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) 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... | Mathlib/Topology/Order.lean | 993 | 994 | |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Order.Filter.Prod
/-!
# Curried Filters
This file provides an operation (`Filter.curry`) on filters which provides the equivalence
`∀ᶠ a in l, ∀ᶠ b... | Mathlib/Order/Filter/Curry.lean | 94 | 103 | |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Partition.Equipartition
/-!
# Turán's theorem
In this file we prove Turán's theorem, the first importan... | @[simp]
lemma turanGraph_zero : turanGraph n 0 = ⊤ := by
ext a b; simp_rw [turanGraph, top_adj, Nat.mod_zero, not_iff_not, Fin.val_inj]
| Mathlib/Combinatorics/SimpleGraph/Turan.lean | 69 | 71 |
/-
Copyright (c) 2023 Hanneke Wiersema. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Hanneke Wiersema, Andrew Yang
-/
import Mathlib.Algebra.Ring.Aut
import Mathlib.NumberTheory.Padics.RingHoms
import Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity
... | (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :
c = ModularCyclotomicCharacter' L n g :=
ModularCyclotomicCharacter.toFun_unique' _ _ _ hc
/-- Given a positive integer `n` and a field `L` containing `n` `n`th roots
of unity, `ModularCyclotomicCharacter n` is a multiplicative homomorphism from the
| Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 198 | 203 |
/-
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.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | V i ⟂ V j :=
hV.pairwise hij
| Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 366 | 368 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.List.Pairwise
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Pairwise relations on a list
This file provides basic results ... | Mathlib/Data/List/Pairwise.lean | 104 | 109 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | rw [pow_succ, ← Nat.div_div_eq_div_mul]
exact Nat.div_mul_le_self _ _
_ = a * b - a / b ^ n := add_tsub_add_eq_tsub_left _ _ _
theorem Nat.geom_sum_le {b : ℕ} (hb : 2 ≤ b) (a n : ℕ) :
∑ i ∈ range n, a / b ^ i ≤ a * b / (b - 1) := by
refine (Nat.le_div_iff_mul_le <| tsub_pos_of_lt hb).2 ?_
rcase... | Mathlib/Algebra/GeomSum.lean | 434 | 441 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | fourierCoeffOn hab f n = 1 / (-2 * π * I * n) *
(fourier (-n) (a : AddCircle (b - a)) * (f b - f a) - (b - a) * fourierCoeffOn hab f' n) :=
fourierCoeffOn_of_hasDerivAt_Ioo hab hn
(fun x hx ↦ hf x hx |>.continuousAt.continuousWithinAt)
(fun x hx ↦ hf x <| mem_Icc_of_Ioo hx)
hf'
| Mathlib/Analysis/Fourier/AddCircle.lean | 506 | 511 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib... | | add _ _ hx hy => rw [map_add, map_add, map_add, map_add, hx, hy, neg_add]
| ι_mul _ _ hx =>
| Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 198 | 199 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | ((f₀ ⊗' g₀) ::: (_root_.Prod.map f₁ g₁)) = ((f₀ ::: f₁) ⊗' (g₀ ::: g₁)) := by
ext i a
cases i
· cases a
rfl
| Mathlib/Data/TypeVec.lean | 563 | 567 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Martingale.Basic
/-!
