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) 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... | x.toQuot
map_zero' := by simp only [← zero_quot, f.map_zero]
| Mathlib/Algebra/RingQuot.lean | 421 | 422 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | rw [union_eq_compl_compl_inter_compl]
exact compl_mem_measurableCylinders (inter_mem_measurableCylinders
(compl_mem_measurableCylinders hs) (compl_mem_measurableCylinders ht))
theorem diff_mem_measurableCylinders (hs : s ∈ measurableCylinders α)
(ht : t ∈ measurableCylinders α) :
s \ t ∈ measurableCyli... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 342 | 356 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma P4_neg_right : P4 x₁ x₂ y ↔ P4 x₁ x₂ (-y) := by
rw [P4, P4, neg_neg, and_comm]
| Mathlib/SetTheory/Surreal/Multiplication.lean | 124 | 125 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | where `(i, j)` indexes the linear map that sends `b j` to `b i`
and sends all other basis vectors to `0`. -/
@[simps! -isSimp repr_apply repr_symm_apply]
noncomputable
abbrev _root_.Basis.end (b : Basis ι R M) : Basis (ι × ι) R (Module.End R M) :=
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 967 | 971 |
/-
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, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.I... | rw [mem_smul_set_iff_inv_smul_mem₀ hR]
rfl
| Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 140 | 142 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
/-!
# The concrete (co)kernels in the category of modules ... | Mathlib/Algebra/Category/ModuleCat/Kernels.lean | 137 | 140 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | convert isTopologicalBasis_of_subbasis hsg
refine le_antisymm (fun t ht ↦ ⟨{t}, by simpa using ht⟩) ?_
rintro _ ⟨g, ⟨hg, hgs⟩, rfl⟩
lift g to Finset (Set α) using hg
exact hsi.finiteInter_mem g hgs
theorem isTopologicalBasis_of_subbasis_of_inter {r : Set (Set α)} (hsg : t = generateFrom r)
(hsi : ∀ ⦃s⦄, ... | Mathlib/Topology/Bases.lean | 93 | 103 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | { toFun := fun x : L => ⁅x, m⁆, map_zero' := zero_lie m, map_add' := fun _ _ => add_lie _ _ _ }
_ _
| Mathlib/Algebra/Lie/Basic.lean | 215 | 217 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,094 | 1,096 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | InnerProductSpace ℝ G :=
InnerProductSpace.rclikeToReal ℂ G
instance : InnerProductSpace ℝ ℂ := InnerProductSpace.complexToReal
@[simp]
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 876 | 881 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... |
theorem Ioo_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioo (pred a) b = Ico a b := by
rw [← Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]
| Mathlib/Order/SuccPred/Basic.lean | 806 | 809 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... |
theorem ext_inner_left {x y : E} (h : ∀ v, ⟪v, x⟫ = ⟪v, y⟫) : x = y := by
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 287 | 288 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mathlib.CategoryTheory.Limits.Op... | /-- If `G` preserves the pushout of `(f,g)`, then the pushout comparison map for `G` at `(f,g)` is
an isomorphism. -/
def PreservesPushout.iso : pushout (G.map f) (G.map g) ≅ G.obj (pushout f g) :=
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 245 | 247 |
/-
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... | rw [← p]
exact Nat.dvd_sub (kx.mul_left _) (ky.mul_left _)
theorem strictMono_y : StrictMono (yn a1)
| _, 0, h => absurd h <| Nat.not_lt_zero _
| m, n + 1, h => by
have : yn a1 m ≤ yn a1 n :=
Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 335 | 342 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions
/-!
