Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 624 | 626 | theorem nodalWeight_eq_eval_nodal_erase_inv :
nodalWeight s v i = (eval (v i) (nodal (s.erase i) v))⁻¹ := by |
rw [eval_nodal, nodalWeight, prod_inv_distrib]
|
/-
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
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 328 | 333 | theorem union_mem_measurableCylinders (hs : s ∈ measurableCylinders α)
(ht : t ∈ measurableCylinders α) :
s ∪ t ∈ measurableCylinders α := by |
rw [union_eq_compl_compl_inter_compl]
exact compl_mem_measurableCylinders (inter_mem_measurableCylinders
(compl_mem_measurableCylinders hs) (compl_mem_measurableCylinders ht))
|
/-
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.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 124 | 125 | theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by |
rw [← Nat.card_uIcc, ← map_subtype_embedding_uIcc, card_map]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
#align_import order.iterate from "leanprover-community/mathlib"@"2258b40dacd2942571... | Mathlib/Order/Iterate.lean | 194 | 199 | theorem iterate_pos_lt_iff_map_lt (h : Commute f g) (hf : Monotone f) (hg : StrictMono g) {x n}
(hn : 0 < n) : f^[n] x < g^[n] x ↔ f x < g x := by |
rcases lt_trichotomy (f x) (g x) with (H | H | H)
· simp only [*, iterate_pos_lt_of_map_lt]
· simp only [*, h.iterate_eq_of_map_eq, lt_irrefl]
· simp only [lt_asymm H, lt_asymm (h.symm.iterate_pos_lt_of_map_lt' hg hf H hn)]
|
/-
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.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,377 | 1,378 | theorem nat_le_lift_iff {n : ℕ} {a : Cardinal.{u}} : n ≤ lift.{v} a ↔ n ≤ a := by |
rw [← lift_natCast.{v,u}, lift_le]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Order.Interval.Finset.Basic
#align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd... | Mathlib/Data/Int/Interval.lean | 185 | 186 | theorem card_fintype_Ioo_of_lt (h : a < b) : (Fintype.card (Set.Ioo a b) : ℤ) = b - a - 1 := by |
rw [card_fintype_Ioo, sub_sub, toNat_sub_of_le h]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 53 | 59 | theorem scaleRoots_ne_zero {p : R[X]} (hp : p ≠ 0) (s : R) : scaleRoots p s ≠ 0 := by |
intro h
have : p.coeff p.natDegree ≠ 0 := mt leadingCoeff_eq_zero.mp hp
have : (scaleRoots p s).coeff p.natDegree = 0 :=
congr_fun (congr_arg (coeff : R[X] → ℕ → R) h) p.natDegree
rw [coeff_scaleRoots_natDegree] at this
contradiction
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topolog... | Mathlib/Analysis/Quaternion.lean | 150 | 150 | theorem coe_real_complex_mul (r : ℝ) (z : ℂ) : (r • z : ℍ) = ↑r * ↑z := by | ext <;> simp
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 100 | 102 | theorem hasseDeriv_one' : hasseDeriv 1 f = derivative f := by |
simp only [hasseDeriv_apply, derivative_apply, ← C_mul_X_pow_eq_monomial, Nat.choose_one_right,
(Nat.cast_commute _ _).eq]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 154 | 159 | theorem integral_fintype_iUnion {ι : Type*} [Fintype ι] {s : ι → Set X}
(hs : ∀ i, MeasurableSet (s i)) (h's : Pairwise (Disjoint on s))
(hf : ∀ i, IntegrableOn f (s i) μ) : ∫ x in ⋃ i, s i, f x ∂μ = ∑ i, ∫ x in s i, f x ∂μ := by |
convert integral_finset_biUnion Finset.univ (fun i _ => hs i) _ fun i _ => hf i
· simp
· simp [pairwise_univ, h's]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 2,081 | 2,088 | theorem Antitone.piecewise_eventually_eq_iInter {β : α → Type*} [Preorder ι] {s : ι → Set α}
[∀ i, DecidablePred (· ∈ s i)] [DecidablePred (· ∈ ⋂ i, s i)]
(hs : Antitone s) (f g : (a : α) → β a) (a : α) :
∀ᶠ i in atTop, (s i).piecewise f g a = (⋂ i, s i).piecewise f g a := by |
classical
convert ← (compl_anti.comp hs).piecewise_eventually_eq_iUnion g f a using 3
· convert congr_fun (Set.piecewise_compl (s _) g f) a
· simp only [(· ∘ ·), ← compl_iInter, Set.piecewise_compl]
|
/-
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
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,289 | 1,300 | theorem lintegral_tsum [Countable β] {f : β → α → ℝ≥0∞} (hf : ∀ i, AEMeasurable (f i) μ) :
∫⁻ a, ∑' i, f i a ∂μ = ∑' i, ∫⁻ a, f i a ∂μ := by |
simp only [ENNReal.tsum_eq_iSup_sum]
rw [lintegral_iSup_directed]
· simp [lintegral_finset_sum' _ fun i _ => hf i]
· intro b
exact Finset.aemeasurable_sum _ fun i _ => hf i
· intro s t
use s ∪ t
constructor
· exact fun a => Finset.sum_le_sum_of_subset Finset.subset_union_left
