Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | obtain hB | hB := le_or_lt B 0
· rw [convexBodySum_volume_eq_zero_of_le_zero K hB, ofReal_eq_zero.mpr hB, zero_pow, mul_zero]
exact finrank_pos.ne'
· suffices volume (convexBodySum K 1) = (convexBodySumFactor K) by
rw [mul_comm]
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 378 | 382 |
/-
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.Complex.Log
/-! # Power funct... | split_ifs with hx hn ha ha <;> rfl
theorem inv_cpow (x : ℂ) (n : ℂ) (hx : x.arg ≠ π) : x⁻¹ ^ n = (x ^ n)⁻¹ := by
rw [inv_cpow_eq_ite, if_neg hx]
/-- `Complex.inv_cpow_eq_ite` with the `ite` on the other side. -/
theorem inv_cpow_eq_ite' (x : ℂ) (n : ℂ) :
(x ^ n)⁻¹ = if x.arg = π then conj (x⁻¹ ^ conj n) else ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 213 | 224 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | · -- i ≠ j: this entry should be 0
rw [updateCol_transpose, det_transpose]
apply det_zero_of_row_eq h
rw [updateRow_self, updateRow_ne (Ne.symm h)]
theorem cramer_row_self (i : n) (h : ∀ j, b j = A j i) : A.cramer b = Pi.single i A.det := by
rw [← transpose_transpose A, det_transpose]
convert cramer_... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 106 | 116 |
/-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.Flat.Basic
import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.RingTheory.QuotSMulTop... | variable (R) in
lemma biUnion_associatedPrimes_eq_compl_regular [IsNoetherianRing R] :
⋃ p ∈ associatedPrimes R M, p = { r : R | IsSMulRegular M r }ᶜ :=
Eq.trans (biUnion_associatedPrimes_eq_zero_divisors R M) <| by
simp_rw [Set.compl_setOf, isSMulRegular_iff_smul_eq_zero_imp_eq_zero,
| Mathlib/RingTheory/Regular/IsSMulRegular.lean | 141 | 145 |
/-
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.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
/-!
# Computational realization of topological spaces (experi... |
theorem ext' [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ}
(H : ∀ a s, s ∈ 𝓝 a ↔ ∃ b, a ∈ F b ∧ F b ⊆ s) : F.toTopsp = T := by
refine TopologicalSpace.ext_nhds fun x ↦ ?_
ext s
rw [mem_nhds_toTopsp, H]
theorem ext [T : TopologicalSpace α] {σ : Type*} {F : Ctop α σ} (H₁ : ∀ a, IsOpen (F a))
| Mathlib/Data/Analysis/Topology.lean | 142 | 149 |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Al... | LinearMap.coe_restrictScalars, AlgHom.toLinearMap_apply]
rw [coeff_rTensorAlgHom_tmul]
simp only [coeff]
exact (finsuppLeft_apply_tmul_apply _ _ _).symm
| Mathlib/RingTheory/TensorProduct/MvPolynomial.lean | 163 | 166 |
/-
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 [← coe_rpow_of_ne_zero, coe_inj, Real.toNNReal_rpow_of_nonneg hx_pos.le]
simp [hx_pos]
theorem ofReal_rpow_of_nonneg {x p : ℝ} (hx_nonneg : 0 ≤ x) (hp_nonneg : 0 ≤ p) :
ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) := by
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 889 | 893 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | simpa using h2 (-u) rfl hw
theorem unitsSMul_cancels_iff (u : ℤˣ) (w : NormalWord d) :
Cancels (-u) (unitsSMul φ u w) ↔ ¬ Cancels u w := by
by_cases h : Cancels u w
· simp only [unitsSMul, h, dite_true, not_true_eq_false, iff_false]
induction w using consRecOn with
| ofGroup => simp [Cancels, unitsSM... | Mathlib/GroupTheory/HNNExtension.lean | 395 | 410 |
/-
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 | 1,627 | 1,629 | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Additive.Corner.Defs
import Mathlib.Combinatorics.SimpleGraph... |
The maximum density of a 3AP-free set in `G` goes to zero as `|G|` tends to infinity. -/
theorem roth_3ap_theorem (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε ≤ card G)
(A : Finset G) (hAε : ε * card G ≤ #A) : ¬ ThreeAPFree (A : Set G) := by
rintro hA
classical
let B : Finset (G × G) := univ.filter fun (... | Mathlib/Combinatorics/Additive/Corner/Roth.lean | 138 | 158 |
/-
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.Indicator
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Algebra.Order.Group.Synonym
import Mathlib.Algebra.O... | · rw [mulIndicator_of_mem hx]
rw [mem_iUnion] at hx
refine le_antisymm ?_ (iSup_le fun i ↦ mulIndicator_le_self' (fun x _ ↦ h1 ▸ bot_le) x)
rcases hx with ⟨i, hi⟩
exact le_iSup_of_le i (ge_of_eq <| mulIndicator_of_mem hi _)
· rw [mulIndicator_of_not_mem hx]
simp only [mem_iUnion, not_exists] at ... | Mathlib/Algebra/Order/Group/Indicator.lean | 158 | 164 |
/-
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.MorphismProperty.Concrete
import Mathlib.CategoryTheory.Sites.LocallyBijective
/-!