# Lemmas on fractions
Let `W : MorphismProperty C`, and objects `X` and `Y` in `C`. In this file,
we introd... | lemma fst (h : LeftFraction₂Rel z₁ z₂) : LeftFractionRel z₁.fst z₂.fst := by
obtain ⟨Z, t₁, t₂, hst, hft, _, ht⟩ := h
exact ⟨Z, t₁, t₂, hst, hft, ht⟩
| Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean | 176 | 178 |
/-
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.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | Mathlib/Data/Nat/ModEq.lean | 466 | 469 | |
/-
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
/... | of the filter `f.lift g`.
| Mathlib/Order/Filter/Lift.lean | 32 | 32 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.Group.Pointwise.Bounds
import Mathlib.Data.Real.Basic
import Mathlib.Ord... | that all values of `f` are at most some nonnegative number `a` to show that `⨆ i, f i ≤ a`.
| Mathlib/Data/Real/Archimedean.lean | 215 | 215 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... |
@[deprecated (since := "2024-12-11")] alias updateColumn := updateCol
variable {M : Matrix m n α} {i : m} {j : n} {b : n → α} {c : m → α}
| Mathlib/Data/Matrix/RowCol.lean | 235 | 238 |
/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... |
/-- **Sperner's theorem**. The size of an antichain in `Finset α` is bounded by the size of the
maximal layer in `Finset α`. This precisely means that `Finset α` is a Sperner order. -/
theorem _root_.IsAntichain.sperner (h𝒜 : IsAntichain (· ⊆ ·) 𝒜.toSet) :
#𝒜 ≤ (Fintype.card α).choose (Fintype.card α / 2) := by... | Mathlib/Combinatorics/SetFamily/LYM.lean | 228 | 245 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | rw [← eqToHom_comp_d C r r']
apply imageSubobject_iso_comp
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 387 | 389 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions... | theorem deriv_arcsin : deriv arcsin = fun x => 1 / √(1 - x ^ 2) := by
funext x
by_cases h : x ≠ -1 ∧ x ≠ 1
· exact (hasDerivAt_arcsin h.1 h.2).deriv
· rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_arcsin.1 h)]
simp only [not_and_or, Ne, Classical.not_not] at h
| Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 93 | 98 |
/-
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, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | section
variable (R M M₂ M₃ M₄)
variable [Semiring R]
variable [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [AddCommMonoid M₄]
variable [Module R M] [Module R M₂] [Module R M₃] [Module R M₄]
| Mathlib/LinearAlgebra/Prod.lean | 674 | 680 |
/-
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... | @[simp]
theorem eLpNorm_exponent_zero {f : α → ε} : eLpNorm f 0 μ = 0 := by simp [eLpNorm]
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 86 | 87 |
/-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Algebra.Spectrum.Basic
/-!
# Unitar... |
variable [CommMonoid R] [StarMul R]
| Mathlib/Algebra/Star/Unitary.lean | 169 | 171 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | Mathlib/Data/Nat/Bitwise.lean | 382 | 382 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | · have : ∀ x : { a // ¬Acc r a }, ∃ y : { a // ¬Acc r a }, r y.1 x.1 := by
rintro ⟨x, hx⟩
cases exists_not_acc_lt_of_not_acc hx with
| intro w h => exact ⟨⟨w, h.1⟩, h.2⟩
choose f h using this
refine fun E =>
by_contradiction fun hx => E.elim' ⟨natGT (fun n => (f^[n] ⟨x, hx⟩).1) fun n =... | Mathlib/Order/OrderIsoNat.lean | 66 | 81 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | simp only [ext_iff_indep, freeOn_ground, freeOn_indep_iff, h.1, true_and]
exact fun I hIX ↦ iff_of_true (h.2.subset hIX) hIX
| Mathlib/Data/Matroid/Constructions.lean | 174 | 175 |
