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 |
|---|---|---|---|---|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
/-!
# Egorov theorem
This file contains the Egorov theorem which states that an almost everywhere convergen... | exact tendsto_const_nhds
rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]
refine tendsto_measure_iInter_atTop
(fun n ↦ (hsm.inter <| notConvergentSeq_measurableSet hf hg).nullMeasurableSet)
(fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)
⟨h.some,... | Mathlib/MeasureTheory/Function/Egorov.lean | 81 | 93 |
/-
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.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# The derivative of func... | DifferentiableWithinAtProp 𝓘(𝕜, E) I' f s x ↔ DifferentiableWithinAt 𝕜 (I' ∘ f) s x := by
simp_rw [DifferentiableWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq]
theorem DifferentiableWithinAtProp_self {f : E → E'} {s :... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 134 | 177 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
import Mathlib.MeasureTh... | {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
(∫⁻ a, (f * g) a ∂μ) ≤ (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) * (∫⁻ a, g a ^ q ∂μ) ^ (1 / q) := by
by_cases hf_zero : ∫⁻ a, f a ^ p ∂μ = 0
· refine Eq.trans_le ?_ (zero_le _)
exact lintegral_mul_eq_zero_of_lintegral_rpow_eq_zero hpq.nonneg hf hf_z... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 150 | 156 |
/-
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.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | Mathlib/Order/Filter/Lift.lean | 435 | 437 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Ker... | · ext; simp [Sigma.desc, coproductIsCoproduct]
· ext; simp [Pi.lift, productIsProduct]
| Mathlib/CategoryTheory/Limits/Opposites.lean | 641 | 642 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... |
@[simp]
theorem congr_refl : congr (.refl : A ≃ₐ[S] A) (.refl : B ≃ₐ[R] B) = .refl :=
AlgEquiv.coe_algHom_injective <| map_id
| Mathlib/RingTheory/TensorProduct/Basic.lean | 1,076 | 1,079 |
/-
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.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | (Nat.cast_inj (R := ℤ)).1
(by rw [Int.ofNat_sub (Int.le_of_ofNat_le_ofNat <| Int.le.intro_sub _ h)]; exact h),
fun h =>
| Mathlib/NumberTheory/PellMatiyasevic.lean | 180 | 182 |
/-
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... | factors uniquely through a morphism `RingQuot r →+* T`.
-/
irreducible_def lift {r : R → R → Prop} :
{ f : R →+* T // ∀ ⦃x y⦄, r x y → f x = f y } ≃ (RingQuot r →+* T) :=
{ toFun := fun f ↦ preLift f.prop
| Mathlib/Algebra/RingQuot.lean | 432 | 436 |
/-
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.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathli... |
lemma IsOpen.exists_msmooth_support_eq_aux {s : Set H} (hs : IsOpen s) :
∃ f : H → ℝ, f.support = s ∧ ContMDiff I 𝓘(ℝ) ∞ f ∧ Set.range f ⊆ Set.Icc 0 1 := by
have h's : IsOpen (I.symm ⁻¹' s) := I.continuous_symm.isOpen_preimage _ hs
rcases h's.exists_smooth_support_eq with ⟨f, f_supp, f_diff, f_range⟩
refine... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 656 | 666 |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | have h₂s :=
sign_eq_of_affineCombination_mem_affineSpan_single_lineMap t.independent hw (Finset.mem_univ _)
(Finset.mem_univ _) (Finset.mem_univ _) h₁₂.symm h₂₃ h₁₃ hr₂0 hr₂1 h₂'
| Mathlib/Analysis/Convex/Between.lean | 561 | 563 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | (Ico ⌈a / (r : ℚ)⌉ ⌈b / (r : ℚ)⌉).map ⟨(· * r), mul_left_injective₀ hr.ne'⟩ := by
ext x
simp only [mem_map, mem_filter, mem_Ico, ceil_le, lt_ceil, div_le_iff₀, lt_div_iff₀,
| Mathlib/Data/Int/CardIntervalMod.lean | 47 | 49 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | @[simp]
theorem Icc_self (a : α) : Icc a a = {a} := by rw [← coe_eq_singleton, coe_Icc, Set.Icc_self]
| Mathlib/Order/Interval/Finset/Basic.lean | 543 | 544 |
/-
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 | 3,201 | 3,211 | |
/-
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... | (op₃ : γ → ι → γ) (a : α) (b : β) (h : ∀ a b i, f (op₁ a i) (op₂ b i) = op₃ (f a b) i) :
foldl op₃ (f a b) l = f (foldl op₁ a l) (foldl op₂ b l) :=
Eq.symm <| by
revert a b
| Mathlib/Data/List/Basic.lean | 853 | 856 |
/-
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... | Mathlib/RingTheory/Polynomial/Basic.lean | 1,276 | 1,281 | |
/-
Copyright (c) 2023 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.Basic
import Mathlib.Topology.ContinuousMap.Bounded.Normed
/-!