# Morphisms of sheaves factor as a locally surjective followed... | · intro
infer_instance
· intro
let data := (locallySurjective J A).factorizationData (locallyInjective J A) φ
have : IsLocallySurjective data.i := data.hi
have : IsLocallyInjective data.p := data.hp
have : Epi data.p := epi_of_epi_fac data.fac
have := mono_of_isLocallyInjective data.p
ha... | Mathlib/CategoryTheory/Sites/EpiMono.lean | 121 | 134 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finite.Card
import Mathlib.Data.Set.Finite.Range
/-!
# ... | section Normalizer
theorem mem_normalizer_fintype {S : Set G} [Finite S] {x : G} (h : ∀ n, n ∈ S → x * n * x⁻¹ ∈ S) :
x ∈ Subgroup.setNormalizer S := by
| Mathlib/Algebra/Group/Subgroup/Finite.lean | 231 | 234 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Sites.CoverLifting
import Mathlib.CategoryTheory.Sites.CoverPreserving
/-! Localization
In this file, given a Grothendieck topology `J` on a cat... | (overEquiv Y).symm S f ↔ S f.left := by
rfl
lemma overEquiv_iff {X : C} {Y : Over X} (S : Sieve Y) {Z : C} (f : Z ⟶ Y.left) :
overEquiv Y S f ↔ S (Over.homMk f : Over.mk (f ≫ Y.hom) ⟶ Y) := by
obtain ⟨S, rfl⟩ := (overEquiv Y).symm.surjective S
simp
@[simp]
lemma functorPushforward_over_map {X Y : C} (f ... | Mathlib/CategoryTheory/Sites/Over.lean | 93 | 103 |
/-
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... | /-- The family of eigenvalues corresponding to a weight. -/
toFun : L → R
| Mathlib/Algebra/Lie/Weights/Basic.lean | 210 | 211 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Nat.Basic
/-!
# Cast of integers (additional theorems)
This ... | -- type had `HasLiftT`
@[simp, norm_cast]
| Mathlib/Data/Int/Cast/Basic.lean | 74 | 76 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 920 | 921 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | essSup (s.piecewise f g) μ = max (essSup f (μ.restrict s)) (essSup g (μ.restrict sᶜ)) := by
simp only [essSup, limsup_piecewise, blimsup_eq_limsup, ae_restrict_eq, hs, hs.compl]; rfl
theorem essSup_indicator_eq_essSup_restrict {s : Set α} {f : α → ℝ≥0∞} (hs : MeasurableSet s) :
essSup (s.indicator f) μ = ess... | Mathlib/MeasureTheory/Function/EssSup.lean | 266 | 270 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | theorem modByMonic_eq_mod (p : R[X]) (hq : Monic q) : p %ₘ q = p % q :=
show p %ₘ q = p %ₘ (q * C (leadingCoeff q)⁻¹) by
simp only [Monic.def.1 hq, inv_one, mul_one, C_1]
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 326 | 328 |
/-
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... | simp_rw [hE, show Indep = M.Indep from funext (by simp [hI])]
exact M.maximality
subset_ground := by
simp_rw [hE, hB]
exact M.subset_ground
| Mathlib/Data/Matroid/Basic.lean | 787 | 791 |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Control.Lawful
import Batteries.Tactic.Congr
import Batteries.Lean.Except
import Batte... | simp only [callCC, StateT.callCC, StateT.run_bind, callCC_bind_right]; ext; rfl
callCC_bind_left := by
| Mathlib/Control/Monad/Cont.lean | 220 | 221 |
/-
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.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,675 | 1,678 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
impo... | exact (continuousMultilinearCurryFin0 𝕜 E F).symm.toContinuousLinearEquiv.map_tsum
· have h'k : (k : ℕ∞) < N := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (Nat.lt_succ_self _)) hk
have A : Summable fun n => iteratedFDeriv 𝕜 k (f n) 0 :=
.of_norm_bounded (v k) (hv k h'k.le) fun n => h'f k n 0 h'k.le
simp... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 195 | 214 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.LocalizerMorphism
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
/-!
