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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 855 | 865 | theorem toNNReal_rpow (x : ℝ≥0∞) (z : ℝ) : x.toNNReal ^ z = (x ^ z).toNNReal := by |
rcases lt_trichotomy z 0 with (H | H | H)
· cases' x with x
· simp [H, ne_of_lt]
by_cases hx : x = 0
· simp [hx, H, ne_of_lt]
· simp [coe_rpow_of_ne_zero hx]
· simp [H]
· cases x
· simp [H, ne_of_gt]
simp [coe_rpow_of_nonneg _ (le_of_lt H)]
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.FieldTheory.Separable
#align_import field_theory.separable_degree from "lea... | Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 135 | 146 | theorem IsSeparableContraction.degree_eq [hF : ExpChar F q] (g : F[X])
(hg : IsSeparableContraction q f g) : g.natDegree = hf.degree := by |
cases hF
· rcases hg with ⟨_, m, hm⟩
rw [one_pow, expand_one] at hm
rw [hf.eq_degree, hm]
· rcases hg with ⟨hg, m, hm⟩
let g' := Classical.choose hf
obtain ⟨hg', m', hm'⟩ := Classical.choose_spec hf
haveI : Fact q.Prime := ⟨by assumption⟩
refine contraction_degree_eq_or_insep q g g' m m' ... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 467 | 478 | theorem coequalizer_isOpen_iff (F : WalkingParallelPair ⥤ TopCat.{u})
(U : Set ((colimit F : _) : Type u)) :
IsOpen U ↔ IsOpen (colimit.ι F WalkingParallelPair.one ⁻¹' U) := by |
rw [colimit_isOpen_iff]
constructor
· intro H
exact H _
· intro H j
cases j
· rw [← colimit.w F WalkingParallelPairHom.left]
exact (F.map WalkingParallelPairHom.left).continuous_toFun.isOpen_preimage _ H
· exact H
|
/-
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.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import al... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 137 | 144 | theorem πSummand_comp_cofan_inj_id_comp_PInfty_eq_PInfty (n : ℕ) :
s.πSummand (IndexSet.id (op [n])) ≫ (s.cofan _).inj (IndexSet.id (op [n])) ≫ PInfty.f n =
PInfty.f n := by |
conv_rhs => rw [← id_comp (PInfty.f n)]
erw [s.decomposition_id, Preadditive.sum_comp]
rw [Fintype.sum_eq_single (IndexSet.id (op [n])), assoc]
rintro A (hA : ¬A.EqId)
rw [assoc, s.cofan_inj_comp_PInfty_eq_zero A hA, comp_zero]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,165 | 2,173 | theorem bound_of_isBigO_cofinite (h : f =O[cofinite] g'') :
∃ C > 0, ∀ ⦃x⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖ := by |
rcases h.exists_pos with ⟨C, C₀, hC⟩
rw [IsBigOWith_def, eventually_cofinite] at hC
rcases (hC.toFinset.image fun x => ‖f x‖ / ‖g'' x‖).exists_le with ⟨C', hC'⟩
have : ∀ x, C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C' := by simpa using hC'
refine ⟨max C C', lt_max_iff.2 (Or.inl C₀), fun x h₀ => ?_⟩
rw [max_m... |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f... | Mathlib/Data/Multiset/Fintype.lean | 176 | 177 | theorem Multiset.toEmbedding_coeEquiv_trans (m : Multiset α) :
m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype _) = m.coeEmbedding := by | ext <;> rfl
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 274 | 278 | theorem condexpInd_ae_eq_condexpIndSMul (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
(hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x : G) :
condexpInd G hm μ s x =ᵐ[μ] condexpIndSMul hm hs hμs x := by |
refine EventuallyEq.trans ?_ (condexpIndL1Fin_ae_eq_condexpIndSMul hm hs hμs x)
simp [condexpInd, condexpIndL1, hs, hμs]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Johan Commelin, Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.Categor... | Mathlib/CategoryTheory/Abelian/Basic.lean | 718 | 721 | theorem epi_snd_of_isLimit [Epi f] {s : PullbackCone f g} (hs : IsLimit s) : Epi s.snd := by |
haveI : Epi (NatTrans.app (limit.cone (cospan f g)).π WalkingCospan.right) :=
Abelian.epi_pullback_of_epi_f f g
apply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,648 | 2,653 | theorem comap_inf_principal_neBot_of_image_mem {f : Filter β} {m : α → β} (hf : NeBot f) {s : Set α}
(hs : m '' s ∈ f) : NeBot (comap m f ⊓ 𝓟 s) := by |
refine ⟨compl_compl s ▸ mt mem_of_eq_bot ?_⟩
rintro ⟨t, ht, hts⟩
rcases hf.nonempty_of_mem (inter_mem hs ht) with ⟨_, ⟨x, hxs, rfl⟩, hxt⟩
exact absurd hxs (hts hxt)
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 154 | 159 | theorem fourier_neg {n : ℤ} {x : AddCircle T} : fourier (-n) x = conj (fourier n x) := by |
induction x using QuotientAddGroup.induction_on'
simp_rw [fourier_apply, toCircle]
rw [← QuotientAddGroup.mk_zsmul, ← QuotientAddGroup.mk_zsmul]
simp_rw [Function.Periodic.lift_coe, ← coe_inv_circle_eq_conj, ← expMapCircle_neg,
neg_smul, mul_neg]
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison, Apurva Nakade
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.SetTheory.Game.PGame
import Mathlib.Tactic.Abel
#align_... | Mathlib/SetTheory/Game/Basic.lean | 953 | 953 | theorem inv_eq_of_equiv_zero {x : PGame} (h : x ≈ 0) : x⁻¹ = 0 := by | classical exact if_pos h
|
/-
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.Lattice