# Centering lemma for stochastic processes
Any `ℕ`-indexed stochastic process which is adapted and integrable can be wri... | unfold h at hω
rw [Pi.sub_apply] at hω
rw [hω, Pi.sub_apply, martingalePart]
simp [hg0]
theorem predictablePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
(hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
(hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : predictablePart... | Mathlib/Probability/Martingale/Centering.lean | 135 | 153 |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... | (actionDiagonalSucc G n).hom.hom f = (f 0, fun i => (f (Fin.castSucc i))⁻¹ * f i.succ) := by
induction n with
| zero => exact Prod.ext rfl (funext fun x => Fin.elim0 x)
| succ n hn =>
refine Prod.ext rfl (funext fun x => ?_)
/- Porting note (https://github.com/leanprover-community/mathlib4/issues/1103... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 111 | 127 |
/-
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... | end NormedSpace
end padicNormE
namespace Padic
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 876 | 880 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... | · dsimp [sSup]
rw [assoc, image.lift_fac, underlyingIso_hom_comp_eq_mk]
instance completeSemilatticeSup {B : C} : CompleteSemilatticeSup (Subobject B) where
sSup := sSup
le_sSup := le_sSup
sSup_le := sSup_le
end Sup
section CompleteLattice
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 638 | 649 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Birthdays... | all_goals
refine lsub_le_iff.2 fun i ↦ ?_
rw [← Order.succ_le_iff]
refine Ordinal.le_iSup (fun _ : Set.Iio _ ↦ _) ⟨_, ?_⟩
apply_rules [birthday_moveLeft_lt, birthday_moveRight_lt]
all_goals
rw [Ordinal.iSup_le_iff]
rintro ⟨i, hi⟩
obtain ⟨j, hj⟩ | ⟨j, hj⟩ := lt... | Mathlib/SetTheory/Game/Birthday.lean | 147 | 174 |
/-
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... | rw [← rpow_mul hx, inv_mul_cancel₀ hy, rpow_one]
theorem pow_rpow_inv_natCast (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ n) ^ (n⁻¹ : ℝ) = x := by
have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 482 | 485 |
/-
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.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | lemma ext (W W' : MorphismProperty C) (h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f ↔ W' f) :
W = W' := by
funext X Y f
rw [h]
| Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 55 | 59 |
/-
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.Group.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | have hl' : ∀ g' ∈ l.1.map ofSubtype, IsSwap g' := fun g' hg' =>
let ⟨g, hg⟩ := List.mem_map.1 hg'
hg.2 ▸ (l.2.2 _ hg.1).of_subtype_isSwap
| Mathlib/GroupTheory/Perm/Sign.lean | 466 | 468 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Group.Torsion
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
/-!
# Lemmas about the ... |
@[to_additive zsmul_left_strictAnti]
lemma zpow_right_strictAnti (ha : a < 1) : StrictAnti fun n : ℤ ↦ a ^ n := by
refine strictAnti_int_of_succ_lt fun n ↦ ?_
rw [zpow_add_one]
exact mul_lt_of_lt_one_right' (a ^ n) ha
| Mathlib/Algebra/Order/Group/Basic.lean | 34 | 39 |
/-
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
... | induction_on s₁ fun s₁ => by
rw [← empty_toFinmap]
| Mathlib/Data/Finmap.lean | 551 | 552 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,171 | 1,175 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,482 | 1,487 | |
/-
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.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
| Mathlib/RingTheory/Coprime/Basic.lean | 94 | 95 |
/-
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 | 314 | 318 | |
/-
Copyright (c) 2021 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.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois.Basic
/-!
# Cyclotomic extens... | (hζ : IsPrimitiveRoot ζ n) :
adjoin A ((cyclotomic n A).rootSet B) =
adjoin A {b : B | ∃ a : ℕ+, a ∈ ({n} : Set ℕ+) ∧ b ^ (a : ℕ) = 1} := by
simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic]
refine le_antisymm (adjoin_mono fun x hx => ?_) (adjoin_le fun x hx => ?_)
· rw [mem_rootSet'] at... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 341 | 346 |
/-
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... | Mathlib/MeasureTheory/Measure/Restrict.lean | 1,025 | 1,035 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Formal power se... | Polynomial.ext fun m => by
simp only [coeff_trunc, AddMonoidHom.map_add, Polynomial.coeff_add]
split_ifs with H
· rfl
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 68 | 71 |
/-
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... | (Multiset.card (p - C a).roots : WithBot ℕ) ≤ degree (p - C a) :=
card_roots <| mt sub_eq_zero.1 fun h => not_le_of_gt hp0 <| h.symm ▸ degree_C_le
_ = degree p := by rw [sub_eq_add_neg, ← C_neg]; exact degree_add_C hp0
theorem card_roots_sub_C' {p : R[X]} {a : R} (hp0 : 0 < degree p) :
Multiset.card ... | Mathlib/Algebra/Polynomial/Roots.lean | 82 | 87 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | [Nonempty s] {n : ℕ} [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = n)
(hc : Cospherical ps) {sx₁ sx₂ : Simplex ℝ P n} (hsx₁ : Set.range sx₁.points ⊆ ps)
(hsx₂ : Set.range sx₂.points ⊆ ps) : sx₁.circumradius = sx₂.circumradius := by
rcases exists_circumradius_eq_of_cospherical_subset h hd... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 654 | 661 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.GradedObject
/-!
