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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 250 | 257 | theorem continuousAt_clog {x : ℂ} (h : x ∈ slitPlane) : ContinuousAt log x := by |
refine ContinuousAt.add ?_ ?_
· refine continuous_ofReal.continuousAt.comp ?_
refine (Real.continuousAt_log ?_).comp Complex.continuous_abs.continuousAt
exact Complex.abs.ne_zero_iff.mpr <| slitPlane_ne_zero h
· have h_cont_mul : Continuous fun x : ℂ => x * I := continuous_id'.mul continuous_const
re... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 256 | 271 | theorem comap_coe_nhdsWithin_Iio_of_Ioo_subset (hb : s ⊆ Iio b)
(hs : s.Nonempty → ∃ a < b, Ioo a b ⊆ s) : comap ((↑) : s → α) (𝓝[<] b) = atTop := by |
nontriviality
haveI : Nonempty s := nontrivial_iff_nonempty.1 ‹_›
rcases hs (nonempty_subtype.1 ‹_›) with ⟨a, h, hs⟩
ext u; constructor
· rintro ⟨t, ht, hts⟩
obtain ⟨x, ⟨hxa : a ≤ x, hxb : x < b⟩, hxt : Ioo x b ⊆ t⟩ :=
(mem_nhdsWithin_Iio_iff_exists_mem_Ico_Ioo_subset h).mp ht
obtain ⟨y, hxy, h... |
/-
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.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 508 | 509 | theorem isWF_insert {a} : IsWF (insert a s) ↔ IsWF s := by |
simp only [← singleton_union, isWF_union, isWF_singleton, true_and_iff]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Triangularizab... | Mathlib/Algebra/Lie/Weights/Basic.lean | 281 | 284 | theorem zero_weightSpace_eq_top_of_nilpotent' [IsNilpotent R L M] :
weightSpace M (0 : L → R) = ⊤ := by |
ext
simp [weightSpace, weightSpaceOf]
|
/-
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 | 466 | 468 | theorem floor_add_one (ha : 0 ≤ a) : ⌊a + 1⌋₊ = ⌊a⌋₊ + 1 := by |
-- Porting note: broken `convert floor_add_nat ha 1`
rw [← cast_one, floor_add_nat ha 1]
|
/-
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.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 305 | 315 | theorem of_diff_eq_zero_of_symmDiff_eq_zero_positive (hu : MeasurableSet u) (hv : MeasurableSet v)
(hsu : 0 ≤[u] s) (hsv : 0 ≤[v] s) (hs : s (u ∆ v) = 0) : s (u \ v) = 0 ∧ s (v \ u) = 0 := by |
rw [restrict_le_restrict_iff] at hsu hsv
on_goal 1 =>
have a := hsu (hu.diff hv) diff_subset
have b := hsv (hv.diff hu) diff_subset
erw [of_union (Set.disjoint_of_subset_left diff_subset disjoint_sdiff_self_right)
(hu.diff hv) (hv.diff hu)] at hs
rw [zero_apply] at a b
constructor
all... |
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | 986 | 988 | theorem of_has_biproduct₂ [HasBinaryBiproduct X Y] :
BicartesianSq (0 : 0 ⟶ X) (0 : 0 ⟶ Y) biprod.inl biprod.inr := by |
convert of_is_biproduct₂ (BinaryBiproduct.isBilimit X Y)
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 614 | 618 | theorem bit1_apply (M : Matrix n n α) (i : n) (j : n) :
(bit1 M) i j = if i = j then bit1 (M i j) else bit0 (M i j) := by |
dsimp [bit1]
by_cases h : i = j <;>
simp [h]
|
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.RepresentationTheory.Action.Basic