# Composition of localization functors
Given two comp... | lemma of_comp (W₃ : MorphismProperty C₁)
[L₁.IsLocalization W₁] [(L₁ ⋙ L₂).IsLocalization W₃]
(hW₁₃ : W₁ ≤ W₃) (hW₂₃ : W₂ = W₃.map L₁) :
L₂.IsLocalization W₂ := by
have : (L₁ ⋙ W₂.Q).IsLocalization W₃ :=
comp L₁ W₂.Q W₁ W₂ W₃ (fun X Y f hf => Localization.inverts W₂.Q W₂ _
(by simpa only [... | Mathlib/CategoryTheory/Localization/Composition.lean | 110 | 122 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
is useful to define the sum over the empty set, so `AddLeftCancelSemigroup` is not enough. -/
| Mathlib/Algebra/Group/Defs.lean | 662 | 663 |
/-
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.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | rw [monomial, Finsupp.single_zero, MvPowerSeries.monomial_zero_eq_C]
rfl
| Mathlib/RingTheory/PowerSeries/Basic.lean | 234 | 235 |
/-
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.Topology.PartialHomeomorph
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Charted spaces
A smooth manifold is a topological space `M`... | apply t1Space_iff_exists_open.2 (fun x y hxy ↦ ?_)
by_cases hy : y ∈ (chartAt H x).source
· refine ⟨(chartAt H x).source ∩ (chartAt H x)⁻¹' ({chartAt H x y}ᶜ), ?_, ?_, by simp⟩
· exact PartialHomeomorph.isOpen_inter_preimage _ isOpen_compl_singleton
· simp only [preimage_compl, mem_inter_iff, mem_chart_so... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 714 | 723 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | nontriviality A
refine minpoly.unique' A _ pb.minpolyGen_monic pb.aeval_minpolyGen fun q hq =>
or_iff_not_imp_left.2 fun hn0 h0 => ?_
exact (pb.dim_le_degree_of_root hn0 h0).not_lt (pb.degree_minpolyGen ▸ hq)
| Mathlib/RingTheory/PowerBasis.lean | 199 | 203 |
/-
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... | -/
abbrev gradedAlgebra : GradedAlgebra (homogeneousSubmodule σ R) :=
weightedGradedAlgebra R (1 : σ → ℕ)
theorem decomposition.decompose'_apply (φ : MvPolynomial σ R) (i : ℕ) :
(decomposition.decompose' φ i : MvPolynomial σ R) = homogeneousComponent i φ :=
| Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 527 | 532 |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | · refine
(Real.continuousAt_arcsin.comp_continuousWithinAt
((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div
continuous_norm.continuousWithinAt ?_)).add
tendsto_const_nhds
lift z to ℝ using him
simpa using hre.ne
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 608 | 614 |
/-
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.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# Lattice operations on multisets
-/
namespace Multiset
variable {α : Type*}
/-! ###... |
section Inf
| Mathlib/Data/Multiset/Lattice.lean | 89 | 90 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Nat.Basic
/-!
