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) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | Mathlib/Order/Filter/NAry.lean | 282 | 286 | |
/-
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.Polynomial.Splits
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.AdjoinRoot
/-!
# Adjoining elements to a field
Some lemmas on ... | (int : IsIntegral R x) (h : Splits (algebraMap R K) (minpoly R x)) (f : K →ₐ[R] L) :
x ∈ f.range :=
show x ∈ Set.range f from Set.image_subset_range _ ((minpoly R x).rootSet K) <| by
rw [image_rootSet h f, mem_rootSet']
| Mathlib/RingTheory/Adjoin/Field.lean | 89 | 92 |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Covariants and contravariants
This file contains general lemma... | @[to_additive]
theorem Group.mulRightReflectLE_of_mulRightMono [Group N] [LE N]
| Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 318 | 319 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | def liftOfRightInverse (hf : Function.RightInverse f_inv f) :
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 738 | 738 |
/-
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... | end List
| Mathlib/Data/List/Basic.lean | 1,323 | 1,324 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# Sums of... | · exact fun _ => by norm_num
· replace hp := H hp (dvd_of_mul_right_dvd hpq)
replace hq := hq (dvd_of_mul_left_dvd hpq)
rw [show 3 = 3 % 4 by norm_num, Ne, ← ZMod.natCast_eq_natCast_iff'] at hp hq ⊢
rw [Nat.cast_mul]
exact help p q hp hq
/-!
### Relation to sums of two squares
-/
/-- If `-1` is a... | Mathlib/NumberTheory/SumTwoSquares.lean | 125 | 138 |
/-
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... | def genWeightSpaceChain : LieSubmodule R L M :=
⨆ k ∈ Ioo p q, genWeightSpace M (k • χ₁ + χ₂)
lemma genWeightSpaceChain_def :
genWeightSpaceChain M χ₁ χ₂ p q = ⨆ k ∈ Ioo p q, genWeightSpace M (k • χ₁ + χ₂) :=
rfl
| Mathlib/Algebra/Lie/Weights/Chain.lean | 104 | 109 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
import Mathlib.Logic.Function.ULift
/-!
# The Yoneda emb... |
lemma CorepresentableBy.isCorepresentable {F : C ⥤ Type v} {X : C} (e : F.CorepresentableBy X) :
F.IsCorepresentable where
has_corepresentation := ⟨X, ⟨e⟩⟩
/-- Alternative constructor for `F.IsCorepresentable`, which takes as an input an
| Mathlib/CategoryTheory/Yoneda.lean | 323 | 328 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.IteratedFDeriv
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Ca... | section General
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E F : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F]
[NormedSpace 𝕜 F] {s t : Set E} {f : E → F} {x : E}
variable (𝕜) in
/-- Definition recording that a function has a symmetric second derivative within a set at
a point... | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | 68 | 172 |
/-
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.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | intro i _
convert abs_real_inner_le_norm (b i) (v i)
simp [b.orthonormal.1 i]
theorem volumeForm_apply_le (v : Fin n → E) : o.volumeForm v ≤ ∏ i : Fin n, ‖v i‖ :=
(le_abs_self _).trans (o.abs_volumeForm_apply_le v)
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 251 | 256 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.Powerset
/-!