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.Rep... | Mathlib/CategoryTheory/Galois/Examples.lean | 104 | 124 | theorem Action.pretransitive_of_isConnected (X : Action FintypeCat (MonCat.of G))
[IsConnected X] : MulAction.IsPretransitive G X.V where
exists_smul_eq x y := by |
/- We show that the `G`-orbit of `x` is a non-initial subobject of `X` and hence by
connectedness, the orbit equals `X.V`. -/
let T : Set X.V := MulAction.orbit G x
have : Fintype T := Fintype.ofFinite T
letI : MulAction G (FintypeCat.of T) := inferInstanceAs <| MulAction G ↑(MulAction.orbit G x)
... |
/-
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
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,346 | 1,349 | theorem moveRight_neg' {x : PGame} (i) :
(-x).moveRight i = -x.moveLeft (toRightMovesNeg.symm i) := by |
cases x
rfl
|
/-
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.BigOperators.Group.Finset
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.big_operators from "leanprover-community/mathlib"... | Mathlib/Data/Set/Pointwise/BigOperators.lean | 58 | 79 | theorem mem_finset_prod (t : Finset ι) (f : ι → Set α) (a : α) :
(a ∈ ∏ i ∈ t, f i) ↔ ∃ (g : ι → α) (_ : ∀ {i}, i ∈ t → g i ∈ f i), ∏ i ∈ t, g i = a := by |
classical
induction' t using Finset.induction_on with i is hi ih generalizing a
· simp_rw [Finset.prod_empty, Set.mem_one]
exact ⟨fun h ↦ ⟨fun _ ↦ a, fun hi ↦ False.elim (Finset.not_mem_empty _ hi), h.symm⟩,
fun ⟨_, _, hf⟩ ↦ hf.symm⟩
rw [Finset.prod_insert hi, Set.mem_mul]
simp_rw [Fins... |
/-
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.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_impor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 80 | 82 | theorem toFunLinear_mul_tmul_mul_aux_1 (p : R[X]) (k : ℕ) (h : Decidable ¬p.coeff k = 0) (a : A) :
ite (¬coeff p k = 0) (a * (algebraMap R A) (coeff p k)) 0 =
a * (algebraMap R A) (coeff p k) := by | classical split_ifs <;> simp [*]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,466 | 1,469 | theorem _root_.Multiset.aestronglyMeasurable_prod' (l : Multiset (α → M))
(hl : ∀ f ∈ l, AEStronglyMeasurable f μ) : AEStronglyMeasurable l.prod μ := by |
rcases l with ⟨l⟩
simpa using l.aestronglyMeasurable_prod' (by simpa using hl)
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathl... | Mathlib/Topology/DenseEmbedding.lean | 169 | 173 | theorem extend_eq' [T2Space γ] {f : α → γ} (di : DenseInducing i)
(hf : ∀ b, ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)) (a : α) : di.extend f (i a) = f a := by |
rcases hf (i a) with ⟨b, hb⟩
refine di.extend_eq_at' b ?_
rwa [← di.toInducing.nhds_eq_comap] at hb
|
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Bicategory.Basic
#align_import category_theory.bicategory.strict from "leanprover-community/mathlib"@"d... | Mathlib/CategoryTheory/Bicategory/Strict.lean | 85 | 88 | theorem eqToHom_whiskerRight {a b c : B} {f g : a ⟶ b} (η : f = g) (h : b ⟶ c) :
eqToHom η ▷ h = eqToHom (congr_arg₂ (· ≫ ·) η rfl) := by |
cases η