import Mathlib.Order.Hom.Lattice
#align_import order.hom.complete_lattice from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a11... | Mathlib/Order/Hom/CompleteLattice.lean | 142 | 143 | theorem map_iInf₂ [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
f (⨅ (i) (j), g i j) = ⨅ (i) (j), f (g i j) := by | simp_rw [map_iInf]
|
/-
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.PartrecCode
import Mathlib.Data.Set.Subsingleton
#align_import computability.halting from "leanprover-community/mathlib"@"a50170a88a4757... | Mathlib/Computability/Halting.lean | 219 | 234 | theorem rice₂ (C : Set Code) (H : ∀ cf cg, eval cf = eval cg → (cf ∈ C ↔ cg ∈ C)) :
(ComputablePred fun c => c ∈ C) ↔ C = ∅ ∨ C = Set.univ := by |
classical exact
have hC : ∀ f, f ∈ C ↔ eval f ∈ eval '' C := fun f =>
⟨Set.mem_image_of_mem _, fun ⟨g, hg, e⟩ => (H _ _ e).1 hg⟩
⟨fun h =>
or_iff_not_imp_left.2 fun C0 =>
Set.eq_univ_of_forall fun cg =>
let ⟨cf, fC⟩ := Set.nonempty_iff_ne_empty.2 C0
(hC _... |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 512 | 517 | theorem HasFDerivWithinAt.mapsTo_tangent_cone {x : E} (h : HasFDerivWithinAt f f' s x) :
MapsTo f' (tangentConeAt 𝕜 s x) (tangentConeAt 𝕜 (f '' s) (f x)) := by |
rintro v ⟨c, d, dtop, clim, cdlim⟩
refine
⟨c, fun n => f (x + d n) - f x, mem_of_superset dtop ?_, clim, h.lim atTop dtop clim cdlim⟩
simp (config := { contextual := true }) [-mem_image, mem_image_of_mem]
|
/-
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.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import ... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 471 | 474 | theorem isUnit_toStalk (x : PrimeSpectrum.Top R) (f : x.asIdeal.primeCompl) :
IsUnit (toStalk R x (f : R)) := by |
erw [← germ_toOpen R (PrimeSpectrum.basicOpen (f : R)) ⟨x, f.2⟩ (f : R)]
exact RingHom.isUnit_map _ (isUnit_to_basicOpen_self R f)
|
/-
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 | 374 | 375 | theorem prod_insert_div {M : Type*} [CommGroup M] [DecidableEq α] (ha : a ∉ s) {f : α → M} :
(∏ x ∈ insert a s, f x) / f a = ∏ x ∈ s, f x := by | simp [ha]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,465 | 1,465 | theorem rTensor_refl_apply : (refl R N).rTensor M y = y := by | rw [rTensor_refl, refl_apply]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 267 | 271 | theorem braiding_leftUnitor_aux₁ (X : C) :
(α_ (𝟙_ C) (𝟙_ C) X).hom ≫
(𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv := by |
coherence
|
/-
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, Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.GCD
/-!
# ℕ and ℤ are normalized... | Mathlib/Algebra/GCDMonoid/Nat.lean | 139 | 145 | theorem exists_unit_of_abs (a : ℤ) : ∃ (u : ℤ) (_ : IsUnit u), (Int.natAbs a : ℤ) = u * a := by |
cases' natAbs_eq a with h h
· use 1, isUnit_one
rw [← h, one_mul]
· use -1, isUnit_one.neg
rw [← neg_eq_iff_eq_neg.mpr h]
simp only [neg_mul, one_mul]
|
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.NumberField.ClassNumber
import Mathlib.NumberTheory.Cyclotomic.Rat
import Mathlib.NumberTheory.Cyclotomic.Embeddings
/-!
# Cyclotomic ... | Mathlib/NumberTheory/Cyclotomic/PID.lean | 30 | 41 | theorem three_pid [IsCyclotomicExtension {3} ℚ K] : IsPrincipalIdealRing (𝓞 K) := by |
apply RingOfIntegers.isPrincipalIdealRing_of_abs_discr_lt
rw [absdiscr_prime 3 K, IsCyclotomicExtension.finrank (n := 3) K
(irreducible_rat (by norm_num)), nrComplexPlaces_eq_totient_div_two 3, totient_prime
PNat.prime_three]
simp only [Int.reduceNeg, PNat.val_ofNat, succ_sub_succ_eq_sub, tsub_zero, ze... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,237 | 2,239 | theorem enumOrd_zero : enumOrd S 0 = sInf S := by |
rw [enumOrd_def]
simp [Ordinal.not_lt_zero]
|
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Lo... | Mathlib/Data/DFinsupp/NeLocus.lean | 155 | 156 | theorem neLocus_sub_right : neLocus (f₁ - g) (f₂ - g) = neLocus f₁ f₂ := by |
simpa only [sub_eq_add_neg] using @neLocus_add_right α N _ _ _ _ _ _
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e1... | Mathlib/FieldTheory/Minpoly/Basic.lean | 258 | 269 | theorem aeval_ne_zero_of_dvdNotUnit_minpoly {a : A[X]} (hx : IsIntegral A x) (hamonic : a.Monic)
(hdvd : DvdNotUnit a (minpoly A x)) : Polynomial.aeval x a ≠ 0 := by |
refine fun ha => (min A x hamonic ha).not_lt (degree_lt_degree ?_)
obtain ⟨_, c, hu, he⟩ := hdvd
have hcm := hamonic.of_mul_monic_left (he.subst <| monic hx)
rw [he, hamonic.natDegree_mul hcm]
-- TODO: port Nat.lt_add_of_zero_lt_left from lean3 core
apply lt_add_of_pos_right
refine (lt_of_not_le fun h =>... |
/-
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 | 1,533 | 1,535 | theorem equivAdjoinSimple_symm_gen (pb : PowerBasis K L) :
pb.equivAdjoinSimple.symm pb.gen = AdjoinSimple.gen K pb.gen := by |
rw [equivAdjoinSimple, equivOfMinpoly_symm, equivOfMinpoly_gen, adjoin.powerBasis_gen]
|
/-