# The action of bifunctors on graded objects
Given a bifunctor `F : C₁ ⥤ C₂ ⥤ C₃` and types `I` and `J`, we construct an obvious... | lemma ι_mapBifunctorMapMap {X₁ X₂ : GradedObject I C₁} (f : X₁ ⟶ X₂)
{Y₁ Y₂ : GradedObject J C₂} (g : Y₁ ⟶ Y₂)
[HasMap (((mapBifunctor F I J).obj X₁).obj Y₁) p]
[HasMap (((mapBifunctor F I J).obj X₂).obj Y₂) p]
(i : I) (j : J) (k : K) (h : p ⟨i, j⟩ = k) :
ιMapBifunctorMapObj F p X₁ Y₁ i j k h ≫ mapB... | Mathlib/CategoryTheory/GradedObject/Bifunctor.lean | 73 | 82 |
/-
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.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... |
theorem splits_iff_comp_splits_of_degree_eq_one {f : K[X]} {p : K[X]} (hd : (p.map i).degree = 1) :
f.Splits i ↔ (f.comp p).Splits i := by
rw [← splits_id_iff_splits, ← splits_id_iff_splits (f := f.comp p), map_comp]
refine ⟨fun h => Splits.comp_of_map_degree_le_one
(le_of_eq (map_id (R := L) ▸ hd)) h, fun... | Mathlib/Algebra/Polynomial/Splits.lean | 168 | 178 |
/-
Copyright (c) 2021 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.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,608 | 1,615 | |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | lt_iff_le_not_le := by decide
max_def := by decide
| Mathlib/Data/Bool/Basic.lean | 154 | 155 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | @[reassoc (attr := simp)]
theorem pentagon_inv_inv_hom_hom_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom =
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv := by
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 227 | 230 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | | ind (a : q.P.A) (f f' : q.P.B a → q.P.W) : (∀ x, Wequiv (f x) (f' x)) → Wequiv ⟨a, f⟩ ⟨a, f'⟩
| abs (a : q.P.A) (f : q.P.B a → q.P.W) (a' : q.P.A) (f' : q.P.B a' → q.P.W) :
abs ⟨a, f⟩ = abs ⟨a', f'⟩ → Wequiv ⟨a, f⟩ ⟨a', f'⟩
| trans (u v w : q.P.W) : Wequiv u v → Wequiv v w → Wequiv u w
| Mathlib/Data/QPF/Univariate/Basic.lean | 169 | 172 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
/-!
# Properties of cyclic permutations constructed... | intro ⟨n, hn⟩ ⟨m, hm⟩
rw [length_toList, ← hc.orderOf] at hm hn
| Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 248 | 249 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b := by
rw [← coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff]
| Mathlib/Order/Interval/Finset/Basic.lean | 88 | 89 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
import Mathlib.Algebra.Ring.Action.Rat
import Mathlib.Tactic.FastInstance
/-!
# Cauchy completion
This fi... |
/-- Show the first 10 items of a representative of this equivalence class of cauchy sequences.
The representative chosen is the one passed in the VM to `Quot.mk`, so two cauchy sequences
converging to the same number may be printed differently.