simp only [id_whiskerRight, eqToHom_refl]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/math... | Mathlib/Probability/Kernel/CondDistrib.lean | 209 | 213 | theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet[mβ.comap X] t) :
∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) := by |
obtain ⟨t', ht', rfl⟩ := ht
rw [set_lintegral_preimage_condDistrib hX hY hs ht']
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 160 | 178 | theorem parallelPair_pullback_initial {X B : C} (π : X ⟶ B)
(c : PullbackCone π π) (hc : IsLimit c) :
(parallelPair (C := (Sieve.ofArrows (fun (_ : Unit) => X) (fun _ => π)).arrows.categoryᵒᵖ)
(Y := op ((Presieve.categoryMk _ (c.fst ≫ π) ⟨_, c.fst, π, ofArrows.mk (), rfl⟩)))
(X := op ((Presieve.category... |
apply Limits.parallelPair_initial_mk
· intro ⟨Z⟩
obtain ⟨_, f, g, ⟨⟩, hh⟩ := Z.property
let X' : (Presieve.ofArrows (fun () ↦ X) (fun () ↦ π)).category :=
Presieve.categoryMk _ π (ofArrows.mk ())
let f' : Z.obj.left ⟶ X'.obj.left := f
exact ⟨(Over.homMk f').op⟩
· intro ⟨Z⟩ ⟨i⟩ ⟨j⟩
let i... |
/-
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.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.comp... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 226 | 234 | theorem IntFractPair.exists_succ_get?_stream_of_gcf_of_get?_eq_some {gp_n : Pair K}
(s_nth_eq : (of v).s.get? n = some gp_n) :
∃ ifp : IntFractPair K, IntFractPair.stream v (n + 1) = some ifp ∧ (ifp.b : K) = gp_n.b := by |
obtain ⟨ifp, stream_succ_nth_eq, gp_n_eq⟩ :
∃ ifp, IntFractPair.stream v (n + 1) = some ifp ∧ Pair.mk 1 (ifp.b : K) = gp_n := by
unfold of IntFractPair.seq1 at s_nth_eq
simpa [Stream'.Seq.get?_tail, Stream'.Seq.map_get?] using s_nth_eq
cases gp_n_eq
simp_all only [Option.some.injEq, exists_eq_left']
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
/-!
# Von Neumann Mean Ergodic Theorem in a Hilbert Space
In ... | Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean | 84 | 103 | theorem ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection (f : E →L[𝕜] E)
(hf : ‖f‖ ≤ 1) (x : E) :
Tendsto (birkhoffAverage 𝕜 f _root_.id · x) atTop
(𝓝 <| orthogonalProjection (LinearMap.eqLocus f 1) x) := by |
/- Due to the previous theorem, it suffices to verify
that the range of `f - 1` is dense in the orthogonal complement
to the submodule of fixed points of `f`. -/
apply (f : E →ₗ[𝕜] E).tendsto_birkhoffAverage_of_ker_subset_closure (f.lipschitz.weaken hf)
· exact orthogonalProjection_mem_subspace_eq_self (K :... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 153 | 154 | theorem toIcoMod_add_toIcoDiv_zsmul (a b : α) : toIcoMod hp a b + toIcoDiv hp a b • p = b := by |
rw [toIcoMod, sub_add_cancel]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Algebra.Hom
#align_import algebra.hom.non_unital_alg from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
/-!
# Morphisms... | Mathlib/Algebra/Algebra/NonUnitalHom.lean | 275 | 277 | theorem coe_mulHom_mk (f : A →ₛₙₐ[φ] B) (h₁ h₂ h₃ h₄) :
((⟨⟨⟨f, h₁⟩, h₂, h₃⟩, h₄⟩ : A →ₛₙₐ[φ] B) : A →ₙ* B) = ⟨f, h₄⟩ := by |
rfl
|
/-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 282 | 285 | theorem commute_subtype_of_commute (i j : ι) (hne : i ≠ j) :