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 | 369 | 382 | theorem isometry_comap_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f)
(H : Monotone m ∨ Surjective f) : comap f (mkMetric m) = mkMetric m := by |
simp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, comap_iSup]
refine surjective_id.iSup_congr id fun ε => surjective_id.iSup_congr id fun hε => ?_
rw [comap_boundedBy _ (H.imp _ id)]
· congr with s : 1
apply extend_congr
· simp [hf.ediam_image]
· intros; simp [hf.injective.subsing... |
/-
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.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 773 | 778 | theorem liftOn_div {P : Sort v} (p q : K[X]) (f : K[X] → K[X] → P) (f0 : ∀ p, f p 0 = f 0 1)
(H' : ∀ {p q p' q'} (_hq : q ≠ 0) (_hq' : q' ≠ 0), q' * p = q * p' → f p q = f p' q')
(H : ∀ {p q p' q'} (_hq : q ∈ K[X]⁰) (_hq' : q' ∈ K[X]⁰), q' * p = q * p' → f p q = f p' q' :=
fun {p q p' q'} hq hq' h => H' (... |
rw [← mk_eq_div, liftOn_mk _ _ f f0 @H']
|
/-
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 Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 222 | 225 | theorem count_le_count_map [DecidableEq β] (l : List α) (f : α → β) (x : α) :
count x l ≤ count (f x) (map f l) := by |
rw [count, count, countP_map]
apply countP_mono_left; simp (config := { contextual := true })
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/ODE/Gronwall.lean | 325 | 352 | theorem ODE_solution_unique_of_mem_Ioo
(ht : t₀ ∈ Ioo a b)
(hf : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t)
(hg : ∀ t ∈ Ioo a b, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t)
(heq : f t₀ = g t₀) :
EqOn f g (Ioo a b) := by |
intros t' ht'
rcases lt_or_le t' t₀ with (h | h)
· have hss : Icc t' t₀ ⊆ Ioo a b :=
fun _ ht'' ↦ ⟨lt_of_lt_of_le ht'.1 ht''.1, lt_of_le_of_lt ht''.2 ht.2⟩
exact ODE_solution_unique_of_mem_Icc_left hv
(ContinuousAt.continuousOn fun _ ht'' ↦ (hf _ <| hss ht'').1.continuousAt)
(fun _ ht'' ↦ (... |
/-
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 Mathlib.Mathport.Rename
#align_import init.meta.well_founded_tactics from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
--... | Mathlib/Init/Meta/WellFoundedTactics.lean | 18 | 18 | theorem Nat.zero_lt_one_add (a : Nat) : 0 < 1 + a := by | simp [Nat.one_add]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 558 | 560 | theorem geom_series_succ (x : R) (h : ‖x‖ < 1) : ∑' i : ℕ, x ^ (i + 1) = ∑' i : ℕ, x ^ i - 1 := by |
rw [eq_sub_iff_add_eq, tsum_eq_zero_add (NormedRing.summable_geometric_of_norm_lt_one x h),
pow_zero, add_comm]
|
/-
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 | 859 | 862 | theorem image_const_mul_Ioi_zero {k : Type*} [LinearOrderedField k] {x : k} (hx : 0 < x) :
(fun y => x * y) '' Ioi (0 : k) = Ioi 0 := by |
erw [(Units.mk0 x hx.ne').mulLeft.image_eq_preimage, preimage_const_mul_Ioi 0 (inv_pos.mpr hx),
zero_div]
|
/-
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 | 72 | 73 | theorem norm_eq_one_of_not_exists_basis (h : ¬∃ s : Finset S, Nonempty (Basis s R S)) (x : S) :
norm R x = 1 := by | rw [norm_apply, LinearMap.det]; split_ifs <;> trivial
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import... | Mathlib/Algebra/Algebra/Operations.lean | 442 | 443 | theorem pow_eq_span_pow_set (n : ℕ) : M ^ n = span R ((M : Set A) ^ n) := by |
rw [← span_pow, span_eq]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.ModelTheory.Basic
#align_import model_theory.language_map from "leanprover-c... | Mathlib/ModelTheory/LanguageMap.lean | 455 | 457 | theorem card_withConstants :
L[[α]].card = Cardinal.lift.{w'} L.card + Cardinal.lift.{max u v} #α := by |
rw [withConstants, card_sum, card_constantsOn]
|
/-
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.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 1,125 | 1,128 | theorem QuadraticForm.isSymm_toMatrix' (Q : QuadraticForm R (n → R)) : Q.toMatrix'.IsSymm := by |
ext i j
rw [toMatrix', Matrix.transpose_apply, LinearMap.toMatrix₂'_apply, LinearMap.toMatrix₂'_apply,
← associated_isSymm, RingHom.id_apply, associated_apply]
|
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_impor... | Mathlib/RingTheory/ChainOfDivisors.lean | 338 | 349 | theorem multiplicity_prime_eq_multiplicity_image_by_factor_orderIso {m p : Associates M}
{n : Associates N} (hn : n ≠ 0) (hp : p ∈ normalizedFactors m) (d : Set.Iic m ≃o Set.Iic n) :
multiplicity p m = multiplicity (↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩)) n := by |
refine le_antisymm (multiplicity_prime_le_multiplicity_image_by_factor_orderIso hp d) ?_
suffices multiplicity (↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩)) n ≤
multiplicity (↑(d.symm (d ⟨p, dvd_of_mem_normalizedFactors hp⟩))) m by
rw [d.symm_apply_apply ⟨p, dvd_of_mem_normalizedFactors hp⟩, Subtype.coe_mk... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 109 | 120 | theorem StrictConvexSpace.of_norm_combo_ne_one
(h :
