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 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 819 | 835 | |
/-
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.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
/-!
# Action of regular elements on a module
We introduce `M`-regular elements, in the... | @[simp]
theorem smul_iff (b : S) (ha : IsSMulRegular M a) : IsSMulRegular M (a • b) ↔ IsSMulRegular M b :=
⟨of_smul _, ha.smul⟩
theorem isLeftRegular [Mul R] {a : R} (h : IsSMulRegular R a) : IsLeftRegular a :=
| Mathlib/Algebra/Regular/SMul.lean | 74 | 78 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | theorem min_div_div_right_of_nonpos (hc : c ≤ 0) (a b : α) : min (a / c) (b / c) = max a b / c :=
Eq.symm <| Antitone.map_max fun _ _ => div_le_div_of_nonpos_of_le hc
| Mathlib/Algebra/Order/Field/Basic.lean | 641 | 642 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... |
@[simp]
theorem midpoint_add_self (x y : V) : midpoint R x y + midpoint R x y = x + y :=
calc
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 188 | 191 |
/-
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.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | simp [LipschitzWith, h.edist_eq]
| Mathlib/Topology/MetricSpace/Isometry.lean | 620 | 622 |
/-
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.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | rw [← Finset.card_union_of_disjoint, Finset.filter_union_right]
· exact s.card_filter_le _
apply Finset.disjoint_filter.mpr
rintro g₁ _ ⟨g₂, h1g₂, h2g₂⟩ ⟨g₃, h1g₃, h2g₃⟩
simp only [mem_preimage] at h1g₃ h1g₂
refine h.le_bot (?_ : g₁⁻¹ ∈ _)
constructor <;> simp only [Set.mem_inv, Set.mem_mul, exists_exists... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 222 | 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, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,216 | 1,216 | |
/-
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.Data.SetLike.Basic
import Mathlib.ModelTheory.Semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure t... | theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
| Mathlib/ModelTheory/Definability.lean | 99 | 102 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Analysis.Filter
import Mathlib.Topology.Bases
import Mathlib.Topology.LocallyFinite
/-!
# Computational realization of topological spaces (experi... | ⟨{ x : Set α // IsOpen x },
{ f := Subtype.val
| Mathlib/Data/Analysis/Topology.lean | 157 | 158 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | proper ZFA prelist. An atom has no members. -/
| Mathlib/SetTheory/Lists.lean | 246 | 246 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | Mathlib/Data/Rat/Defs.lean | 515 | 519 | |
/-
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.... | (∑ i ∈ range n, x ^ i) * (1 - x) = 1 - x ^ n := by
have := congr_arg Neg.neg (geom_sum_mul x n)
rw [neg_sub, ← mul_neg, neg_sub] at this
exact this
| Mathlib/Algebra/GeomSum.lean | 240 | 244 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 485 | 489 | |
/-
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 toIcoDiv_zsmul_add (a b : α) (m : ℤ) : toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b := by
| Mathlib/Algebra/Order/ToIntervalMod.lean | 218 | 219 |
/-
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.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | · simp [d₂_eq_zero _ c₁₂ i₁ i₂ i₁₂ h]
@[reassoc]
lemma mapMap_D₁ (i₁₂ i₁₂' : I₁₂) :
GradedObject.mapMap (toGradedObjectMap φ) _ i₁₂ ≫ L.D₁ c₁₂ i₁₂ i₁₂' =
| Mathlib/Algebra/Homology/TotalComplex.lean | 373 | 377 |
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
/-!
# Gram-Schmidt Orthogonalization and Orthonor... | simp [gramSchmidtNormed, inner_smul_left, inner_smul_right, gramSchmidt_orthogonal 𝕜 f hij]
theorem span_gramSchmidtNormed (f : ι → E) (s : Set ι) :
span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s) := by
refine span_eq_span
| Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 275 | 279 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Subalgebras over Comm... |
end Center
| Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 861 | 863 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | simp only [add_quot, add_assoc]
zero_add := by
rintro ⟨⟨⟩⟩
simp [add_quot, ← zero_quot, zero_add]
add_zero := by
| Mathlib/Algebra/RingQuot.lean | 262 | 266 |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | x = y := by
apply ext'; intro i
induction' i with i i_ih
· cases x.approx 0
cases y.approx 0
constructor
· apply ext_aux x y x
· rw [← agree_iff_agree']
apply x.consistent
· rw [← agree_iff_agree', i_ih]
apply y.consistent
introv H'
dsimp only [iselect] at H
cases H'
... | Mathlib/Data/PFunctor/Univariate/M.lean | 510 | 527 |
/-
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.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-!