# Cast of integers (additional theorems)
This ... |
end Nat
| Mathlib/Data/Int/Cast/Basic.lean | 39 | 41 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | refine hW.le_bot ⟨hx, fun y hy ↦ ?_⟩
specialize b₁ ⟨y, hy⟩
simp_rw [domRestrict₁₂_apply] at b₁
rw [hB.ortho_comm]
exact b₁
/-- An orthogonal basis with respect to a left-separating bilinear map has no self-orthogonal
elements. -/
theorem IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft [Nontrivial R]
{B... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 759 | 769 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Disjointed
import Mathlib.Algebra.Order.Ring.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!... | lemma card_box : ∀ {n}, n ≠ 0 → #(box n : Finset (ℤ × ℤ)) = 8 * n
| n + 1, _ => by
simp_rw [Prod.card_box_succ, card_Icc, sub_neg_eq_add]
norm_cast
refine tsub_eq_of_eq_add ?_
zify
ring
@[simp] lemma mem_box : ∀ {n}, x ∈ box n ↔ max x.1.natAbs x.2.natAbs = n
| 0 => by simp [Prod.ext_iff]
| Mathlib/Order/Interval/Finset/Box.lean | 88 | 97 |
/-
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.Ring.Int.Units
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
/-!
# fintype instanc... |
theorem Nat.card_eq_card_units_add_one [GroupWithZero α] [Finite α] :
Nat.card α = Nat.card αˣ + 1 := by
rw [Nat.card_units, tsub_add_cancel_of_le Nat.card_pos]
| Mathlib/Data/Fintype/Units.lean | 42 | 46 |
/-
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.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import... |
end ZMod
section ZModChar
variable {C : Type v} [CommMonoid C]
section ZModCharDef
/-- We can define an additive character on `ZMod n` when we have an `n`th root of unity `ζ : C`. -/
def zmodChar (n : ℕ) [NeZero n] {ζ : C} (hζ : ζ ^ n = 1) : AddChar (ZMod n) C where
| Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 123 | 134 |
/-
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 | 379 | 393 | |
/-
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... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 239 | 243 | |
/-
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... | @[simp]
theorem sign_neg (θ : Angle) : (-θ).sign = -θ.sign := by
simp_rw [sign, sin_neg, Left.sign_neg]
theorem sign_antiperiodic : Function.Antiperiodic sign (π : Angle) := fun θ => by
rw [sign, sign, sin_add_pi, Left.sign_neg]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 712 | 718 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | @[to_additive (attr := simp) mod_card_zsmul]
theorem zpow_mod_card (a : G) (n : ℤ) : a ^ (n % Fintype.card G : ℤ) = a ^ n := by
rw [eq_comm, ← zpow_mod_orderOf, ← Int.emod_emod_of_dvd n
(Int.natCast_dvd_natCast.2 orderOf_dvd_card), zpow_mod_orderOf]
| Mathlib/GroupTheory/OrderOfElement.lean | 927 | 931 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | obtain ⟨_|_, hi⟩ := i <;> simp
lemma map_comp {i j k : Fin (n + 1 + 1)} (hij : i ≤ j) (hjk : j ≤ k) :
| Mathlib/CategoryTheory/ComposableArrows.lean | 339 | 341 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | rw [zero_smul_set hs, ← singleton_zero]
by_cases h : finrank ℝ E = 0
· haveI : Subsingleton E := finrank_zero_iff.1 h
simp only [h, one_mul, ENNReal.ofReal_one, abs_one, Subsingleton.eq_univ_of_nonempty hs,
pow_zero, Subsingleton.eq_univ_of_nonempty (singleton_nonempty (0 : E))]
| Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 368 | 372 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | @[simp]
theorem kernelSubobjectIsoComp_hom_arrow {X' : C} (f : X' ⟶ X) [IsIso f] (g : X ⟶ Y) [HasKernel g] :
(kernelSubobjectIsoComp f g).hom ≫ (kernelSubobject g).arrow =
(kernelSubobject (f ≫ g)).arrow ≫ f := by
| Mathlib/CategoryTheory/Subobject/Limits.lean | 181 | 184 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Na... | exact B.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one hf.contDiffOn hg.contDiffOn
uniqueDiffOn_univ (mem_univ x) hn hB
section
variable {𝕜' : Type*} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' F]
[IsScalarTower 𝕜 𝕜' F]
| Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 237 | 244 |
/-
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... | Mathlib/Data/List/Basic.lean | 3,485 | 3,486 | |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | reflTransGen_of_succ _ (fun i _ => K i) fun i _ => by
rw [inter_comm]
exact K i
/-- The iUnion of connected sets indexed by a type with an archimedean successor (like `ℕ` or `ℤ`)
such that any two neighboring sets meet is connected. -/
theorem IsConnected.iUnion_of_chain [Nonempty β] {s : β → Set α} ... | Mathlib/Topology/Connected/Basic.lean | 211 | 217 |
/-
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... | Mathlib/Order/LiminfLimsup.lean | 1,247 | 1,251 | |
/-
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
-/
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Ring.Hom.Defs
i... | cases a <;> cases b <;> aesop
| Mathlib/Algebra/Order/Ring/WithTop.lean | 57 | 57 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Homology
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# Homolo... | @[simp]
lemma opcyclesMap'_sub :
opcyclesMap' (φ - φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ -
opcyclesMap' φ' h₁ h₂ := by
simp only [sub_eq_add_neg, opcyclesMap'_add, opcyclesMap'_neg]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 241 | 245 |
/-
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... | exact image2_union_right
| Mathlib/Data/Finset/NAry.lean | 155 | 156 |
/-
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... | exact i h
/-- Let `R` satisfy the strong rank condition. If `m` elements of a free rank `n` `R`-module are
linearly independent, then `m ≤ n`. -/
theorem Basis.card_le_card_of_linearIndependent_aux {R : Type*} [Semiring R] [StrongRankCondition R]
(n : ℕ) {m : ℕ} (v : Fin m → Fin n → R) : LinearIndependent R v ... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 294 | 299 |
/-
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.MellinTransform
/-!