# The antidiagonal on a multiset.
The antidiagonal of a multiset `s` consists of all pairs `(t₁, t₂)`
such that `t₁ + t₂ =... | theorem antidiagonal_zero : @antidiagonal α 0 = {(0, 0)} :=
rfl
| Mathlib/Data/Multiset/Antidiagonal.lean | 61 | 62 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | theorem rdropWhile_eq_self_iff : rdropWhile p l = l ↔ ∀ hl : l ≠ [], ¬p (l.getLast hl) := by
simp [rdropWhile, reverse_eq_iff, getLast_eq_getElem, Nat.pos_iff_ne_zero]
variable (p) (l)
theorem dropWhile_idempotent : dropWhile p (dropWhile p l) = dropWhile p l := by
simp only [dropWhile_eq_self_iff]
exact fun h ... | Mathlib/Data/List/DropRight.lean | 144 | 160 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | Mathlib/Data/Real/Sqrt.lean | 527 | 531 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.Analysis.RCLike.Basic
/-! # Further lemmas about `RCLike` -/
open scoped Finset
variable {K ... | instance rclike_to_real : FiniteDimensional ℝ K := ⟨{1, I}, by simp [span_one_I]⟩
variable (K E)
variable [NormedAddCommGroup E] [NormedSpace K E]
| Mathlib/Analysis/RCLike/Lemmas.lean | 53 | 56 |
/-
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 | 935 | 936 | |
/-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
/-!
# Canonical homomorphism from a finite famil... | · rw [one_pow]
exact mul_one _
· intro j hj
simp only [Finset.mem_erase] at hj
simp [hj]
/--
The universal property of `MonoidHom.noncommPiCoprod`
Given monoid morphisms `φᵢ : Nᵢ → M` whose images pairwise commute,
there exists a unique monoid morphism `φ : Πᵢ Nᵢ → M` that induces the `φᵢ`,
and it is ... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 125 | 137 |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | obtain ⟨l, hl⟩ := IsNilpotent.nilpotent R L M₂
let lcs_eq_bot {m n} (N : LieSubmodule R L M) (le : m ≤ n) (hn : lowerCentralSeries R L N m = ⊥) :
lowerCentralSeries R L N n = ⊥ := by
simpa [hn] using antitone_lowerCentralSeries R L N le
| Mathlib/Algebra/Lie/Nilpotent.lean | 312 | 315 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | theorem comap_op_one :
comap (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (1 : Submodule R Aᵐᵒᵖ) = 1 := by
ext
simp
| Mathlib/Algebra/Algebra/Operations.lean | 378 | 381 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Filippo A. E. Nuccio, Sam van Gool
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Order.Irreducible
import Mathlib.Order.Upp... | @[simp] lemma supIrredLowerSet_symm_apply (s : {s : LowerSet α // SupIrred s}) [Fintype s] :
supIrredLowerSet.symm s = (s.1 : Set α).toFinset.sup id := by
classical
obtain ⟨s, hs⟩ := s
obtain ⟨a, rfl⟩ := supIrred_iff_of_finite.1 hs
cases nonempty_fintype α
have : LocallyFiniteOrder α := Fintype.toLocallyF... | Mathlib/Order/Birkhoff.lean | 154 | 161 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,852 | 1,854 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... | ↑(Zsqrtd.norm (x % y)) = Complex.normSq (x - y * (x / y : ℤ[i]) : ℂ) := by simp [mod_def]
_ = Complex.normSq (y : ℂ) * Complex.normSq (x / y - (x / y : ℤ[i]) : ℂ) := by
rw [← normSq_mul, mul_sub, mul_div_cancel₀ _ this]
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 206 | 208 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | l).symm
| Mathlib/SetTheory/Ordinal/Exponential.lean | 211 | 211 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | by_cases Hi : I ≤ f i
· exact Or.inr (Or.inr ⟨i, Finset.mem_insert_self i t, Hi⟩)
have : ¬I ⊓ f a ⊓ f b ⊓ t.inf f ≤ f i := by
simp only [hp.1.inf_le, hp.1.inf_le', not_or]
exact ⟨⟨⟨Hi, Ha⟩, Hb⟩, Ht⟩
rcases Set.not_subset.1 this with ⟨r, ⟨⟨⟨hrI, hra⟩, hrb⟩, hr⟩, hri⟩
| Mathlib/RingTheory/Ideal/Operations.lean | 1,020 | 1,025 |
/-
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.Measure.Typeclasses.SFinite
/-!
