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) 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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | theorem mem_nthRootsFinset {n : ℕ} (h : 0 < n) (a : R) {x : R} :
x ∈ nthRootsFinset n a ↔ x ^ (n : ℕ) = a := by
classical
rw [nthRootsFinset_def, mem_toFinset, mem_nthRoots h]
@[simp]
theorem nthRootsFinset_zero (a : R) : nthRootsFinset 0 a = ∅ := by
classical simp [nthRootsFinset_def]
theorem map_mem_nthRo... | Mathlib/Algebra/Polynomial/Roots.lean | 324 | 338 |
/-
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... | theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} :
span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by
simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]
constructor
· intro h zI hzI
exact h x (dvd_refl x) zI hzI
· rintro h _ ⟨z, rfl⟩ zI hzI
rw [mul_comm x z, mul_assoc]
exact J.mul_... | Mathlib/RingTheory/Ideal/Operations.lean | 446 | 457 |
/-
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... | theorem OneOneReducible.to_many_one {α β} [Primcodable α] [Primcodable β] {p : α → Prop}
{q : β → Prop} : p ≤₁ q → p ≤₀ q
| Mathlib/Computability/Reduce.lean | 89 | 90 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | induction n generalizing s with
| zero =>
intro hs
refine (Gamma_pos_of_pos ?_).ne'
rwa [Nat.cast_zero, neg_zero] at hs
| succ _ n_ih =>
intro hs'
have : Gamma (s + 1) ≠ 0 := by
apply n_ih
· intro m
specialize hs (1 + m)
contrapose! hs
rw [← eq_sub_iff_add_e... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 508 | 521 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angle between complex numbers
This file rel... | lemma angle_one_right (hx : x ≠ 0) : angle x 1 = |x.arg| := by simp [angle_eq_abs_arg, hx]
| Mathlib/Analysis/Complex/Angle.lean | 46 | 46 |
/-
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.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... |
namespace Monotone
/-- Pointwise supremum of two monotone functions is a monotone function. -/
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
(hg : Monotone g) :
Monotone (f ⊔ g) := fun _ _ h => sup_le_sup (hf h) (hg h)
| Mathlib/Order/Lattice.lean | 853 | 859 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... |
theorem toOuterMeasure_apply : p.toOuterMeasure s = ∑' x, s.indicator p x :=
tsum_congr fun x => smul_dirac_apply (p x) x s
| Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 136 | 138 |
/-
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.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... |
end NClique
/-! ### Graphs without cliques -/
section CliqueFree
variable {m n : ℕ}
/-- `G.CliqueFree n` means that `G` has no `n`-cliques. -/
def CliqueFree (n : ℕ) : Prop :=
∀ t, ¬G.IsNClique n t
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 302 | 315 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | @[deprecated (since := "2025-04-22")] alias setLaverage_lt_top := setLAverage_lt_top
theorem laverage_add_measure :
⨍⁻ x, f x ∂(μ + ν) =
μ univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂μ + ν univ / (μ univ + ν univ) * ⨍⁻ x, f x ∂ν := by
| Mathlib/MeasureTheory/Integral/Average.lean | 158 | 162 |
/-
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... | refine ⟨fun h a b has hbs ↦ ?_,
fun h ⟨x, hx⟩ ↦ ⟨x, ((singleton_subset_iff.2 hx).antisymm' (fun y hy ↦ h _ _ hy hx))⟩⟩
obtain ⟨x, rfl⟩ := h ⟨_, has⟩
rw [(has : a = x), (hbs : b = x)]
theorem encard_le_one_iff_subsingleton : s.encard ≤ 1 ↔ s.Subsingleton := by
rw [encard_le_one_iff, Set.Subsingleton]
taut... | Mathlib/Data/Set/Card.lean | 315 | 322 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,018 | 1,021 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... |
* `0 + b = b`
| Mathlib/Tactic/Ring/Basic.lean | 364 | 365 |
/-
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.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | · have : Nonempty {w : InfinitePlace K // IsComplex w} := ⟨⟨w, hw⟩⟩
exact nontrivial_prod_right
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 209 | 211 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | (∃ x', x' ∈ I ∧ algebraMap R P x' = algebraMap R P x) ↔ x ∈ I :=
⟨fun ⟨_, hx', Eq⟩ => IsLocalization.injective _ h Eq ▸ hx', fun h => ⟨x, h, rfl⟩⟩
theorem coeIdeal_injective' (h : S ≤ nonZeroDivisors R) :
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 298 | 301 |
/-
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 | 1,632 | 1,637 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 718 | 720 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Yury Kudryashov
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
/-!