/-
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.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.Powe... | IsLowerSet (𝒜.nonMemberSubfamily a : Set (Finset α)) := fun s t hts => by
simp_rw [mem_coe, mem_nonMemberSubfamily]
exact And.imp (h hts) (mt <| @hts _)
| Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 35 | 38 |
/-
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... | map_smul' := fun r p => LinearMap.ext fun q => (Submodule.Quotient.eq _).mpr <| by
change h (r • p) ⊗ₜ[R] q - r • h p ⊗ₜ[R] q ∈ range (rTensor Q f)
rw [TensorProduct.smul_tmul', ← TensorProduct.sub_tmul]
apply le_comap_range_rTensor f
rw [exact_iff] at hfg
simp only [← hfg, mem_ker, ma... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 315 | 329 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.BoundedOrder.Lattice
import Mathlib.Data.List.Induction
import Mathlib.Order.MinMax
import Ma... | m ∈ l ∧ (∀ a ∈ l, f m ≤ f a) ∧ ∀ a ∈ l, f a ≤ f m → l.idxOf m ≤ l.idxOf a :=
@mem_argmax_iff _ βᵒᵈ _ _ _ _ _
theorem argmin_eq_some_iff :
argmin f l = some m ↔
m ∈ l ∧ (∀ a ∈ l, f m ≤ f a) ∧ ∀ a ∈ l, f a ≤ f m → l.idxOf m ≤ l.idxOf a :=
mem_argmin_iff
end LinearOrder
section MaximumMinimum
| Mathlib/Data/List/MinMax.lean | 233 | 244 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | /-- For maps between vector spaces, `MDifferentiableWithinAt` and `DifferentiableWithinAt`
coincide -/
theorem mdifferentiableWithinAt_iff_differentiableWithinAt :
| Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 60 | 62 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | exact hR₂'
theorem epi_of_epi_of_epi_of_epi (hR₂ : R₂.Exact) (hR₁' : Epi (R₁.map' 1 2))
(h₀ : Epi (app' φ 0)) (h₁ : Epi (app' φ 2)) :
Epi (app' φ 1) := by
let ψ : mk₃ (R₁.map' 0 1) (R₁.map' 1 2) (0 : _ ⟶ R₁.obj' 0) ⟶
mk₃ (R₂.map' 0 1) (R₂.map' 1 2) (0 : _ ⟶ R₁.obj' 0) := homMk₃ (app' φ 0) (app' φ 1)
... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 207 | 219 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
import Mathlib.Tactic.Bound... | _ = _ := by simp only [← pow_succ, ← two_mul, ← mul_assoc]; rfl
· obtain hab | hba := le_total a b
| Mathlib/Algebra/Order/Ring/Basic.lean | 217 | 218 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | classical
rw [Associates.mk_ne_zero, finprod_def]
split_ifs
· rw [Finset.prod_ne_zero_iff]
intro v _
apply pow_ne_zero _ v.ne_bot
· exact one_ne_zero
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 149 | 156 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option α`
In this file we define
* `Option.t... | Mathlib/Data/Finset/Option.lean | 165 | 166 | |
/-
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
/... | theorem sInter_lift'_sets (hh : Monotone h) : ⋂₀ { s | s ∈ f.lift' h } = ⋂ s ∈ f, h s :=
(sInter_lift_sets (monotone_principal.comp hh)).trans <| iInter₂_congr fun _ _ => csInf_Ici
theorem lift'_le {f : Filter α} {g : Set α → Set β} {h : Filter β} {s : Set α} (hs : s ∈ f)
(hg : 𝓟 (g s) ≤ h) : f.lift' g ≤ h :=
... | Mathlib/Order/Filter/Lift.lean | 223 | 228 |
/-
Copyright (c) 2022 Filippo A. E. Nuccio Mortarino Majno di Capriglio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Junyan Xu
-/
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Path
/-!
# H-s... | Mathlib/Topology/Homotopy/HSpaces.lean | 263 | 264 | |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... | Mathlib/Order/OmegaCompletePartialOrder.lean | 876 | 877 | |
/-
Copyright (c) 2023 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Mario Carneiro, Robert Y. Lewis, Patrick Massot
-/
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.Qify
/-!