· exact fun ... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysi... | Mathlib/Analysis/Convex/Caratheodory.lean | 52 | 98 | theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndependent 𝕜 ((↑) : t → E))
{x : E} (m : x ∈ convexHull 𝕜 (↑t : Set E)) :
∃ y : (↑t : Set E), x ∈ convexHull 𝕜 (↑(t.erase y) : Set E) := by |
simp only [Finset.convexHull_eq, mem_setOf_eq] at m ⊢
obtain ⟨f, fpos, fsum, rfl⟩ := m
obtain ⟨g, gcombo, gsum, gpos⟩ := exists_nontrivial_relation_sum_zero_of_not_affine_ind h
replace gpos := exists_pos_of_sum_zero_of_exists_nonzero g gsum gpos
clear h
let s := @Finset.filter _ (fun z => 0 < g z) (fun _ =... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 162 | 163 | theorem integral_empty : ∫ x in ∅, f x ∂μ = 0 := by |
rw [Measure.restrict_empty, integral_zero_measure]
|
/-
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.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 180 | 181 | theorem pbind_map (f : α → β) (x : Option α) (g : ∀ b : β, b ∈ x.map f → Option γ) :
pbind (Option.map f x) g = x.pbind fun a h ↦ g (f a) (mem_map_of_mem _ h) := by | cases x <;> rfl
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Sets.Opens
#align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589b... | Mathlib/Topology/PartialHomeomorph.lean | 924 | 925 | theorem trans_of_set' {s : Set Y} (hs : IsOpen s) :
e.trans (ofSet s hs) = e.restr (e.source ∩ e ⁻¹' s) := by | rw [trans_ofSet, restr_source_inter]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Finset.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Category... | Mathlib/CategoryTheory/Limits/Lattice.lean | 170 | 177 | theorem pushout_eq_sup [SemilatticeSup α] [OrderBot α] (x y z : α) (f : z ⟶ x) (g : z ⟶ y) :
pushout f g = x ⊔ y :=
calc
pushout f g = colimit (span f g) := rfl
_ = Finset.univ.sup (span f g).obj := by | rw [finite_colimit_eq_finset_univ_sup]
_ = z ⊔ (x ⊔ (y ⊔ ⊥)) := rfl
_ = z ⊔ (x ⊔ y) := by rw [sup_bot_eq]
_ = x ⊔ y := sup_eq_right.mpr (le_sup_of_le_left f.le)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Scott Morrison
-/
import Mathlib.CategoryTheory.Comma.Basic
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Category... | Mathlib/CategoryTheory/Comma/StructuredArrow.lean | 392 | 394 | theorem hom_ext (h : IsUniversal f) {c : C} {η η' : f.right ⟶ c}
(w : f.hom ≫ T.map η = f.hom ≫ T.map η') : η = η' := by |
rw [h.hom_desc η, h.hom_desc η', w]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 375 | 375 | theorem image_neg_Ici : Neg.neg '' Ici a = Iic (-a) := by | simp
|
/-
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.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#al... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 110 | 112 | theorem PInfty_idem : (PInfty : K[X] ⟶ _) ≫ PInfty = PInfty := by |
ext n
exact PInfty_f_idem n
|
/-
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.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4... | Mathlib/Data/Finset/Powerset.lean | 41 | 44 | theorem coe_powerset (s : Finset α) :
(s.powerset : Set (Finset α)) = ((↑) : Finset α → Set α) ⁻¹' (s : Set α).powerset := by |
ext
simp
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,376 | 1,383 | theorem iUnion_range_eq_iUnion (C : ι → Set α) {f : ∀ x : ι, β → C x}
(hf : ∀ x : ι, Surjective (f x)) : ⋃ y : β, range (fun x : ι => (f x y).val) = ⋃ x, C x := by |
ext x; rw [mem_iUnion, mem_iUnion]; constructor
· rintro ⟨y, i, rfl⟩
exact ⟨i, (f i y).2⟩
· rintro ⟨i, hx⟩
cases' hf i ⟨x, hx⟩ with y hy
exact ⟨y, i, congr_arg Subtype.val hy⟩
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 238 | 245 | theorem mem_canonicalEquiv_apply {I : FractionalIdeal S P} {x : P'} :
x ∈ canonicalEquiv S P P' I ↔
∃ y ∈ I,
IsLocalization.map P' (RingHom.id R) (fun y (hy : y ∈ S) => show RingHom.id R y ∈ S from hy)
(y : P) =
x := by |
rw [canonicalEquiv, mapEquiv_apply, mem_map]
exact ⟨fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩, fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩⟩
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 803 | 805 | theorem exists_subset_range_and_iff {f : α → β} {p : Set β → Prop} :
(∃ s, s ⊆ range f ∧ p s) ↔ ∃ s, p (f '' s) := by |
rw [← exists_range_iff, range_image]; rfl
|
/-
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
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 1,009 | 1,010 | theorem iSupLift_comp_inclusion {i : ι} (h : K i ≤ T) :
(iSupLift K dir f hf T hT).comp (inclusion h) = f i := by | ext; simp
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 523 | 525 | theorem integral_cases (f : ℝ → E) (a b) :