# Restriction of a measure to a sub-σ-algebra
## Main definitions
* `MeasureTheory.Measure.trim`: restric... | simp [Measure.trim]
| Mathlib/MeasureTheory/Measure/Trim.lean | 37 | 38 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,283 | 1,284 | |
/-
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
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Bases.Finite
/-!
# (Co)product of a family of f... | is a pricipal filter, see `pi_principal`. -/
theorem mem_pi_principal {t : Set ((i : ι) → α i)} :
t ∈ pi (fun i ↦ 𝓟 (s i)) ↔ ∃ I : Set ι, I.Finite ∧ I.pi s ⊆ t :=
(hasBasis_pi (fun i ↦ hasBasis_principal _)).mem_iff.trans <| by simp
/-- The indexed product of a (possibly, infinite) family of principal filters
i... | Mathlib/Order/Filter/Pi.lean | 142 | 153 |
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | HasFDerivAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') x ↔ HasFDerivAt f f' x := by
simp_rw [← hasFDerivWithinAt_univ, iso.comp_hasFDerivWithinAt_iff]
| Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 116 | 118 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.Algebra.Group.ULift
import Mathlib.Data.Int.ModEq
import M... | (hR : ringChar R ≠ 2) : (2 : R) ≠ 0 := by
rw [Ne, (by norm_cast : (2 : R) = (2 : ℕ)), ringChar.spec, Nat.dvd_prime Nat.prime_two]
exact mt (or_iff_left hR).mp CharP.ringChar_ne_one
-- We have `CharP.neg_one_ne_one`, which assumes `[Ring R] (p : ℕ) [CharP R p] [Fact (2 < p)]`.
-- This is a version using `ringCh... | Mathlib/Algebra/CharP/Basic.lean | 110 | 116 |
/-
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.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | lift b to ℝ≥0 using hb
induction a
· simp
· simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]
exact le_max_left _ _
| Mathlib/Data/ENNReal/Operations.lean | 412 | 416 |
/-
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... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 1,117 | 1,125 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... | ∫⁻ b in s, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
= ∫⁻ x in s ×ˢ t, f x ∂(κ a) := by
conv_rhs => rw [← κ.disintegrate κ.condKernel]
rw [Kernel.setLIntegral_compProd _ _ _ hf hs ht]
lemma setLIntegral_condKernel_univ_right (hf : Measurable f) (a : α) {s : Set β}
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 66 | 71 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | def polarBilin : BilinMap R M N :=
LinearMap.mk₂ R (polar Q) (polar_add_left Q) (polar_smul_left Q) (polar_add_right Q)
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 301 | 302 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... |
/-- If `f` is continuous on the circle `|z - c| = R`, `R > 0`, the `‖f z‖` is less than or equal to
`C : ℝ` on this circle, and this norm is strictly less than `C` at some point `z` of the circle,
| Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 365 | 367 |
/-
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.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | simp [completedHurwitzZetaEven₀]
@[simp]
lemma completedCosZeta_neg (a : UnitAddCircle) (s : ℂ) :
completedCosZeta (-a) s = completedCosZeta a s := by
simp [completedCosZeta]
@[simp]
lemma completedCosZeta₀_neg (a : UnitAddCircle) (s : ℂ) :
completedCosZeta₀ (-a) s = completedCosZeta₀ a s := by
| Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 349 | 358 |
/-
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... | simp_rw [Subtype.ext_iff, MonoidHom.id_apply, coe_mul, mul_assoc, mul_right_inj]
rw [smul_apply_eq_smul_apply_inv_smul, smul_eq_mul_unop, MulOpposite.unop_op, mul_left_inj,
← Subtype.ext_iff, Equiv.apply_eq_iff_eq, inv_smul_eq_iff]
exact left_eq_mul.mpr ((QuotientGroup.eq_one_iff _).mpr h.2)
theorem eq_one_o... | Mathlib/GroupTheory/SchurZassenhaus.lean | 82 | 90 |
/-
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.ObjectProperty.Shift
import Mathlib.CategoryTheory.Triangulated.Pretriangulated
/-!