# `rify` tactic
The `... | @[rify_simps] lemma ratCast_le (a b : ℚ) : a ≤ b ↔ (a : ℝ) ≤ (b : ℝ) := by simp
| Mathlib/Tactic/Rify.lean | 78 | 78 |
/-
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 | 1,181 | 1,183 | |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | exact cons₂ _ (append (Sublist.map _ h) ih)
protected theorem Subperm.sym2 {xs ys : List α} (h : xs <+~ ys) : xs.sym2 <+~ ys.sym2 := by
obtain ⟨xs', hx, h⟩ := h
exact hx.sym2.symm.subperm.trans h.sym2.subperm
theorem length_sym2 {xs : List α} : xs.sym2.length = Nat.choose (xs.length + 1) 2 := by
induction x... | Mathlib/Data/List/Sym.lean | 207 | 223 |
/-
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 | 547 | 547 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | (D : Matrix n n R) {a b : α} (hab : a < b) :
BlockTriangular (fromBlocks A B 0 D) (Sum.elim (fun _ => a) fun _ => b) := by
rintro (c | c) (d | d) hcd <;> first | simp [hab.not_lt] at hcd ⊢
/-! ### Determinant -/
variable [CommRing R] [DecidableEq m] [Fintype m] [DecidableEq n] [Fintype n]
theorem equiv_bl... | Mathlib/LinearAlgebra/Matrix/Block.lean | 187 | 198 |
/-
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... | ℕ → α →ₛ β :=
hf.choose
protected theorem tendsto_approx {_ : MeasurableSpace α} (hf : StronglyMeasurable f) :
∀ x, Tendsto (fun n => hf.approx n x) atTop (𝓝 (f x)) :=
hf.choose_spec
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 162 | 167 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... |
@[simp]
theorem filter_lt (n m l : ℕ) :
((Ico n m).filter fun x => x < l) = Ico n (min m l) := by
rcases le_total m l with hml | hlm
· rw [min_eq_left hml, filter_lt_of_top_le hml]
| Mathlib/Data/List/Intervals.lean | 143 | 148 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | -- Porting note: added implicit argument
(hB₁ : associated' (N := N) Q ≠ 0) :
∃ x, Q x ≠ 0 := by
rw [← not_forall]
intro h
apply hB₁
rw [(QuadraticMap.ext h : Q = 0), LinearMap.map_zero]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 972 | 978 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,665 | 1,675 | |
/-
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.Logic.OpClass
import Mathlib.Order.Lattice
/-!
# `max` and `min`
This file proves basic properties about maxima and minima on a `LinearOrder`.
## Ta... |
theorem Max.right_comm (a b c : α) : max (max a b) c = max (max a c) b := by
rw [max_assoc, max_comm b, max_assoc]
theorem MonotoneOn.map_max (hf : MonotoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (max a b) =
| Mathlib/Order/MinMax.lean | 180 | 184 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | ↑(a ^ k) = (a : α) ^ k :=
map_pow SignType.castHom _ _
@[simp, norm_cast] lemma coe_zpow {α} [GroupWithZero α] [HasDistribNeg α] (a : SignType) (k : ℤ) :
| Mathlib/Data/Sign.lean | 279 | 282 |
/-
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.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.Invarian... |
variable [InvariantBasisNumber R]
/-- The dimension theorem: if `v` and `v'` are two bases, their index types
have the same cardinalities. -/
theorem mk_eq_mk_of_basis (v : Basis ι R M) (v' : Basis ι' R M) :
Cardinal.lift.{w'} #ι = Cardinal.lift.{w} #ι' := by
classical
haveI := nontrivial_of_invariantBasisNum... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 58 | 83 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... | Mathlib/Data/List/Intervals.lean | 231 | 241 | |
/-
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... | rw [wcovBy_eq_reflGen_covBy, reflTransGen_reflGen]
end Relation
namespace Prod
| Mathlib/Order/Cover.lean | 495 | 500 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.List
/-!