(∫ x in a..b, f x ∂μ) ∈ ({∫ x in Ι a b, f x ∂μ, -∫ x in Ι a b, f x ∂μ} : Set E) := by |
rw [intervalIntegral_eq_integral_uIoc]; split_ifs <;> simp
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 135 | 142 | theorem Ideal.mul_add_mem_pow_succ_inj (P : Ideal S) {i : ℕ} (a d d' e e' : S) (a_mem : a ∈ P ^ i)
(e_mem : e ∈ P ^ (i + 1)) (e'_mem : e' ∈ P ^ (i + 1)) (h : d - d' ∈ P) :
a * d + e - (a * d' + e') ∈ P ^ (i + 1) := by |
have : a * d - a * d' ∈ P ^ (i + 1) := by
simp only [← mul_sub]
exact Ideal.mul_mem_mul a_mem h
convert Ideal.add_mem _ this (Ideal.sub_mem _ e_mem e'_mem) using 1
ring
|
/-
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.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 929 | 942 | theorem ae_tendsto_average [NormedSpace ℝ E] [CompleteSpace E] {f : α → E}
(hf : LocallyIntegrable f μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ⨍ y in a, f y ∂μ) (v.filterAt x) (𝓝 (f x)) := by |
filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x
rw [tendsto_iff_norm_sub_tendsto_zero]
refine squeeze_zero' (eventually_of_forall fun a => norm_nonneg _) ?_ hx
filter_upwards [h'x, v.eventually_measure_lt_top x, v.eventually_filterAt_integrableOn x hf]
with a ha... |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 174 | 178 | theorem add_tsub_add_le_tsub_add_tsub : a + b - (c + d) ≤ a - c + (b - d) := by |
rw [add_comm c, tsub_le_iff_left, add_assoc, ← tsub_le_iff_left, ← tsub_le_iff_left]
refine (tsub_le_tsub_right add_tsub_le_assoc c).trans ?_
rw [add_comm a, add_comm (a - c)]
exact add_tsub_le_assoc
|
/-
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.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 766 | 769 | theorem toFinset_range [DecidableEq α] [Fintype β] (f : β → α) [Fintype (Set.range f)] :
(Set.range f).toFinset = Finset.univ.image f := by |
ext
simp
|
/-
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.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import... | Mathlib/RingTheory/Filtration.lean | 492 | 501 | theorem Ideal.iInf_pow_eq_bot_of_isDomain [IsNoetherianRing R] [IsDomain R] (h : I ≠ ⊤) :
⨅ i : ℕ, I ^ i = ⊥ := by |
rw [eq_bot_iff]
intro x hx
by_contra hx'
have := Ideal.mem_iInf_smul_pow_eq_bot_iff I x
simp_rw [smul_eq_mul, ← Ideal.one_eq_top, mul_one] at this
obtain ⟨r, hr⟩ := this.mp hx
have := mul_right_cancel₀ hx' (hr.trans (one_mul x).symm)
exact I.eq_top_iff_one.not.mp h (this ▸ r.prop)
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import ... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 90 | 96 | theorem reverse_apply (x : CliffordAlgebra (0 : QuadraticForm R Unit)) :
reverse (R := R) x = x := by |
induction x using CliffordAlgebra.induction with
| algebraMap r => exact reverse.commutes _
| ι x => rw [ι_eq_zero, LinearMap.zero_apply, reverse.map_zero]
| mul x₁ x₂ hx₁ hx₂ => rw [reverse.map_mul, mul_comm, hx₁, hx₂]
| add x₁ x₂ hx₁ hx₂ => rw [reverse.map_add, hx₁, hx₂]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2f... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 74 | 81 | theorem pow_div_two_eq_neg_one_or_one {a : ZMod p} (ha : a ≠ 0) :
a ^ (p / 2) = 1 ∨ a ^ (p / 2) = -1 := by |
cases' Prime.eq_two_or_odd (@Fact.out p.Prime _) with hp2 hp_odd
· subst p; revert a ha; intro a; fin_cases a
· tauto
· simp
rw [← mul_self_eq_one_iff, ← pow_add, ← two_mul, two_mul_odd_div_two hp_odd]
exact pow_card_sub_one_eq_one ha
|
/-
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.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"... | Mathlib/Order/SuccPred/Limit.lean | 480 | 480 | theorem not_isPredLimit [NoMaxOrder α] : ¬IsPredLimit a := by | simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,474 | 1,475 | theorem round_add_int (x : α) (y : ℤ) : round (x + y) = round x + y := by |
rw [round, round, Int.fract_add_int, Int.floor_add_int, Int.ceil_add_int, ← apply_ite₂, ite_self]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,288 | 2,319 | theorem splitAt_eq_take_drop (n : ℕ) (l : List α) : splitAt n l = (take n l, drop n l) := by |
by_cases h : n < l.length <;> rw [splitAt, go_eq_take_drop]
· rw [if_pos h]; rfl
· rw [if_neg h, take_all_of_le <| le_of_not_lt h, drop_eq_nil_of_le <| le_of_not_lt h]
where
go_eq_take_drop (n : ℕ) (l xs : List α) (acc : Array α) : splitAt.go l xs n acc =
if n < xs.length then (acc.toList ++ take n xs, d... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 457 | 458 | theorem tan_arccos (x : ℝ) : tan (arccos x) = √(1 - x ^ 2) / x := by |
