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) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | theorem succ_le_iff_eq_top : succ a ≤ a ↔ a = ⊤ :=
| Mathlib/Order/SuccPred/Basic.lean | 368 | 368 |
/-
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... |
variable {k : Type*} {V : Type*} (P : Type*) [CommRing k] [AddCommGroup V] [Module k V]
variable [AffineSpace V P] {ι : Type*} (s : Finset ι)
-- TODO: define `affineMap.proj`, `affineMap.fst`, `affineMap.snd`
/-- A weighted sum, as an affine map on the points involved. -/
def weightedVSubOfPoint (w : ι → k) : (ι → P)... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 1,132 | 1,138 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 396 | 404 | |
/-
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 ... | Mathlib/GroupTheory/Submonoid/Inverses.lean | 208 | 210 | |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | theorem associates_irreducible : Irreducible <| Associates.mk v.asIdeal :=
Associates.irreducible_mk.mpr v.irreducible
/-- An equivalence between the height one and maximal spectra for rings of Krull dimension 1. -/
def equivMaximalSpectrum (hR : ¬IsField R) : HeightOneSpectrum R ≃ MaximalSpectrum R where
toFun v ... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 976 | 981 |
/-
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.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | theorem succ_top : succ (⊤ : α) = ⊤ := by
| Mathlib/Order/SuccPred/Basic.lean | 365 | 365 |
/-
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, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Round
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
/-... | @[norm_num ⌊_⌋]
def evalIntFloor : NormNumExt where eval {u αZ} e := do
match u, αZ, e with
| 0, ~q(ℤ), ~q(@Int.floor $α $instR $instO $instF $x) =>
match ← derive x with
| .isBool .. => failure
| .isNat _ _ pb => do
let _i ← synthInstanceQ q(IsStrictOrderedRing $α)
assertInstancesCommute
... | Mathlib/Data/Rat/Floor.lean | 128 | 163 |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
import Mathlib.Algebra.Regular.Basic
/-!
# Product of regular elements
## TODO
Move to `Mathlib.Algebra.BigOper... | Mathlib/Algebra/Regular/Pow.lean | 54 | 58 | |
/-
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... | lemma d_comp_XIsoOfEq_inv (K : HomologicalComplex V c) {p₂ p₃ : ι} (h : p₃ = p₂) (p₁ : ι) :
K.d p₁ p₂ ≫ (K.XIsoOfEq h).inv = K.d p₁ p₃ := by subst h; simp
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 139 | 141 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | IsLocalization.mk' R[T;T⁻¹] p (⟨X^n, n, rfl⟩ : Submonoid.powers (X : R[X])) =
toLaurent p * T (-n) := by
rw [←IsUnit.mul_left_inj (isUnit_T n), mul_T_assoc, neg_add_cancel, T_zero, mul_one]
exact mk'_mul_T p n
| Mathlib/Algebra/Polynomial/Laurent.lean | 522 | 526 |
/-
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 | 797 | 798 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Chris Hughes
-/
import Mathlib.Data.List.Nodup
/-!
# List duplicates
## Main definitions
* `List.Duplicate x l : Prop` is an inductive property that holds when `x`... | theorem Duplicate.not_nodup (h : x ∈+ l) : ¬Nodup l := fun H =>
nodup_iff_forall_not_duplicate.mp H _ h
theorem duplicate_iff_two_le_count [DecidableEq α] : x ∈+ l ↔ 2 ≤ count x l := by
simp [replicate_succ, duplicate_iff_sublist, le_count_iff_replicate_sublist]
instance decidableDuplicate [DecidableEq α] (x : α)... | Mathlib/Data/List/Duplicate.lean | 117 | 126 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Oliver Nash
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.D... | (univBall c r).symm.continuousOn.mono <| ball_subset_univBall_target c r
| Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 149 | 150 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
/-!