# t-structures on triangulated categories
This files introdu... | variable (t : TStructure C)
lemma exists_triangle (A : C) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁) :
∃ (X Y : C) (_ : t.le n₀ X) (_ : t.ge n₁ Y) (f : X ⟶ A) (g : A ⟶ Y)
(h : Y ⟶ X⟦(1 : ℤ)⟧), Triangle.mk f g h ∈ distTriang C := by
obtain ⟨X, Y, hX, hY, f, g, h, mem⟩ := t.exists_triangle_zero_one (A⟦n₀⟧)
let T := (Tr... | Mathlib/CategoryTheory/Triangulated/TStructure/Basic.lean | 75 | 86 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... |
end Set
| Mathlib/Data/ENNReal/Basic.lean | 723 | 724 |
/-
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, Johannes Hölzl, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... | b * a < c :=
calc
b * a < b * 1 := mul_lt_mul_left' ha b
_ = b := mul_one b
| Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 579 | 582 |
/-
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.Algebra.Pi
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearIndepe... | apply linearIndependent_iUnion_finite hs'
intro j J _ hiJ
have h₀ :
| Mathlib/LinearAlgebra/StdBasis.lean | 55 | 57 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | end NumberField
open Fintype Module
variable (K)
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 598 | 602 |
/-
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.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | (hv : Pairwise fun i j => ⟪v i, v j⟫ = 0) : |o.volumeForm v| = ∏ i : Fin n, ‖v i‖ := by
rcases n with - | n
· refine o.eq_or_eq_neg_of_isEmpty.elim ?_ ?_ <;> rintro rfl <;> simp
haveI : FiniteDimensional ℝ E := .of_fact_finrank_eq_succ n
have hdim : finrank ℝ E = Fintype.card (Fin n.succ) := by simpa using ... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 262 | 275 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... |
/-- Extract the expression from a normal form. -/
def NormalExpr.e : NormalExpr → Expr
| Mathlib/Tactic/Abel.lean | 145 | 147 |
/-
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.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... |
/-- A real Lipschitz function into a finite dimensional real vector space is differentiable
almost everywhere. For the general Rademacher theorem assuming
that the source space is finite dimensional, see `LipschitzWith.ae_differentiableAt`. -/
theorem LipschitzWith.ae_differentiableAt_real {C : ℝ≥0} {f : ℝ → V} (h : L... | Mathlib/Analysis/BoundedVariation.lean | 107 | 124 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,406 | 2,407 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Yury Kudryashov
-/
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.LocallyFinite
import Mathlib.Topology.Maps.Proper.Basic
import Mathlib.Topology.UniformSpace.Un... | (φ : Π i, C(δ i, α)) (h_proper : ∀ i, IsProperMap (φ i))
(h_lf : LocallyFinite fun i ↦ range (φ i)) (h_cover : ⋃ i, range (φ i) = univ) :
(inferInstanceAs <| UniformSpace C(α, β)) = ⨅ i, .comap (comp · (φ i)) inferInstance := by
-- We check the analogous result for `UniformOnFun` using
| Mathlib/Topology/UniformSpace/CompactConvergence.lean | 340 | 343 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Sophie Morel
-/
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
imp... | -/
@[simps]
def toCocone [DecidableEq J] {A : Type w} [AddCommGroup A] (f : Quot F →+ A) : Cocone F where
pt := AddCommGrp.of A
| Mathlib/Algebra/Category/Grp/Colimits.lean | 180 | 183 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
/-!