∀ (x : H i) (y : H j), Commute ((H i).subtype x) ((H j).subtype y) := by |
rintro ⟨x, hx⟩ ⟨y, hy⟩
exact hcomm hne x y hx hy
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 728 | 731 | theorem orderOf_dvd_of_mem_zpowers (h : y ∈ Subgroup.zpowers x) : orderOf y ∣ orderOf x := by |
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp h
rw [orderOf_dvd_iff_pow_eq_one]
exact zpow_pow_orderOf
|
/-
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 | 774 | 778 | theorem wbtw_smul_vadd_smul_vadd_of_nonpos_of_le (x : P) (v : V) {r₁ r₂ : R} (hr₁ : r₁ ≤ 0)
(hr₂ : r₂ ≤ r₁) : Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) := by |
convert wbtw_smul_vadd_smul_vadd_of_nonneg_of_le x (-v) (Left.nonneg_neg_iff.2 hr₁)
(neg_le_neg_iff.2 hr₂) using 1 <;>
rw [neg_smul_neg]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8e... | Mathlib/Algebra/Polynomial/Expand.lean | 212 | 221 | theorem coeff_contract {p : ℕ} (hp : p ≠ 0) (f : R[X]) (n : ℕ) :
(contract p f).coeff n = f.coeff (n * p) := by |
simp only [contract, coeff_monomial, sum_ite_eq', finset_sum_coeff, mem_range, not_lt,
ite_eq_left_iff]
intro hn
apply (coeff_eq_zero_of_natDegree_lt _).symm
calc
f.natDegree < f.natDegree + 1 := Nat.lt_succ_self _
_ ≤ n * 1 := by simpa only [mul_one] using hn
_ ≤ n * p := mul_le_mul_of_nonneg_... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 67 | 69 | theorem col_smul [SMul R α] (x : R) (v : m → α) : col (x • v) = x • col v := by |
ext
rfl
|
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Algebra.InfiniteSum.Ring
import Mathlib.Topology.Instances.Real
import Mathlib.Topology.Metr... | Mathlib/Topology/Instances/NNReal.lean | 179 | 182 | theorem summable_coe {f : α → ℝ≥0} : (Summable fun a => (f a : ℝ)) ↔ Summable f := by |
constructor
· exact fun ⟨a, ha⟩ => ⟨⟨a, ha.nonneg fun x => (f x).2⟩, hasSum_coe.1 ha⟩
· exact fun ⟨a, ha⟩ => ⟨a.1, hasSum_coe.2 ha⟩
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
Some proofs and docs came from `algebra/commute` (c) Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Init.Logic
import Mathlib.Tactic.Cases
#alig... | Mathlib/Algebra/Group/Semiconj/Defs.lean | 97 | 97 | theorem one_right (a : M) : SemiconjBy a 1 1 := by | rw [SemiconjBy, mul_one, one_mul]
|
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespace List
variable ... | Mathlib/Data/List/ReduceOption.lean | 19 | 21 | theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by |
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,106 | 1,111 | theorem vectorSpan_image_eq_span_vsub_set_right_ne (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s) :
vectorSpan k (p '' s) = Submodule.span k ((· -ᵥ p i) '' (p '' (s \ {i}))) := by |
conv_lhs =>
rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), ← Set.insert_eq_of_mem hi,
← Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]
simp [Submodule.span_insert_eq_span]
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.nakayama from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
/-!