# Diffeomorphisms
This file implem... |
section
| Mathlib/Geometry/Manifold/Diffeomorph.lean | 547 | 549 |
/-
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.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... |
@[simp]
theorem sup_lt_iff : b ⊔ c < a ↔ b < a ∧ c < a :=
⟨fun h => ⟨le_sup_left.trans_lt h, le_sup_right.trans_lt h⟩,
fun h => sup_ind (p := (· < a)) b c h.1 h.2⟩
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) :=
| Mathlib/Order/Lattice.lean | 678 | 684 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | · rw [ha, monomial_zero_right, mirror_zero]
| Mathlib/Algebra/Polynomial/Mirror.lean | 44 | 44 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 936 | 938 | |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
/-!
# Roots of cyclotomic polynomials.
We gather results abou... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 213 | 223 | |
/-
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... |
lemma div_sq_cancel (a b : G₀) : a ^ 2 * b / a = a * b := by
obtain rfl | ha := eq_or_ne a 0
· simp
| Mathlib/Algebra/GroupWithZero/Basic.lean | 449 | 452 |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | (zero : f 0 = 0) : SMulPosReflectLT α β where
elim b hb a₁ a₂ h := by
simp only [← hf, ← lt_iff_lt_of_le_iff_le' hf hf, zero, smul] at *
| Mathlib/Algebra/Order/Module/Defs.lean | 1,118 | 1,120 |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Patrick Stevens
-/
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Tactic.Common
import Mathlib.Algebra.Field.Defs
... | Mathlib/Data/Nat/Cast/Field.lean | 53 | 58 | |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | u ~[l] v ↔ ∃ (φ : α → β) (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v := by
rw [IsEquivalent, isLittleO_iff_exists_eq_mul]
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 176 | 177 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
/-!
# Type Spaces
This file defines the space of complete types over a first-order theory.
(Note that types in model theor... |
variable {T} {v : α → M}
@[simp]
theorem mem_typeOf {φ : L[[α]].Sentence} :
| Mathlib/ModelTheory/Types.lean | 165 | 169 |
/-
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-... | theorem IsMaximal.mem_of_models (h : T.IsMaximal) {φ : L.Sentence} (hφ : T ⊨ᵇ φ) : φ ∈ T := by
refine (h.mem_or_not_mem φ).resolve_right fun con => ?_
rw [models_iff_not_satisfiable, Set.union_singleton, Set.insert_eq_of_mem con] at hφ
exact hφ h.1
| Mathlib/ModelTheory/Satisfiability.lean | 427 | 430 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | x ∈ S ↔ ∃ s : R, x = s • generator S := by
simp_rw [@eq_comm _ x, ← mem_span_singleton, span_singleton_generator]
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 104 | 106 |
/-
Copyright (c) 2023 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Order.Filter.Germ.Basic
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.Normed.Module.Basic
/-! ... | intro hx
apply ((hf hx).and <| (eventually_nhdsSet_iff_forall.mp h x hx).eventually_nhds).mono
rintro y ⟨hy, h'y⟩
rwa [Germ.coe_eq.mpr h'y]
theorem forall_restrictGermPredicate_iff {P : ∀ x : X, Germ (𝓝 x) Y → Prop} :
(∀ x, RestrictGermPredicate P A x f) ↔ ∀ᶠ x in 𝓝ˢ A, P x f := by
| Mathlib/Topology/Germ.lean | 104 | 110 |
/-
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.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... |
theorem intCast_eq_intCast_iff (a b : ℤ) (c : ℕ) : (a : ZMod c) = (b : ZMod c) ↔ a ≡ b [ZMOD c] :=
CharP.intCast_eq_intCast (ZMod c) c
theorem intCast_eq_intCast_iff' (a b : ℤ) (c : ℕ) : (a : ZMod c) = (b : ZMod c) ↔ a % c = b % c :=
ZMod.intCast_eq_intCast_iff a b c
theorem val_intCast {n : ℕ} (a : ℤ) [NeZero n... | Mathlib/Data/ZMod/Basic.lean | 477 | 485 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.Pointwise
/-!