# Stabilisation of gcf Computations Under Termination
## Summary
We show that the continuants and co... | | succ n IH =>
cases s_head_eq : s.head with
| none => simp only [convs'Aux, s_head_eq]
| some gp_head =>
have : s.tail.TerminatedAt n := by
simp only [Stream'.Seq.TerminatedAt, s.get?_tail, terminatedAt_n]
have := IH this
rw [convs'Aux] at this
simp [this, Nat.add_eq, add_... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 45 | 58 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Relation
/-!
# Parallel computation
Parallel computation of a computable sequence of computations by
a diagonal enumeration.
The imp... | obtain ⟨a, e⟩ := e
have : corec parallel.aux1 (l, S) = return a := by
apply destruct_eq_pure
simp only [parallel.aux1, rmap, corec_eq]
rw [e]
rw [this]
exact ret_terminates a
intro l S c m T
revert l S
apply @terminatesRecOn _ _ c T _ _
· intro a l S m
apply lem1
induct... | Mathlib/Data/Seq/Parallel.lean | 57 | 119 |
/-
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.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... |
/-- If `x` and `y` are distinct letters and `w₁ w₂` are words such that `xw₁` reduces to `yw₂`, then
`w₁` reduces to `x⁻¹yw₂`. -/
@[to_additive "If `x` and `y` are distinct letters and `w₁ w₂` are words such that `x + w₁` reduces
to `y + w₂`, then `w₁` reduces to `-x + y + w₂`."]
theorem inv_of_red_of_ne {x1 b1 x2 b... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 288 | 305 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | y ^ 2 + W'.a₁ * x * y + W'.a₃ * y - (x ^ 3 + W'.a₂ * x ^ 2 + W'.a₄ * x + W'.a₆) := by
simp only [polynomial]
eval_simp
rw [add_mul, ← add_assoc]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 176 | 180 |
/-
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
/-... | Mathlib/Order/Bounds/Basic.lean | 945 | 947 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison
-/
import Mathlib.CategoryTheory.Equivalence
/-!
# Opposite categories
We provide a category instance on `Cᵒᵖ`.
The morphisms `X ⟶ Y` are defined to be the... | In informal mathematics no distinction is made between these.
-/
@[simps]
| Mathlib/CategoryTheory/Opposites.lean | 168 | 170 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.RingTheory.Congruence.Defs
/-!
# Congruence relations on rings
This fi... | Mathlib/RingTheory/Congruence/Basic.lean | 559 | 560 | |
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.IsPrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
import Mathlib.Data.Z... | theorem coe_factor_orderIso_map_eq_one_iff {m u : Associates M} {n : Associates N} (hu' : u ≤ m)
(d : Set.Iic m ≃o Set.Iic n) : (d ⟨u, hu'⟩ : Associates N) = 1 ↔ u = 1 :=
⟨fun hu => by
rw [show u = (d.symm ⟨d ⟨u, hu'⟩, (d ⟨u, hu'⟩).prop⟩) by
simp only [Subtype.coe_eta, OrderIso.symm_apply_apply, Subty... | Mathlib/RingTheory/ChainOfDivisors.lean | 224 | 231 |
/-
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.Asymptotics
/-!
