u.snd i) := by
c... | Mathlib/Data/QPF/Univariate/Basic.lean | 101 | 114 |
/-
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.Algebra.BigOperators.Group.List.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Data.Fintype.EquivFin
import... | | nil =>
simp only [not_mem_nil] at h
exact List.mem_singleton.2 (Equiv.ext fun x => Decidable.byContradiction <| h x)
| cons a l IH =>
by_cases hfa : f a = a
· refine mem_append_left _ (IH fun x hx => mem_of_ne_of_mem ?_ (h x hx))
rintro rfl
exact hx hfa
have hfa' : f (f a) ≠ f a := mt (fun h... | Mathlib/Data/Fintype/Perm.lean | 47 | 74 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
theorem two_zsmul_toReal_eq_two_mul_add_two_pi {θ : Angle} :
((2 : ℤ) • θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2 := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 592 | 594 |
/-
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.Order.Archimedean.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib... | classical
have : update f i 1 ≤ update g i 1 := update_le_update_iff.mpr ⟨rfl.le, fun i _ ↦ h i⟩
have : 1 / f i * a₁ ≤ 1 / g i * a₂ := hasProd_le this (hf.update i 1) (hg.update i 1)
simpa only [one_div, mul_inv_cancel_left] using mul_lt_mul_of_lt_of_le hi this
| Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 220 | 223 |
/-
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... | ∃ c : Sphere P, ∀ sx : Simplex ℝ P n, Set.range sx.points ⊆ ps → sx.circumsphere = c := by
haveI : Nonempty (⊤ : AffineSubspace ℝ P) := Set.univ.nonempty
rw [← finrank_top, ← direction_top ℝ V P] at hd
refine exists_circumsphere_eq_of_cospherical_subset ?_ hd hc
exact Set.subset_univ _
/-- Two n-simplices ... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 750 | 766 |
/-
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.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# Measurability of the integral against a kernel
Th... |
open ProbabilityTheory.Kernel
namespace MeasureTheory
variable [NormedSpace ℝ E]
omit [IsSFiniteKernel κ] in
theorem StronglyMeasurable.integral_kernel ⦃f : β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x ↦ ∫ y, f y ∂κ x := by
classical
by_cases hE : CompleteSpace E; swap
· simp [integral, h... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 42 | 99 |
/-
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... | ⟨fun _ ↦ Module.End.iSup_maxGenEigenspace_eq_top _⟩
instance (N : LieSubmodule K L M) [IsTriangularizable K L M] : IsTriangularizable K L N := by
refine ⟨fun y ↦ ?_⟩
rw [← N.toEnd_restrict_eq_toEnd y]
exact Module.End.genEigenspace_restrict_eq_top _ (IsTriangularizable.maxGenEigenspace_eq_top y)
/-- For a tri... | Mathlib/Algebra/Lie/Weights/Basic.lean | 757 | 790 |
/-
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
... | rw [show Sigma.map p = (biproduct.isoCoproduct _).inv ≫ biproduct.map p ≫
(biproduct.isoCoproduct _).hom by aesop]
infer_instance
/-- Two biproducts which differ by an equivalence in the indexing type,
and up to isomorphism in the factors, are isomorphic.
Unfortunately there are two natural ways to define eac... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 619 | 630 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | theorem foldl_concat
(f : β → α → β) (b : β) (x : α) (xs : List α) :
List.foldl f b (xs ++ [x]) = f (List.foldl f b xs) x := by
simp only [List.foldl_append, List.foldl]
| Mathlib/Data/List/Basic.lean | 820 | 824 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,171 | 1,173 | |
/-
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... | (hf : Disjoint f (Filter.cocompact Y)) :
𝓝ˢ s ×ˢ f ≤ 𝓝ˢ (s ×ˢ Set.univ) := by
obtain ⟨K, hKf, hK⟩ := (disjoint_cocompact_right f).mp hf
| Mathlib/Topology/Compactness/Compact.lean | 624 | 626 |
/-
Copyright (c) 2024 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Equiv
import Mathlib.Topology.Algebra.ContinuousAffineMap
/-!
# Conti... | rintro ⟨e, econt, einv_cont⟩ ⟨e', e'cont, e'inv_cont⟩ H
congr
| Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 65 | 67 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Partial derivatives of polynomials
This file defi... | pderiv i (f ^ n) = n * f ^ (n - 1) * pderiv i f := by
rw [(pderiv i).leibniz_pow f n, nsmul_eq_mul, smul_eq_mul, mul_assoc]
| Mathlib/Algebra/MvPolynomial/PDeriv.lean | 111 | 112 |
/-
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.Fintype.Card
import Mathlib.Order.UpperLower.Basic
/-!