# Submonoid of inverses
Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid ... | This is a `MonoidHom` when `M` is commutative. -/
@[to_additive
"The function from `S.leftAdd` to `S` sending an element to its right additive
inverse in `S`. This is an `AddMonoidHom` when `M` is commutative."]
noncomputable def fromLeftInv : S.leftInv → S := fun x ↦ x.prop.choose
@[to_additive (attr := simp)]
... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 87 | 94 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Results about `minpoly... | if σ₁ x = σ₂ x then σ₁ (aeval x (derivative <| minpoly R x)) else 0 := by
apply eval₂_minpolyDiv_of_eval₂_eq_zero
rw [AlgHom.comp_algebraMap, ← σ₂.comp_algebraMap, ← eval₂_map, ← RingHom.coe_coe, eval₂_hom,
eval_map, ← aeval_def, minpoly.aeval, map_zero]
lemma eval_minpolyDiv_of_aeval_eq_zero [IsDomain S... | Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean | 68 | 76 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | lhs
rw [← num_divInt_den p, ← num_divInt_den q]
| Mathlib/Data/Rat/Defs.lean | 340 | 341 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... |
lemma truncatedSup_union_left (hs : a ∈ lowerClosure s) (ht : a ∉ lowerClosure t) :
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 159 | 160 |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.Linea... | exact (finrank_top K L).ge
exact this ▸ minpoly.irreducible (IsIntegral.of_finite K α)
| Mathlib/FieldTheory/KummerExtension.lean | 512 | 514 |
/-
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.HurwitzZetaEven
import Mathlib.NumberTheory.LSeries.HurwitzZetaOdd
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
/-!
# The... | /-- Expression for `hurwitzZeta a 1` as a limit. (Mathematically `hurwitzZeta a 1` is
undefined, but our construction assigns some value to it; this lemma is mostly of interest for
determining what that value is). -/
lemma tendsto_hurwitzZeta_sub_one_div_nhds_one (a : UnitAddCircle) :
Tendsto (fun s ↦ hurwitzZeta a... | Mathlib/NumberTheory/LSeries/HurwitzZeta.lean | 89 | 96 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... |
@[fun_prop]
theorem DifferentiableWithinAt.const_sub (hf : DifferentiableWithinAt 𝕜 f s x) (c : F) :
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 671 | 673 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Monoid with zero and group with zero homomorphisms
This file defines homomorphi... | map_zero' := by dsimp; rw [map_zero, zero_mul] }
end MonoidWithZeroHom
| Mathlib/Algebra/GroupWithZero/Hom.lean | 215 | 218 |
/-
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
... | Differentiable ℂ (cosZeta a) :=
fun _ ↦ differentiableAt_cosZeta a (Or.inr ha)
| Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 723 | 725 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | lemma add'_assoc (f₁ f₂ f₃ : L.obj X ⟶ L.obj Y) :
add' W (add' W f₁ f₂) f₃ = add' W f₁ (add' W f₂ f₃) := by
obtain ⟨α, h₁, h₂, h₃⟩ := exists_leftFraction₃ L W f₁ f₂ f₃
rw [add'_eq W f₁ f₂ α.forgetThd h₁ h₂, add'_eq W f₂ f₃ α.forgetFst h₂ h₃,
add'_eq W _ _ (LeftFraction₂.mk (α.f + α.f') α.f'' α.s α.hs) rfl h... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 169 | 176 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,403 | 1,405 | |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | intro o
induction' o using limitRecOn with o IH o l IH
· intro h₁
refine ⟨0, le_antisymm ?_ h₁⟩
rw [derivFamily_zero]
exact nfpFamily_le_fp (fun i => (H i).monotone) (Ordinal.zero_le _) ha
· intro h₁
rcases le_or_lt a (derivFamily f o) with h | h
· exact IH h
refine ⟨... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 183 | 193 |
/-
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.Nat
import Mathlib.Data.PNat.Defs
/-!