# Integration of bounded continuous functions
In this file,... | lemma integral_const_sub (f : X →ᵇ ℝ) (c : ℝ) :
∫ x, (const X c - f) x ∂μ = μ.real Set.univ • c - ∫ x, f x ∂μ := by
simp [integral_sub (integrable_const c) (f.integrable _)]
| Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean | 128 | 130 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,515 | 1,517 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | Mathlib/Data/List/Nodup.lean | 410 | 413 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | theorem map₂_pure : map₂ m (pure a) (pure b) = pure (m a b) := by rw [map₂_pure_right, map_pure]
| Mathlib/Order/Filter/NAry.lean | 124 | 125 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 440 | 442 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Tactic.Inhabit
/-!
# Types with a unique term
In this file we define a typeclass `Unique`,
which expresses that a type h... | | ⟨⟨x⟩, h⟩, ⟨⟨y⟩, _⟩ => by congr; rw [h x, h y]
| Mathlib/Logic/Unique.lean | 133 | 134 |
/-
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.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | calc
inductiveLimitDist f x z = dist (leRecOn hx (f _) x.2 : X m) (leRecOn hz (f _) z.2 : X m) :=
inductiveLimitDist_eq_dist I x z m hx hz
_ ≤ dist (leRecOn hx (f _) x.2 : X m) (leRecOn hy (f _) y.2 : X m) +
dist (leRecOn hy (f _) y.2 : X m) (leRecOn hz (f _) z.2 : X m) :=
(d... | Mathlib/Topology/MetricSpace/Gluing.lean | 567 | 585 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.SelfAdjoint
/-... |
theorem coe_iSup_of_directed [Nonempty ι] {S : ι → NonUnitalStarSubalgebra R A}
| Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 1,009 | 1,010 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathl... | · simp
rw [MvPowerSeries.inv_eq_iff_mul_eq_one, ← map_mul, inv_mul_cancel₀ hr, map_one]
simpa using hr
@[simp]
| Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 283 | 287 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 558 | 561 | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | exact m.map_toEnumFinset_fst
@[simp]
theorem card_coe (m : Multiset α) : Fintype.card m = Multiset.card m := by
| Mathlib/Data/Multiset/Fintype.lean | 206 | 209 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Tactic.Lift... | Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean | 314 | 315 | |
/-
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
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | ⟨fun x y ↦ ⟨min x y, min_rec' _ x.2 y.2⟩⟩ (fun (a : Subtype p) ↦ (a : α))
| Mathlib/Order/Basic.lean | 1,042 | 1,042 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,125 | 1,132 | |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... |
theorem logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x := by
rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]
exact log_lt_log hx hxy
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 276 | 280 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 842 | 847 | |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | @[simp]
theorem realize_bot : (⊥ : L.Formula α).Realize v ↔ False :=
Iff.rfl
@[simp]
theorem realize_top : (⊤ : L.Formula α).Realize v ↔ True :=
BoundedFormula.realize_top
@[simp]
theorem realize_inf : (φ ⊓ ψ).Realize v ↔ φ.Realize v ∧ ψ.Realize v :=
BoundedFormula.realize_inf
@[simp]
theorem realize_imp : (φ.... | Mathlib/ModelTheory/Semantics.lean | 509 | 528 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | theorem lt_iff' : μ < ν ↔ μ ≤ ν ∧ ∃ s, μ s < ν s :=
lt_iff_le_not_le.trans <| and_congr Iff.rfl <| by simp only [le_iff', not_forall, not_le]
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 964 | 966 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file deals with the following version... | theorem cutExpand_add_single {a' a : α} (s : Multiset α) (h : r a' a) :
CutExpand r (s + {a'}) (s + {a}) :=
| Mathlib/Logic/Hydra.lean | 85 | 86 |
/-
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... | @[reassoc (attr := simp)]
lemma map_hom_ofInv_id (s : Y ⟶ X) (hs : W s) (L : C ⥤ D) (hL : W.IsInvertedBy L) :
L.map s ≫ (ofInv s hs).map L hL = 𝟙 _ := by
letI := hL _ hs
simp [map]
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 159 | 163 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... |