∀ x y : E,
‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
StrictConvexSpace ℝ E := by |
refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex ?_)
simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,
frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]
intro x hx y hy hne
rcases h x y hx hy hne with ⟨a, b, ha,... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
#align_import model_theory.substructures from "lea... | Mathlib/ModelTheory/Substructures.lean | 140 | 144 | theorem Term.realize_mem {α : Type*} (t : L.Term α) (xs : α → M) (h : ∀ a, xs a ∈ S) :
t.realize xs ∈ S := by |
induction' t with a n f ts ih
· exact h a
· exact Substructure.fun_mem _ _ _ ih
|
/-
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.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 162 | 164 | theorem coord_reindex (i : ι') : (b.reindex e).coord i = b.coord (e.symm i) := by |
ext
classical simp [AffineBasis.coord]
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 210 | 214 | theorem upperCentralSeries_mono : Monotone (upperCentralSeries G) := by |
refine monotone_nat_of_le_succ ?_
intro n x hx y
rw [mul_assoc, mul_assoc, ← mul_assoc y x⁻¹ y⁻¹]
exact mul_mem hx (Normal.conj_mem (upperCentralSeries_normal G n) x⁻¹ (inv_mem hx) y)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 124 | 128 | theorem isPreconnected_sUnion (x : α) (c : Set (Set α)) (H1 : ∀ s ∈ c, x ∈ s)
(H2 : ∀ s ∈ c, IsPreconnected s) : IsPreconnected (⋃₀ c) := by |
apply isPreconnected_of_forall x
rintro y ⟨s, sc, ys⟩
exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩
|
/-
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.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 1,215 | 1,218 | theorem toSet_image (f : ZFSet → ZFSet) [H : Definable 1 f] (x : ZFSet) :
(image f x).toSet = f '' x.toSet := by |
ext
simp
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 319 | 323 | theorem one_le_norm_mul_norm_symm [RingHomIsometric σ₁₂] [Nontrivial E] (e : E ≃SL[σ₁₂] F) :
1 ≤ ‖(e : E →SL[σ₁₂] F)‖ * ‖(e.symm : F →SL[σ₂₁] E)‖ := by |
rw [mul_comm]
convert (e.symm : F →SL[σ₂₁] E).opNorm_comp_le (e : E →SL[σ₁₂] F)
rw [e.coe_symm_comp_coe, ContinuousLinearMap.norm_id]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 559 | 562 | theorem support_X_pow [Nontrivial R] (s : σ) (n : ℕ) :
(X s ^ n : MvPolynomial σ R).support = {Finsupp.single s n} := by |
classical
rw [X_pow_eq_monomial, support_monomial, if_neg (one_ne_zero' R)]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 363 | 369 | theorem iSup_lt_lift {ι} {f : ι → Cardinal} {c : Cardinal}
(hι : Cardinal.lift.{v, u} #ι < c.ord.cof)
(hf : ∀ i, f i < c) : iSup.{max u v + 1, u + 1} f < c := by |
rw [← ord_lt_ord, iSup_ord (Cardinal.bddAbove_range.{u, v} _)]
refine sup_lt_ord_lift hι fun i => ?_
rw [ord_lt_ord]
apply hf
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 589 | 590 | theorem inv_pow_le_inv_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹ := by |
convert one_div_pow_le_one_div_pow_of_le a1 mn using 1 <;> simp
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-communit... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 132 | 141 | theorem GammaIntegral_ofReal (s : ℝ) :
GammaIntegral ↑s = ↑(∫ x : ℝ in Ioi 0, Real.exp (-x) * x ^ (s - 1)) := by |
have : ∀ r : ℝ, Complex.ofReal' r = @RCLike.ofReal ℂ _ r := fun r => rfl
rw [GammaIntegral]
conv_rhs => rw [this, ← _root_.integral_ofReal]
refine setIntegral_congr measurableSet_Ioi ?_
intro x hx; dsimp only
conv_rhs => rw [← this]
rw [ofReal_mul, ofReal_cpow (mem_Ioi.mp hx).le]
simp
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.convex.... | Mathlib/Analysis/Convex/StrictConvexBetween.lean | 56 | 72 | theorem Collinear.wbtw_of_dist_eq_of_dist_le {p p₁ p₂ p₃ : P} {r : ℝ}
(h : Collinear ℝ ({p₁, p₂, p₃} : Set P)) (hp₁ : dist p₁ p = r) (hp₂ : dist p₂ p ≤ r)
(hp₃ : dist p₃ p = r) (hp₁p₃ : p₁ ≠ p₃) : Wbtw ℝ p₁ p₂ p₃ := by |
rcases h.wbtw_or_wbtw_or_wbtw with (hw | hw | hw)
· exact hw
· by_cases hp₃p₂ : p₃ = p₂
· simp [hp₃p₂]
have hs : Sbtw ℝ p₂ p₃ p₁ := ⟨hw, hp₃p₂, hp₁p₃.symm⟩
have hs' := hs.dist_lt_max_dist p
rw [hp₁, hp₃, lt_max_iff, lt_self_iff_false, or_false_iff] at hs'
exact False.elim (hp₂.not_lt hs')
·... |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomia... | Mathlib/Algebra/Polynomial/Reverse.lean | 173 | 208 | theorem reflect_mul_induction (cf cg : ℕ) :
∀ N O : ℕ,
∀ f g : R[X],
f.support.card ≤ cf.succ →
g.support.card ≤ cg.succ →
f.natDegree ≤ N →
g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g := by |
induction' cf with cf hcf
--first induction (left): base case
· induction' cg with cg hcg
-- second induction (right): base case
· intro N O f g Cf Cg Nf Og
rw [← C_mul_X_pow_eq_self Cf, ← C_mul_X_pow_eq_self Cg]
simp_rw [mul_assoc, X_pow_mul, mul_assoc, ← pow_add (X : R[X]), reflect_C_mul,
... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis
#align_import algebra.direct_sum.module from "leanprover-commun... | Mathlib/Algebra/DirectSum/Module.lean | 363 | 383 | theorem IsInternal.collectedBasis_coe (h : IsInternal A) {α : ι → Type*}
(v : ∀ i, Basis (α i) R (A i)) : ⇑(h.collectedBasis v) = fun a : Σi, α i ↦ ↑(v a.1 a.2) := by |
funext a
-- Porting note: was
-- simp only [IsInternal.collectedBasis, toModule, coeLinearMap, Basis.coe_ofRepr,
-- Basis.repr_symm_apply, DFinsupp.lsum_apply_apply, DFinsupp.mapRange.linearEquiv_apply,
-- DFinsupp.mapRange.linearEquiv_symm, DFinsupp.mapRange_single, Finsupp.total_single,
-- LinearEq... |
/-
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.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 689 | 707 | theorem lt_nmul_iff₃ :
d < a ⨳ b ⨳ c ↔
∃ a' < a, ∃ b' < b, ∃ c' < c,
d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤
a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c' := by |
refine ⟨fun h => ?_, ?_⟩
· rcases lt_nmul_iff.1 h with ⟨e, he, c', hc, H₁⟩
rcases lt_nmul_iff.1 he with ⟨a', ha, b', hb, H₂⟩
refine ⟨a', ha, b', hb, c', hc, ?_⟩
have := nadd_le_nadd H₁ (nmul_nadd_le H₂ hc.le)
simp only [nadd_nmul, nadd_assoc] at this
rw [nadd_left_comm, nadd_left_comm d, nadd_l... |
/-
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.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 77 | 79 | theorem image₂_subset (hs : s ⊆ s') (ht : t ⊆ t') : image₂ f s t ⊆ image₂ f s' t' := by |
rw [← coe_subset, coe_image₂, coe_image₂]
exact image2_subset hs ht
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Ker.lean | 148 | 155 | theorem ker_eq_bot_of_injective {f : F} (hf : Injective f) : ker f = ⊥ := by |
have : Disjoint ⊤ (ker f) := by
-- Porting note: `← map_zero f` should work here, but it needs to be directly applied to H.
rw [disjoint_ker]
intros _ _ H
rw [← map_zero f] at H
exact hf H
simpa [disjoint_iff_inf_le]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.TypeTags
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Ring.Hom.Defs
#align_i... | Mathlib/Deprecated/Group.lean | 404 | 406 | theorem IsAddGroupHom.sub {α β} [AddGroup α] [AddCommGroup β] {f g : α → β} (hf : IsAddGroupHom f)
(hg : IsAddGroupHom g) : IsAddGroupHom fun a => f a - g a := by |
simpa only [sub_eq_add_neg] using hf.add hg.neg
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#ali... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 73 | 76 | theorem dart_card_eq_sum_degrees : Fintype.card G.Dart = ∑ v, G.degree v := by |
haveI := Classical.decEq V
simp only [← card_univ, ← dart_fst_fiber_card_eq_degree]
exact card_eq_sum_card_fiberwise (by simp)
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 105 | 115 | theorem finite_of_norm_le (B : ℝ) : {x : K | IsIntegral ℤ x ∧ ∀ φ : K →+* A, ‖φ x‖ ≤ B}.Finite := by |
let C := Nat.ceil (max B 1 ^ finrank ℚ K * (finrank ℚ K).choose (finrank ℚ K / 2))
have := bUnion_roots_finite (algebraMap ℤ K) (finrank ℚ K) (finite_Icc (-C : ℤ) C)
refine this.subset fun x hx => ?_; simp_rw [mem_iUnion]
have h_map_ℚ_minpoly := minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx.1
refine ⟨_... |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 369 | 372 | theorem geom_sum_Ico' [DivisionRing α] {x : α} (hx : x ≠ 1) {m n : ℕ} (hmn : m ≤ n) :
∑ i ∈ Finset.Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x) := by |
simp only [geom_sum_Ico hx hmn]
convert neg_div_neg_eq (x ^ m - x ^ n) (1 - x) using 2 <;> abel
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 389 | 390 | theorem rev_lt_iff {i j : Fin n} : rev i < j ↔ rev j < i := by |
rw [← rev_lt_rev, rev_rev]
|
/-
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.Countable
import Mathlib.Order.Disjointed
import Mathlib.Tactic.Measurability
#align_import measure_theory.measurable_space_d... | Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 513 | 515 | theorem measurableSet_iInf {ι} {m : ι → MeasurableSpace α} {s : Set α} :
MeasurableSet[iInf m] s ↔ ∀ i, MeasurableSet[m i] s := by |
rw [iInf, measurableSet_sInf, forall_mem_range]
|
/-
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.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,094 | 1,096 | theorem isOpen_extend_preimage {s : Set E} (hs : IsOpen s) :
IsOpen (f.source ∩ f.extend I ⁻¹' s) := by |
rw [← extend_source f I]; exact isOpen_extend_preimage' f I hs
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 862 | 865 | theorem integrableOn_Ioi_deriv_of_nonneg' (hderiv : ∀ x ∈ Ici a, HasDerivAt g (g' x) x)
(g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x) (hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by |
refine integrableOn_Ioi_deriv_of_nonneg ?_ (fun x hx => hderiv x hx.out.le) g'pos hg
exact (hderiv a left_mem_Ici).continuousAt.continuousWithinAt
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.PreservesLocallyBijec... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 204 | 212 | theorem hasSheafCompose : J.HasSheafCompose F where
isSheaf P hP := by |