# Countable sets
I... | alias ⟨_, Countable.sUnion⟩ := Countable.sUnion_iff
@[simp]
theorem countable_union {s t : Set α} : (s ∪ t).Countable ↔ s.Countable ∧ t.Countable := by
simp [union_eq_iUnion, and_comm]
| Mathlib/Data/Set/Countable.lean | 222 | 226 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | simp_rw [mixedEmbedding.norm_apply, normAtPlace_smul, mul_pow, prod_mul_distrib,
prod_pow_eq_pow_sum, sum_mult_eq]
theorem norm_real (c : ℝ) :
mixedEmbedding.norm ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = |c| ^ finrank ℚ K := by
rw [show ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = c • 1 by ext <;> simp, ... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 445 | 450 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | show #(swap x y).support = #⟨x ::ₘ y ::ₘ 0, by simp [hxy]⟩ from
congr_arg card <| by simp [support_swap hxy, *, Finset.ext_iff]
@[simp]
theorem card_support_eq_two {f : Perm α} : #f.support = 2 ↔ IsSwap f := by
constructor <;> intro h
· obtain ⟨x, t, hmem, hins, ht⟩ := card_eq_succ.1 h
obtain ⟨y, rfl⟩ :=... | Mathlib/GroupTheory/Perm/Support.lean | 612 | 628 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... |
variable (𝕜 E G) in
/-- Continuous bilinear map realizing `(f, z) ↦ f.smulRight z`. -/
def smulRightL : ContinuousMultilinearMap 𝕜 E 𝕜 →L[𝕜] G →L[𝕜] ContinuousMultilinearMap 𝕜 E G :=
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 797 | 800 |
/-
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... | (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.left_ne_zero_of_oangle_eq_pi_div_two h))]
/-- A side of a right-angled triangle divided by the sine of the opposite angle equals the
hypotenuse, version subtracting vectors. -/
theorem norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 416 | 423 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin, Kim Morrison
-/
import Mathlib.Analysis.Normed.Group.SemiNormedGrp
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.CategoryTheory.Limits.Sha... | Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean | 412 | 419 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | __ := Equiv.arrowProdEquivProdArrow γ _ _
measurable_toFun := by
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 504 | 505 |
/-
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.HomologicalComplexLimits
import Mathlib.Algebra.Homology.Additive
/-! Binary biproducts of homological complexes
In this file, it is shown tha... | @[reassoc (attr := simp)]
lemma biprod_lift_snd_f (α : M ⟶ K) (β : M ⟶ L) (i : ι) :
(biprod.lift α β).f i ≫ (biprod.snd : K ⊞ L ⟶ L).f i = β.f i := by
rw [← comp_f, biprod.lift_snd]
| Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean | 103 | 106 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | theorem adjugate_subsingleton [Subsingleton n] (A : Matrix n n α) : adjugate A = 1 := by
ext i j
simp [Subsingleton.elim i j, adjugate_apply, det_eq_elem_of_subsingleton _ i, one_apply]
| Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 282 | 285 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
/-!