# Intersecting families
This file defines intersecting families and proves their basic proper... | · exact h _ hc
| Mathlib/Combinatorics/SetFamily/Intersecting.lean | 61 | 61 |
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | rw [eq_C_of_natDegree_le_zero (le_trans hs sC), ← dvd_content_iff_C_dvd] at rs
apply rs rprim.dvd
have hcancel := natDegree_cancelLeads_lt_of_natDegree_le_natDegree hs (lt_of_not_ge sC)
rw [sdeg] at hcancel
apply Nat.find_min con hcancel
refine
⟨_, rfl, ⟨dvd_cancelLeads_of_dvd_of_dvd p... | Mathlib/RingTheory/Polynomial/Content.lean | 409 | 417 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Subgroup.exists_ri... | rw [← diff_inv, smul_diff', diff_self, one_mul, inv_pow, inv_inv]
_ = 1 ^ H.index := (Quotient.exact' hα).trans (one_pow H.index).symm
theorem exists_smul_eq (hH : Nat.Coprime (Nat.card H) H.index) (α β : H.QuotientDiff) :
∃ h : H, h • α = β :=
Quotient.inductionOn' α
(Quotient.inductionOn'... | Mathlib/GroupTheory/SchurZassenhaus.lean | 93 | 100 |
/-
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.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.Invarian... |
theorem rank_span {v : ι → M} (hv : LinearIndependent R v) :
Module.rank R ↑(span R (range v)) = #(range v) := by
haveI := nontrivial_of_invariantBasisNumber R
| Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 369 | 372 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | | add _ _ _ _ hz₁ hz₂ => rw [lie_add]; exact add_mem hz₁ hz₂
lemma lie_mem_genWeightSpaceChain_of_genWeightSpace_eq_bot_left [LieRing.IsNilpotent H]
(hp : genWeightSpace M (p • α + χ) = ⊥)
{x : L} (hx : x ∈ rootSpace H (-α))
{y : M} (hy : y ∈ genWeightSpaceChain M α χ p q) :
⁅x, y⁆ ∈ genWeightSpaceCh... | Mathlib/Algebra/Lie/Weights/Chain.lean | 156 | 179 |
/-
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.Algebra.Module.Basic
import Mathlib.Algebra.Module.LinearMap.Defs
import Mathlib.RingTheory.HahnSeries.Basic
import Mathlib.Algebra.BigOperators.Group.... | (hxy : x.orderTop < y.orderTop) : (x + y).orderTop = x.orderTop := by
have hx : x ≠ 0 := ne_zero_iff_orderTop.mpr hxy.ne_top
let g : Γ := Set.IsWF.min x.isWF_support (support_nonempty_iff.2 hx)
have hcxyne : (x + y).coeff g ≠ 0 := by
| Mathlib/RingTheory/HahnSeries/Addition.lean | 177 | 180 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | List.nodup_append_comm]
simp [List.getLast_eq_getElem]
theorem Nodup.erase_getElem [DecidableEq α] {l : List α} (hl : l.Nodup)
(i : Nat) (h : i < l.length) : l.erase l[i] = l.eraseIdx ↑i := by
induction l generalizing i with
| nil => simp
| cons a l IH =>
cases i with
| zero => simp
... | Mathlib/Data/List/Nodup.lean | 259 | 269 |
/-
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.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Set.Image
/-!
# Support of a func... | (mulSupport fun x => (f x, g x)) = mulSupport f ∪ mulSupport g :=
Set.ext fun x => by
simp only [mulSupport, not_and_or, mem_union, mem_setOf_eq, Prod.mk_eq_one, Ne]
| Mathlib/Algebra/Group/Support.lean | 182 | 184 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,205 | 1,205 | |
/-
Copyright (c) 2021 Eric Weiser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Ring.Subring.Pointwise
/-!
# Pointwise actions on subalgebras.
If `R'` acts on an `R`-algebra `A`... | show y * z ∈ S ⊔ T
exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)
/-- As submodules, subalgebras are idempotent. -/
@[simp]
theorem isIdempotentElem_toSubmodule (S : Subalgebra R A) :
| Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | 27 | 32 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | apply le_antisymm
· intro x hx
simp only [IsLocalRing.mem_maximalIdeal, mem_nonunits] at hx
rwa [Ideal.mem_span_singleton, ← norm_lt_one_iff_dvd]
· rw [Ideal.span_le, Set.singleton_subset_iff]
exact p_nonnunit
theorem prime_p : Prime (p : ℤ_[p]) := by
rw [← Ideal.span_singleton_prime, ← maximalIdea... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 437 | 445 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | theorem coeff_zero_eq_constantCoeff_apply (φ : MvPowerSeries σ R) :
coeff R (0 : σ →₀ ℕ) φ = constantCoeff σ R φ :=
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 442 | 443 |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Common
/-!
The type `List.Vector` represents lists with fixed length.