# Strong reducibility and degrees.
This file defines the notions of computable many-one reduction and one-one
reduc... | (h : Computable e) : (q ∘ e) ≤₁ q :=
OneOneReducible.mk _ h e.injective
theorem OneOneReducible.of_equiv_symm {α β} [Primcodable α] [Primcodable β] {e : α ≃ β}
(q : β → Prop) (h : Computable e.symm) : q ≤₁ (q ∘ e) := by
| Mathlib/Computability/Reduce.lean | 94 | 98 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... | exact ((hasFDerivAt_jacobiTheta₂ z hτ).comp z (hasFDerivAt_prodMk_left z τ)).hasDerivAt
rwa [← step1.tsum_eq] at step3
lemma continuousAt_jacobiTheta₂' (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
ContinuousAt (fun p : ℂ × ℂ ↦ jacobiTheta₂' p.1 p.2) (z, τ) := by
obtain ⟨T, hT, hτ'⟩ := exists_between hτ
obtain ⟨S, h... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 358 | 373 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... | Mathlib/Data/Sym/Basic.lean | 680 | 690 | |
/-
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... | monoidal_coherence
| Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 42 | 43 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... | · exact Or.inr not_terminatedAt_pred_n
fib_le_of_contsAux_b this
have zero_lt_B : 0 < B := B_ineq.trans_lt' <| cast_pos.2 <| fib_pos.2 n.succ_pos
have : 0 ≤ pB := (cast_nonneg _).trans pB_ineq
have : 0 < ifp.fr :=
ifp_fr_ne_zero.lt_of_le' <| IntFractPair.nth_stream_fr_nonneg stream_nth_e... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 381 | 451 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Floris van Doorn
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# `norm_num` plugin for ... |
lemma Multiset.range_succ' {n nn n' : ℕ} (pn : NormNum.IsNat n nn) (pn' : nn = Nat.succ n') :
| Mathlib/Tactic/NormNum/BigOperators.lean | 178 | 179 |
/-
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.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... |
end Map
section Seq
| Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 101 | 105 |
/-
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.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 316 | 321 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | Mathlib/Order/BooleanAlgebra.lean | 750 | 750 | |
/-
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... | theorem UniformContinuous.prodMk_right {f : α × β → γ} (h : UniformContinuous f) (a) :
UniformContinuous fun b => f (a, b) :=
h.comp (uniformContinuous_const.prodMk uniformContinuous_id)
@[deprecated (since := "2025-03-10")]
| Mathlib/Topology/UniformSpace/Basic.lean | 751 | 755 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
import Mathlib.Topology.FiberBundl... |
theorem mem_trivialization_at_source (b : B) (x : E b) :
⟨b, x⟩ ∈ (a.pretrivializationAt b).source :=
a.toFiberPrebundle.mem_pretrivializationAt_source b x
| Mathlib/Topology/VectorBundle/Basic.lean | 808 | 812 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | (x + y).num * (x.denom * y.denom) = (x.num * y.denom + x.denom * y.num) * (x + y).denom :=
(num_mul_eq_mul_denom_iff (mul_ne_zero (denom_ne_zero x) (denom_ne_zero y))).mpr <| by
conv_lhs => rw [← num_div_denom x, ← num_div_denom y]
rw [div_add_div, RingHom.map_mul, RingHom.map_add, RingHom.map_mul, RingHo... | Mathlib/FieldTheory/RatFunc/Basic.lean | 910 | 913 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | rw [div_eq_inv_mul, mul_smul, h, inv_smul_smul₀]
rwa [inner_self_ne_zero]
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 678 | 679 |
/-
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
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,206 | 2,208 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
/-!... |
end AlgebraicTopology
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 194 | 198 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | change s • (1 : S) ⊗ₜ[R] x = s ⊗ₜ[R] x
rw [TensorProduct.smul_tmul']
congr 1