# Slop... | theorem StrictConvexOn.secant_strict_mono_aux2 (hf : StrictConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s)
(hz : z ∈ s) (hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) < (f z - f x) / (z - x) := by
have hxy' : 0 < y - x := by linarith
have hxz' : 0 < z - x := by linarith
rw [div_lt_div_iff₀ hxy' hxz']
linari... | Mathlib/Analysis/Convex/Slope.lean | 283 | 299 |
/-
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... | instance instIsTopologicalAddGroup : IsTopologicalAddGroup 𝓢(E, F) :=
(schwartzSeminormFamily ℝ E F).addGroupFilterBasis.isTopologicalAddGroup
| Mathlib/Analysis/Distribution/SchwartzSpace.lean | 500 | 502 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... | end OrderedAddCommMonoid
section LinearOrderedAddCommMonoid
variable [AddCommMonoid M] [LinearOrder M] [CanonicallyOrderedAdd M]
{w : σ → M} (φ : MvPolynomial σ R)
| Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 546 | 551 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,141 | 1,142 | |
/-
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 | 353 | 355 | |
/-
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.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | refine h.congr_fun (fun x hx ↦ ?_) measurableSet_Ioi
rw [Complex.deriv_ofReal_cpow_const (ht.trans hx).ne' (fun h ↦ (Complex.zero_re ▸ h ▸ hs).false)]
theorem integrableOn_Ioi_deriv_norm_ofReal_cpow {s : ℂ} {t : ℝ} (ht : 0 < t) (hs : s.re ≤ 0):
| Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 200 | 203 |
/-
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... | simp_rw [mul_pow, pow_mul, Complex.I_sq, mul_comm]
/-- The power series expansion of `Complex.sin`. -/
theorem Complex.hasSum_sin (z : ℂ) :
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 68 | 71 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | (ι_sq_scalar _ _).trans <| by simp
theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by
| Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 69 | 71 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Valuation.Integers
/-!
# Integral elements over the ring of integers of a valuation
The r... | let ⟨p, hpm, hpx⟩ := hx
le_of_not_lt fun hvx : 1 < v x => by
rw [hpm.as_sum, eval₂_add, eval₂_pow, eval₂_X, eval₂_finset_sum, add_eq_zero_iff_eq_neg] at hpx
replace hpx := congr_arg v hpx; refine ne_of_gt ?_ hpx
rw [v.map_neg, v.map_pow]
refine v.map_sum_lt' (zero_lt_one.trans_le (one_le_pow_of_one_... | Mathlib/RingTheory/Valuation/Integral.lean | 31 | 42 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,214 | 1,218 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.Typeclasse... | (hf.continuousAt (hU.mem_nhds ha)).prodMk_nhds (hg.continuousAt (hU.mem_nhds ha))
(isClosed_diagonal.isOpen_compl.mem_nhds ha')
replace := (this.eq_empty_of_measure_zero h).le
exact fun x hx => Classical.not_not.1 fun h => this ⟨hx, h⟩
/-- If two continuous functions are a.e. equal, then they are equ... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 119 | 130 |
/-
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.... | · left; exact isUnit_of_mul_eq_one_right _ _ hk
· left; exact ha₁
· rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]
left; exact hprod
· rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
· rw [hc]; rw [hc, mul_assoc] at hprod
... | Mathlib/Algebra/BigOperators/Associated.lean | 103 | 111 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | refine ⟨fun hp i hi ↦ IsRelPrime.prod_right_iff.mpr fun j hj ↦ ?_, fun hp ↦ ?_⟩
· rw [Finset.mem_sdiff, Finset.mem_singleton] at hj
obtain ⟨hj, ji⟩ := hj
exact @hp ⟨i, hi⟩ ⟨j, hj⟩ fun h ↦ ji (congrArg Subtype.val h).symm
· rintro ⟨i, hi⟩ ⟨j, hj⟩ h
apply IsRelPrime.prod_right_iff.mp (hp i hi)
exact... | Mathlib/RingTheory/Coprime/Lemmas.lean | 261 | 275 |
/-
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.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | theorem bind_powerset_len {α : Type*} (S : Multiset α) :
(bind (Multiset.range (card S + 1)) fun k => S.powersetCard k) = S.powerset := by
induction S using Quotient.inductionOn
| Mathlib/Data/Multiset/Powerset.lean | 270 | 272 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | rintro _ ⟨_, ⟨x,hx,rfl⟩, rfl⟩; exact invFunOn_apply_mem hx
| Mathlib/Data/Set/Function.lean | 951 | 952 |
/-
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, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | variable [StrongRankCondition R] [Module.Free R M] [Module.Free R M'] [Module.Free R M₁]
open Module.Free
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 149 | 151 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Algebra.Module... | Mathlib/Algebra/Module/Submodule/Basic.lean | 458 | 459 | |
/-
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.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.Cover.Open
/-!