# Finite intervals of positive naturals
This file proves that `ℕ+` is a `LocallyFiniteOrder` ... | Mathlib/Data/PNat/Interval.lean | 108 | 109 | |
/-
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 toTopologicalSpace_prod {α} {β} [u : UniformSpace α] [v : UniformSpace β] :
@UniformSpace.toTopologicalSpace (α × β) instUniformSpaceProd =
@instTopologicalSpaceProd α β u.toTopologicalSpace v.toTopologicalSpace :=
| Mathlib/Topology/UniformSpace/Basic.lean | 762 | 764 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 612 | 617 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | μ (f '' s) ≤ ∑' n, μ (f '' (s ∩ u n)) := by
conv_lhs => rw [A, image_iUnion]
exact measure_iUnion_le _
_ ≤ ∑' n, ∫⁻ x in s ∩ u n, ENNReal.ofReal |(f' x).det| ∂μ := by
apply ENNReal.tsum_le_tsum fun n => ?_
apply
addHaar_image_le_lintegral_abs_det_fderiv_aux2 μ (hs.inter (u_meas n... | Mathlib/MeasureTheory/Function/Jacobian.lean | 893 | 921 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | {M N : Matrix m m R} (hM : BlockTriangular M b)
(hN : BlockTriangular N b) : BlockTriangular (M * N) b := by
intro i j hij
| Mathlib/LinearAlgebra/Matrix/Block.lean | 175 | 177 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | Mathlib/Data/List/Zip.lean | 177 | 180 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.Matrix.RowCol
/-!
# Dot product of two vectors
This file conta... | (Finset.sum_eq_zero_iff_of_nonneg fun i _ => mul_self_nonneg (v i)).trans <| by
simp [funext_iff]
| Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 95 | 97 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... | mul_left_mem_nonZeroDivisors_eq_zero_iff c.prop
lemma mul_mem_nonZeroDivisors : a * b ∈ M₀⁰ ↔ a ∈ M₀⁰ ∧ b ∈ M₀⁰ where
mp h := by
constructor <;> intro x h' <;> apply h
· rw [← mul_assoc, h', zero_mul]
· rw [mul_comm a b, ← mul_assoc, h', zero_mul]
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 244 | 250 |
/-
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.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | See also `DenseRange.separableSpace`. -/
theorem SeparableSpace.of_denseRange {ι : Sort _} [Countable ι] (u : ι → α) (hu : DenseRange u) :
SeparableSpace α :=
⟨⟨range u, countable_range u, hu⟩⟩
| Mathlib/Topology/Bases.lean | 338 | 341 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... |
theorem isUnit_of_self_mul_dvd_separable {p q : R[X]} (hp : p.Separable) (hq : q * q ∣ p) :
IsUnit q := by
obtain ⟨p, rfl⟩ := hq
apply isCoprime_self.mp
have : IsCoprime (q * (q * p))
(q * (derivative q * p + derivative q * p + q * derivative p)) := by
| Mathlib/FieldTheory/Separable.lean | 160 | 166 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.MorphismProperty.Concrete
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Lim... | Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean | 176 | 179 | |
/-
Copyright (c) 2022 Yaël Dillies, Ella Yu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Ella Yu
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Fintype.Prod
import Mathlib.Algebra.Group.Poin... | end Mul
open scoped Combinatorics.Additive
| Mathlib/Combinatorics/Additive/Energy.lean | 154 | 157 |
/-
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.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | constructor
· intro h
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 131 | 132 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | have h' : ∑ i ∈ s', w' i = 0 := by
rw [← h, Finset.sum_indicator_subset _ (Finset.subset_insert i0 s)]
have hs' : s'.weightedVSub p w' = p1 -ᵥ p i0 := by
rw [hs]
exact (Finset.weightedVSub_indicator_subset _ _ (Finset.subset_insert i0 s)).symm
let w0 : ι → k := Function.update (Function.co... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 1,008 | 1,030 |
/-
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
/-!... |
variable {C}
@[simp]
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 145 | 148 |
/-
Copyright (c) 2019 Zhouhang Zhou. 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.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 818 | 819 | |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.NumberTheory.NumberField.Discriminant.Defs
/-!