# Continu... | theorem continuousAt_cpow {p : ℂ × ℂ} (hp_fst : p.fst ∈ slitPlane) :
ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) p := by
rw [continuousAt_congr (cpow_eq_nhds' <| slitPlane_ne_zero hp_fst)]
refine continuous_exp.continuousAt.comp ?_
exact
| Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 74 | 78 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Data.Set.Order
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.Interval.Set.LinearOrder
import Mathlib.Tac... | lemma Icc_subset_uIcc : Icc a b ⊆ [[a, b]] := Icc_subset_Icc inf_le_left le_sup_right
| Mathlib/Order/Interval/Set/UnorderedInterval.lean | 88 | 88 |
/-
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.Comma.StructuredArrow.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Equivalence
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal... | noncomputable def LeftExtension.isUniversalPostcomp₁Equiv (ex : LeftExtension L' F) :
ex.IsUniversal ≃ ((LeftExtension.postcomp₁ G e.inv F).obj ex).IsUniversal := by
apply IsInitial.isInitialIffObj (LeftExtension.postcomp₁ G e.inv F)
/-- Given an isomorphism `e : L ⋙ G ≅ L'`, a right extension of `F` along `L'` ... | Mathlib/CategoryTheory/Functor/KanExtension/Basic.lean | 351 | 361 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,405 | 1,409 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Pointwise.Interval
impor... | EqOn f 1 S ∨ EqOn f (-1) S := by
have : DiscreteTopology ({1, -1} : Set 𝕜) := Finite.instDiscreteTopology
have hmaps : MapsTo f S {1, -1} := by
simpa only [EqOn, Pi.one_apply, Pi.pow_apply, sq_eq_one_iff] using hsq
simpa using hS.eqOn_const_of_mapsTo hf hmaps
/-- If `f, g` are functions `α → 𝕜`, both c... | Mathlib/Topology/Algebra/Field.lean | 155 | 162 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | @[simp]
theorem cos_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x + n * (2 * π)) = cos x :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 340 | 341 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... |
@[simp]
lemma equation_zero : W'.Equation 0 0 ↔ W'.a₆ = 0 := by
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 199 | 201 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | { refl := fun (x, y) ↦ Rel.refl x y, symm := Rel.symm, trans := Rel.trans }
| Mathlib/Data/Sym/Sym2.lean | 73 | 74 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | rw [rpow_mul, rpow_natCast]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 135 | 136 |
/-
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.Single
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# The homology of single complexes
The main definition in this file ... | (singleObjXSelf c j A).hom := by
have := ((single C c j).obj A).isIso_iCycles j _ rfl (by simp)
rw [← cancel_epi (((single C c j).obj A).iCycles j),
← HomologicalComplex.homology_π_ι_assoc, homologyι_singleObjOpcyclesSelfIso_inv,
homologyπ_singleObjHomologySelfIso_hom, singleObjCyclesSelfIso_hom]
var... | Mathlib/Algebra/Homology/SingleHomology.lean | 120 | 127 |
/-
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.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | theorem Iio_succ (a : α) : Iio (succ a) = Iic a :=
Iio_succ_of_not_isMax <| not_isMax _
| Mathlib/Order/SuccPred/Basic.lean | 472 | 473 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | exact ENNReal.mul_lt_top hX.2 hY.2
/-- If the product of two independent real-valued random variables is integrable and
the second one is not almost everywhere zero, then the first one is integrable. -/
theorem IndepFun.integrable_left_of_integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
[NormedDivisi... | Mathlib/Probability/Integration.lean | 158 | 175 |
/-
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... | dsimp only
rw [dist_eq_norm_vsub V (s.affineCombination ℝ p w₁) (s.affineCombination ℝ p w₂), ←
@inner_self_eq_norm_mul_norm ℝ, Finset.affineCombination_vsub]
have h : (∑ i ∈ s, (w₁ - w₂) i) = 0 := by
simp_rw [Pi.sub_apply, Finset.sum_sub_distrib, h₁, h₂, sub_self]
exact inner_weightedVSub p h p h
-- P... | Mathlib/Geometry/Euclidean/Basic.lean | 78 | 87 |
/-
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... | apply Nat.mod_eq_of_lt (tsub_lt_of_lt (val_lt _))
| Mathlib/Data/ZMod/Basic.lean | 1,005 | 1,006 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... |
/-- The angle between a nonnegative multiple of a vector and that vector is 0. -/
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 271 | 272 |
/-
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... | let ⟨a, b, h1⟩ := H1
let ⟨c, d, h2⟩ := H2
⟨a * c, a * x * d + b * c * y + b * d * z,
calc a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z
| Mathlib/RingTheory/Coprime/Basic.lean | 102 | 105 |
/-
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.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | Mathlib/Data/Finsupp/Basic.lean | 1,739 | 1,750 | |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
/-!