# Abstract functional equations for Mellin transforms
This file formalises a general version of an argument used to pro... | namespace WeakFEPair
/-!
## Auxiliary results I: lemmas on asymptotics
-/
/-- As `x → 0`, we have `f x = x ^ (-P.k) • constant` up to a rapidly decaying error. -/
lemma hf_zero (P : WeakFEPair E) (r : ℝ) :
(fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) =O[𝓝[>] 0] (· ^ r) := by
have := (P.hg_top (-(r + P.k))).c... | Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean | 135 | 156 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | end L1
section Function
variable [CompleteSpace F] {T T' T'' : Set α → E →L[ℝ] F} {C C' C'' : ℝ} {f g : α → E}
variable (μ T)
open Classical in
/-- Extend `T : Set α → E →L[ℝ] F` to `(α → E) → F` (for integrable functions `α → E`). We set it to
0 if the function is not integrable. -/
def setToFun (hT : DominatedFinM... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 606 | 631 |
/-
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.ClosedImmersion
import Mathlib.AlgebraicGeometry.PullbackCarrier
import Mathlib.Topology.LocalAtTarget
/-!
# Universally closed ... |
instance universallyClosed_snd {X Y Z : Scheme} (f : X ⟶ Z) (g : Y ⟶ Z) [hf : UniversallyClosed f] :
UniversallyClosed (pullback.snd f g) :=
MorphismProperty.pullback_snd f g hf
instance universallyClosed_isLocalAtTarget : IsLocalAtTarget @UniversallyClosed := by
rw [universallyClosed_eq]
| Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 88 | 94 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | theorem U_eval_neg_one (n : ℤ) : (U R n).eval (-1) = n.negOnePow * (n + 1) := by
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp; norm_num
| add_two n ih1 ih2 =>
simp only [U_add_two, eval_sub, eval_mul, eval_ofNat, eval_X, mul_neg, mul_one, ih1,
Int.cast_add, Int.cast_... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 241 | 250 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
alias ⟨IndepFun.measure_inter_preimage_eq_mul, _⟩ := indepFun_iff_measure_inter_preimage_eq_mul
theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Type*}
| Mathlib/Probability/Independence/Basic.lean | 567 | 570 |
/-
Copyright (c) 2022 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# The filter of small sets
This file defines the ... | (comap f l).smallSets = comap (image f) l.smallSets := by
refine (gc_map_comap _).u_comm_of_l_comm (gc_map_comap _) bind_smallSets_gc bind_smallSets_gc ?_
simp [Function.comp_def, map_bind, bind_map]
| Mathlib/Order/Filter/SmallSets.lean | 125 | 128 |
/-
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.Finset.Card
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# The pow... | rw [← hu.2]
exact Subset.trans (erase_insert_subset a u) hu.1
· have : ¬∃ u : Finset α, u ⊆ s ∧ insert a u = t := by simp [Ne.symm (ne_insert_of_not_mem _ _ h)]
simp [Finset.erase_eq_of_not_mem h, this]
| Mathlib/Data/Finset/Powerset.lean | 93 | 96 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | theorem compLinearMap_apply (g : MultilinearMap R M₁' M₂) (f : ∀ i, M₁ i →ₗ[R] M₁' i)
(m : ∀ i, M₁ i) : g.compLinearMap f m = g fun i => f i (m i) :=
rfl
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 359 | 362 |
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... |
end completeTheory
end Language
| Mathlib/ModelTheory/Satisfiability.lean | 453 | 456 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... |
theorem mem_nonZeroDivisors_of_leadingCoeff {p : R[X]} (h : p.leadingCoeff ∈ R⁰) : p ∈ R[X]⁰ :=
mem_nonzeroDivisors_of_coeff_mem _ h