# Maps on the unit circle
In this file we prove some basic lemmas ... | noncomputable def homeomorphCircle (hT : T ≠ 0) : AddCircle T ≃ₜ Circle :=
(homeomorphAddCircle T (2 * π) hT (by positivity)).trans homeomorphCircle'
theorem homeomorphCircle_apply (hT : T ≠ 0) (x : AddCircle T) :
homeomorphCircle hT x = toCircle x := by
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 179 | 183 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... |
@[deprecated (since := "2024-10-24")]
alias dvd_ord_compl_of_dvd_not_dvd := dvd_ordCompl_of_dvd_not_dvd
/-- If `n` is a nonzero natural number and `p ≠ 1`, then there are natural numbers `e`
| Mathlib/Data/Nat/Factorization/Basic.lean | 262 | 266 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Action.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Order.Module.Defs
import Mathli... | Mathlib/Algebra/Order/Module/OrderedSMul.lean | 169 | 171 | |
/-
Copyright (c) 2023 Yaël Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
lemma UniformConcaveOn.strictConcaveOn (hf : UniformConcaveOn s φ f) (hφ : ∀ r, r ≠ 0 → 0 < φ r) :
StrictConcaveOn ℝ s f := by
refine ⟨hf.1, fun x hx y hy hxy a b ha hb hab ↦ (hf.2 hx hy ha.le hb.le hab).trans_lt' <|
lt_add_of_pos_right _ ?_⟩
rw [← sub_ne_zero, ← norm_pos_iff] at hxy
have := hφ _ hxy.ne'... | Mathlib/Analysis/Convex/Strong.lean | 77 | 83 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Hom.CompTypeclasses
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Notation.Prod
import Mathlib.Algebra.Ring.Act... | (f : F) (c : M) (x : X) : f (c • x) = c • f x :=
map_smulₛₗ f c x
@[to_additive]
instance : MulActionSemiHomClass (X →ₑ[φ] Y) φ X Y where
| Mathlib/GroupTheory/GroupAction/Hom.lean | 149 | 153 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,368 | 1,369 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... | rintro z h₁ h₂
refine (h₁ i₀ (mem_univ _)).2.not_lt ?_
simpa only [Function.update_self] using (h₂ i₀ (mem_univ _)).1
end PiPreorder
section PiPartialOrder
variable [DecidableEq ι] [∀ i, PartialOrder (α i)]
| Mathlib/Order/Interval/Set/Pi.lean | 101 | 109 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
/-!
# GCD... | /-! ### GCD -/
| Mathlib/Algebra/GCDMonoid/Multiset.lean | 104 | 106 |
/-
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 | 297 | 299 | |
/-
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... | end SecondCountableStronglyMeasurable
/-- A function is strongly measurable if and only if it is measurable and has separable
range. -/
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 625 | 628 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... | simp
@[simp] lemma primeFactors_one : primeFactors 1 = ∅ := by
| Mathlib/Data/Nat/PrimeFin.lean | 59 | 61 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | protected theorem sup {f g : ι → Ω → ℝ} (hf : Submartingale f ℱ μ) (hg : Submartingale g ℱ μ) :
Submartingale (f ⊔ g) ℱ μ := by
refine ⟨fun i => @StronglyMeasurable.sup _ _ _ _ (ℱ i) _ _ _ (hf.adapted i) (hg.adapted i),
fun i j hij => ?_, fun i => Integrable.sup (hf.integrable _) (hg.integrable _)⟩
refine E... | Mathlib/Probability/Martingale/Basic.lean | 234 | 240 |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | theorem rTensor_exact : Exact (rTensor Q f) (rTensor Q g) := by
rw [rTensor_exact_iff_lTensor_exact]
exact lTensor_exact Q hfg hg
/-- Right-exactness of tensor product (`rTensor`) -/
| Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 389 | 393 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.... | [∀ a : α, DecidablePred (Associated a)] {s : Multiset α} (n : α) (h : ∀ a ∈ s, Prime a)
(div : ∀ a ∈ s, a ∣ n) (uniq : ∀ a, s.countP (Associated a) ≤ 1) : s.prod ∣ n := by
| Mathlib/Algebra/BigOperators/Associated.lean | 120 | 121 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Idempotents.Karoubi
/-!