# Eigenvalues are the roots of the minimal polynomial.
## Tags
e... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 128 | 130 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | WellFounded r ↔ IsEmpty (((· > ·) : ℕ → ℕ → Prop) ↪r r) := by
constructor
· rintro ⟨h⟩
| Mathlib/Order/OrderIsoNat.lean | 84 | 86 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | | 0 => ⟨0, (P.xNextIso rfl).hom ≫ zero, by simpa using comm_zero⟩
| n + 1 =>
let I := mkCoinductiveAux₁ e zero one comm_one succ n
⟨I.1 ≫ (Q.xPrevIso rfl).inv, (P.xNextIso rfl).hom ≫ I.2.1, by simpa using I.2.2⟩
| Mathlib/Algebra/Homology/Homotopy.lean | 614 | 618 |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... | iterate 6 exfalso; linarith
· congr
| Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 205 | 206 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | apply leftMoves_add_cases x <;>
intro i
| Mathlib/SetTheory/Game/Nim.lean | 266 | 267 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... |
/-- If `f : X → Y` is continuous and `Y` is equipped with the Borel sigma algebra, then
the push-forward of finite measures `f* : FiniteMeasure X → FiniteMeasure Y` is continuous
(in the topologies of weak convergence of measures). -/
lemma continuous_map {f : Ω → Ω'} (f_cont : Continuous f) :
Continuous (fun ν ↦ ... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 789 | 801 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... |
instance invertibleT (n : ℤ) : Invertible (T n : R[T;T⁻¹]) where
invOf := T (-n)
invOf_mul_self := by rw [← T_add, neg_add_cancel, T_zero]
| Mathlib/Algebra/Polynomial/Laurent.lean | 219 | 222 |
/-
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.Group.Commute.Defs
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Algebra.Opposites
import Mathlib.Tactic.MinImports
import Mathlib.Tactic.Spread
/... | Mathlib/Algebra/Group/Opposite.lean | 257 | 257 | |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | @[reassoc (attr := simp)]
theorem limit.isoLimitCone_hom_π {F : J ⥤ C} [HasLimit F] (t : LimitCone F) (j : J) :
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 224 | 225 |
/-
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.Probability.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | · rfl
· exact hm
end
section CondIndep
lemma condIndep_iff_condIndepSets (m' m₁ m₂ : MeasurableSpace Ω) {mΩ : MeasurableSpace Ω}
| Mathlib/Probability/Independence/Conditional.lean | 256 | 263 |
/-
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... | Mathlib/MeasureTheory/Integral/Average.lean | 822 | 875 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring.Finset... |
/-- The values of an additive character on an additive group are units. -/
lemma val_isUnit {A M} [AddGroup A] [Monoid M] (φ : AddChar A M) (a : A) : IsUnit (φ a) :=
| Mathlib/Algebra/Group/AddChar.lean | 369 | 371 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,479 | 1,483 | |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Stopping
import Mathlib.Tactic.AdaptationNote
/-!