# Gluing Schemes
Given a family of gluing data of schemes, we m... | gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y = 𝟙 _ := by
apply pullback.hom_ext <;> simp_rw [Category.id_comp, Category.assoc]
· apply glued_cover_cocycle_fst
| Mathlib/AlgebraicGeometry/Gluing.lean | 302 | 304 |
/-
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... | · exact (image2_subset_right inter_subset_left).trans ht₁
· exact (image2_subset_right inter_subset_right).trans ht₂
/-- The other direction does not hold because of the `h`-`h` cross terms on the RHS. -/
theorem map₂_distrib_le_right {m : δ → γ → ε} {n : α → β → δ} {m₁ : α → γ → α'} {m₂ : β → γ → β'}
| Mathlib/Order/Filter/NAry.lean | 223 | 227 |
/-
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 | 356 | 363 | |
/-
Copyright (c) 2024 Yaël Dillies, Kin Yau James Wong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kin Yau James Wong, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
... | instance condKernelCountable.instIsCondKernel [∀ a, IsMarkovKernel (κCond a)]
(h_atom : ∀ x y, x ∈ measurableAtom y → κCond x = κCond y) (κ : Kernel α (β × Ω))
[IsSFiniteKernel κ] [∀ a, (κ a).IsCondKernel (κCond a)] :
κ.IsCondKernel (condKernelCountable κCond h_atom) := by
constructor
ext a s hs
conv_... | Mathlib/Probability/Kernel/Disintegration/Basic.lean | 209 | 217 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | theorem eraseLead_monomial (i : ℕ) (r : R) : eraseLead (monomial i r) = 0 := by
classical
by_cases hr : r = 0
· subst r
| Mathlib/Algebra/Polynomial/EraseLead.lean | 127 | 130 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | simp_rw [← LinearMap.det_toMatrix B, ← LinearMap.det_toMatrix b]
have : ((LinearMap.toMatrix B B) (LinearMap.pi fun i ↦ f i ∘ₗ LinearMap.proj i)) =
Matrix.blockDiagonal (fun i ↦ LinearMap.toMatrix b b (f i)) := by
ext ⟨i₁, i₂⟩ ⟨j₁, j₂⟩
unfold B
simp_rw [LinearMap.toMatrix_apply', Matrix.blockDiago... | Mathlib/LinearAlgebra/Determinant.lean | 353 | 358 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | rw [hd.cycleType, ← extendDomain_mul, (hd.extendDomain f).cycleType, hσ, hτ]
theorem cycleType_ofSubtype {p : α → Prop} [DecidablePred p] {g : Perm (Subtype p)} :
cycleType (ofSubtype g) = cycleType g :=
cycleType_extendDomain (Equiv.refl (Subtype p))
theorem mem_cycleType_iff {n : ℕ} {σ : Perm α} :
n ∈... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 310 | 318 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Grou... | e.inv j ≫ e.hom j = 𝟙 _ := by
rw [← GradedObject.categoryOfGradedObjects_comp, e.inv_hom_id,
GradedObject.categoryOfGradedObjects_id]
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/GradedObject.lean | 114 | 118 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Notation
/-!