theorem splits_of_comp (j : L →+* F) {f : K[X]} (h : Splits (j.comp i) f)
(roots_mem_range : ∀ a ∈ (f.map (j.comp i)).roots, a ∈ j.range) : Splits i f := by
choose lift lift_eq using roots_mem_range
rw [splits_iff_exists_multiset]
refine ⟨(f.map (j.comp i)).roots.pmap lift fun _ ↦ id, map_injective _ j.injec... | Mathlib/Algebra/Polynomial/Splits.lean | 495 | 500 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
/-!
# Leading terms of Witt vector multiplication
Th... | (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) -
x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) := by
simp only [← peval_polyOfInterest']
obtain ⟨f₀, hf₀⟩ := exists_restrict_to_vars k (polyOfInterest_vars p n)
have : ∀ (a : Multiset (Fin 2)) (b : Multiset ℕ), a ×ˢ b = a.product b := fu... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 235 | 246 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | noncomputable def whiskerLeft {X Y Z : Mon_ C} (M : Bimod X Y) {N₁ N₂ : Bimod Y Z} (f : N₁ ⟶ N₂) :
M.tensorBimod N₁ ⟶ M.tensorBimod N₂ where
hom :=
colimMap
(parallelPairHom _ _ _ _ (_ ◁ f.hom) (_ ◁ f.hom)
(by rw [whisker_exchange])
(by
simp only [Category.assoc, tensor_whisker... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 355 | 371 |
/-
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.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | [NoZeroSMulDivisors R S] {p : R[X]} {s : S} (hpzero : p ≠ 0) (hp : aeval s p = 0) :
aeval s p.primPart = 0 := by
rw [eq_C_content_mul_primPart p, map_mul, aeval_C] at hp
have hcont : p.content ≠ 0 := fun h => hpzero (content_eq_zero_iff.1 h)
replace hcont := Function.Injective.ne (FaithfulSMul.algebraMap_... | Mathlib/RingTheory/Polynomial/Content.lean | 278 | 283 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | Mathlib/Data/Nat/PartENat.lean | 764 | 766 | |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... | rw [← (posSemidef_lapMatrix ℝ G).toLinearMap₂'_zero_iff, star_trivial,
lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable]
section
variable [DecidableEq G.ConnectedComponent]
lemma mem_ker_toLin'_lapMatrix_of_connectedComponent {G : SimpleGraph V} [DecidableRel G.Adj]
[DecidableEq G.ConnectedCom... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 125 | 137 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | @[deprecated (since := "2025-04-22")] alias toReal_setLaverage := toReal_setLAverage
/-! ### First moment method -/
| Mathlib/MeasureTheory/Integral/Average.lean | 474 | 476 |
/-
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.Data.Nat.Prime.Factorial
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Lemmas of Gauss and Eisenstein
... |
/-- The given sum is the number of integer points in the triangle formed by the diagonal of the
rectangle `(0, p/2) × (0, q/2)`. -/
private theorem sum_Ico_eq_card_lt {p q : ℕ} :
∑ a ∈ Ico 1 (p / 2).succ, a * q / p =
#{x ∈ Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ | x.2 * p ≤ x.1 * q} :=
if hp0 : p = 0 th... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 157 | 164 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | Mathlib/Algebra/Group/Defs.lean | 1,329 | 1,330 | |
/-
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.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | (v.pmap f (by
simp only [Nat.succ_eq_add_one, toList_cons, List.mem_cons, forall_eq_or_imp] at hp
exact hp.2)) := rfl
| Mathlib/Data/Vector/Basic.lean | 80 | 82 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Data.ENNReal.Action
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.MetricSpace.Algebra
... |
variable [NormedDivisionRing α] [SeminormedAddCommGroup β]
| Mathlib/Analysis/Normed/MulAction.lean | 111 | 112 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | theorem eq_div_of_mul_eq_left {a b c : R} (hb : b ≠ 0) (h : a * b = c) : a = c / b := by
rw [← h, mul_div_cancel_right₀ _ hb]
| Mathlib/Algebra/EuclideanDomain/Basic.lean | 84 | 85 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
import Mathlib.Topology.Homotopy.Contractible
import Mathlib.CategoryTheory.PUnit
import Math... | Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean | 94 | 98 | |
/-
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.Finset.Pi
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite.Basic
/-!