# Hitting time
Given a stochastic process, the hitting time provides th... | exact min_rec' (fun j => u j ω ∈ s) hk₁_s hk₂_s
obtain ⟨k, hk₁, hk₂⟩ := h''
refine le_trans ?_ (hk₁.2.trans (min_le_right _ _))
exact hitting_le_of_mem hk₁.1 (hk₁.2.trans (min_le_left _ _)) hk₂
theorem hitting_le_iff_of_lt [WellFoundedLT ι] {m : ι} (i : ι) (hi : i < m) :
hitting u s n m ω ≤ i ↔ ∃... | Mathlib/Probability/Process/HittingTime.lean | 161 | 173 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... |
@[deprecated IsNilpotent.nilpotent (since := "2025-01-07")]
theorem exists_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent L M] :
∃ k, lowerCentralSeries R L M k = ⊥ :=
IsNilpotent.nilpotent R L M
@[simp] lemma iInf_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent L M] :
| Mathlib/Algebra/Lie/Nilpotent.lean | 279 | 285 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... |
theorem tendsto_add_one_pow_atTop_atTop_of_pos
[Semiring α] [LinearOrder α] [IsStrictOrderedRing α] [Archimedean α] {r : α}
(h : 0 < r) : Tendsto (fun n : ℕ ↦ (r + 1) ^ n) atTop atTop :=
tendsto_atTop_atTop_of_monotone' (pow_right_mono₀ <| le_add_of_nonneg_left h.le) <|
not_bddAbove_iff.2 fun _ ↦ Set.exi... | Mathlib/Analysis/SpecificLimits/Basic.lean | 148 | 163 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | integralSum f vol (π.biUnionTagged πi) = ∑ J ∈ π.boxes, integralSum f vol (πi J) := by
refine (π.sum_biUnion_boxes _ _).trans <| sum_congr rfl fun J hJ => sum_congr rfl fun J' hJ' => ?_
rw [π.tag_biUnionTagged hJ hJ']
theorem integralSum_biUnion_partition (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F)
| Mathlib/Analysis/BoxIntegral/Basic.lean | 83 | 87 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 880 | 882 | |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | theorem modByMonicHom_root_pow (h : IsAdjoinRootMonic S f) {n : ℕ} (hdeg : n < natDegree f) :
h.modByMonicHom (h.root ^ n) = X ^ n := by
nontriviality R
| Mathlib/RingTheory/IsAdjoinRoot.lean | 345 | 347 |
/-
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... | rcases hi with ⟨j, hj⟩
rw [← hj, s₁.dist_circumcenter_eq_circumradius j]
| Mathlib/Geometry/Euclidean/Circumcenter.lean | 407 | 408 |
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | liftAux R f a = liftAux R f b := by
induction h with
| lie_self a' => simp only [liftAux_map_mul, NonUnitalAlgHom.map_zero, lie_self]
| leibniz_lie a' b' c' =>
simp only [liftAux_map_mul, liftAux_map_add, sub_add_cancel, lie_lie]
| smul b' _ h₂ => simp only [liftAux_map_smul, h₂]
| add_right c' _ h₂ =... | Mathlib/Algebra/Lie/Free.lean | 197 | 206 |
/-
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.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Sub.Basic
impo... | (h : AddLECancellable (b * c)) : (a - b) * c = a * c - b * c := by
obtain (hab | hba) := total_of (· ≤ ·) a b
· rw [tsub_eq_zero_iff_le.2 hab, zero_mul, tsub_eq_zero_iff_le.2 (mul_le_mul_right' hab c)]
· apply h.eq_tsub_of_add_eq
rw [← add_mul, tsub_add_cancel_of_le hba]
| Mathlib/Algebra/Order/Ring/Canonical.lean | 106 | 110 |
/-
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... | integral_Ioi_of_hasDerivAt_of_tendsto hcont hderiv
(integrableOn_Ioi_deriv_of_nonneg hcont hderiv g'pos hg) hg
/-- When a function has a limit at infinity `l`, and its derivative is nonnegative, then the
integral of the derivative on `(a, +∞)` is `l - g a` (and the derivative is integrable, see
`integrable_on_Io... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 814 | 820 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.BigOperators.Option
import Ma... |
end Fintype
end Pi
-- TODO: this is a basic theorem about `Fintype.card`,
-- and ideally could be moved to `Mathlib.Data.Fintype.Card`.
| Mathlib/Data/Fintype/BigOperators.lean | 191 | 196 |
/-
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.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | | succ k ih => rw [pow_succ, pow_succ, ih, ← toMatrix_mul]
theorem Matrix.toLin_mul [Finite l] [DecidableEq m] (A : Matrix l m R) (B : Matrix m n R) :
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 644 | 646 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... |
@[simp] theorem maximal_ge_iff : Maximal (y ≤ ·) x ↔ y ≤ x ∧ IsMax x :=
| Mathlib/Order/Minimal.lean | 187 | 188 |
/-
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 Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Un... | lemma cast_comm (q : ℚ≥0) (a : α) : q * a = a * q := cast_commute _ _
| Mathlib/Data/Rat/Cast/Defs.lean | 47 | 48 |
/-
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 | 270 | 272 | |
/-
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.RingTheory.Valuation.Basic
/-!