| Mathlib/Algebra/Polynomial/RingDivision.lean | 124 | 126 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | {G : Subgroup Kˣ} [Fintype G] [Decidable (G = ⊥)] :
∑ x : G, (x.val : K) = if G = ⊥ then 1 else 0 := by
by_cases G_bot : G = ⊥
· subst G_bot
simp only [univ_unique, sum_singleton, ↓reduceIte, Units.val_eq_one, OneMemClass.coe_eq_one]
rw [Set.default_coe_singleton]
rfl
· simp only [G_bot, ite_f... | Mathlib/FieldTheory/Finite/Basic.lean | 167 | 175 |
/-
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... | · exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
end NonUnitalNonAssocRing
| Mathlib/Algebra/Ring/Ext.lean | 176 | 180 |
/-
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.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | refine ⟨fun a => ?_⟩
obtain ⟨n, hn⟩ := h (a.swap : _ →r _).toOrderHom
exact hn n.succ n.lt_succ_self.le ((RelEmbedding.map_rel_iff _).2 n.lt_succ_self)
theorem WellFoundedGT.monotone_chain_condition' [Preorder α] [h : WellFoundedGT α] (a : ℕ →o α) :
∃ n, ∀ m, n ≤ m → ¬a n < a m :=
wellFoundedGT_iff_m... | Mathlib/Order/OrderIsoNat.lean | 216 | 224 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomologicalFunctor
import Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
import Mathlib.Algebra.Homology.HomologySeque... | simp [descShortComplex]
@[reassoc (attr := simp)]
lemma inl_v_descShortComplex_f (i j : ℤ) (h : i + (-1) = j) :
| Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean | 56 | 59 |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Ma... | def piCurry {ι : Type*} {κ : ι → Type*} (α : ∀ i, κ i → Type*)
[∀ i k, AddCommMonoid (α i k)] [∀ i k, Module R (α i k)] :
| Mathlib/LinearAlgebra/Pi.lean | 458 | 459 |
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Bhavik Mehta, Eric Wieser
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import ... |
See also `Finset.antidiagonal_congr`. -/
lemma antidiagonal_congr' (hp : p ∈ antidiagonal n) (hq : q ∈ antidiagonal n) :
p = q ↔ p.2 = q.2 := by
rw [← Prod.swap_inj]
exact antidiagonal_congr (swap_mem_antidiagonal.2 hp) (swap_mem_antidiagonal.2 hq)
| Mathlib/Algebra/Order/Antidiag/Prod.lean | 116 | 122 |
/-
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.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | (Option.casesOn (Sum.casesOn (f a) (fun _ => Option.none) Option.some : Option γ)
(some (Sum.casesOn (f a) (fun b => some (g a b)) fun _ => Option.none)) fun c =>
(h a c).map Option.some :
Part (Option σ)) :=
| Mathlib/Computability/Partrec.lean | 700 | 703 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.List.Nodup
/-!
# Antidiagonals in ℕ × ℕ as lists
This file defines the antidiagonals of ℕ × ℕ as lists: the `n`-th antidiagonal is the list of
p... | antidiagonal (n + 1) = (antidiagonal n).map (Prod.map id Nat.succ) ++ [(n + 1, 0)] := by
simp only [antidiagonal, range_succ, Nat.add_sub_cancel_left, map_append, append_assoc,
Nat.sub_self, singleton_append, map_map, map]
congr 1
apply map_congr_left
simp +contextual [le_of_lt, Nat.sub_add_comm]
| Mathlib/Data/List/NatAntidiagonal.lean | 68 | 73 |
/-
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... | rw [totient_prime_pow (prime_of_mem_primeFactors hp) h]
/-- Euler's product formula for the totient function. -/
theorem totient_mul_prod_primeFactors (n : ℕ) :
(φ n * ∏ p ∈ n.primeFactors, p) = n * ∏ p ∈ n.primeFactors, (p - 1) := by
by_cases hn : n = 0; · simp [hn]
rw [totient_eq_prod_factorization hn]
n... | Mathlib/Data/Nat/Totient.lean | 265 | 272 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Order.Fin.Tuple
/-!