| Mathlib/Algebra/Order/CauSeq/Completion.lean | 248 | 252 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: the quadratic for... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 374 | 378 | |
/-
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... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,031 | 3,033 | |
/-
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.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 472 | 495 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,463 | 1,464 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | lemma pow_eq_one_iff_cases : a ^ n = 1 ↔ n = 0 ∨ a = 1 ∨ a = -1 ∧ Even n := by
simp [← pow_eq_pow_iff_cases]
| Mathlib/Algebra/Order/Ring/Abs.lean | 226 | 227 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | Fintype.prod_equiv (finCongr <| Fin.val_succ i)
(fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Fin.is_lt _).le j))
(fun j => (Fin.cons x xs : _ → ℕ) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j))
fun j => rfl
simp_rw [this]
| Mathlib/Algebra/BigOperators/Fin.lean | 362 | 366 |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... |
theorem bindOnSupport_pure (p : PMF α) : (p.bindOnSupport fun a _ => pure a) = p := by
simp only [PMF.bind_pure, PMF.bindOnSupport_eq_bind]
@[simp]
theorem bindOnSupport_bindOnSupport (p : PMF α) (f : ∀ a ∈ p.support, PMF β)
(g : ∀ b ∈ (p.bindOnSupport f).support, PMF γ) :
| Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 246 | 252 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | tsub_le_iff_right.2 <|
calc
a ≤ a - b + b := le_tsub_add
| Mathlib/Algebra/Order/Sub/Defs.lean | 145 | 147 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | rw [MeasurableEquiv.piCongrLeft_apply_apply e x i]
rw [this, pi_pi, Finset.prod_equiv e.symm]
· simp only [Finset.mem_univ, implies_true]
| Mathlib/MeasureTheory/Constructions/Pi.lean | 409 | 411 |
/-
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... | Injective (lift n f) ↔ ∀ m, f.1 m = 0 → (m : ZMod n) = 0 := by
simp only [← AddMonoidHom.ker_eq_bot_iff, eq_bot_iff, SetLike.le_def,
ZMod.intCast_surjective.forall, ZMod.lift_coe, AddMonoidHom.mem_ker, AddSubgroup.mem_bot]
end lift
| Mathlib/Data/ZMod/Basic.lean | 1,140 | 1,144 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 3,135 | 3,146 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | (hsx₁ : Set.range sx₁.points ⊆ ps) (hsx₂ : Set.range sx₂.points ⊆ ps) :
sx₁.circumradius = sx₂.circumradius := by
rcases exists_circumradius_eq_of_cospherical hd hc with ⟨r, hr⟩
rw [hr sx₁ hsx₁, hr sx₂ hsx₂]
/-- All n-simplices among cospherical points in an n-dimensional
subspace have the same circumcente... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 674 | 680 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | Mathlib/Algebra/Module/Defs.lean | 502 | 502 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | @[simp]
lemma δ_comp_ofHom {n : ℤ} (z₁ : Cochain F G n) (f : G ⟶ K) (m : ℤ) :
δ n m (z₁.comp (Cochain.ofHom f) (add_zero n)) =
(δ n m z₁).comp (Cochain.ofHom f) (add_zero m) := by
rw [← Cocycle.ofHom_coe, δ_comp_zero_cocycle]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 716 | 722 |
/-
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.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | apply Measurable.prod <;> rw [measurable_pi_iff] <;> rintro i <;> apply measurable_pi_apply
measurable_invFun := by
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 587 | 588 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | constructor <;> intro h
· rw [Set.mem_range] at h
rcases h with ⟨⟨x, hx⟩, h⟩
use x
rw [← h]
exact f.mem_graph ⟨x, hx⟩
obtain ⟨x, h⟩ := h
rw [mem_graph_iff] at h
obtain ⟨x, h⟩ := h
rw [Set.mem_range]
use x
simp only at h
rw [h.2]
theorem mem_domain_iff_of_eq_graph {f g : E →ₗ.[R] F} (h... | Mathlib/LinearAlgebra/LinearPMap.lean | 799 | 820 |
/-
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... |
@[simp]
| Mathlib/Data/Real/Sqrt.lean | 149 | 150 |
/-