# Fintype instances for pi types
-/
assert_not_exists Order... | ((piFinset fun _i : α ↦ t).image fun f ↦ f a) = t := by
obtain rfl | ht := t.eq_empty_or_nonempty
· haveI : Nonempty α := ⟨a⟩
simp
· exact eval_image_piFinset (fun _ ↦ t) a fun _ _ ↦ ht
| Mathlib/Data/Fintype/Pi.lean | 100 | 105 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite.Range
import Mathlib.Order.Partition.Finpartition
/-!
# Equivalenc... | Mathlib/Data/Setoid/Partition.lean | 483 | 487 | |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Ackermann function
In this file, we def... | theorem ack_pos : ∀ m n, 0 < ack m n
| 0, n => by simp
| m + 1, 0 => by
rw [ack_succ_zero]
apply ack_pos
| m + 1, n + 1 => by
rw [ack_succ_succ]
apply ack_pos
| Mathlib/Computability/Ackermann.lean | 97 | 105 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | /-- right projection of a `prod` vector -/
def prod.snd : ∀ {n} {α β : TypeVec.{u} n}, α ⊗ β ⟹ β
| succ _, α, β, Fin2.fs i => @prod.snd _ (drop α) (drop β) i
| Mathlib/Data/TypeVec.lean | 428 | 430 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | ## The Gamma doubling formula
As a fun application of the Bohr-Mollerup theorem, we prove the Gamma-function doubling formula
(for positive real `s`). The idea is that `2 ^ s * Gamma (s / 2) * Gamma (s / 2 + 1 / 2)` is
log-convex and satisfies the Gamma functional equation, so it must actually be a constant
multiple o... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 358 | 365 |
/-
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.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... | Mathlib/Geometry/Euclidean/MongePoint.lean | 727 | 732 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | one_mul := Algebra.TensorProduct.one_mul
mul_one := Algebra.TensorProduct.mul_one
natCast_zero := AddMonoidWithOne.natCast_zero
| Mathlib/RingTheory/TensorProduct/Basic.lean | 392 | 394 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.... | ∃ g : SL(2, R), g 0 j = a ∧ g 1 j = b := by
obtain ⟨u, v, h⟩ := hab
refine match j with
| 0 => ⟨⟨!![a, -v; b, u], ?_⟩, rfl, rfl⟩
| 1 => ⟨⟨!![v, a; -u, b], ?_⟩, rfl, rfl⟩ <;>
· rw [Matrix.det_fin_two_of, ← h]
| Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 396 | 401 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... | theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 := by
classical
refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)
split_ifs with h <;> simp only [h, zero_le', le_rfl]
theorem apply_ne_top (p : PMF α) (a : α) : p a ≠ ∞ :=
ne_of_lt (lt_of_le_of_lt (p.coe_le_one a) ENNReal.one_lt_top)
th... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 115 | 133 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | theorem ne_of_isReal_isComplex {w w' : InfinitePlace K} (h : IsReal w) (h' : IsComplex w') :
w ≠ w' := fun h_eq ↦ not_isReal_iff_isComplex.mpr h' (h_eq ▸ h)
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 406 | 407 |
/-
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, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | simp only [le_iff_lt_or_eq, reduceCtorEq, or_false] at hmas
exact (List.not_lt_nil _ hmas).elim
| cons a' as =>
rw [List.chain'_cons] at ha
refine gt_of_gt_of_ge ha.1 ?_
rw [le_iff_lt_or_eq] at hmas
rcases hmas with hmas | hmas
· by_contra! hh
rw [← not_le] at hmas