# Extension of functors to the idempotent completion
In this file, we construct an extension `functorExtension₁`
of func... | Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | 249 | 257 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Group.Subgroup.Map
import Mathlib.Algebra.Module.Submodule.... | have : ∀ i, (p i).map f ≤ p i := by
rintro i - ⟨v, hv, rfl⟩
exact hf i v hv
suffices (iInf p).map f ≤ iInf p by exact fun v hv => this ⟨v, hv, rfl⟩
| Mathlib/Algebra/Module/Submodule/Map.lean | 462 | 465 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,370 | 1,372 | |
/-
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... | (hf hb).not_isMin ha
theorem StrictAnti.isMin_of_apply (hf : StrictAnti f) (ha : IsMax (f a)) : IsMin a :=
| Mathlib/Order/Monotone/Basic.lean | 277 | 279 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | simp_rw [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_mul h, Multiset.toFinset_add]
exact Finset.card_union_le _ _
theorem natSepDegree_mul_eq_iff (g : F[X]) :
(f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ (f = 0 ∧ g = 0) ∨ IsCoprime f g := by
| Mathlib/FieldTheory/SeparableDegree.lean | 425 | 429 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | variable [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] {s : Set E}
/-- Alternative definition of set convexity, using division. -/
| Mathlib/Analysis/Convex/Basic.lean | 513 | 515 |
/-
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 | 3,015 | 3,018 | |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.NormNum.Ineq
/-!
# `norm_num` extension for integer div/mod and divides
This file adds support f... | (ha : IsInt a a') (hb : IsNat b b')
(hm : q * b' = m) (h : r + m = a') (h₂ : Nat.blt r b' = true) :
IsNat (a % b) r := ⟨by
obtain ⟨⟨rfl⟩, ⟨rfl⟩⟩ := ha, hb
simp only [← h, ← hm, Int.add_mul_emod_self_right]
rw [Int.emod_eq_of_lt] <;> [simp; simpa using h₂]⟩
| Mathlib/Tactic/NormNum/DivMod.lean | 90 | 96 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.AbstractCompletion
/-!
# Hausdorff completions of uniform spaces
The goal is to construct a left-adjoint to the... | SeparationQuotient.mk_eq_mk.1 h
end
end CauchyFilter
| Mathlib/Topology/UniformSpace/Completion.lean | 274 | 279 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... |
/-- If `f : ℝ → ℝ` is strictly convex on `S` and differentiable at `x ∈ S`, then the slope of any
secant line with left endpoint at `x` is strictly greater than the derivative of `f` at `x`. -/
lemma lt_slope_of_hasDerivAt (hfc : StrictConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
| Mathlib/Analysis/Convex/Deriv.lean | 771 | 774 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | s.WOppSide y (pointReflection R x y) :=
(wbtw_pointReflection R _ _).wOppSide₁₃ hx
theorem sOppSide_pointReflection {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) :
s.SOppSide y (pointReflection R x y) := by
| Mathlib/Analysis/Convex/Side.lean | 722 | 726 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | private theorem candidates_dist_inl (fA : f ∈ candidates X Y) (x y : X) :
f (inl x, inl y) = dist x y :=
fA.1.1.1.1.1 x y
private theorem candidates_dist_inr (fA : f ∈ candidates X Y) (x y : Y) :
f (inr x, inr y) = dist x y :=
fA.1.1.1.1.2 x y
| Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 129 | 136 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... |
/-- Any type `α` has decidable equality classically. -/
| Mathlib/Logic/Basic.lean | 700 | 701 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | {p' : Filter ι'} (h : TendstoUniformly F f p) (h' : TendstoUniformly F' f' p') :
TendstoUniformly (fun (i : ι × ι') a => (F i.1 a, F' i.2 a)) (fun a => (f a, f' a)) (p ×ˢ p') :=
(h.prodMap h').comp fun a => (a, a)
@[deprecated (since := "2025-03-10")]
alias TendstoUniformly.prod := TendstoUniformly.prodMk
| Mathlib/Topology/UniformSpace/UniformConvergence.lean | 283 | 289 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
/-- The eigenspaces of a self-adjoint operator are mutually orthogonal. -/
theorem orthogonalFamily_eigenspaces (hT : T.IsSymmetric) :
| Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 76 | 79 |
/-
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.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
/-!