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.... | /-- Extension for the `positivity` tactic: `ENNReal.ofNNReal`. -/
| Mathlib/Data/ENNReal/Basic.lean | 745 | 745 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | map_target' p hp := by simp_all
left_inv' p hp := by simp_all
| Mathlib/Logic/Equiv/PartialEquiv.lean | 760 | 761 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... |
theorem StrictAnti.isMax_of_apply (hf : StrictAnti f) (ha : IsMin (f a)) : IsMax a :=
of_not_not fun h ↦
| Mathlib/Order/Monotone/Basic.lean | 273 | 275 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | @[simp]
lemma homologyMap_add : homologyMap (φ + ψ) i = homologyMap φ i + homologyMap ψ i := by
dsimp [homologyMap]
rw [← ShortComplex.homologyMap_add]
rfl
@[simp]
lemma homologyMap_sub : homologyMap (φ - ψ) i = homologyMap φ i - homologyMap ψ i := by
dsimp [homologyMap]
rw [← ShortComplex.homologyMap_sub]
... | Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 704 | 717 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,707 | 1,711 | |
/-
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.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... | ⟨fun _ => mem_nhdsGT_iff_exists_Ioo_subset' h⟩
@[deprecated (since := "2024-12-22")]
alias nhdsWithin_Ioi_basis' := nhdsGT_basis_of_exists_gt
lemma nhdsGT_basis [NoMaxOrder α] (a : α) : (𝓝[>] a).HasBasis (a < ·) (Ioo a) :=
| Mathlib/Topology/Order/LeftRightNhds.lean | 82 | 87 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 416 | 439 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... |
@[simp]
theorem toIcoMod_add_left' (a b : α) : toIcoMod hp (p + a) b = p + toIcoMod hp a b := by
| Mathlib/Algebra/Order/ToIntervalMod.lean | 413 | 415 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | simp only [eqToHom_trans]
@[reassoc (attr := simp)]
lemma XIsoOfEq_hom_comp_d (K : HomologicalComplex V c) {p₁ p₂ : ι} (h : p₁ = p₂) (p₃ : ι) :
(K.XIsoOfEq h).hom ≫ K.d p₂ p₃ = K.d p₁ p₃ := by subst h; simp
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 124 | 129 |
/-
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, Kim Morrison, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... | simp
@[simp]
theorem finRange_map_get (l : List α) : (finRange l.length).map l.get = l :=
| Mathlib/Data/List/FinRange.lean | 58 | 61 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | def IsSymmetric.toSelfAdjoint (hT : IsSymmetric T) : selfAdjoint (E →L[𝕜] E) :=
⟨⟨T, hT.continuous⟩, ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.mpr hT⟩
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 295 | 297 |
/-
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 ... | change ((coyonedaOpColimitIsoLimitCoyoneda' F).hom ≫ _).app A = _
rw [coyonedaOpColimitIsoLimitCoyoneda'_hom_comp_π]
@[reassoc (attr := simp)]
lemma colimitHomIsoLimitYoneda'_inv_comp_π [HasLimitsOfShape I (Type u₂)] (A : C) (i : I) :
(colimitHomIsoLimitYoneda' F A).inv ≫ (coyoneda.map (colimit.ι F ⟨i⟩).op).ap... | Mathlib/CategoryTheory/Limits/IndYoneda.lean | 122 | 127 |
/-
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.Polynomial.AlgebraMap
import Mathlib.RingTheory.IsTensorProduct
/-!
# Base change of polynomial algebras
Given `[CommSemiring R] [Semiring A] [Al... | simp only [toFunLinear_tmul_apply, toFunBilinear_apply_eq_sum]
ext k
simp_rw [coeff_sum, coeff_monomial, sum_def, Finset.sum_ite_eq', mem_support_iff, Ne]
| Mathlib/RingTheory/PolynomialAlgebra.lean | 80 | 82 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | @[simp]
lemma Ψ_zero : W.Ψ 0 = 0 := by
rw [← Nat.cast_zero, Ψ_ofNat, preΨ'_zero, C_0, zero_mul]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 321 | 323 |
/-
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.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Sequence of measurable functions associated to a sequence o... | exact h.1
(h_ss hx i).symm
theorem aeSeq_eq_mk_of_mem_aeSeqSet (hf : ∀ i, AEMeasurable (f i) μ) {x : α}
(hx : x ∈ aeSeqSet hf p) (i : ι) : aeSeq hf p i x = (hf i).mk (f i) x := by