have hP' : Presheaf.IsSheaf K (e.inverse.op ⋙ P ⋙ F) := by
change Presheaf.IsSheaf K ((_ ⋙ _) ⋙ _)
apply HasSheafCompose.isSheaf
exact e.inverse.op_comp_isSheaf K J ⟨P, hP⟩
replace hP' : Presheaf.IsSheaf J (e.functor.op ⋙ e.inverse.op ⋙ P ⋙ F) :=
e.functor.op_comp_isSheaf _ _ ⟨_, hP'⟩
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 589 | 589 | theorem sub_self (a : Ordinal) : a - a = 0 := by | simpa only [add_zero] using add_sub_cancel a 0
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.pfunctor.multivariate.M from ... | Mathlib/Data/PFunctor/Multivariate/M.lean | 223 | 233 | theorem M.bisim_lemma {α : TypeVec n} {a₁ : (mp P).A} {f₁ : (mp P).B a₁ ⟹ α} {a' : P.A}
{f' : (P.B a').drop ⟹ α} {f₁' : (P.B a').last → M P α}
(e₁ : M.dest P ⟨a₁, f₁⟩ = ⟨a', splitFun f' f₁'⟩) :
∃ (g₁' : _)(e₁' : PFunctor.M.dest a₁ = ⟨a', g₁'⟩),
f' = M.pathDestLeft P e₁' f₁ ∧
f₁' = fun x : (las... |
generalize ef : @splitFun n _ (append1 α (M P α)) f' f₁' = ff at e₁
let he₁' := PFunctor.M.dest a₁;
rcases e₁' : he₁' with ⟨a₁', g₁'⟩;
rw [M.dest_eq_dest' _ e₁'] at e₁
cases e₁; exact ⟨_, e₁', splitFun_inj ef⟩
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Init.Order.LinearOrder
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.Spread
import Mathlib.Tactic.Conv... | Mathlib/Order/Basic.lean | 1,071 | 1,073 | theorem min_def_lt (x y : α) : min x y = if x < y then x else y := by |
rw [min_comm, min_def, ← ite_not]
simp only [not_le]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.Separation
import Mathlib.Topology.LocallyConstant.Basic
#align_import topology.discrete_quotient f... | Mathlib/Topology/DiscreteQuotient.lean | 330 | 333 | theorem map_comp (h1 : LEComap g B C) (h2 : LEComap f A B) :
map (g.comp f) (h1.comp h2) = map g h1 ∘ map f h2 := by |
ext ⟨⟩
rfl
|
/-
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.Analysis.SpecialFunctions.Pow.Real
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.LinearAlgebra.Matrix.AbsoluteValue
import Mathlib.NumberTheory.... | Mathlib/NumberTheory/ClassNumber/Finite.lean | 287 | 319 | theorem exists_mk0_eq_mk0 [IsDedekindDomain S] [Algebra.IsAlgebraic R L] (I : (Ideal S)⁰) :
∃ J : (Ideal S)⁰,
ClassGroup.mk0 I = ClassGroup.mk0 J ∧
algebraMap _ _ (∏ m ∈ finsetApprox bS adm, m) ∈ (J : Ideal S) := by |
set M := ∏ m ∈ finsetApprox bS adm, m
have hM : algebraMap R S M ≠ 0 := prod_finsetApprox_ne_zero bS adm
obtain ⟨b, b_mem, b_ne_zero, b_min⟩ := exists_min abv I
suffices Ideal.span {b} ∣ Ideal.span {algebraMap _ _ M} * I.1 by
obtain ⟨J, hJ⟩ := this
refine ⟨⟨J, ?_⟩, ?_, ?_⟩
· rw [mem_nonZeroDivisors... |
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 1,596 | 1,604 | theorem infinite_basis_le_maximal_linearIndependent' {ι : Type w} (b : Basis ι R M) [Infinite ι]
{κ : Type w'} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) :
Cardinal.lift.{w'} #ι ≤ Cardinal.lift.{w} #κ := by |
let Φ := fun k : κ => (b.repr (v k)).support
have w₁ : #ι ≤ #(Set.range Φ) := by
apply Cardinal.le_range_of_union_finset_eq_top
exact union_support_maximal_linearIndependent_eq_range_basis b v i m
have w₂ : Cardinal.lift.{w'} #(Set.range Φ) ≤ Cardinal.lift.{w} #κ := Cardinal.mk_range_le_lift
exact (Car... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 79 | 85 | theorem IsCompact.inter_right (hs : IsCompact s) (ht : IsClosed t) : IsCompact (s ∩ t) := by |
intro f hnf hstf
obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f :=
hs (le_trans hstf (le_principal_iff.2 inter_subset_left))
have : x ∈ t := ht.mem_of_nhdsWithin_neBot <|
hx.mono <| le_trans hstf (le_principal_iff.2 inter_subset_right)
exact ⟨x, ⟨hsx, this⟩, hx⟩
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 260 | 264 | theorem affineCombination_eq_centerMass {ι : Type*} {t : Finset ι} {p : ι → E} {w : ι → R}
(hw₂ : ∑ i ∈ t, w i = 1) : t.affineCombination R p w = centerMass t w p := by |
rw [affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ w _ hw₂ (0 : E),
Finset.weightedVSubOfPoint_apply, vadd_eq_add, add_zero, t.centerMass_eq_of_sum_1 _ hw₂]
simp_rw [vsub_eq_sub, sub_zero]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 351 | 353 | theorem hnot_symmDiff_self : (￢a) ∆ a = ⊤ := by |
rw [eq_top_iff, symmDiff, hnot_sdiff, sup_sdiff_self]
exact Codisjoint.top_le codisjoint_hnot_left
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 831 | 833 | theorem mem_iInf_iff {f : ι → LowerSet α} : (a ∈ ⨅ i, f i) ↔ ∀ i, a ∈ f i := by |
rw [← SetLike.mem_coe, coe_iInf]
exact mem_iInter
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#... | Mathlib/Order/Interval/Set/Pi.lean | 180 | 181 | theorem image_update_Ico_right (f : ∀ i, α i) (i : ι) (b : α i) :
update f i '' Ico (f i) b = Ico f (update f i b) := by | simpa using image_update_Ico f i (f i) b
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.... | Mathlib/MeasureTheory/Group/Measure.lean | 729 | 770 | theorem measure_univ_of_isMulLeftInvariant [WeaklyLocallyCompactSpace G] [NoncompactSpace G]