rw [arccos, tan_pi_div_two_sub, tan_arcsin, inv_div]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,582 | 2,583 | theorem pmap_eq_nil {p : α → Prop} {f : ∀ a, p a → β} {l H} : pmap f l H = [] ↔ l = [] := by |
rw [← length_eq_zero, length_pmap, length_eq_zero]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanpro... | Mathlib/ModelTheory/Satisfiability.lean | 233 | 252 | theorem exists_elementaryEmbedding_card_eq_of_ge (M : Type w') [L.Structure M] [iM : Infinite M]
(κ : Cardinal.{w}) (h1 : Cardinal.lift.{w} L.card ≤ Cardinal.lift.{max u v} κ)
(h2 : Cardinal.lift.{w} #M ≤ Cardinal.lift.{w'} κ) :
∃ N : Bundled L.Structure, Nonempty (M ↪ₑ[L] N) ∧ #N = κ := by |
obtain ⟨N0, hN0⟩ := (L.elementaryDiagram M).exists_large_model_of_infinite_model κ M
rw [← lift_le.{max u v}, lift_lift, lift_lift] at h2
obtain ⟨N, ⟨NN0⟩, hN⟩ :=
exists_elementaryEmbedding_card_eq_of_le (L[[M]]) N0 κ
(aleph0_le_lift.1 ((aleph0_le_lift.2 (aleph0_le_mk M)).trans h2))
(by
s... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 293 | 299 | theorem mem_factors_mul_of_coprime {a b : ℕ} (hab : Coprime a b) (p : ℕ) :
p ∈ (a * b).factors ↔ p ∈ a.factors ∪ b.factors := by |
rcases a.eq_zero_or_pos with (rfl | ha)
· simp [(coprime_zero_left _).mp hab]
rcases b.eq_zero_or_pos with (rfl | hb)
· simp [(coprime_zero_right _).mp hab]
rw [mem_factors_mul ha.ne' hb.ne', List.mem_union_iff]
|
/-
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.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 745 | 748 | theorem algebraMap_apply {r : R} :
algebraMap R (MvPowerSeries σ A) r = C σ A (algebraMap R A r) := by |
change (MvPowerSeries.map σ (algebraMap R A)).comp (C σ R) r = _
simp
|
/-
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.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
... | Mathlib/RingTheory/SimpleModule.lean | 190 | 191 | theorem eq_bot_or_exists_simple_le (N : Submodule R M) : N = ⊥ ∨ ∃ m ≤ N, IsSimpleModule R m := by |
simpa only [isSimpleModule_iff_isAtom, and_comm] using eq_bot_or_exists_atom_le _
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 1,227 | 1,230 | theorem Integrable.ofReal {f : α → ℝ} (hf : Integrable f μ) :
Integrable (fun x => (f x : 𝕜)) μ := by |
rw [← memℒp_one_iff_integrable] at hf ⊢
exact hf.ofReal
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Contraction.lean | 245 | 256 | theorem lTensorHomEquivHomLTensor_toLinearMap :
(lTensorHomEquivHomLTensor R M P Q).toLinearMap = lTensorHomToHomLTensor R M P Q := by |
classical -- Porting note: missing decidable for choosing basis
let e := congr (LinearEquiv.refl R P) (dualTensorHomEquiv R M Q)
have h : Function.Surjective e.toLinearMap := e.surjective
refine (cancel_right h).1 ?_
ext f q m
dsimp [e, lTensorHomEquivHomLTensor]
simp only [lTensorHomEquivHomLTensor, dua... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 743 | 746 | theorem insert_toList_zoom {t : RBNode α} (ht : Balanced t c n)
(e : zoom (cmp v) t = (t', p)) :
(t.insert cmp v).toList = p.withList (t'.setRoot v).toList := by |
rw [← setBlack_toList, ← Path.zoom_insert ht e, setBlack_toList, Path.insert_toList]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 897 | 905 | theorem Basis.iUnion_basis_iUnion {ι : Type _} (X I : ι → Set α) (hI : ∀ i, M.Basis (I i) (X i))
(h_ind : M.Indep (⋃ i, I i)) : M.Basis (⋃ i, I i) (⋃ i, X i) := by |
refine h_ind.basis_of_forall_insert
(iUnion_subset (fun i ↦ (hI i).subset.trans (subset_iUnion _ _))) ?_
rintro e ⟨⟨_, ⟨⟨i, hi, rfl⟩, (hes : e ∈ X i)⟩⟩, he'⟩
rw [mem_iUnion, not_exists] at he'
refine ((hI i).insert_dep ⟨hes, he' _⟩).superset (insert_subset_insert (subset_iUnion _ _)) ?_
rw [insert_subset... |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 594 | 601 | theorem Convex.exists_mem_add_smul_eq (h : Convex 𝕜 s) {x y : E} {p q : 𝕜} (hx : x ∈ s) (hy : y ∈ s)
(hp : 0 ≤ p) (hq : 0 ≤ q) : ∃ z ∈ s, (p + q) • z = p • x + q • y := by |
rcases _root_.em (p = 0 ∧ q = 0) with (⟨rfl, rfl⟩ | hpq)
· use x, hx
simp
· replace hpq : 0 < p + q := (add_nonneg hp hq).lt_of_ne' (mt (add_eq_zero_iff' hp hq).1 hpq)
refine ⟨_, convex_iff_div.1 h hx hy hp hq hpq, ?_⟩
simp only [smul_add, smul_smul, mul_div_cancel₀ _ hpq.ne']
|
/-
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.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 239 | 243 | theorem res_injective : Injective (@res α) := by |
intro x y h
ext n
apply res_eq_res.mp _ (Nat.lt_succ_self _)
rw [h]
|
/-
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.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
/-!