# The M construction as a multivariate polyn... | obtain ⟨g₁', e₁', rfl, rfl⟩ := M.bisim_lemma P e₁
obtain ⟨g₂', e₂', e₃, rfl⟩ := M.bisim_lemma P e₂
cases h'.symm.trans e₁'
cases h'.symm.trans e₂')
| root x a f h' i c =>
exact congr_fun (congr_fun e₃ i) c
| child x a f h' i c p IH =>
exact IH _ _ (h'' _)
theorem M.bisim₀ {α : TypeVec n} (R... | Mathlib/Data/PFunctor/Multivariate/M.lean | 236 | 267 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.Group.Idempotent
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.Order.Hom.BoundedLattic... |
theorem add_eq_zero' : a + b = 0 ↔ a = b :=
calc
a + b = 0 ↔ a = -b := add_eq_zero_iff_eq_neg
_ ↔ a = b := by rw [neg_eq]
| Mathlib/Algebra/Ring/BooleanRing.lean | 77 | 81 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... |
theorem LowerSemicontinuousAt.comp_continuousAt {f : α → β} {g : ι → α} {x : ι}
(hf : LowerSemicontinuousAt f (g x)) (hg : ContinuousAt g x) :
LowerSemicontinuousAt (fun x ↦ f (g x)) x :=
fun _ lt ↦ hg.eventually (hf _ lt)
theorem LowerSemicontinuousAt.comp_continuousAt_of_eq {f : α → β} {g : ι → α} {y : α}... | Mathlib/Topology/Semicontinuous.lean | 387 | 400 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib... | (hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
(bound_integrable : Integrable (bound : α → ℝ) μ)
(h_diff : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' a) x₀) :
Integrable F' μ ∧ HasDerivAt (fun x ↦ ∫ a, F x a ∂μ) (∫ a, F' a ∂μ) x₀ := by
... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 258 | 284 |
/-
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.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A schem... | theorem mem_basicOpen'' {U : X.Opens} (f : Γ(X, U)) (x : X) :
x ∈ X.basicOpen f ↔ ∃ (m : x ∈ U), IsUnit (X.presheaf.germ U x m f) :=
Iff.rfl
| Mathlib/AlgebraicGeometry/Scheme.lean | 530 | 533 |
/-
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.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 659 | 660 | |
/-
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 | 758 | 759 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... |
theorem coe_strictMono : StrictMono (fun (a : α) => (a : WithBot α)) := fun _ _ => coe_lt_coe.2
theorem coe_mono : Monotone (fun (a : α) => (a : WithBot α)) := fun _ _ => coe_le_coe.2
| Mathlib/Order/WithBot.lean | 282 | 286 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... |
@[deprecated (since := "2025-03-05")] alias
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 993 | 994 |
/-
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 | 986 | 988 | |
/-
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.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... |
theorem permutationsAux2_fst (t : α) (ts : List α) (r : List β) :
∀ (ys : List α) (f : List α → β), (permutationsAux2 t ts r ys f).1 = ys ++ ts
| [], _ => rfl
| Mathlib/Data/List/Permutation.lean | 56 | 59 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | by_contradiction fun ne =>
have : b₁ ∈ t a₁ ∩ t a₂ := ⟨h₁, b_eq.symm ▸ h₂⟩
| Mathlib/Data/Set/Lattice.lean | 1,308 | 1,309 |
/-
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.Homotopy
import Mathlib.Algebra.Homology.Linear
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.... | Mathlib/Algebra/Homology/HomotopyCategory.lean | 297 | 298 | |
/-
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 only [comp_v _ _ h p _ q rfl (by omega), neg_v, neg_comp]
@[simp]
protected lemma smul_comp {n₁ n₂ n₁₂ : ℤ} (k : R) (z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂)
(h : n₁ + n₂ = n₁₂) : (k • z₁).comp z₂ h = k • (z₁.comp z₂ h) := by
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 321 | 325 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 338 | 342 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 782 | 785 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | rather than typeclasses, with coercions back to `X ⟶ Y`.