... | Mathlib/Data/List/Chain.lean | 435 | 458 |
/-
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 | 736 | 740 | |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... | rw [mul_add, mul_one, ← mul_assoc, Units.mul_inv, one_mul]
end Units
namespace RingHom
| Mathlib/Algebra/Ring/Units.lean | 123 | 128 |
/-
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... | end IicFTC
section UnivFTC
variable {E : Type*} {f f' : ℝ → E} {m n : E} [NormedAddCommGroup E]
[NormedSpace ℝ E]
/-- **Fundamental theorem of calculus-2**, on the whole real line
When a function has a limit `m` at `-∞` and `n` at `+∞`, and its derivative is integrable, then the
integral of the derivative is `n - ... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 984 | 999 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.... | toFun := Coe.coe
map_one' := one_coe
map_mul' := mul_coe
| Mathlib/Data/PNat/Basic.lean | 209 | 212 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | Mathlib/Data/Part.lean | 758 | 759 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.IsPoly
/-!
## Multiplication by `n` in the ring of Witt vectors
In this file we show that multiplication by `n` in the ring of ... | · rw [wittMulN, ← bind₁_bind₁, wittAdd, wittStructureInt_prop]
simp only [map_add, Nat.cast_succ, bind₁_X_right]
| Mathlib/RingTheory/WittVector/MulP.lean | 67 | 68 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | LinearIsometryEquiv.ext <| o.rightAngleRotation_neg_orientation
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 291 | 291 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 778 | 791 | |
/-
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.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
/-!
# Homotopy between paths
In this file,... | variable {p₀ p₁ p₂ : Path x₀ x₁}
/-- Given a path `p`, we can define a `Homotopy p p` by `F (t, x) = p x`.
| Mathlib/Topology/Homotopy/Path.lean | 89 | 91 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.Identities
/-!
# Witt vectors over a domain
This file builds to the proof `WittVector.instIsDomain`,
an instance that says i... | let n := Nat.find hex
use n, x.shift n
refine ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp ?_⟩
exact Nat.find_min hex hi
/-!
## Witt vectors over a domain
If `R` is an integral domain, then so is `𝕎 R`.
This argument is adapted from
<https://math.stackexchange.com/questions/4117247/ring... | Mathlib/RingTheory/WittVector/Domain.lean | 88 | 98 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
/-!
# Surreal numbers
The basic theory of surreal numbers, built on top ... | /-- A small family of surreals is bounded below. -/
lemma bddBelow_range_of_small {ι : Type*} [Small.{u} ι] (f : ι → Surreal.{u}) :
BddBelow (Set.range f) := by
induction' f using Quotient.induction_on_pi with f
let g : ι → PGame.{u} := Subtype.val ∘ f
have hg (i) : (g i).Numeric := Subtype.prop _
conv in (... | Mathlib/SetTheory/Surreal/Basic.lean | 418 | 433 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,958 | 1,959 | |
/-
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... | /-- For the single implications with fewer assumptions, see `one_div_le_one_div_of_le` and
| Mathlib/Algebra/Order/Field/Basic.lean | 122 | 122 |
/-
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.MeasureTheory.Measure.AbsolutelyContinuous
/-!