exact mul_one _
variable {R M N S}
/-- The base change of `M` along `R → S` is linearly equivalent to `S ⊗[R] M`. -/
| Mathlib/RingTheory/IsTensorProduct.lean | 210 | 217 |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | theorem superpolynomialDecay_mul_param_pow_iff (hk : Tendsto k l atTop) (n : ℕ) :
SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f := by
simpa [mul_comm f] using superpolynomialDecay_param_pow_mul_iff f hk n
| Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 267 | 269 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | rcases IH with (IH | IH)
· rcases cases_head IH with (rfl | ⟨e, be, ec⟩)
· exact Or.inr (single bd)
· cases U bd be
exact Or.inl ec
· exact Or.inr (IH.tail bd)
end ReflTransGen
namespace TransGen
| Mathlib/Logic/Relation.lean | 360 | 369 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | protected theorem inf_comm (a b : CauSeq α abs) : a ⊓ b = b ⊓ a := Subtype.ext (inf_comm _ _)
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 789 | 790 |
/-
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... | (by ext s; simp)
@[simp]
lemma algebraTensorAlgEquiv_tmul (a : A) (p : MvPolynomial σ R) :
algebraTensorAlgEquiv R A (a ⊗ₜ p) = a • MvPolynomial.map (algebraMap R A) p := by
simp [algebraTensorAlgEquiv, Algebra.smul_def]
rfl
@[simp]
lemma algebraTensorAlgEquiv_symm_X (s : σ) :
(algebraTensorAlgEquiv R A... | Mathlib/RingTheory/TensorProduct/MvPolynomial.lean | 215 | 225 |
/-
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.NumberTheory.LSeries.RiemannZeta
import Mathlib.NumberTheory.Harmonic.GammaDeriv
/-!
# Asymptotics of `ζ s` as `s → 1`
The goal of this file is to ev... |
/-- The topological sum of `ZetaAsymptotics.term (n + 1) 1` over all `n : ℕ` is `1 - γ`. This is
proved by directly evaluating the sum of the first `N` terms and using the limit definition of `γ`.
-/
lemma term_tsum_one : HasSum (fun n ↦ term (n + 1) 1) (1 - γ) := by
rw [hasSum_iff_tendsto_nat_of_nonneg (fun n ↦ ter... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 122 | 134 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic... | have claim1 : ∀ A : Set X, IsClosed A → A ∈ F → x ∈ A := by
intro A hA h
by_contra H
rw [le_nhds_iff] at cond
| Mathlib/Topology/Category/Compactum.lean | 275 | 278 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... | exact ⟨n, Nat.pos_of_ne_zero (nonZeroDivisors.coe_ne_zero _), hn⟩
theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
AddMonoid.IsTorsion M ↔ Module.IsTorsion ℤ M := by
refine ⟨fun h x => ?_, fun h x => ?_⟩
· obtain ⟨n, h0, hn⟩ := (h x).exists_nsmul_eq_zero
exact
⟨⟨n, mem_nonZeroDivisors_of_ne... | Mathlib/Algebra/Module/Torsion.lean | 890 | 902 |
/-
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.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | rcases w₂ with ⟨⟨_, _, _⟩, _, _⟩
simp_all
open Subgroup.IsComplement
| Mathlib/GroupTheory/PushoutI.lean | 277 | 281 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | rfl
theorem preimage_iterate_eq {f : α → α} {n : ℕ} : Set.preimage f^[n] = (Set.preimage f)^[n] := by
induction n with
| zero => simp
| succ n ih => rw [iterate_succ, iterate_succ', preimage_comp_eq, ih]
theorem preimage_preimage {g : β → γ} {f : α → β} {s : Set γ} :
f ⁻¹' (g ⁻¹' s) = (fun x => g (f x)) ⁻... | Mathlib/Data/Set/Image.lean | 139 | 147 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | rcases exists_null_pairwise_disjoint_diff hd with ⟨u, hum, hu₀, hud⟩
exact
⟨fun i => t i \ u i, fun i => diff_subset.trans (ht_sub _), fun i =>
| Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 226 | 228 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equaliz... | obtain ⟨s, hs', hs⟩ :=
(isCompactOpen_iff_eq_finset_affine_union _).mp
⟨hSU _ _ Set.subset_union_left S.2 hS Set.subset_union_right U.1.2
U.2.isCompact,
| Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 272 | 275 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | insertNthEquiv (fun _ ↦ α) (last n) = snocEquiv (fun _ ↦ α) := by ext; simp