simp only [aeSeq, hx, if_true]
| Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 50 | 56 |
/-
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... | exact eq_tsub_of_add_eq rfl
theorem val_add_le {n : ℕ} (a b : ZMod n) : (a + b).val ≤ a.val + b.val := by
cases n
| Mathlib/Data/ZMod/Basic.lean | 645 | 648 |
/-
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.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | Setoid.symm (CauSeq.equiv_lim (newton_cau_seq hnorm)) _ hε
private theorem soln_deriv_norm : ‖F.derivative.eval soln‖ = ‖F.derivative.eval a‖ :=
norm_deriv_eq (newton_seq_deriv_norm hnorm)
| Mathlib/NumberTheory/Padics/Hensel.lean | 322 | 326 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | BddBelow s → BddBelow t → BddBelow (s ∪ t) :=
@BddAbove.union αᵒᵈ _ _ _ _
| Mathlib/Order/Bounds/Basic.lean | 325 | 327 |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... | refine ⟨fun h ε hε ↦ ?_, fun h ε hε ↦ ?_⟩
· have hf : (fun i => (μ { x | ε ≤ dist (f i x) (g x) }).toNNReal) =
ENNReal.toNNReal ∘ (fun i => (μ { x | ε ≤ dist (f i x) (g x) })) := rfl
rw [hf, ENNReal.tendsto_toNNReal_iff' (hfin ε)]
exact h ε hε
· rw [← ENNReal.tendsto_toNNReal_iff ENNReal.zero_ne_t... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 69 | 81 |
/-
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... | sup_inf_sdiff _ _ := Prod.ext (sup_inf_sdiff _ _) (sup_inf_sdiff _ _)
inf_inf_sdiff _ _ := Prod.ext (inf_inf_sdiff _ _) (inf_inf_sdiff _ _)
| Mathlib/Order/BooleanAlgebra.lean | 465 | 466 |
/-
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.Group.Action.Pointwise.Finset
import Mathlib.GroupTheory.QuotientGroup.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Stabiliz... |
section CommGroup
variable [CommGroup G] {s t : Set G} {a : G}
| Mathlib/Algebra/Pointwise/Stabilizer.lean | 121 | 124 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | (v.support.choose (fun a₁ => f a₁ = a₂)
(by
rcases Finset.mem_map.1 h with ⟨a, ha, rfl⟩
exact ExistsUnique.intro a ⟨ha, rfl⟩ fun b ⟨_, hb⟩ => f.injective hb))
else 0
mem_support_toFun a₂ := by
| Mathlib/Data/Finsupp/Defs.lean | 373 | 378 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,011 | 1,015 | |
/-
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 | 535 | 539 | |
/-
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
... | theorem keys_erase (a : α) (s : Finmap β) : (erase a s).keys = s.keys.erase a :=
induction_on s fun s => by simp
| Mathlib/Data/Finmap.lean | 381 | 382 |
/-
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.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | rw [h.cycleFactorsFinset_mul_eq_union]
simp only [Finset.union_val, Function.comp_apply]
rw [← Multiset.add_eq_union_iff_disjoint.mpr _, Multiset.map_add]
simp only [Finset.disjoint_val, Disjoint.disjoint_cycleFactorsFinset h]
theorem Disjoint.cycleType_noncommProd {ι : Type*} {k : ι → Perm α} {s : Finset ι}
... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 238 | 264 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... | theorem projIic_surjective : Surjective (projIic b) := fun x => ⟨x, projIic_coe x⟩
| Mathlib/Order/Interval/Set/ProjIcc.lean | 119 | 120 |
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... |
theorem degree_map_eq_of_injective (hf : Injective f) (p : R[X]) : degree (p.map f) = degree p :=
| Mathlib/Algebra/Polynomial/Monic.lean | 388 | 389 |
/-
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... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,094 | 1,096 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | * `Finsupp.mapRange.addMonoidHom`
* `Finsupp.mapRange.addEquiv`
* `Finsupp.mapRange.linearMap`
* `Finsupp.mapRange.linearEquiv`
| Mathlib/Data/Finsupp/Defs.lean | 287 | 290 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 700 | 702 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 712 | 714 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Order.SuccPred.Archimedean
/-!