# Nakayama... | Mathlib/RingTheory/Nakayama.lean | 137 | 140 | theorem le_of_le_smul_of_le_jacobson_bot {R M} [CommRing R] [AddCommGroup M] [Module R M]
{I : Ideal R} {N N' : Submodule R M} (hN' : N'.FG)
(hIJ : I ≤ jacobson ⊥) (hNN : N' ≤ N ⊔ I • N') : N' ≤ N := by |
rw [← sup_eq_left, sup_eq_sup_smul_of_le_smul_of_le_jacobson hN' hIJ hNN, bot_smul, sup_bot_eq]
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
... | Mathlib/Algebra/MvPolynomial/Supported.lean | 135 | 141 | theorem exists_restrict_to_vars (R : Type*) [CommRing R] {F : MvPolynomial σ ℤ}
(hF : ↑F.vars ⊆ s) : ∃ f : (s → R) → R, ∀ x : σ → R, f (x ∘ (↑) : s → R) = aeval x F := by |
rw [← mem_supported, supported_eq_range_rename, AlgHom.mem_range] at hF
cases' hF with F' hF'
use fun z ↦ aeval z F'
intro x
simp only [← hF', aeval_rename]
|
/-
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 | 591 | 592 | theorem mul_swap_eq_iff {i j : α} {σ : Perm α} : σ * swap i j = σ ↔ i = j := by |
rw [mul_right_eq_self, swap_eq_one_iff]
|
/-
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 | 2,525 | 2,527 | theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (l H) :
map g (pmap f l H) = pmap (fun a h => g (f a h)) l H := by |
induction l <;> [rfl; simp only [*, pmap, map]]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f0242... | Mathlib/Data/PNat/Prime.lean | 103 | 106 | theorem eq_one_of_lt_two {n : ℕ+} : n < 2 → n = 1 := by |
intro h; apply le_antisymm; swap
· apply PNat.one_le
· exact PNat.lt_add_one_iff.1 h
|
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.fiber_bundle.constructions from "leanprove... | Mathlib/Topology/FiberBundle/Constructions.lean | 300 | 302 | theorem Pullback.continuous_proj (f : B' → B) : Continuous (π F (f *ᵖ E)) := by |
rw [continuous_iff_le_induced, Pullback.TotalSpace.topologicalSpace, pullbackTopology_def]
exact inf_le_left
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 157 | 161 | theorem contDiffWithinAtProp_mono_of_mem (n : ℕ∞) ⦃s x t⦄ ⦃f : H → H'⦄ (hts : s ∈ 𝓝[t] x)
(h : ContDiffWithinAtProp I I' n f s x) : ContDiffWithinAtProp I I' n f t x := by |
refine h.mono_of_mem ?_
refine inter_mem ?_ (mem_of_superset self_mem_nhdsWithin inter_subset_right)
rwa [← Filter.mem_map, ← I.image_eq, I.symm_map_nhdsWithin_image]
|
/-
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.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 106 | 106 | theorem condexp_of_not_le (hm_not : ¬m ≤ m0) : μ[f|m] = 0 := by | rw [condexp, dif_neg hm_not]
|
/-
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
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 988 | 990 | theorem lf_iff_lt_or_fuzzy {x y : PGame} : x ⧏ y ↔ x < y ∨ x ‖ y := by |
simp only [lt_iff_le_and_lf, Fuzzy, ← PGame.not_le]
tauto
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | Mathlib/Topology/Inseparable.lean | 282 | 284 | theorem inseparable_iff_closure_eq : (x ~ᵢ y) ↔ closure ({x} : Set X) = closure {y} := by |
simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, ← subset_antisymm_iff,
eq_comm]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 442 | 444 | theorem map_punctured_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝[≠] x) = 𝓝[≠] (h x) := by |
convert h.embedding.map_nhdsWithin_eq ({x}ᶜ) x
rw [h.image_compl, Set.image_singleton]
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.cha... | Mathlib/RepresentationTheory/Character.lean | 64 | 65 | theorem char_tensor (V W : FdRep k G) : (V ⊗ W).character = V.character * W.character := by |
ext g; convert trace_tensorProduct' (V.ρ g) (W.ρ g)