Presumably we could write `X ↪ Y` and `X ↠ Y`.
-/
| Mathlib/CategoryTheory/Iso.lean | 460 | 463 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | mem_essImage f := by
have := Localization.essSurj L W
obtain ⟨X, ⟨eX⟩⟩ : ∃ (X : C), Nonempty (L.obj X ≅ f.left) :=
⟨_, ⟨L.objObjPreimageIso f.left⟩⟩
obtain ⟨Y, ⟨eY⟩⟩ : ∃ (Y : C), Nonempty (L.obj Y ≅ f.right) :=
⟨_, ⟨L.objObjPreimageIso f.right⟩⟩
obtain ⟨φ, hφ⟩ := Localization.exists_leftFr... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 767 | 780 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.AlgebraicGeometry.Limits
import Mathlib.CategoryTheory.MorphismPrope... | attribute [instance] AffineTargetMorphismProperty.IsLocal.respectsIso
open AffineTargetMorphismProperty in
instance (P : MorphismProperty Scheme) [IsLocalAtTarget P] : (of P).IsLocal where
respectsIso := inferInstance
to_basicOpen _ _ H := IsLocalAtTarget.restrict H _
of_basicOpenCover {_ Y} _ _ _ hs := IsLocalA... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 410 | 417 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory
import Mathlib.Algebra.Category.ModuleCat.Colimits
/-!
# Homology and exactness of short complexes of modules
In... | lemma moduleCat_exact_iff_ker_sub_range :
S.Exact ↔ LinearMap.ker S.g.hom ≤ LinearMap.range S.f.hom := by
rw [moduleCat_exact_iff]
aesop
lemma moduleCat_exact_iff_range_eq_ker :
S.Exact ↔ LinearMap.range S.f.hom = LinearMap.ker S.g.hom := by
rw [moduleCat_exact_iff_ker_sub_range]
| Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean | 48 | 55 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Field.Equiv
import Mathlib.Algebra.Field.Subfield.Basic
import Mathlib.Algebra.Order.Ring... | ∃ x y : A, y ∈ nonZeroDivisors A ∧ algebraMap _ _ x / algebraMap _ _ y = z :=
let ⟨x, ⟨y, hy⟩, h⟩ := mk'_surjective (nonZeroDivisors A) z
⟨x, y, hy, by rwa [mk'_eq_div] at h⟩
theorem isUnit_map_of_injective (hg : Function.Injective g) (y : nonZeroDivisors A) :
| Mathlib/RingTheory/Localization/FractionRing.lean | 180 | 184 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | /-- The cartesian product of `C^n` functions is `C^n`. -/
theorem ContDiff.prodMk {f : E → F} {g : E → G} (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
ContDiff 𝕜 n fun x : E => (f x, g x) :=
contDiffOn_univ.1 <| hf.contDiffOn.prodMk hg.contDiffOn
@[deprecated (since := "2025-03-09")]
alias ContDiff.prod := C... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 530 | 537 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | le_antisymm
(fun s hs =>
let ⟨s₁, hs₁, s₂, hs₂, h⟩ := mem_prod_iff.mp hs
| Mathlib/Order/Filter/Prod.lean | 316 | 318 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | obtain ⟨k, h⟩ := (mem_genWeightSpace _ _ _).mp hm x
refine ⟨k, ?_⟩
simpa [h] using LinearMap.congr_fun (Module.End.commute_pow_left_of_commute this k) m
variable {f}
| Mathlib/Algebra/Lie/Weights/Basic.lean | 521 | 526 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod... |
variable [NonAssocSemiring S] [NonAssocSemiring T]
| Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 39 | 40 |
/-
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... | Mathlib/Order/Interval/Finset/Nat.lean | 347 | 352 | |
/-
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... | induction s using Quot.inductionOn
simpa using List.formPerm_eq_self_of_not_mem _ _ hx
theorem formPerm_apply_mem_eq_next (s : Cycle α) (h : Nodup s) (x : α) (hx : x ∈ s) :
| Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 159 | 162 |
/-
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... | obtain ⟨a, b, ab, rfl⟩ := hσ
obtain ⟨c, d, cd, rfl⟩ := hτ
by_cases ac : a = c
· subst ac
exact swap_mul_swap_same_mem_closure_three_cycles ab cd
have h' : swap a b * swap c d = swap a b * swap a c * (swap c a * swap c d) := by
simp [swap_comm c a, mul_assoc]
rw [h']
exact
mul_mem (swap_mul_swa... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 673 | 685 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | exact Real.norm_of_nonneg <| by positivity
_ ≤ r i n := hn' i
lemma bi_min_div_two_lt_one : b (min_bi b) / 2 < 1 := by
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 191 | 194 |
/-
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.CategoryTheory.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... | Mathlib/Algebra/Homology/ImageToKernel.lean | 432 | 436 | |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Preserves.Shapes... | apply PullbackCone.mono_of_isLimitMkIdId _ (isLimitOfIsLimitPullbackConeMap F _ this)
instance (priority := 100) reflectsMonomorphisms_of_reflectsLimitsOfShape
[ReflectsLimitsOfShape WalkingCospan F] : F.ReflectsMonomorphisms where
reflects f _ := reflects_mono_of_reflectsLimit F f
| Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 45 | 49 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... |
theorem cardinal_lift_le_rank {ι : Type w} {v : ι → M}
(hv : LinearIndependent R v) :
Cardinal.lift.{v} #ι ≤ Cardinal.lift.{w} (Module.rank R M) := by
rw [Module.rank]
refine le_trans ?_ (lift_le.mpr <| le_ciSup (bddAbove_range _) ⟨_, hv.linearIndepOn_id⟩)
| Mathlib/LinearAlgebra/Dimension/Basic.lean | 79 | 84 |
/-
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.CategoryTheory.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... | Epi (imageToKernel f (0 : B ⟶ C) (by simp)) := by
simp only [imageToKernel_zero_right]
haveI := epi_image_of_epi f
rw [← imageSubobject_arrow]
infer_instance
| Mathlib/Algebra/Homology/ImageToKernel.lean | 147 | 152 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | | 0, ~q(ℝ), ~q(Gamma $a) =>
match ← core q(inferInstance) q(inferInstance) a with
| .positive pa =>
assertInstancesCommute
pure (.positive q(Gamma_pos_of_pos $pa))
| .nonnegative pa =>
assertInstancesCommute
pure (.nonnegative q(Gamma_nonneg_of_nonneg $pa))
| _ => pure .none
| Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 487 | 495 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Roots of natural numbers, rounded up and down
This file defines the flooring and... | /-- The RHS is a noncomputable version of `Nat.ceilRoot` with better order theoretical
properties. -/
lemma ceilRoot_def :
ceilRoot n a = if n = 0 ∨ a = 0 then 0 else (a.factorization ⌈/⌉ n).prod (· ^ ·) := by
unfold ceilRoot
split_ifs with h <;>
simp [Finsupp.ceilDiv_def, prod_mapRange_index pow_zero, Nat.... | Mathlib/Data/Nat/Factorization/Root.lean | 112 | 118 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | emultiplicity q p ≤ 1 := by
contrapose! hq
apply isUnit_of_self_mul_dvd_separable hsep
rw [← sq]
apply pow_dvd_of_le_emultiplicity
exact Order.add_one_le_of_lt hq
/-- A separable polynomial is square-free.
See `PerfectField.separable_iff_squarefree` for the converse when the coefficients are a perfect
f... | Mathlib/FieldTheory/Separable.lean | 175 | 186 |
/-
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.Order.Floor.Semiring
import Mathlib.Data.Nat.Log
/-!