# Vitali families
On a metric space `X` with a measure `μ`, consider for each `x : X` a family o... | (v.tendsto_filterAt_iff.mp tendsto_id).2 ε hε
theorem eventually_filterAt_measurableSet (x : X) : ∀ᶠ t in v.filterAt x, MeasurableSet t := by
| Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 234 | 236 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | lie_mem := fun {x m} h ↦ by
suffices ⁅x, f m⁆ ∈ N' by simp [this]
apply N'.lie_mem h }
@[simp]
theorem toSubmodule_comap :
(N'.comap f : Submodule R M) = (N' : Submodule R M').comap (f : M →ₗ[R] M') :=
| Mathlib/Algebra/Lie/Submodule.lean | 807 | 813 |
/-
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... | hitting u s n m ω ≤ i ↔ ∃ j ∈ Set.Icc n i, u j ω ∈ s := by
constructor <;> intro h'
· exact ⟨hitting u s n m ω, ⟨le_hitting_of_exists h_exists, h'⟩, hitting_mem_set h_exists⟩
· have h'' : ∃ k ∈ Set.Icc n (min m i), u k ω ∈ s := by
obtain ⟨k₁, hk₁_mem, hk₁_s⟩ := h_exists
| Mathlib/Probability/Process/HittingTime.lean | 154 | 158 |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
/-!
# Topological Kőnig's lemma
A topological version of Kőnig's lemma i... | end TopologicalKonig
end TopCat
| Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 129 | 145 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | F.map coprod.inr ≫ inv (coprodComparison F A B) = coprod.inr := by simp [IsIso.inv_comp_eq]
/-- If the coproduct comparison morphism is an iso, its inverse is natural. -/
@[reassoc]
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,186 | 1,189 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... |
section LT
| Mathlib/Order/Cover.lean | 187 | 188 |
/-
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... | Supermartingale f 𝒢 μ := by
rw [← neg_neg f]
refine (submartingale_of_setIntegral_le_succ hadp.neg (fun i => (hint i).neg) ?_).neg
simpa only [integral_neg, Pi.neg_apply, neg_le_neg_iff]
| Mathlib/Probability/Martingale/Basic.lean | 371 | 374 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | exact finrank_add_finrank_orthogonal' <| by
simp [finrank_span_singleton hv, _i.elim, add_comm]
end Submodule
open Module Submodule
/-- An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of t... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,113 | 1,122 |
/-
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... | calc
∏ i : Fin n, f i = ∏ i : { x // x ∈ range n }, f i :=
Fintype.prod_equiv (Fin.equivSubtype.trans (Equiv.subtypeEquivRight (by simp))) _ _ (by simp)
_ = ∏ i ∈ range n, f i := by rw [← attach_eq_univ, prod_attach]
| Mathlib/Data/Fintype/BigOperators.lean | 210 | 213 |
/-
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... | · intro h; refine ⟨fun x y => ?_⟩; (suffices [x] = [y] by simpa using this); apply h; rfl
· intros hα l1 l2 hl
| Mathlib/Data/List/Basic.lean | 100 | 101 |
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 767 | 772 | |
/-
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.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | split_ifs
exacts [le_rfl, zero_le_one]
lemma defaultRatCDF_le_one (q : ℚ) : defaultRatCDF q ≤ 1 := by
unfold defaultRatCDF
| Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 172 | 176 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Data.Complex.Trigonometric
/-!
# Trigonometric functions as sums of infinite series
In t... | div_self Complex.I_ne_zero, mul_comm _ ((-1 : ℂ) ^ _), mul_one_div, mul_div_assoc, mul_assoc]
theorem Complex.cos_eq_tsum' (z : ℂ) :
Complex.cos z = ∑' n : ℕ, (z * Complex.I) ^ (2 * n) / ↑(2 * n)! :=
(Complex.hasSum_cos' z).tsum_eq.symm
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 75 | 79 |
/-
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... | /-- The proposition that `i` is a left descent of `w`; that is, $\ell(s_i w) < \ell(w)$. -/
def IsLeftDescent (w : W) (i : B) : Prop := ℓ (s i * w) < ℓ w
/-- The proposition that `i` is a right descent of `w`; that is, $\ell(w s_i) < \ell(w)$. -/
def IsRightDescent (w : W) (i : B) : Prop := ℓ (w * s i) < ℓ w
| Mathlib/GroupTheory/Coxeter/Length.lean | 271 | 275 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... |
variable [RCLike 𝕜]
variable [NormedAddCommGroup D] [NormedSpace ℝ D]
| Mathlib/Analysis/Distribution/SchwartzSpace.lean | 1,071 | 1,073 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 546 | 548 | |
/-
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.Geometry.Euclidean.Sphere.Basic
/-!