# Formal (multivariate) power series - Truncation
`MvPowerSeries... | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 43 | 46 | theorem coeff_truncFun (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
(truncFun n φ).coeff m = if m < n then coeff R m φ else 0 := by |
classical
simp [truncFun, MvPolynomial.coeff_sum]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
imp... | Mathlib/FieldTheory/Adjoin.lean | 1,049 | 1,052 | theorem bot_eq_top_of_finrank_adjoin_le_one [FiniteDimensional F E]
(h : ∀ x : E, finrank F F⟮x⟯ ≤ 1) : (⊥ : IntermediateField F E) = ⊤ := by |
apply bot_eq_top_of_finrank_adjoin_eq_one
exact fun x => by linarith [h x, show 0 < finrank F F⟮x⟯ from finrank_pos]
|
/-
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.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 606 | 606 | theorem sInf_add {s : Set ℝ≥0∞} : sInf s + a = ⨅ b ∈ s, b + a := by | simp [sInf_eq_iInf, iInf_add]
|
/-
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... | Mathlib/Order/OrderIsoNat.lean | 251 | 259 | theorem WellFounded.iSup_eq_monotonicSequenceLimit [CompleteLattice α]
(h : WellFounded ((· > ·) : α → α → Prop)) (a : ℕ →o α) :
iSup a = monotonicSequenceLimit a := by |
refine (iSup_le fun m => ?_).antisymm (le_iSup a _)
rcases le_or_lt m (monotonicSequenceLimitIndex a) with hm | hm
· exact a.monotone hm
· cases' WellFounded.monotone_chain_condition'.1 h a with n hn
have : n ∈ {n | ∀ m, n ≤ m → a n = a m} := fun k hk => (a.mono hk).eq_of_not_lt (hn k hk)
exact (Nat.sI... |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatoric... | Mathlib/Combinatorics/HalesJewett.lean | 175 | 176 | theorem apply_none {α ι} (l : Line α ι) (x : α) (i : ι) (h : l.idxFun i = none) : l x i = x := by |
simp only [Option.getD_none, h, l.apply]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 664 | 686 | theorem hausdorffMeasure_zero_singleton (x : X) : μH[0] ({x} : Set X) = 1 := by |
apply le_antisymm
· let r : ℕ → ℝ≥0∞ := fun _ => 0
let t : ℕ → Unit → Set X := fun _ _ => {x}
have ht : ∀ᶠ n in atTop, ∀ i, diam (t n i) ≤ r n := by
simp only [t, r, imp_true_iff, eq_self_iff_true, diam_singleton, eventually_atTop,
nonpos_iff_eq_zero, exists_const]
simpa [t, liminf_const]... |
/-
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.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 881 | 899 | theorem wbtw_iff_sameRay_vsub {x y z : P} : Wbtw R x y z ↔ SameRay R (y -ᵥ x) (z -ᵥ y) := by |
refine ⟨Wbtw.sameRay_vsub, fun h => ?_⟩
rcases h with (h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩)
· rw [vsub_eq_zero_iff_eq] at h
simp [h]
· rw [vsub_eq_zero_iff_eq] at h
simp [h]
· refine
⟨r₂ / (r₁ + r₂),
⟨div_nonneg hr₂.le (add_nonneg hr₁.le hr₂.le),
div_le_one_of_le (le_add_of_nonneg_l... |
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 600 | 600 | theorem Icc_diff_Ioc_self (h : a ≤ b) : Icc a b \ Ioc a b = {a} := by | simp [← coe_inj, h]
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
#align_import line... | Mathlib/LinearAlgebra/Charpoly/Basic.lean | 95 | 96 | theorem pow_eq_aeval_mod_charpoly (k : ℕ) : f ^ k = aeval f (X ^ k %ₘ f.charpoly) := by |
rw [← aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]
|
/-
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.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 417 | 430 | theorem sum_properDivisors_dvd (h : (∑ x ∈ n.properDivisors, x) ∣ n) :
∑ x ∈ n.properDivisors, x = 1 ∨ ∑ x ∈ n.properDivisors, x = n := by |
cases' n with n
· simp
· cases' n with n
· simp at h
· rw [or_iff_not_imp_right]
intro ne_n
have hlt : ∑ x ∈ n.succ.succ.properDivisors, x < n.succ.succ :=
lt_of_le_of_ne (Nat.le_of_dvd (Nat.succ_pos _) h) ne_n
symm
rw [← mem_singleton, eq_properDivisors_of_subset_of_sum_e... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Mario Carneiro, Martin Dvorak
-/
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac... | Mathlib/Data/List/Join.lean | 123 | 129 | theorem drop_take_succ_eq_cons_get (L : List α) (i : Fin L.length) :
(L.take (i + 1)).drop i = [get L i] := by |
induction' L with head tail ih
· exact (Nat.not_succ_le_zero i i.isLt).elim
rcases i with ⟨_ | i, hi⟩
· simp
· simpa using ih ⟨i, Nat.lt_of_succ_lt_succ hi⟩
|
/-
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.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-com... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 170 | 174 | theorem det_smul_coords_eq_cramer_coords (x : P) :
(b.toMatrix b₂).det • b₂.coords x = (b.toMatrix b₂)ᵀ.cramer (b.coords x) := by |
have hu := b.isUnit_toMatrix b₂
rw [Matrix.isUnit_iff_isUnit_det] at hu
rw [← b.toMatrix_inv_vecMul_toMatrix, Matrix.det_smul_inv_vecMul_eq_cramer_transpose _ _ hu]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 153 | 155 | theorem bodd_negOfNat (n : ℕ) : bodd (negOfNat n) = n.bodd := by |
cases n <;> simp (config := {decide := true})
rfl
|
/-
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.Init.Align
import Mathlib.Topology.PartialHomeomorph
#align_import geometry.manifold.charted_space from "leanprover-community/mathlib"@"431589bc... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 787 | 790 | theorem modelProd_range_prod_id {H : Type*} {H' : Type*} {α : Type*} (f : H → α) :