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 615 | 627 | theorem Ideal.dvd_iff_le {I J : Ideal A} : I ∣ J ↔ J ≤ I :=
⟨Ideal.le_of_dvd, fun h => by
by_cases hI : I = ⊥
· have hJ : J = ⊥ := by | rwa [hI, ← eq_bot_iff] at h
rw [hI, hJ]
have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI
have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=
le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))
(le_of_eq (inv_mul_cancel hI'))
obta... |
/-
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 | 706 | 707 | theorem compl_mem_coclosedCompact : sᶜ ∈ coclosedCompact X ↔ IsCompact (closure s) := by |
rw [mem_coclosedCompact_iff, compl_compl]
|
/-
Copyright (c) 2021 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import dat... | Mathlib/Data/Nat/Choose/Central.lean | 131 | 138 | theorem succ_dvd_centralBinom (n : ℕ) : n + 1 ∣ n.centralBinom := by |
have h_s : (n + 1).Coprime (2 * n + 1) := by
rw [two_mul, add_assoc, coprime_add_self_right, coprime_self_add_left]
exact coprime_one_left n
apply h_s.dvd_of_dvd_mul_left
apply Nat.dvd_of_mul_dvd_mul_left zero_lt_two
rw [← mul_assoc, ← succ_mul_centralBinom_succ, mul_comm]
exact mul_dvd_mul_left _ (t... |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 1,162 | 1,170 | theorem continuousAt_zero' [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E}
{r : ℝ} (hp : p.closedBall 0 r ∈ (𝓝 0 : Filter E)) : ContinuousAt p 0 := by |
refine continuousAt_zero_of_forall' fun ε hε ↦ ?_
obtain ⟨k, hk₀, hk⟩ : ∃ k : 𝕜, 0 < ‖k‖ ∧ ‖k‖ * r < ε := by
rcases le_or_lt r 0 with hr | hr
· use 1; simpa using hr.trans_lt hε
· simpa [lt_div_iff hr] using exists_norm_lt 𝕜 (div_pos hε hr)
rw [← set_smul_mem_nhds_zero_iff (norm_pos_iff.1 hk₀), smu... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 293 | 293 | theorem cons_product : (a ::ₘ s) ×ˢ t = map (Prod.mk a) t + s ×ˢ t := by | simp [SProd.sprod, product]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 142 | 144 | theorem tendstoUniformly_iff_tendstoUniformlyOnFilter :
TendstoUniformly F f p ↔ TendstoUniformlyOnFilter F f p ⊤ := by |
rw [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, principal_univ]
|
/-
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 | 197 | 200 | theorem of_diff {M : Type*} [AddCommGroup M] [TopologicalSpace M] [T2Space M]
{v : VectorMeasure α M} {A B : Set α} (hA : MeasurableSet A) (hB : MeasurableSet B)
(h : A ⊆ B) : v (B \ A) = v B - v A := by |
rw [← of_add_of_diff hA hB h, add_sub_cancel_left]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 575 | 576 | theorem mem_ball_self (h : 0 < ε) : x ∈ ball x ε := by |
rwa [mem_ball, edist_self]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Logic.UnivLE
import Mathlib.Logic.Small.Basic
#align_import catego... | Mathlib/CategoryTheory/EssentiallySmall.lean | 257 | 259 | theorem essentiallySmall_iff_of_thin {C : Type u} [Category.{v} C] [Quiver.IsThin C] :
EssentiallySmall.{w} C ↔ Small.{w} (Skeleton C) := by |
simp [essentiallySmall_iff, CategoryTheory.locallySmall_of_thin]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 71 | 74 | theorem isPositive_zero : IsPositive (0 : E →L[𝕜] E) := by |
refine ⟨isSelfAdjoint_zero _, fun x => ?_⟩
change 0 ≤ re ⟪_, _⟫
rw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 409 | 419 | theorem factorization_le_iff_dvd {d n : ℕ} (hd : d ≠ 0) (hn : n ≠ 0) :
d.factorization ≤ n.factorization ↔ d ∣ n := by |
constructor
· intro hdn
set K := n.factorization - d.factorization with hK
use K.prod (· ^ ·)
rw [← factorization_prod_pow_eq_self hn, ← factorization_prod_pow_eq_self hd,