# Ring of integers under a given valuation
The elements with valuation less than or equal to 1.
TODO: Define characteristic pre... | (fun hy : algebraMap O F y = 0 =>
have hx : x = 0 :=
hv.1 <|
(algebraMap O F).map_zero.symm ▸ (v.zero_iff.1 <| le_zero_iff.1 (v.map_zero ▸ hy ▸ h))
hx.symm ▸ dvd_zero y)
fun hy : algebraMap O F y ≠ 0 =>
have : v ((algebraMap O F y)⁻¹ * algebraMap O F x) ≤ 1 := by
rw [← v.... | Mathlib/RingTheory/Valuation/Integers.lean | 113 | 125 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | all_goals omega
theorem Nat.decreasing_induction_of_infinite
(h : ∀ n, P (n + 1) → P n) (hP : { x | P x }.Infinite) (n : ℕ) : P n :=
Nat.decreasing_induction_of_not_bddAbove h (mt BddAbove.finite hP) n
theorem Nat.cauchy_induction' (seed : ℕ) (h : ∀ n, P (n + 1) → P n) (hs : P seed)
(hi : ∀ x, seed ≤ x ... | Mathlib/Order/Interval/Finset/Nat.lean | 294 | 302 |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,
Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser
-/
import Mathlib.NumberTheory.FLT.Basic
import... | exact dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT Int.prime_three h3a h3c HF
· refine H (-b) (-c) a ha (by simp [h3b]) (by simp [h3c]) h3a hbc ?_
rw [add_eq_zero_iff_eq_neg, ← (show Odd 3 by decide).neg_pow] at HF
rw [← HF]
ring
open Finset in
private lemma fermatLastTheoremThree_of_dvd_a_of_gcd_eq_one_o... | Mathlib/NumberTheory/FLT/Three.lean | 93 | 113 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | rw [disjoint_comm]
exact H1.symm.mul_left H2.symm
| Mathlib/GroupTheory/Perm/Support.lean | 104 | 106 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum... | end ADEInequality
| Mathlib/NumberTheory/ADEInequality.lean | 251 | 256 |
/-
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
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | Mathlib/Data/List/Range.lean | 168 | 169 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Alg... |
@[to_additive]
lemma prod_map_le_prod_map {s : Multiset ι} (f : ι → α) (g : ι → α) (h : ∀ i, i ∈ s → f i ≤ g i) :
(s.map f).prod ≤ (s.map g).prod :=
prod_le_prod_of_rel_le <| rel_map.2 <| rel_refl_of_refl_on h
| Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean | 55 | 60 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... |
/-- If the angle between two vectors is nonzero, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ 0 := by
rintro rfl; simp at h
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 78 | 82 |
/-
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.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... |
theorem hasMFDerivAt_id (x : M) :
HasMFDerivAt I I (@id M) x (ContinuousLinearMap.id 𝕜 (TangentSpace I x)) := by
refine ⟨continuousAt_id, ?_⟩
have : ∀ᶠ y in 𝓝[range I] (extChartAt I x) x, (extChartAt I x ∘ (extChartAt I x).symm) y = y := by
apply Filter.mem_of_superset (extChartAt_target_mem_nhdsWithin x... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 122 | 129 |
/-
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.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | (this.comp (quasiMeasurePreserving_inv_of_right_invariant μ))
simp_rw [Function.comp_def, mul_inv_rev, inv_inv] at this
exact this
end QuasiMeasurePreserving
| Mathlib/MeasureTheory/Group/Prod.lean | 459 | 464 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,262 | 2,263 | |
/-
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.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... | protected def lid (M : Type v) (ι : Type* := PUnit) [AddCommMonoid M] [Module R M] [Unique ι] :
(⨁ _ : ι, M) ≃ₗ[R] M :=
{ DirectSum.id M ι, toModule R ι M fun _ ↦ LinearMap.id with }