(μ : Measure G) [IsOpenPosMeasure μ] [μ.IsMulLeftInvariant] : μ univ = ∞ := by |
/- Consider a closed compact set `K` with nonempty interior. For any compact set `L`, one may
find `g = g (L)` such that `L` is disjoint from `g • K`. Iterating this, one finds
infinitely many translates of `K` which are disjoint from each other. As they all have the
same positive mass, it follows that t... |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data... | Mathlib/Data/List/GetD.lean | 154 | 154 | theorem getI_zero_eq_headI : l.getI 0 = l.headI := by | cases l <;> rfl
|
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 283 | 286 | theorem liftIsometry_apply_apply (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
liftIsometry 𝕜 E F f x = lift f.toMultilinearMap x := by |
simp only [liftIsometry, LinearIsometryEquiv.coe_mk, liftEquiv_apply,
LinearMap.mkContinuous_apply]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 106 | 106 | theorem singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by | simp [h]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92... | Mathlib/LinearAlgebra/LinearPMap.lean | 823 | 844 | theorem neg_graph (f : E →ₗ.[R] F) :
(-f).graph =
f.graph.map ((LinearMap.id : E →ₗ[R] E).prodMap (-(LinearMap.id : F →ₗ[R] F))) := by |
ext x; cases' x with x_fst x_snd
constructor <;> intro h
· rw [mem_graph_iff] at h
rcases h with ⟨y, hy, h⟩
rw [LinearPMap.neg_apply] at h
rw [Submodule.mem_map]
simp only [mem_graph_iff, LinearMap.prodMap_apply, LinearMap.id_coe, id,
LinearMap.neg_apply, Prod.mk.inj_iff, Prod.exists, exist... |
/-
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.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 72 | 76 | theorem measure_biUnion_le {I : Set ι} (μ : F) (hI : I.Countable) (s : ι → Set α) :
μ (⋃ i ∈ I, s i) ≤ ∑' i : I, μ (s i) := by |
have := hI.to_subtype
rw [biUnion_eq_iUnion]
apply measure_iUnion_le
|
/-
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.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 489 | 490 | theorem sin_neg : sin (-x) = -sin x := by |
simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 124 | 126 | theorem parent'_lt (self : UnionFind) (i : Fin self.size) :
(self.arr.get i).parent < self.size := by |
simp only [← parentD_eq, parentD_lt, Fin.is_lt, Array.data_length]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,900 | 1,906 | theorem transfer_append {w : V} (q : G.Walk v w) (hpq) :
(p.append q).transfer H hpq =
(p.transfer H fun e he => hpq _ (by simp [he])).append
(q.transfer H fun e he => hpq _ (by simp [he])) := by |
induction p with
| nil => simp
| cons _ _ ih => simp only [Walk.transfer, cons_append, cons.injEq, heq_eq_eq, true_and, ih]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 1,007 | 1,011 | theorem Function.LeftInverse.map_nhds_eq {f : α → β} {g : β → α} {x : β}
(h : Function.LeftInverse f g) (hf : ContinuousWithinAt f (range g) (g x))
(hg : ContinuousAt g x) : map g (𝓝 x) = 𝓝[range g] g x := by |
simpa only [nhdsWithin_univ, image_univ] using
(h.leftInvOn univ).map_nhdsWithin_eq (h x) (by rwa [image_univ]) hg.continuousWithinAt
|
/-
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.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprove... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 127 | 140 | theorem eqId_iff_eq : A.EqId ↔ A.1 = Δ := by |
constructor
· intro h
dsimp at h
rw [h]
rfl
· intro h
rcases A with ⟨_, ⟨f, hf⟩⟩
simp only at h
subst h
refine ext _ _ rfl ?_
haveI := hf
simp only [eqToHom_refl, comp_id]
exact eq_id_of_epi f
|
/-
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.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Spe... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 73 | 115 | theorem MeasureTheory.measure_lt_one_eq_integral_div_gamma {p : ℝ} (hp : 0 < p) :
μ {x : E | g x < 1} =
.ofReal ((∫ (x : E), Real.exp (- (g x) ^ p) ∂μ) / Real.Gamma (finrank ℝ E / p + 1)) := by |
-- We copy `E` to a new type `F` on which we will put the norm defined by `g`
letI F : Type _ := E
letI : NormedAddCommGroup F :=
{ norm := g
dist := fun x y => g (x - y)
dist_self := by simp only [_root_.sub_self, h1, forall_const]
dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
di... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 912 | 913 | theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b := by |
rw [insert_eq, union_comm, Ico_union_right h]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 185 | 192 | theorem angle_eq_angle_of_angle_eq_pi (p1 : P) {p2 p3 p4 : P} (h : ∠ p2 p3 p4 = π) :
∠ p1 p2 p3 = ∠ p1 p2 p4 := by |
unfold angle at *
rcases angle_eq_pi_iff.1 h with ⟨_, ⟨r, ⟨hr, hpr⟩⟩⟩
rw [eq_comm]
convert angle_smul_right_of_pos (p1 -ᵥ p2) (p3 -ᵥ p2) (add_pos (neg_pos_of_neg hr) zero_lt_one)
rw [add_smul, ← neg_vsub_eq_vsub_rev p2 p3, smul_neg, neg_smul, ← hpr]
simp
|