← Finsupp.prod_add_index' pow_zero pow_add, hK, add_tsub_cancel_of_le hdn]
· rintro ⟨c, rfl⟩
rw [factorization_mul hd... |
/-
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.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 414 | 424 | theorem prop_of_isGood {l : Products I} (J : I → Prop) [∀ j, Decidable (J j)]
(h : l.isGood (π C J)) : ∀ a, a ∈ l.val → J a := by |
intro i hi
by_contra h'
apply h
suffices eval (π C J) l = 0 by
rw [this]
exact Submodule.zero_mem _
ext ⟨_, _, _, rfl⟩
rw [eval_eq, if_neg fun h ↦ ?_, LocallyConstant.zero_apply]
simpa [Proj, h'] using h i hi
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 1,970 | 1,972 | theorem erase_cons_of_ne {a b : α} {s : Finset α} (ha : a ∉ s) (hb : a ≠ b) :
erase (cons a s ha) b = cons a (erase s b) fun h => ha <| erase_subset _ _ h := by |
simp only [cons_eq_insert, erase_insert_of_ne hb]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 421 | 461 | theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
(hx : x ∈ ball (0 : E) (1 + D)) : 0 < y D x := by |
simp only [mem_ball_zero_iff] at hx
refine (integral_pos_iff_support_of_nonneg (w_mul_φ_nonneg D x) ?_).2 ?_
· have F_comp : HasCompactSupport (w D) := w_compact_support E Dpos
have B : LocallyIntegrable (φ : E → ℝ) μ :=
(locallyIntegrable_const _).indicator measurableSet_closedBall
have C : Contin... |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 41 | 41 | theorem rotate_nil (n : ℕ) : ([] : List α).rotate n = [] := by | simp [rotate]
|
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
... | Mathlib/NumberTheory/Pell.lean | 329 | 336 | theorem sign_y_zpow_eq_sign_of_x_pos_of_y_pos {a : Solution₁ d} (hax : 0 < a.x) (hay : 0 < a.y)
(n : ℤ) : (a ^ n).y.sign = n.sign := by |
rcases n with ((_ | n) | n)
· rfl
· rw [Int.ofNat_eq_coe, zpow_natCast]
exact Int.sign_eq_one_of_pos (y_pow_succ_pos hax hay n)
· rw [zpow_negSucc]
exact Int.sign_eq_neg_one_of_neg (neg_neg_of_pos (y_pow_succ_pos hax hay n))
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctio... | Mathlib/Analysis/MeanInequalities.lean | 712 | 722 | theorem Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : Summable fun i => f i ^ p) (hg_sum : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by |
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After leanprover/lean4#2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 663 | 670 | theorem Filter.HasBasis.equicontinuousAt_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → Set X}
{p₂ : κ₂ → Prop} {s₂ : κ₂ → Set (α × α)} {F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).HasBasis p₁ s₁)
(hα : (𝓤 α).HasBasis p₂ s₂) :
EquicontinuousAt F x₀ ↔
∀ k₂, p₂ k₂ → ∃ k₁, p₁ k₁ ∧ ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, ... |
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
hX.tendsto_iff (UniformFun.hasBasis_nhds_of_basis ι α _ hα)]
rfl
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 210 | 213 | theorem quasiSeparated_over_affine_iff {X Y : Scheme} (f : X ⟶ Y) [IsAffine Y] :
QuasiSeparated f ↔ QuasiSeparatedSpace X.carrier := by |
rw [quasiSeparated_eq_affineProperty,
QuasiSeparated.affineProperty_isLocal.affine_target_iff f, QuasiSeparated.affineProperty]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topolo... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 78 | 89 | theorem cauchySeq_range_of_norm_bounded {f : ℕ → E} (g : ℕ → ℝ)
(hg : CauchySeq fun n => ∑ i ∈ range n, g i) (hf : ∀ i, ‖f i‖ ≤ g i) :
CauchySeq fun n => ∑ i ∈ range n, f i := by |
refine Metric.cauchySeq_iff'.2 fun ε hε => ?_