TODO: The API of `List.Vector` i... | Mathlib/Data/Vector/Defs.lean | 268 | 270 | |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Compactness.Bases
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Separation.Profinite
import Mathlib.Topology.Sets.Closeds
/... | cases nonempty_fintype (Clopens X)
cases nonempty_fintype (Clopens Y)
exact .of_surjective _ surjective_finset_sup_prod
lemma countable_iff_secondCountable [T2Space X]
[TotallyDisconnectedSpace X] : Countable (Clopens X) ↔ SecondCountableTopology X := by
refine ⟨fun h ↦ ⟨{s : Set X | IsClopen s}, ?_, ?_⟩, ... | Mathlib/Topology/ClopenBox.lean | 77 | 92 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
theorem uIcc_eq_union : [[a, b]] = Icc a b ∪ Icc b a :=
coe_injective <| by
| Mathlib/Order/Interval/Finset/Basic.lean | 1,046 | 1,048 |
/-
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.Hom
import Mathlib.Algebra.GroupWithZero.Action.Prod
/-!
# Morphisms of non-unital algebras
This file defines morphisms between two types, ... | { f with map_smul' := map_smul f }
| Mathlib/Algebra/Algebra/NonUnitalHom.lean | 444 | 445 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 1,030 | 1,033 | |
/-
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.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | | oadd e n a, 0, h => by injection h
| 0, oadd e n a, h => by injection h
| oadd e₁ n₁ a₁, oadd e₂ n₂ a₂, h => by
| Mathlib/SetTheory/Ordinal/Notation.lean | 158 | 160 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | Mathlib/Data/List/Zip.lean | 250 | 251 | |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,211 | 2,216 | |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Mario Carneiro, Reid Barton, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Func... | Mathlib/Topology/Sheaves/Presheaf.lean | 475 | 480 | |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... | changeLevel (dvd_trans hm hd) χ = changeLevel hd (changeLevel hm χ) := by
simp [changeLevel_def, MonoidHom.comp_assoc, ZMod.unitsMap_comp]
lemma changeLevel_eq_cast_of_dvd {m : ℕ} (hm : n ∣ m) (a : Units (ZMod m)) :
(changeLevel hm χ) a = χ (ZMod.cast (a : ZMod m)) := by
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 91 | 95 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... | variable (A K) in
lemma map_equiv_traceDual [IsDomain A] [IsFractionRing B L] [IsDomain B]
[FaithfulSMul A B] (I : Submodule B (FractionRing B)) :
(traceDual A (FractionRing A) I).map (FractionRing.algEquiv B L) =
traceDual A K (I.map (FractionRing.algEquiv B L)) := by
show Submodule.map (FractionRing.a... | Mathlib/RingTheory/DedekindDomain/Different.lean | 121 | 128 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... | section bits
theorem vecAlt0_vecAppend (v : Fin n → α) :
vecAlt0 rfl (vecAppend rfl v v) = v ∘ (fun n ↦ n + n) := by
ext i
simp_rw [Function.comp, vecAlt0, vecAppend_eq_ite]
split_ifs with h <;> congr
· rw [Fin.val_mk] at h
exact (Nat.mod_eq_of_lt h).symm
· rw [Fin.val_mk, not_lt] at h
| Mathlib/Data/Fin/VecNotation.lean | 338 | 347 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | theorem Part.bind_comp (f : β →. γ) (g : α →. β) (a : Part α) :
a.bind (f.comp g) = (a.bind g).bind f := by
ext c
| Mathlib/Data/PFun.lean | 514 | 516 |
/-
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... | have := @integral_sin_pow_mul_cos_pow_odd a b 0 1
norm_num at this; exact this
/-- Simplification of the integral of `sin x ^ m * cos x ^ n`, case `m` is odd. -/
theorem integral_sin_pow_odd_mul_cos_pow (m n : ℕ) :
(∫ x in a..b, sin x ^ (2 * m + 1) * cos x ^ n) = ∫ u in cos b..cos a, u^n * (↑1 - u ^ 2) ^ m :=
... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 760 | 771 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Ackermann function
In this file, we def... | induction' n with n IH
· simp
· simpa [mul_succ]
| Mathlib/Computability/Ackermann.lean | 82 | 85 |
/-
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.Topology.Bases
import Mathlib.Topology.Compactness.LocallyCompact
import Mathlib.Topology.Compactness.LocallyFinite
/... |
/-- A topological space is σ-compact iff `univ` is σ-compact. -/
lemma isSigmaCompact_univ_iff : IsSigmaCompact (univ : Set X) ↔ SigmaCompactSpace X :=
⟨fun h => ⟨h⟩, fun h => h.1⟩
| Mathlib/Topology/Compactness/SigmaCompact.lean | 147 | 151 |
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