# Positive & negative parts
Mathematical structures posse... | Mathlib/Algebra/Order/Group/PosPart.lean | 163 | 163 | |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... |
variable {M}
variable [LieRing.IsNilpotent L]
variable [NoZeroSMulDivisors ℤ R] [NoZeroSMulDivisors R M] [IsNoetherian R M]
variable (α : L → R) (β : Weight R L M)
/-- This is the largest `n : ℕ` such that `i • α + β` is a weight for all `0 ≤ i ≤ n`. -/
noncomputable
def chainTopCoeff : ℕ :=
letI := Classical.propD... | Mathlib/Algebra/Lie/Weights/Chain.lean | 246 | 258 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Setoid.Basic
import Mathlib.Order.Atoms
impor... |
variable [DecidableEq α] {a b c : α}
section Bind
variable {P : Finpartition a} {Q : ∀ i ∈ P.parts, Finpartition i}
/-- Given a finpartition `P` of `a` and finpartitions of each part of `P`, this yields the
finpartition of `a` obtained by juxtaposing all the subpartitions. -/
@[simps]
| Mathlib/Order/Partition/Finpartition.lean | 356 | 365 |
/-
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.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... |
theorem toReal_pos_iff_ne_top (p : ℝ≥0∞) [Fact (1 ≤ p)] : 0 < p.toReal ↔ p ≠ ∞ :=
⟨fun h hp =>
| Mathlib/Data/ENNReal/Real.lean | 363 | 365 |
/-
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
/-!
# Idempotence of the Karoubi envelope
In this file, we construct the equivalence of categories
`KaroubiKaroubi.equivalen... | simpa only [hom_ext_iff, comp_f] using P.idem
@[reassoc]
| Mathlib/CategoryTheory/Idempotents/KaroubiKaroubi.lean | 32 | 34 |
/-
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.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
/-!
# Division of `MvPolynomial` by monomials
## Main definitions
* `MvPolynomial... | Mathlib/Algebra/MvPolynomial/Division.lean | 244 | 247 | |
/-
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... | WithBot.coe_lt_coe, ← @not_le ℕ, max_eq_sup_coe, Nat.cast_withBot, Ne, not_imp_not]
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 506 | 507 |
/-
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.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... |
/-- The order of a formal power series is at least `n` if
the `i`th coefficient is `0` for all `i < n`. -/
| Mathlib/RingTheory/PowerSeries/Order.lean | 99 | 101 |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | @[simp]
theorem inv_simple (i : B) : (s i)⁻¹ = s i :=
(eq_inv_of_mul_eq_one_right (cs.simple_mul_simple_self i)).symm
@[simp]
theorem simple_mul_simple_pow (i i' : B) : (s i * s i') ^ M i i' = 1 := by
have : (FreeGroup.of i * FreeGroup.of i') ^ M i i' ∈ M.relationsSet := ⟨(i, i'), rfl⟩
| Mathlib/GroupTheory/Coxeter/Basic.lean | 218 | 224 |
/-
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.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | @[simp] protected lemma inv_mul_le_one (a : ℝ≥0∞) : a⁻¹ * a ≤ 1 := by simp [mul_comm]
| Mathlib/Data/ENNReal/Inv.lean | 371 | 372 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | (h₁ : IsTopologicalBasis (t := t₁) B₁) (h₂ : IsTopologicalBasis (t := t₂) B₂) :
IsTopologicalBasis (t := t₁ ⊓ t₂) (image2 (· ∩ ·) B₁ B₂) := by
| Mathlib/Topology/Bases.lean | 244 | 245 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.HurwitzZetaEven
import Mathlib.NumberTheory.LSeries.HurwitzZetaOdd
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
/-!
# The... | lemma differentiableAt_hurwitzZeta_sub_one_div (a : UnitAddCircle) :
DifferentiableAt ℂ (fun s ↦ hurwitzZeta a s - 1 / (s - 1) / Gammaℝ s) 1 := by
simp only [hurwitzZeta, add_sub_right_comm]
exact (differentiableAt_hurwitzZetaEven_sub_one_div a).add (differentiable_hurwitzZetaOdd a 1)
| Mathlib/NumberTheory/LSeries/HurwitzZeta.lean | 84 | 87 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | · rw [← hp', hperm.symm.prod_eq']
refine hd'.imp ?_
exact Disjoint.commute
· rintro ⟨hc, hd, hp⟩
refine List.toFinset_eq_of_perm _ _ ?_
refine list_cycles_perm_list_cycles ?_ hc' hc hd' hd
rw [hp, hp']
| Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 511 | 517 |
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