(range fun p : ModelProd H H' ↦ (f p.1, p.2)) = range f ×ˢ (univ : Set H') := by |
rw [prod_range_univ_eq]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.Unifo... | Mathlib/Topology/Instances/Real.lean | 195 | 198 | theorem Periodic.compact_of_continuous [TopologicalSpace α] {f : ℝ → α} {c : ℝ} (hp : Periodic f c)
(hc : c ≠ 0) (hf : Continuous f) : IsCompact (range f) := by |
rw [← hp.image_uIcc hc 0]
exact isCompact_uIcc.image hf
|
/-
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.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#al... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 138 | 140 | theorem PInfty_comp_QInfty : (PInfty : K[X] ⟶ _) ≫ QInfty = 0 := by |
ext n
apply PInfty_f_comp_QInfty_f
|
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algeb... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 254 | 256 | theorem ι_ne_one [Nontrivial R] (x : M) : ι R x ≠ 1 := by |
rw [← (algebraMap R (ExteriorAlgebra R M)).map_one, Ne, ι_eq_algebraMap_iff]
exact one_ne_zero ∘ And.right
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,211 | 1,212 | theorem nonempty_iInter {f : ι → Set α} : (⋂ i, f i).Nonempty ↔ ∃ x, ∀ i, x ∈ f i := by |
simp [nonempty_iff_ne_empty, iInter_eq_empty_iff]
|
/-
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
#align_import data.typevec from "leanprover-... | Mathlib/Data/TypeVec.lean | 674 | 676 | theorem dropFun_diag {α} : dropFun (@prod.diag (n + 1) α) = prod.diag := by |
ext i : 2
induction i <;> simp [dropFun, *] <;> rfl
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 474 | 479 | theorem norm_dvd_iff {x : S} (hx : Prime (Algebra.norm ℤ x)) {y : ℤ} :
Algebra.norm ℤ x ∣ y ↔ x ∣ y := by |
rw [← Ideal.mem_span_singleton (y := x), ← eq_intCast (algebraMap ℤ S), ← Ideal.mem_comap,
← Ideal.span_singleton_absNorm, Ideal.mem_span_singleton, Ideal.absNorm_span_singleton,
Int.natAbs_dvd]
rwa [Ideal.absNorm_span_singleton, ← Int.prime_iff_natAbs_prime]
|
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,593 | 2,594 | theorem ofAdd_multiset_prod (s : Multiset α) : ofAdd s.sum = (s.map ofAdd).prod := by |
simp [ofAdd]; rfl
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 74 | 75 | theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 0 ↔ A i j = 1 := by | rw [← apply_ne_one_iff h, Classical.not_not]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 267 | 284 | theorem prod_embeddings_eq_finrank_pow [Algebra L F] [IsScalarTower K L F] [IsAlgClosed E]
[IsSeparable K F] [FiniteDimensional K F] (pb : PowerBasis K L) :
∏ σ : F →ₐ[K] E, σ (algebraMap L F pb.gen) =
((@Finset.univ _ (PowerBasis.AlgHom.fintype pb)).prod
fun σ : L →ₐ[K] E => σ pb.gen) ^ finrank L... |
haveI : FiniteDimensional L F := FiniteDimensional.right K L F
haveI : IsSeparable L F := isSeparable_tower_top_of_isSeparable K L F
letI : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
rw [Fintype.prod_equiv algHomEquivSigma (fun σ : F →ₐ[K] E => _) fun σ => σ.1 pb.gen,
← Finset.univ_sigma_univ, Fin... |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,686 | 1,687 | theorem inf_subgroupOf_left (H K : Subgroup G) : (K ⊓ H).subgroupOf K = H.subgroupOf K := by |
rw [inf_comm, inf_subgroupOf_right]
|
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
import Mathlib.Topology.MetricSpace.PiNat
#align_import topology.metric_space.cantor_scheme from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb25... | Mathlib/Topology/MetricSpace/CantorScheme.lean | 83 | 86 | theorem map_mem (x : (inducedMap A).1) (n : ℕ) : (inducedMap A).2 x ∈ A (res x n) := by |
have := x.property.some_mem
rw [mem_iInter] at this
exact this n
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
#align_import c... | Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 602 | 603 | theorem condition (a) : I.fst a ≫ K.π (I.fstFrom a) = I.snd a ≫ K.π (I.sndFrom a) := by |
rw [← K.snd_app_right, ← K.fst_app_right]
|
/-
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
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 103 | 104 | theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ := by |
rw [IntegrableOn, Measure.restrict_univ]
|
/-
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.Computability.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 388 | 469 | theorem rec_computable {α σ} [Primcodable α] [Primcodable σ] {c : α → Code} (hc : Computable c)
{z : α → σ} (hz : Computable z) {s : α → σ} (hs : Computable s) {l : α → σ} (hl : Computable l)
{r : α → σ} (hr : Computable r) {pr : α → Code × Code × σ × σ → σ} (hpr : Computable₂ pr)
{co : α → Code × Code × σ ... |
-- TODO(Mario): less copy-paste from previous proof
intros _ _ _ _ F
let G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p =>
letI a := p.1.1; letI IH := p.1.2; letI n := p.2.1; letI m := p.2.2
(IH.get? m).bind fun s =>
(IH.get? m.unpair.1).bind fun s₁ =>
(IH.get? m.unpair.2).map fun s₂ =>
cond n... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e... | Mathlib/Data/DList/Defs.lean | 62 | 66 | theorem ofList_toList (l : DList α) : DList.ofList (DList.toList l) = l := by |