# ... | (-1) ^ ((p ^ (k + 1) : ℕ).totient / 2) * p ^ ((p : ℕ) ^ k * ((p - 1) * (k + 1) - 1)) := by
haveI hne := IsCyclotomicExtension.neZero' (p ^ (k + 1)) K L
haveI mf : Module.Finite K L := finiteDimensional {p ^ (k + 1)} K L
haveI se : Algebra.IsSeparable K L := (isGalois (p ^ (k + 1)) K L).to_isSeparable
rw [... | Mathlib/NumberTheory/Cyclotomic/Discriminant.lean | 62 | 122 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 427 | 428 | |
/-
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 | 2,558 | 2,559 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... |
theorem ascPochhammer_nat_eq_natCast_descFactorial (S : Type*) [Semiring S] (a b : ℕ) :
(ascPochhammer S b).eval (a : S) = (a + b - 1).descFactorial b := by
norm_cast
rw [ascPochhammer_nat_eq_descFactorial]
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 164 | 169 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... |
theorem oadd_lt_oadd_2 {e o₁ o₂ : ONote} {n₁ n₂ : ℕ+} (h₁ : NF (oadd e n₁ o₁)) (h : (n₁ : ℕ) < n₂) :
oadd e n₁ o₁ < oadd e n₂ o₂ := by
simp only [lt_def, repr]
refine lt_of_lt_of_le ((add_lt_add_iff_left _).2 h₁.snd'.repr_lt) (le_trans ?_ (le_add_right _ _))
rwa [← mul_succ,Ordinal.mul_le_mul_iff_left (opow_... | Mathlib/SetTheory/Ordinal/Notation.lean | 267 | 275 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | (liftOn s fun t => Prod.map id AList.toFinmap (AList.extract a t)) fun s₁ s₂ p => by
simp [perm_lookup p, toFinmap_eq, perm_erase p]
| Mathlib/Data/Finmap.lean | 479 | 481 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
/-!
# Functions invariant under (quasi)ergodic map
In this file we prove tha... | theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
{s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
(h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
(hg_eq : g ∘ f =ᵐ[μ] g) :
∃ c, g =ᵐ[μ] const α c := by
refin... | Mathlib/Dynamics/Ergodic/Function.lean | 27 | 35 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
theorem IsCoprime.prod_right : (∀ i ∈ t, IsCoprime x (s i)) → IsCoprime x (∏ i ∈ t, s i) := by
simpa only [isCoprime_comm] using IsCoprime.prod_left (R := R)
| Mathlib/RingTheory/Coprime/Lemmas.lean | 61 | 66 |
/-
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.Complex
import Qq
/-! # P... |
@[simp] lemma rpow_rpow_inv (hx : 0 ≤ x) (hy : y ≠ 0) : (x ^ y) ^ y⁻¹ = x := by
rw [← rpow_mul hx, mul_inv_cancel₀ hy, rpow_one]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 477 | 479 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | theorem log_natCast_nonneg (n : ℕ) : 0 ≤ log n := by
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 207 | 207 |
/-
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.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | @[reassoc (attr := simp)]
lemma coyonedaOpColimitIsoLimitCoyoneda_inv_comp_π (i : I) :
(coyonedaOpColimitIsoLimitCoyoneda F).inv ≫ coyoneda.map (colimit.ι F i).op =
limit.π (F.op.comp coyoneda) ⟨i⟩ := by
| Mathlib/CategoryTheory/Limits/IndYoneda.lean | 55 | 58 |
/-
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.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L`... |
theorem lie_le_left : ⁅I, J⁆ ≤ I := by rw [lie_comm]; exact lie_le_right I J
theorem lie_le_inf : ⁅I, J⁆ ≤ I ⊓ J := by rw [le_inf_iff]; exact ⟨lie_le_left I J, lie_le_right J I⟩
| Mathlib/Algebra/Lie/IdealOperations.lean | 135 | 138 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Algebra.Order.Ring.Int
/-!