| Mathlib/Algebra/DirectSum/Module.lean | 180 | 182 |
/-
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 | 706 | 706 | |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 781 | 784 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 858 | 863 | |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | ∃ n : ℕ, Submodule.span R s ≤ degreeLT R n := by
rcases span_le_degreeLE_of_finite s_fin with ⟨n, _⟩
exact ⟨n + 1, by rwa [degreeLT_succ_eq_degreeLE]⟩
/-- If `R` is a nontrivial ring, the polynomials `R[X]` are not finite as an `R`-module. When `R` is
a field, this is equivalent to `R[X]` being an infinite-dim... | Mathlib/RingTheory/Polynomial/Basic.lean | 240 | 249 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... |
@[simp] theorem lmarginal_univ [Fintype δ] {f : (∀ i, X i) → ℝ≥0∞} :
∫⋯∫⁻_univ, f ∂μ = fun _ => ∫⁻ x, f x ∂Measure.pi μ := by
let e : { j // j ∈ Finset.univ } ≃ δ := Equiv.subtypeUnivEquiv mem_univ
ext1 x
simp_rw [lmarginal, measurePreserving_piCongrLeft μ e |>.lintegral_map_equiv, updateFinset_def]
simp
| Mathlib/MeasureTheory/Integral/Marginal.lean | 190 | 196 |
/-
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.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
impor... |
@[simp]
theorem t'_iij (i j : D.J) : D.t' i i j = (pullbackSymmetry _ _).hom := by
have eq₁ := D.t_fac i i j
have eq₂ := (IsIso.eq_comp_inv (D.f i i)).mpr (@pullback.condition _ _ _ _ _ _ (D.f i j) _)
rw [D.t_id, Category.comp_id, eq₂] at eq₁
have eq₃ := (IsIso.eq_comp_inv (D.f i i)).mp eq₁
rw [Category.asso... | Mathlib/CategoryTheory/GlueData.lean | 77 | 85 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.... | theorem sum_conjClasses_card_eq_card [Fintype <| ConjClasses G] [Fintype G]
[∀ x : ConjClasses G, Fintype x.carrier] :
∑ x : ConjClasses G, x.carrier.toFinset.card = Fintype.card G := by
suffices (Σ x : ConjClasses G, x.carrier) ≃ G by simpa using (Fintype.card_congr this)
simpa [carrier_eq_preimage_mk] usi... | Mathlib/GroupTheory/ClassEquation.lean | 31 | 35 |
/-
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.Order.Group.Indicator
import Mathlib.Analysis.Normed.Affine.AddTorsor
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analy... | (fun _ => zero_le_one) _⟩
| succ n ihn =>
simp only [approx, mem_Icc]
refine ⟨midpoint_le_midpoint ?_ (ihn _).1, midpoint_le_midpoint (ihn _).2 ?_⟩ <;>
apply approx_le_approx_of_U_sub_C
exacts [subset_closure, subset_closure]
theorem approx_le_succ (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ ... | Mathlib/Topology/UrysohnsLemma.lean | 209 | 217 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... |
protected theorem ContinuousOn.arcsin (h : ContinuousOn f s) : ContinuousOn (arcsin <| f ·) s :=
fun x hx ↦ (h x hx).arcsin
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 462 | 465 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | obtain ⟨g₀, hg₀⟩ := exists_max_im z
-- then among those, minimize re
obtain ⟨g, hg, hg'⟩ := exists_row_one_eq_and_min_re z (bottom_row_coprime g₀)
refine ⟨g, ?_⟩
| Mathlib/NumberTheory/Modular.lean | 414 | 417 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... |
/-- A positive solution is a generator (up to sign) of the group of all solutions to the
Pell equation `x^2 - d*y^2 = 1` if and only if it is a fundamental solution. -/
theorem pos_generator_iff_fundamental (a : Solution₁ d) :
(1 < a.x ∧ 0 < a.y ∧ ∀ b : Solution₁ d, ∃ n : ℤ, b = a ^ n ∨ b = -a ^ n) ↔ IsFundamental... | Mathlib/NumberTheory/Pell.lean | 661 | 687 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,123 | 1,128 | |
/-
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.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
/-!