cases' l with app inv
simp only [ofList, toList, mk.injEq]
funext x
rw [(inv x)]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.bas... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 509 | 510 | theorem derivWithin_congr_set' (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
derivWithin f s x = derivWithin f t x := by | simp only [derivWithin, fderivWithin_congr_set' y h]
|
/-
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.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.eq... | Mathlib/Order/Partition/Equipartition.lean | 89 | 100 | theorem IsEquipartition.card_large_parts_eq_mod (hP : P.IsEquipartition) :
(P.parts.filter fun p ↦ p.card = s.card / P.parts.card + 1).card = s.card % P.parts.card := by |
have z := P.sum_card_parts
rw [← sum_filter_add_sum_filter_not (s := P.parts)
(p := fun x ↦ x.card = s.card / P.parts.card + 1),
hP.filter_ne_average_add_one_eq_average,
sum_const_nat (m := s.card / P.parts.card + 1) (by simp),
sum_const_nat (m := s.card / P.parts.card) (by simp),
← hP.filter... |
/-
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.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"... | Mathlib/Data/Set/Sigma.lean | 23 | 28 | theorem range_sigmaMk (i : ι) : range (Sigma.mk i : α i → Sigma α) = Sigma.fst ⁻¹' {i} := by |
apply Subset.antisymm
· rintro _ ⟨b, rfl⟩
simp
· rintro ⟨x, y⟩ (rfl | _)
exact mem_range_self y
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
/-! # Measurability of the line derivative
We prove in `me... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 92 | 99 | theorem stronglyMeasurable_lineDeriv_uncurry (hf : Continuous f) :
StronglyMeasurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) := by |
borelize 𝕜
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
exact (stronglyMeasurable_deriv_with_param this).comp_measurable measurable_prod_mk_right
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import ana... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 107 | 108 | theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := by |
rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv]
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 184 | 187 | theorem Module.rank_le_one_iff_top_isPrincipal [Module.Free K V] :
Module.rank K V ≤ 1 ↔ (⊤ : Submodule K V).IsPrincipal := by |
haveI := Module.Free.of_equiv (topEquiv (R := K) (M := V)).symm
rw [← Submodule.rank_le_one_iff_isPrincipal, rank_top]
|
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 217 | 221 | theorem ext' (x y : M F) (H : ∀ i : ℕ, x.approx i = y.approx i) : x = y := by |
cases x
cases y
congr with n
apply H
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 766 | 769 | theorem join_cons (s : WSeq α) (S) : join (cons s S) = think (append s (join S)) := by |
simp only [join, think]
dsimp only [(· <$> ·)]
simp [join, cons, append]
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,227 | 1,234 | theorem mk_finsupp_lift_of_infinite' (α : Type u) (β : Type v) [Nonempty α] [Zero β] [Infinite β] :
#(α →₀ β) = max (lift.{v} #α) (lift.{u} #β) := by |
cases fintypeOrInfinite α
· rw [mk_finsupp_lift_of_fintype]
have : ℵ₀ ≤ (#β).lift := aleph0_le_lift.2 (aleph0_le_mk β)
rw [max_eq_right (le_trans _ this), power_nat_eq this]
exacts [Fintype.card_pos, lift_le_aleph0.2 (lt_aleph0_of_finite _).le]
· apply mk_finsupp_lift_of_infinite
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Bhavik Mehta
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/... | Mathlib/Probability/CondCount.lean | 65 | 67 | theorem finite_of_condCount_ne_zero {s t : Set Ω} (h : condCount s t ≠ 0) : s.Finite := by |
by_contra hs'
simp [condCount, cond, Measure.count_apply_infinite hs'] at h
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Generator
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
#align_import category_theory.preadditive.generator from "leanprover-com... | Mathlib/CategoryTheory/Preadditive/Generator.lean | 62 | 66 | theorem isSeparator_iff_faithful_preadditiveCoyonedaObj (G : C) :
IsSeparator G ↔ (preadditiveCoyonedaObj (op G)).Faithful := by |
rw [isSeparator_iff_faithful_preadditiveCoyoneda, preadditiveCoyoneda_obj]
exact ⟨fun h => Functor.Faithful.of_comp _ (forget₂ _ AddCommGroupCat.{v}),
fun h => Functor.Faithful.comp _ _⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 219 | 220 | theorem prime_mul_iff {a b : ℕ} : Nat.Prime (a * b) ↔ a.Prime ∧ b = 1 ∨ b.Prime ∧ a = 1 := by |
simp only [iff_self_iff, irreducible_mul_iff, ← irreducible_iff_nat_prime, Nat.isUnit_iff]
|
/-
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.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 84 | 88 | theorem isPiSystem_Ioo_rat :
IsPiSystem (⋃ (a : ℚ) (b : ℚ) (_ : a < b), {Ioo (a : ℝ) (b : ℝ)}) := by |
convert isPiSystem_Ioo ((↑) : ℚ → ℝ) ((↑) : ℚ → ℝ)
ext x
simp [eq_comm]
|
/-
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.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 311 | 311 | theorem smul_im (a : ℤ) (b : ℤ√d) : (↑a * b).im = a * b.im := by | simp
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
/-!