# Collections ... | @[simp]
theorem antidiagonalTuple_one (n : ℕ) : antidiagonalTuple 1 n = [![n]] := by
simp_rw [antidiagonalTuple, antidiagonal, List.range_succ, List.map_append, List.map_singleton,
Nat.sub_self, List.flatMap_append, List.flatMap_singleton, List.flatMap_map]
conv_rhs => rw [← List.nil_append [![n]]]
congr 1
| Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 120 | 125 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.Adjunction
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-!
# The pullback of a shift by a monoid morphism
Given a shift by a... | lemma pullbackShiftFunctorZero'_inv_app :
(shiftFunctorZero _ A).inv.app X = (shiftFunctorZero' C (φ 0) (by rw [map_zero])).inv.app X ≫
(pullbackShiftIso C φ 0 (φ 0) rfl).inv.app X := by
rw [pullbackShiftFunctorZero_inv_app]
simp only [Functor.id_obj, pullbackShiftIso, eqToIso.inv, eqToHom_app, shiftFunct... | Mathlib/CategoryTheory/Shift/Pullback.lean | 93 | 103 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum... | Mathlib/NumberTheory/ADEInequality.lean | 259 | 268 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,038 | 1,039 | |
/-
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... | Mathlib/Topology/UniformSpace/Basic.lean | 1,318 | 1,321 | |
/-
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
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... |
theorem restrict_union₀ (h : AEDisjoint μ s t) (ht : NullMeasurableSet t μ) :
μ.restrict (s ∪ t) = μ.restrict s + μ.restrict t := by
simp [← restrict_union_add_inter₀ s ht, restrict_zero_set h]
| Mathlib/MeasureTheory/Measure/Restrict.lean | 243 | 247 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.AbstractCompletion
/-!
# Hausdorff completions of uniform spaces
The goal is to construct a left-adjoint to the... | instance [h : Nonempty α] : Nonempty (CauchyFilter α) :=
h.recOn fun a => Nonempty.intro <| CauchyFilter.pureCauchy a
section Extend
open Classical in
| Mathlib/Topology/UniformSpace/Completion.lean | 215 | 220 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.Sheaf.ChangeOfRings
import Mathlib.CategoryTheory.Sites.LocallySurjective
/-!
# The associated sheaf of a presheaf of modules
In thi... | protected lemma smul_add : smul α φ r (m + m') = smul α φ r m + smul α φ r m' := by
let S := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve φ m ⊓ Presheaf.imageSieve φ m'
have hS : S ∈ J X.unop := by
refine J.intersection_covering (J.intersection_covering ?_ ?_) ?_
all_goals apply Presheaf.imageSieve_mem
a... | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean | 238 | 249 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | intro xs
by_cases Hx : x ∈ ⋃ i : ↥s × ↥s, T i
· rw [← hA] at Hx
obtain ⟨p, pA, xp⟩ : ∃ p : ↥s × ↥s, p ∈ A ∧ x ∈ T p := by
simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx
apply h'x p pA ⟨xs, xp⟩
· exact False.elim (hx ⟨xs, Hx⟩)
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 659 | 686 |
/-
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.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | rfl
| Mathlib/Order/Interval/Multiset.lean | 194 | 194 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... |
@[simp]
| Mathlib/Algebra/GroupWithZero/Basic.lean | 226 | 227 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Prime.Pow
import Mathlib.NumberTheory.Divisors
/-!