# Identically distributed random variables
Two random variable... | theorem eLpNorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν)
(p : ℝ≥0∞) : eLpNorm f p μ = eLpNorm g p ν := by
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp only [h_top, eLpNorm, eLpNormEssSup, ENNReal.top_ne_zero, eq_self_iff_true, if_true,
if_false]
... | Mathlib/Probability/IdentDistrib.lean | 190 | 206 |
/-
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... | hf.comp₂ (· • ·) hg
theorem EventuallyEq.sup [Max β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f')
(hg : g =ᶠ[l] g') : (fun x => f x ⊔ g x) =ᶠ[l] fun x => f' x ⊔ g' x :=
hf.comp₂ (· ⊔ ·) hg
theorem EventuallyEq.inf [Min β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f')
(hg : g =ᶠ[l] g') : (fu... | Mathlib/Order/Filter/Basic.lean | 973 | 983 |
/-
Copyright (c) 2016 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.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | if and only if it is `(f · = ·)` for some function `f`. -/
lemma forall_existsUnique_iff {r : α → β → Prop} :
(∀ a, ∃! b, r a b) ↔ ∃ f : α → β, ∀ {a b}, r a b ↔ f a = b := by
| Mathlib/Logic/Function/Basic.lean | 932 | 934 |
/-
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.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Set.UnorderedInterval
/-!
# Locally... | ⟨h.toFinset.sup' ((Finite.toFinset_nonempty h).mpr hs) id, fun x hx ↦ Finset.le_sup' id
((Finite.mem_toFinset h).mpr hx)⟩⟩,
fun ⟨h₀, h₁⟩ ↦ BddBelow.finite_of_bddAbove h₀ h₁⟩
| Mathlib/Order/Interval/Finset/Defs.lean | 1,219 | 1,221 |
/-
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.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | theorem isOpen_iff {s : Set (CofiniteTopology X)} : IsOpen s ↔ s.Nonempty → sᶜ.Finite :=
Iff.rfl
| Mathlib/Topology/Constructions.lean | 256 | 258 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... |
section Group
/-! ### Nonsingular points -/
variable (W') in
/-- A nonsingular point on a Weierstrass curve `W` in affine coordinates. This is either the unique
point at infinity `WeierstrassCurve.Affine.Point.zero` or a nonsingular affine point
`WeierstrassCurve.Affine.Point.some (x, y)` satisfying the Weierstrass ... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 608 | 624 |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | theorem Measurable.mono {ma ma' : MeasurableSpace α} {mb mb' : MeasurableSpace β} {f : α → β}
(hf : @Measurable α β ma mb f) (ha : ma ≤ ma') (hb : mb' ≤ mb) : @Measurable α β ma' mb' f :=
| Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 182 | 183 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 1,065 | 1,068 | |
/-
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.Algebra.Order.Ring.Canonical
import Mathlib.Order.Partition.Equipartition
/-!
# Equitabilising a partition
This file allows t... | /-- We can find equipartitions of arbitrary size. -/
theorem exists_equipartition_card_eq (hn : n ≠ 0) (hs : n ≤ #s) :
∃ P : Finpartition s, P.IsEquipartition ∧ #P.parts = n := by
rw [← pos_iff_ne_zero] at hn
have : (n - #s % n) * (#s / n) + #s % n * (#s / n + 1) = #s := by
| Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean | 190 | 194 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... | @[deprecated (since := "2025-03-10")]
protected alias prod_mk_left := LipschitzWith.prodMk_left
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 244 | 245 |
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