# Number field discriminant
This file defines the discrimi... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 105 | 151 | theorem exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
∃ a ∈ (I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧
|Algebra.norm ℚ (a:K)| ≤ FractionalIdeal.absNorm I.1 * (4 / π) ^ NrComplexPlaces K *
(finrank ℚ K).factorial / (finrank ℚ K) ^ (finrank ℚ K) * Real.sqrt |discr... |
-- The smallest possible value for `exists_ne_zero_mem_ideal_of_norm_le`
let B := (minkowskiBound K I * (convexBodySumFactor K)⁻¹).toReal ^ (1 / (finrank ℚ K : ℝ))
have h_le : (minkowskiBound K I) ≤ volume (convexBodySum K B) := by
refine le_of_eq ?_
rw [convexBodySum_volume, ← ENNReal.ofReal_pow (by pos... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 81 | 83 | theorem product_image_snd [DecidableEq β] (ht : s.Nonempty) : (s ×ˢ t).image Prod.snd = t := by |
ext i
simp [mem_image, ht.exists_mem]
|
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
/-!
# Perfect fields and rings
In this f... | Mathlib/FieldTheory/Perfect.lean | 339 | 342 | theorem roots_expand_image_iterateFrobenius [DecidableEq R] :
(expand R (p ^ n) f).roots.toFinset.image (iterateFrobenius R p n) = f.roots.toFinset := by |
rw [Finset.image_toFinset, roots_expand_pow_map_iterateFrobenius,
(roots f).toFinset_nsmul _ (expChar_pow_pos R p n).ne']
|
/-
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.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 112 | 117 | theorem finite_mulSupport_coe {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => (v.asIdeal : FractionalIdeal R⁰ K) ^
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)).Finite := by |
rw [mulSupport]
simp_rw [Ne, zpow_natCast, ← FractionalIdeal.coeIdeal_pow, FractionalIdeal.coeIdeal_eq_one]
exact finite_mulSupport hI
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 160 | 169 | theorem _root_.TendstoLocallyUniformlyOn.deriv (hf : TendstoLocallyUniformlyOn F f φ U)
(hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hU : IsOpen U) :
TendstoLocallyUniformlyOn (deriv ∘ F) (deriv f) φ U := by |
rw [tendstoLocallyUniformlyOn_iff_forall_isCompact hU]
rcases φ.eq_or_neBot with rfl | hne
· simp only [TendstoUniformlyOn, eventually_bot, imp_true_iff]
rintro K hKU hK
obtain ⟨δ, hδ, hK4, h⟩ := exists_cthickening_tendstoUniformlyOn hf hF hK hU hKU
refine h.congr_right fun z hz => cderiv_eq_deriv hU (hf.d... |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... | Mathlib/RingTheory/LocalProperties.lean | 647 | 680 | theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType := by |
rw [RingHom.ofLocalizationSpan_iff_finite]
introv R hs H
-- mirrors the proof of `finite_ofLocalizationSpan`
letI := f.toAlgebra
letI := fun r : s => (Localization.awayMap f r).toAlgebra
have : ∀ r : s,
IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) :=
... |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 583 | 587 | theorem map_zero (f : α → β) : (0 : VectorMeasure α M).map f = 0 := by |
by_cases hf : Measurable f
· ext i hi
rw [map_apply _ hf hi, zero_apply, zero_apply]
· exact dif_neg hf
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 86 | 88 | theorem count_add' (a b : ℕ) : count p (a + b) = count (fun k ↦ p (k + b)) a + count p b := by |
rw [add_comm, count_add, add_comm]
simp_rw [add_comm b]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 108 | 109 | theorem volume_ball (a r : ℝ) : volume (Metric.ball a r) = ofReal (2 * r) := by |
rw [ball_eq_Ioo, volume_Ioo, ← sub_add, add_sub_cancel_left, two_mul]
|
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