end InsertNth
| Mathlib/Data/Fin/Tuple/Basic.lean | 979 | 981 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | simpa using ih
| Mathlib/GroupTheory/Coxeter/Length.lean | 267 | 267 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | inf_le_right _ _ _ := mem_inf_of_right
le_inf := fun _ _ _ h₁ h₂ _s ⟨_a, ha, _b, hb, hs⟩ => hs.symm ▸ inter_mem (h₁ ha) (h₂ hb)
le_sSup _ _ h₁ _ h₂ := h₂ h₁
sSup_le _ _ h₁ _ h₂ _ h₃ := h₁ _ h₃ h₂
sInf_le _ _ h₁ _ h₂ := by rw [← Filter.sSup_lowerBounds]; exact fun _ h₃ ↦ h₃ h₁ h₂
le_sInf _ _ h₁ _ h₂ := by rw... | Mathlib/Order/Filter/Basic.lean | 231 | 238 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Peter Nelson
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determina... | rfl
/- Let `W` be obtained from the matrix by subtracting `r = (v 0) / (w 0)` times each column
from the next column, starting from the penultimate column. This doesn't change the determinant.-/
set r := v 0 / w 0 with hr
| Mathlib/LinearAlgebra/Vandermonde.lean | 165 | 168 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | Galois object. -/
lemma exists_hom_from_galois_of_fiber (X : C) (x : F.obj X) :
∃ (A : C) (f : A ⟶ X) (a : F.obj A), IsGalois A ∧ F.map f a = x := by
obtain ⟨A, a, h1, h2⟩ := exists_galois_representative F X
obtain ⟨f, hf⟩ := h2.surjective x
exact ⟨A, f, a, h1, hf⟩
| Mathlib/CategoryTheory/Galois/Decomposition.lean | 294 | 299 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,612 | 1,614 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... | @[nontriviality, simp]
lemma iIndepFun.of_subsingleton [Subsingleton ι] {β : ι → Type*} {m : ∀ i, MeasurableSpace (β i)}
{f : ∀ i, Ω → β i} [IsProbabilityMeasure μ] : iIndepFun f μ :=
Kernel.iIndepFun.of_subsingleton
protected lemma iIndepFun.iIndep {m : ∀ i, MeasurableSpace (κ i)} {f : ∀ x : ι, Ω → κ x}
| Mathlib/Probability/Independence/Basic.lean | 226 | 231 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Logic.Equiv.Basic
/-!
# Multiplicative ... | Mathlib/Algebra/Group/Equiv/Basic.lean | 575 | 577 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... |
end Const
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 312 | 313 |
/-
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.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | rw [completedHurwitzZetaEven, WeakFEPair.Λ, sub_div, sub_div]
congr 1
· change completedHurwitzZetaEven₀ a s - (1 / (s / 2)) • (if a = 0 then 1 else 0) / 2 =
| Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 305 | 307 |
/-
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.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... |
theorem fderivWithin_mul_const (hxs : UniqueDiffWithinAt 𝕜 s x)
(hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝔸') :
fderivWithin 𝕜 (fun y => c y * d) s x = d • fderivWithin 𝕜 c s x :=
(hc.hasFDerivWithinAt.mul_const d).fderivWithin hxs
| Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 502 | 506 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... | (hf : Continuous f) (hg : Continuous g) (hspec : ∀ x, f x ⤳ g x) :
Continuous (s.piecewise f g) := by
| Mathlib/Topology/Inseparable.lean | 188 | 189 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Determinant
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Ri... |
/-- Shows that all denominators are nonnegative. -/
theorem zero_le_of_den : 0 ≤ (of v).dens n := by
rw [den_eq_conts_b, nth_cont_eq_succ_nth_contAux]; exact zero_le_of_contsAux_b
theorem le_of_succ_succ_get?_contsAux_b {b : K}
(nth_partDen_eq : (of v).partDens.get? n = some b) :
| Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 275 | 281 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | lt_of_lt_of_le (WithBot.coe_lt_coe.2 n.lt_succ_self) <|
Finset.le_sup <| of_mem_support_derivative hp
theorem degree_derivative_le {p : R[X]} : p.derivative.degree ≤ p.degree :=