# Intervals in ℤ
This file defines integer ranges. `range m n` is the set of integers greater than `m` and strictly
... |
instance decidableLELT (P : Int → Prop) [DecidablePred P] (m n : ℤ) :
Decidable (∀ r, m ≤ r → r < n → P r) :=
decidable_of_iff (∀ r ∈ range m n, P r) <| by simp only [mem_range_iff, and_imp]
| Mathlib/Data/Int/Range.lean | 29 | 32 |
/-
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 measure_subtype_coe_le_comap (hs : NullMeasurableSet s μ) (t : Set s) :
μ (((↑) : s → α) '' t) ≤ μ.comap Subtype.val t :=
le_comap_apply _ _ Subtype.coe_injective (fun _ =>
MeasurableSet.nullMeasurableSet_subtype_coe hs) _
| Mathlib/MeasureTheory/Measure/Restrict.lean | 727 | 731 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | rw [cyclotomic']
have prim_root_two : primitiveRoots 2 R = {(-1 : R)} := by
simp only [Finset.eq_singleton_iff_unique_mem, mem_primitiveRoots two_pos]
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 78 | 80 |
/-
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.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | Continuous[t₁, t₃] f → Continuous[t₁ ⊓ t₂, t₃] f :=
continuous_le_dom inf_le_left
theorem continuous_inf_dom_right {t₁ t₂ : TopologicalSpace α} {t₃ : TopologicalSpace β} :
| Mathlib/Topology/Order.lean | 688 | 691 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | (is_real_TFAE z).out 0 2
| Mathlib/Analysis/RCLike/Basic.lean | 354 | 354 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | f.locallyFinite.exists_finset_nhd_support_subset hso ho x
/-- If `f` is a partition of unity that is subordinate to a family of open sets `U i` and
`g : ι → X → E` is a family of functions such that each `g i` is continuous on `U i`, then the sum
`fun x ↦ ∑ᶠ i, f i x • g i x` is a continuous function. -/
| Mathlib/Topology/PartitionOfUnity.lean | 297 | 301 |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | /-- a relation is bisimilar if it meets the `BisimO` test -/
def IsBisimulation :=
∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → BisimO R (destruct s₁) (destruct s₂)
-- If two streams are bisimilar, then they are equal
theorem eq_of_bisim (bisim : IsBisimulation R) {s₁ s₂} (r : s₁ ~ s₂) : s₁ = s₂ := by
apply Subtype.eq
apply Stream'.eq_... | Mathlib/Data/Seq/Seq.lean | 376 | 405 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | ⟨fun h =>
(mem_or_eq_of_mem_set h).elim (nodup_cons.1 hl).1 fun hba => ha (hba ▸ mem_cons_self),
| Mathlib/Data/List/Nodup.lean | 354 | 355 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.Torsion
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Data.NNRat.Order
import Mathlib.Gro... |
/-- The multichoose function is the quotient of ascending Pochhammer evaluation by the corresponding
factorial. When applied to natural numbers, `multichoose k n` describes choosing a multiset of `n`
items from a type of size `k`, i.e., choosing with replacement. -/
| Mathlib/RingTheory/Binomial.lean | 83 | 86 |
/-
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.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.... | rw [Ideal.span_le, Set.singleton_subset_iff]
exact b_mem
rw [Ideal.dvd_iff_le, Ideal.mul_le]
intro r' hr' a ha
rw [Ideal.mem_span_singleton] at hr' ⊢
obtain ⟨q, r, r_mem, lt⟩ := exists_mem_finset_approx' bS adm a b_ne_zero
apply @dvd_of_mul_left_dvd _ _ q
simp only [Algebra.smul_def] at lt
rw [←
... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 288 | 320 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | · obtain ⟨a, ha⟩ := hf
exact ⟨f a, f.injective.ne ha.1, a, ha.1, ha.1⟩
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 728 | 729 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... |
theorem Module.finrank_le_one_iff_top_isPrincipal [Module.Free K V] [Module.Finite K V] :
finrank K V ≤ 1 ↔ (⊤ : Submodule K V).IsPrincipal := by
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 212 | 214 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | @[reassoc (attr := simp)]
theorem biproduct.toSubtype_fromSubtype [DecidablePred p] :
biproduct.toSubtype f p ≫ biproduct.fromSubtype f p =
biproduct.map fun j => if p j then 𝟙 (f j) else 0 := by
ext1 i
by_cases h : p i
| Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 789 | 794 |
/-
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
/-!... | cases n with
| zero => simp only [P_f_0_eq]
| succ n =>