# Second intersection of a sphere and a line
This file defines and proves basic results about the second intersection... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 175 | 183 | |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... | rw [clog_of_two_le hb hn]
exact zero_lt_succ _
| Mathlib/Data/Nat/Log.lean | 255 | 256 |
/-
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.Data.List.Nodup
import Mathlib.Data.Set.Prod
/-!
# Finite products of types
This file defines the product of types over a list. For `l : List ι` ... | simp
· rw [TProd.elim_of_ne _ hji, snd_mk, elim_mk is]
@[ext]
theorem ext :
| Mathlib/Data/Prod/TProd.lean | 99 | 103 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | · field_simp [ofReal_ne_zero.mpr pi_ne_zero]
ring_nf
simp only [I_sq]
ring
theorem _root_.fourierIntegral_gaussian_pi (hb : 0 < b.re) :
(𝓕 fun (x : ℝ) ↦ cexp (-π * b * x ^ 2)) =
fun t : ℝ ↦ 1 / b ^ (1 / 2 : ℂ) * cexp (-π / b * t ^ 2) := by
simpa only [mul_zero, zero_mul, add_zero] using fourie... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 228 | 246 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Data.Complex.Trigonometric
/-!
# Trigonometric functions as sums of infinite series
In t... |
end Complex
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 130 | 132 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | Mathlib/Data/List/DropRight.lean | 258 | 263 | |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
/-!
This file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal
... | def HomogeneousIdeal.radical (I : HomogeneousIdeal 𝒜) : HomogeneousIdeal 𝒜 :=
⟨I.toIdeal.radical, I.isHomogeneous.radical⟩
@[simp]
| Mathlib/RingTheory/GradedAlgebra/Radical.lean | 172 | 175 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Properties of LUB and GLB in an order topology
-/
open Set Filter TopologicalSpa... | exact ⟨v, fun a b hlt ↦ hmono.reflect_lt <| hu hlt, hux, hlim⟩
theorem IsGLB.exists_seq_strictAnti_tendsto_of_not_mem {t : Set α} {x : α}
[IsCountablyGenerated (𝓝 x)] (htx : IsGLB t x) (not_mem : x ∉ t) (ht : t.Nonempty) :
∃ u : ℕ → α, StrictAnti u ∧ (∀ n, x < u n) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ n, u n ∈ t :=... | Mathlib/Topology/Order/IsLUB.lean | 254 | 262 |
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir, Yury Kudryashov
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Order.Filter.Ring
import Mathli... | instance isTotal [LE β] [IsTotal β (· ≤ ·)] : IsTotal β* (· ≤ ·) :=
⟨fun f g =>
inductionOn₂ f g fun _f _g => eventually_or.1 <| Eventually.of_forall fun _x => total_of _ _ _⟩
| Mathlib/Order/Filter/FilterProduct.lean | 82 | 84 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 602 | 607 | |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | (hf : SuperpolynomialDecay l k f) : SuperpolynomialDecay l k (k⁻¹ * f) := by
simpa using hf.param_zpow_mul hk (-1)
theorem SuperpolynomialDecay.param_inv_mul (hk : Tendsto k l atTop)
(hf : SuperpolynomialDecay l k f) : SuperpolynomialDecay l k (f * k⁻¹) :=
(hf.inv_param_mul hk).congr fun _ => mul_comm _ _
... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 239 | 249 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... |
@[reassoc]
theorem associator_naturality_middle (X : C) {Y Y' : C} (f : Y ⟶ Y') (Z : C) :
(X ◁ f) ▷ Z ≫ (α_ X Y' Z).hom = (α_ X Y Z).hom ≫ X ◁ f ▷ Z := by simp
| Mathlib/CategoryTheory/Monoidal/Category.lean | 446 | 449 |
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