letI := Classical.decEq R
| Mathlib/Algebra/Polynomial/Derivative.lean | 167 | 171 |
/-
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... | Mathlib/Data/Fin/VecNotation.lean | 579 | 581 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,391 | 2,392 | |
/-
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.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | theorem isComplement'_of_coprime [Finite G]
(h1 : Nat.card H * Nat.card K = Nat.card G)
(h2 : Nat.Coprime (Nat.card H) (Nat.card K)) : IsComplement' H K :=
isComplement'_of_card_mul_and_disjoint h1 (disjoint_iff.mpr (inf_eq_bot_of_coprime h2))
theorem isComplement'_stabilizer {α : Type*} [MulAction G α] (a :... | Mathlib/GroupTheory/Complement.lean | 820 | 825 |
/-
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, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | @[deprecated (since := "2024-10-28")]
alias IsUniformInducing.inducing := IsUniformInducing.isInducing
@[deprecated (since := "2024-10-28")] alias UniformInducing.inducing := IsUniformInducing.isInducing
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 110 | 114 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | -- we can't invoke the aux lemmas above without obtaining the decidable instance we are
-- already building; but now we've left the data, so we can do this non-constructively
-- without sacrificing computability.
let _inst (f : Perm α) : DecidableRel (SameCycle f) := Classical.decRel _
... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 244 | 255 |
/-
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... | Ioo_subset_Ioo le_rfl h
| Mathlib/Order/Interval/Finset/Basic.lean | 184 | 185 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Joseph Myers
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
/-!
# Bounds on specific values of the exponential
-/
namesp... | Mathlib/Data/Complex/ExponentialBounds.lean | 74 | 75 | |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.ToIntervalMod
import Mathli... |
lemma GrowsPolynomially.of_isEquivalent {f g : ℝ → ℝ} (hg : GrowsPolynomially g)
(hf : f ~[atTop] g) : GrowsPolynomially f := by
have : f = g + (f - g) := by ext; simp
rw [this]
| Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | 710 | 714 |
/-
Copyright (c) 2024 Frédéric Marbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Marbach
-/
import Mathlib.Algebra.Lie.Derivation.AdjointAction
import Mathlib.Algebra.Lie.Killing
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Derivations of ... | ((killingForm R 𝔻).restrict 𝕀).Nondegenerate := by
convert LieAlgebra.IsKilling.killingForm_nondegenerate R 𝕀
exact killingForm_restrict_range_ad R L
/-- The range of the adjoint action on a finite-dimensional Killing Lie algebra is full. -/
@[simp]
lemma range_ad_eq_top : 𝕀 = ⊤ := by
rw [← LieSubalgebra... | Mathlib/Algebra/Lie/Derivation/Killing.lean | 89 | 96 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Data.Set.L... | classical
induction s using Finset.induction_on with
| empty => exact ⟨Not.elim one_ne_zero, fun ⟨_, H, _⟩ => by simp at H⟩
| insert _ _ ha ih => rw [prod_insert ha, mul_eq_zero, exists_mem_insert, ih]
| Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean | 47 | 51 |
/-
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.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... |
instance decidableLT [LT α] [DecidableLT α] : DecidableLT (WithBot α)
| Mathlib/Order/WithBot.lean | 394 | 395 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | trans sin (π / 2 ^ 5)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 742 | 742 |
/-
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 ncard_powerset (s : Set α) (hs : s.Finite := by toFinite_tac) :
(𝒫 s).ncard = 2 ^ s.ncard := by
have h := Cardinal.mk_powerset s
| Mathlib/Data/Set/Card.lean | 587 | 591 |
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