# Integer logarithms in a field with respect to a natural base
This file defines two `ℤ`-value... | rw [log_of_right_le_one _ (hr.trans zero_le_one),
Nat.clog_of_right_le_one ((Nat.ceil_eq_zero.mpr <| inv_nonpos.2 hr).trans_le zero_le_one),
Int.ofNat_zero, neg_zero]
| Mathlib/Data/Int/Log.lean | 87 | 90 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | protected theorem gen_mem_nhds (f : α →ᵤ[𝔖] β) (hs : s ∈ 𝔖) {V : Set (β × β)} (hV : V ∈ 𝓤 β) :
{g | ∀ x ∈ s, (toFun 𝔖 f x, toFun 𝔖 g x) ∈ V} ∈ 𝓝 f := by
rw [UniformOnFun.nhds_eq]
apply_rules [mem_iInf_of_mem, mem_principal_self]
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 769 | 773 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 166 | 166 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... |
open List
namespace RelEmbedding
variable {α β : Type*} {ra : α → α → Prop} {rb : β → β → Prop}
@[simp]
theorem sorted_listMap (e : ra ↪r rb) {l : List α} : (l.map e).Sorted rb ↔ l.Sorted ra := by
| Mathlib/Data/List/Sort.lean | 251 | 259 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | smul_mono h le_rfl
instance : CovariantClass (Submodule R A) (Submodule R M) HSMul.hSMul LE.le :=
| Mathlib/Algebra/Algebra/Operations.lean | 143 | 145 |
/-
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 | 1,590 | 1,592 | |
/-
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.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Order.Interval.Finset.Basic
/-!
# Finite... | omega
| Mathlib/Data/Int/Interval.lean | 137 | 138 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | simpa only [conjTranspose_conjTranspose] using posSemidef_conjTranspose_mul_self Aᴴ
lemma eigenvalues_conjTranspose_mul_self_nonneg (A : Matrix m n 𝕜) [DecidableEq n] (i : n) :
0 ≤ (isHermitian_transpose_mul_self A).eigenvalues i :=
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 284 | 287 |
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | @[simp, mfld_simps]
theorem subtypeRestr_coe :
((e.subtypeRestr hs : PartialHomeomorph s Y) : s → Y) = Set.restrict ↑s (e : X → Y) :=
rfl
@[simp, mfld_simps]
| Mathlib/Topology/PartialHomeomorph.lean | 1,299 | 1,304 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | represent the same element of `W`. -/
def IsReduced (ω : List B) : Prop := ℓ (π ω) = ω.length
@[simp]
| Mathlib/GroupTheory/Coxeter/Length.lean | 208 | 211 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 305 | 306 | |
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... |
theorem DifferentiableWithinAt.hasGradientWithinAt (h : DifferentiableWithinAt 𝕜 f s x) :
HasGradientWithinAt f (gradientWithin f s x) s x := by
rw [hasGradientWithinAt_iff_hasFDerivWithinAt, gradientWithin,
(toDual 𝕜 F).apply_symm_apply (fderivWithin 𝕜 f s x)]
| Mathlib/Analysis/Calculus/Gradient/Basic.lean | 127 | 131 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | nhds_eq_comap_uniformity x := by
simp only [nhds_induced, nhds_eq_comap_uniformity, comap_comap, Function.comp_def]
| Mathlib/Topology/UniformSpace/Basic.lean | 388 | 390 |
/-
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.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | (ENNReal.ofReal (rnDerivAux κ (κ + η) a x) /
ENNReal.ofReal (1 - rnDerivAux κ (κ + η) a x)) ∂(κ + η) a
_ = ∫⁻ x in s, ENNReal.ofReal (rnDerivAux κ (κ + η) a x) ∂(κ + η) a := by
refine setLIntegral_congr_fun hsm (ae_of_all _ fun x hx ↦ ?_)
rw [ofNNReal_toNNReal, ← ENNReal.ofReal_div_of_pos, d... | Mathlib/Probability/Kernel/RadonNikodym.lean | 356 | 389 |
/-
Copyright (c) 2022 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.Algebra.Module.Submodule.Invariant
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.RingTheory.Finit... | simp_rw [RingHom.id_apply, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
LinearMap.comp_apply, LinearEquiv.coe_toLinearMap] at h ⊢
simp_rw [pow_succ, ← mul_assoc, mul_smul _ X, ← hf, ← of_symm_X_smul, ← h]
/-- Construct an `R[X]`-linear equivalence out of `AEval R M a` from a `R`-linear map out of `... | Mathlib/Algebra/Polynomial/Module/AEval.lean | 96 | 101 |
/-
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.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | (fun _ ih => by apply even <;> exact ih _ <| by linarith only)
(fun _ ih => by apply odd <;> exact ih _ <| by linarith only) n
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 137 | 139 |
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