simp only [P_succ, comp_add, comp_id, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
add_eq_left]
exact (HigherFacesVanish.of_P q n).comp_Hσ_eq_zero (Nat.succ_le_iff.mp hqn)
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 36 | 42 |
/-
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... | (hq : ∀ᶠ x in f, p x → q x) : ∀ᶠ x in f, q x :=
mp_mem hp hq
theorem Eventually.mono {p q : α → Prop} {f : Filter α} (hp : ∀ᶠ x in f, p x)
(hq : ∀ x, p x → q x) : ∀ᶠ x in f, q x :=
| Mathlib/Order/Filter/Basic.lean | 612 | 616 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | /-- Hölder's inequality, as an inequality on the `ℒp` seminorm of a scalar product `φ • f`. -/
theorem eLpNorm_smul_le_mul_eLpNorm {p q r : ℝ≥0∞} {f : α → E} (hf : AEStronglyMeasurable f μ)
{φ : α → 𝕜} (hφ : AEStronglyMeasurable φ μ) [hpqr : HolderTriple p q r] :
eLpNorm (φ • f) r μ ≤ eLpNorm φ p μ * eLpNorm f... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 318 | 321 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | Instead, we ask that the composition of `decode : ℕ → Option α` with
`encode : Option α → ℕ` is primitive recursive. Said composition is
| Mathlib/Computability/Primrec.lean | 124 | 125 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | theorem nhds_top_basis_Ici [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]
[Nontrivial α] [DenselyOrdered α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) Ici :=
| Mathlib/Topology/Order/Basic.lean | 324 | 325 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | theorem nfpFamily_lt_ord_lift {ι} {f : ι → Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c)
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 290 | 290 |
/-
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... | Mathlib/Algebra/Polynomial/RingDivision.lean | 864 | 876 | |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | def pdf {_ : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω) (μ : Measure E := by volume_tac) :
E → ℝ≥0∞ :=
(map X ℙ).rnDeriv μ
theorem pdf_def {_ : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E} {X : Ω → E} :
pdf X ℙ μ = (map X ℙ).rnDeriv μ := rfl
| Mathlib/Probability/Density.lean | 122 | 128 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines various properties of ... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 410 | 415 | |
/-
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... | Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk, Word.equivPair_smul_same,
Word.equivPair_tail_eq_inv_smul, Word.rcons_eq_smul, ← of_apply_eq_base φ i,
MonoidHom.apply_ofInjective_symm, equiv_fst_eq_mul_inv, mul_assoc, map_mul, map_inv,
Word.prod_smul, ofCoprodI_of, inv_mul_cancel_left, mul_inv_cancel_left]
@[si... | Mathlib/GroupTheory/PushoutI.lean | 536 | 541 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 905 | 913 | |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | Mathlib/Data/Option/Basic.lean | 390 | 391 | |
/-
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.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | (by rw [ofReal_im, log_im, arg_ofReal_of_nonneg hx])
@[simp, norm_cast]
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 60 | 62 |
/-
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.... | def succAbove (p : Fin (n + 1)) (i : Fin n) : Fin (n + 1) :=
if castSucc i < p then i.castSucc else i.succ
| Mathlib/Data/Fin/Basic.lean | 929 | 931 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... |
section
| Mathlib/Data/Set/Lattice.lean | 393 | 395 |
/-
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, f (b, ω) ∂(ρ.condKernel b) ∂ρ.fst = ∫ x in s ×ˢ Set.univ, f x ∂ρ := by
rw [← setIntegral_condKernel hs MeasurableSet.univ hf]; simp_rw [Measure.restrict_univ]
lemma setIntegral_condKernel_univ_left {t : Set Ω} (ht : MeasurableSet t)
(hf : IntegrableOn f (Set.univ ×ˢ t) ρ) :
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 209 | 213 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Order.ScottContinuity
import Mathlib.Topology.Order.UpperLowerSetTopology
/-!
# Scott topology
This file introduces the Scott topology on a p... | fun _d ⟨_b, hb⟩ _ _a ha hds ↦ hs (ha.1 hb) <| hds hb
lemma IsLowerSet.dirSupInacc (hs : IsLowerSet s) : DirSupInacc s := hs.compl.dirSupClosed.of_compl
| Mathlib/Topology/Order/ScottTopology.lean | 115 | 117 |
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