# Prime powers a... | · rintro ⟨p, k, hp, hk, rfl⟩
refine ⟨p, ⟨hp, dvd_pow_self _ hk.ne'⟩, ?_⟩
rintro q ⟨hq, hq'⟩
exact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')
rintro ⟨p, ⟨hp, hn⟩, hq⟩
rcases eq_or_ne n 0 with (rfl | hn₀)
· cases (hq 2 ⟨Nat.prime_two, dvd_zero 2⟩).trans (hq 3 ⟨Nat.prime_three, dvd_ze... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 89 | 108 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | exact mem_iUnion₂.mpr ⟨y, hy₁, hy₂.le⟩
/-- The closed thickening of a compact set `E` is the union of the balls `Metric.closedBall x δ`
over `x ∈ E`.
| Mathlib/Topology/MetricSpace/Thickening.lean | 528 | 532 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... |
end norm_num_ext
end LucasLehmer
/-!
This implementation works successfully to prove `(2^4423 - 1).Prime`,
and all the Mersenne primes up to this point appear in [Archive/Examples/MersennePrimes.lean].
These can be calculated nearly instantly, and `(2^9689 - 1).Prime` only fails due to deep
recursion.
(Note by kmil... | Mathlib/NumberTheory/LucasLehmer.lean | 599 | 617 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,646 | 1,648 | |
/-
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 | 2,405 | 2,406 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathli... | @[simp]
lemma num_eq_zero (x : K) : IsFractionRing.num A x = 0 ↔ x = 0 :=
| Mathlib/RingTheory/Localization/NumDen.lean | 93 | 94 |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | /-- Two measures which are finite on closed-open intervals are equal if they agree on all
open-closed intervals. -/
theorem ext_of_Ioc {α : Type*} [TopologicalSpace α] {_m : MeasurableSpace α}
[SecondCountableTopology α] [ConditionallyCompleteLinearOrder α] [OrderTopology α]
[BorelSpace α] [NoMinOrder α] (μ ν :... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 408 | 413 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | split_ifs with h
· conv_rhs => rw [← coeff_mul_monomial _ s]
congr with t
| Mathlib/Algebra/MvPolynomial/Basic.lean | 703 | 705 |
/-
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.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | ∀ x y : Y, f (Sum.inr x, Sum.inr y) = dist x y) ∧
∀ x y, f (x, y) = f (y, x)) ∧
∀ x y z, f (x, z) ≤ f (x, y) + f (y, z)) ∧
∀ x, f (x, x) = 0) ∧
| Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 76 | 79 |
/-
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.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 807 | 808 | |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | rcases H2 with (⟨hl, rfl⟩ | ⟨hr, rfl⟩) <;> rcases H1 with (h | ⟨_, h₂⟩)
· exact le_trans (Nat.le_add_left _ _) (le_trans h (Nat.le_add_left _ _))
· exact
| Mathlib/Data/Ordmap/Ordset.lean | 316 | 318 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... | AntivaryOn (f ^ n) g s := fun _i hi _j hj hij ↦ pow_le_pow_left' (hfg hi hj hij) _
| Mathlib/Algebra/Order/Monovary.lean | 62 | 62 |
/-
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... | /-! ### Normal ordinal functions -/
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 359 | 360 |
/-
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 | 1,476 | 1,477 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,174 | 1,176 | |
/-
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.Group.Action.Pointwise.Finset
import Mathlib.Data.Real.Basic
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Tactic.Positivity.Finset
/-!
... | variable {A B : Set G}
/-- **Ruzsa's covering lemma** for sets. See also `Finset.ruzsa_covering_mul`. -/
@[to_additive "**Ruzsa's covering lemma** for sets. See also `Finset.ruzsa_covering_add`."]
lemma ruzsa_covering_mul (hA : A.Finite) (hB : B.Finite) (hB₀ : B.Nonempty)
(hK : Nat.card (A * B) ≤ K * Nat.card B) :... | Mathlib/Combinatorics/Additive/RuzsaCovering.lean | 62 | 75 |
/-
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
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | (tsub_le_iff_right.1 <| le_iInf fun _ => tsub_le_iff_right.2 <| iInf_le _ _)
| Mathlib/Data/ENNReal/Real.lean | 429 | 429 |
/-
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.NumberTheory.LegendreSymbol.JacobiSymbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... | change 2 ∣ (a : ℤ).gcd b at h
rw [hf, ← even_iff_two_dvd] at h
exact Nat.not_even_one h
| Mathlib/Tactic/NormNum/LegendreSymbol.lean | 103 | 105 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | | _, ⟨_, rfl⟩ => ⟨_, ⟨_, rfl⟩, rfl⟩,
| Mathlib/Data/Part.lean | 379 | 379 |
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