# Intervals `Ixx (f x) (f (Order.succ x))`
In... | rcases le_total n m with hnm | hmn
· rw [Ico_eq_empty_of_le hnm, Ioc_eq_empty_of_le (hf hnm), biUnion_empty]
· refine Succ.rec ?_ ?_ hmn
· simp only [Ioc_self, Ico_self, biUnion_empty]
· intro k hmk ihk
rw [← Ioc_union_Ioc_eq_Ioc (hf hmk) (hf <| le_succ _), union_comm, ← ihk]
by_cases hk : IsM... | Mathlib/Order/SuccPred/IntervalSucc.lean | 38 | 48 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
/-!
# Leading terms of Witt vector multiplication
Th... | rw [rename_monomial, vars_monomial, Finsupp.mapDomain_single]
· apply Subset.trans Finsupp.support_single_subset
simpa using mem_range.mp hx
· apply pow_ne_zero
exact mod_cast hp.out.ne_zero
/-- This is the polynomial whose degree we want to get a handle on. -/
def polyOfInterest (n : ℕ) : 𝕄 :... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 99 | 110 |
/-
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.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | /-- The `vectorSpan` of a finite affinely independent family has dimension one less than its
cardinality. -/
lemma AffineIndependent.finrank_vectorSpan_add_one [Fintype ι] [Nonempty ι] {p : ι → P}
(hi : AffineIndependent k p) : finrank k (vectorSpan k (Set.range p)) + 1 = Fintype.card ι := by
rw [hi.finrank_vecto... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 128 | 133 |
/-
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-... | (empty_theory_categorical.{0} ℵ₀ _).isComplete ℵ₀ _ le_rfl (by simp)
⟨by
haveI : Language.empty.Structure ℕ := emptyStructure
exact ((model_infiniteTheory_iff Language.empty).2 (inferInstanceAs (Infinite ℕ))).bundled⟩
fun M => (model_infiniteTheory_iff Language.empty).1 M.is_model
| Mathlib/ModelTheory/Satisfiability.lean | 494 | 498 |
/-
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.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... |
theorem aeval_algEquiv (f : A ≃ₐ[R] B) (x : A) : aeval (f x) = (f : A →ₐ[R] B).comp (aeval x) :=
aeval_algHom (f : A →ₐ[R] B) x
| Mathlib/Algebra/Polynomial/AlgebraMap.lean | 356 | 359 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 956 | 959 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | lemma shiftFunctor_of_induced (a : A) :
letI := HasShift.induced F A s i
shiftFunctor D a = s a := by
rfl
| Mathlib/CategoryTheory/Shift/Induced.lean | 157 | 161 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | measurableSet i := hφ.measurableSet (u i)
section AECoverUnionInterCountable
variable {α ι : Type*} [Countable ι] [MeasurableSpace α] {μ : Measure α}
theorem AECover.biUnion_Iic_aecover [Preorder ι] {φ : ι → Set α} (hφ : AECover μ atTop φ) :
AECover μ atTop fun n : ι => ⋃ (k) (_h : k ∈ Iic n), φ k :=
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 327 | 334 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | ContinuousLinearMap.zero_apply]
theorem setToL1_congr_left (T T' : Set α → E →L[ℝ] F) {C C' : ℝ}
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 427 | 429 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
/-!
# One-dimensional de... | hf.const_sub c
theorem HasStrictDerivAt.const_sub (c : F) (hf : HasStrictDerivAt f f' x) :
| Mathlib/Analysis/Calculus/Deriv/Add.lean | 363 | 365 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | rw [EventuallyEq, eventually_atTop]
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 85 | 85 |
/-
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
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory... | A constructor for lax monoidal functors whose axioms are described by `tensorHom` instead of
`whiskerLeft` and `whiskerRight`.
-/
def ofTensorHom : F.LaxMonoidal where
ε' := ε'
| Mathlib/CategoryTheory/Monoidal/Functor.lean | 178 | 182 |
/-
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.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | (𝓝 (∫ x in A, density κ ν a x s ∂(ν a))) := by
refine tendsto_setIntegral_of_L1' (μ := ν a) (fun x ↦ density κ ν a x s)
(integrable_density hκν a hs) (F := fun i x ↦ densityProcess κ ν i a x s) (l := atTop)
(Eventually.of_forall (fun n ↦ integrable_densityProcess hκν _ _ hs)) ?_ A
refine (tendsto_con... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 503 | 507 |
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