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import... | Mathlib/GroupTheory/PGroup.lean | 310 | 314 | theorem to_sup_of_normal_right {H K : Subgroup G} (hH : IsPGroup p H) (hK : IsPGroup p K)
[K.Normal] : IsPGroup p (H ⊔ K : Subgroup G) := by |
rw [← QuotientGroup.ker_mk' K, ← Subgroup.comap_map_eq]
apply (hH.map (QuotientGroup.mk' K)).comap_of_ker_isPGroup
rwa [QuotientGroup.ker_mk']
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Bas... | Mathlib/Algebra/Regular/Basic.lean | 235 | 238 | theorem not_isRightRegular_zero_iff : ¬IsRightRegular (0 : R) ↔ Nontrivial R := by |
rw [nontrivial_iff, not_iff_comm, isRightRegular_zero_iff_subsingleton, subsingleton_iff]
push_neg
exact Iff.rfl
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic... | Mathlib/GroupTheory/Perm/Basic.lean | 278 | 282 | theorem sigmaCongrRightHom_injective {α : Type*} {β : α → Type*} :
Function.Injective (sigmaCongrRightHom β) := by |
intro x y h
ext a b
simpa using Equiv.congr_fun h ⟨a, b⟩
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 99 | 107 | theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ := by |
refine le_antisymm ?_ ?_
· refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw [adjointAux_apply, LinearIsometryEquiv.norm_map]
exact toSesqForm_apply_norm_le
· nth_rw 1 [← adjointAux_adjointAux A]
refine ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) fun x => ?_
rw... |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
#align_import ring_theory.graded_algebra.radical from "leanprover-community/mathlib"@"f1944b30c97c5... | Mathlib/RingTheory/GradedAlgebra/Radical.lean | 172 | 175 | theorem Ideal.IsHomogeneous.radical {I : Ideal A} (h : I.IsHomogeneous 𝒜) :
I.radical.IsHomogeneous 𝒜 := by |
rw [h.radical_eq]
exact Ideal.IsHomogeneous.sInf fun _ => And.left
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanpr... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 188 | 194 | theorem lift_spec [HasReverse V'] (φ : Prefunctor V V') :
Symmetrify.of.comp (Symmetrify.lift φ) = φ := by |
fapply Prefunctor.ext
· rintro X
rfl
· rintro X Y f
rfl
|
/-
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
#align_import algebra.grou... | Mathlib/Algebra/Order/Ring/Basic.lean | 229 | 230 | theorem one_lt_pow_iff_of_nonneg (ha : 0 ≤ a) (hn : n ≠ 0) : 1 < a ^ n ↔ 1 < a := by |
simpa only [one_pow] using pow_lt_pow_iff_left (zero_le_one' R) ha hn
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 621 | 638 | theorem mem_iInf {ι} {s : ι → Filter α} {U : Set α} :
(U ∈ ⨅ i, s i) ↔ ∃ I : Set ι, I.Finite ∧ ∃ V : I → Set α, (∀ i, V i ∈ s i) ∧ U = ⋂ i, V i := by |
constructor
· rw [iInf_eq_generate, mem_generate_iff]
rintro ⟨t, tsub, tfin, tinter⟩
rcases eq_finite_iUnion_of_finite_subset_iUnion tfin tsub with ⟨I, Ifin, σ, σfin, σsub, rfl⟩
rw [sInter_iUnion] at tinter
set V := fun i => U ∪ ⋂₀ σ i with hV
have V_in : ∀ i, V i ∈ s i := by
rintro i
... |
/-
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 | 1,302 | 1,304 | theorem ceil_add_le (a b : α) : ⌈a + b⌉ ≤ ⌈a⌉ + ⌈b⌉ := by |
rw [ceil_le, Int.cast_add]
exact add_le_add (le_ceil _) (le_ceil _)
|
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 639 | 655 | theorem reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal (M : Matrix p p 𝕜)
(e : p ≃ n)
(H :
∃ (L L' : List (TransvectionStruct n 𝕜)) (D : n → 𝕜),
(L.map toMatrix).prod * Matrix.reindexAlgEquiv 𝕜 e M * (L'.map toMatrix).prod =
diagonal D) :
∃ (L L' : List (Transvect... |
rcases H with ⟨L₀, L₀', D₀, h₀⟩
refine ⟨L₀.map (reindexEquiv e.symm), L₀'.map (reindexEquiv e.symm), D₀ ∘ e, ?_⟩
have : M = reindexAlgEquiv 𝕜 e.symm (reindexAlgEquiv 𝕜 e M) := by
simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id,
Equiv.symm_comp_self, reindexAlgEquiv_app... |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 214 | 220 | theorem coe_of_s_get?_rat_eq :
(((of q).s.get? n).map (Pair.map (↑)) : Option <| Pair K) = (of v).s.get? n := by |
simp only [of, IntFractPair.seq1, Stream'.Seq.map_get?, Stream'.Seq.get?_tail]
simp only [Stream'.Seq.get?]
rw [← IntFractPair.coe_stream'_rat_eq v_eq_q]
rcases succ_nth_stream_eq : IntFractPair.stream q (n + 1) with (_ | ⟨_, _⟩) <;>
simp [Stream'.map, Stream'.get, succ_nth_stream_eq]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 461 | 466 | theorem sum_tmul {α : Type*} (s : Finset α) (m : α → M) (n : N) :
(∑ a ∈ s, m a) ⊗ₜ[R] n = ∑ a ∈ s, m a ⊗ₜ[R] n := by |
classical
induction' s using Finset.induction with a s has ih h
· simp
· simp [Finset.sum_insert has, add_tmul, ih]
|
/-
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.Order.RelClasses
import Mathlib.Data.List.Basic
#align_import data.list.lex from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c7... | Mathlib/Data/List/Lex.lean | 235 | 238 | theorem cons_le_cons [LinearOrder α] (a : α) {l l' : List α} (h : l' ≤ l) :
a :: l' ≤ a :: l := by |
rw [le_iff_lt_or_eq] at h ⊢
exact h.imp .cons (congr_arg _)
|
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