refine (Metric.cauchySeq_iff'.1 hg ε hε).imp fun N hg n hn => ?_
specialize hg n hn
rw [dist_eq_norm, ← sum_Ico_eq_sub _ hn] at hg ⊢
calc
‖∑ k ∈ Ico N n, f k‖ ≤ ∑ k ∈ _, ‖f k‖ := norm_sum_le _ _
_ ≤ ∑ k ∈ _, g k := sum_le_sum fun x _ => hf x
_ ≤ ‖∑ k ... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 130 | 130 | theorem singleton_bind : bind {a} f = f a := by | simp [bind]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "le... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 397 | 431 | theorem isClosed_setOf_isCompactOperator {𝕜₁ 𝕜₂ : Type*} [NontriviallyNormedField 𝕜₁]
[NormedField 𝕜₂] {σ₁₂ : 𝕜₁ →+* 𝕜₂} {M₁ M₂ : Type*} [SeminormedAddCommGroup M₁]
[AddCommGroup M₂] [NormedSpace 𝕜₁ M₁] [Module 𝕜₂ M₂] [UniformSpace M₂] [UniformAddGroup M₂]
[ContinuousConstSMul 𝕜₂ M₂] [T2Space M₂] [... |
refine isClosed_of_closure_subset ?_
rintro u hu
rw [mem_closure_iff_nhds_zero] at hu
suffices TotallyBounded (u '' Metric.closedBall 0 1) by
change IsCompactOperator (u : M₁ →ₛₗ[σ₁₂] M₂)
rw [isCompactOperator_iff_isCompact_closure_image_closedBall (u : M₁ →ₛₗ[σ₁₂] M₂) zero_lt_one]
exact isCompact_... |
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology... | Mathlib/Topology/TietzeExtension.lean | 314 | 417 | theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
{e : X → Y} (he : ClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f := by |
inhabit X
-- Put `a = ⨅ x, f x` and `b = ⨆ x, f x`
obtain ⟨a, ha⟩ : ∃ a, IsGLB (range f) a := ⟨_, isGLB_ciInf f.isBounded_range.bddBelow⟩
obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b := ⟨_, isLUB_ciSup f.isBounded_range.bddAbove⟩
-- Then `f x ∈ [a, b]` for all `x`
have hmem : ∀ x, f x ∈ Icc a b := fun x => ⟨ha.... |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semicon... | Mathlib/Data/Int/GCD.lean | 401 | 403 | theorem lcm_comm (i j : ℤ) : lcm i j = lcm j i := by |
rw [Int.lcm, Int.lcm]
exact Nat.lcm_comm _ _
|
/-
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 | 155 | 157 | theorem tsub_le_tsub_add_tsub : a - c ≤ a - b + (b - c) := by |
rw [tsub_le_iff_left, ← add_assoc, add_right_comm]
exact le_add_tsub.trans (add_le_add_right le_add_tsub _)
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#... | Mathlib/Data/Complex/Basic.lean | 715 | 715 | theorem normSq_neg (z : ℂ) : normSq (-z) = normSq z := by | simp [normSq]
|
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 782 | 787 | theorem Measurable.iInf_Prop {α} [MeasurableSpace α] [ConditionallyCompleteLattice α]
(p : Prop) {f : δ → α} (hf : Measurable f) : Measurable fun b => ⨅ _ : p, f b := by |
simp_rw [ciInf_eq_ite]
split_ifs with h
· exact hf
· exact measurable_const
|
/-
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 | 335 | 336 | theorem preimage_const_sub_Ioc : (fun x => a - x) ⁻¹' Ioc b c = Ico (a - c) (a - b) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, inter_comm]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 142 | 146 | theorem perm_comp_perm : (Perm ∘r Perm : List α → List α → Prop) = Perm := by |
funext a c; apply propext
constructor
· exact fun ⟨b, hab, hba⟩ => Perm.trans hab hba
· exact fun h => ⟨a, Perm.refl a, h⟩
|
/-
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
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 750 | 752 | theorem Associated.of_mul_right [CancelCommMonoidWithZero α] {a b c d : α} :
a * b ~ᵤ c * d → b ~ᵤ d → b ≠ 0 → a ~ᵤ c := by |
rw [mul_comm a, mul_comm c]; exact Associated.of_mul_left
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Ma... | Mathlib/RingTheory/WittVector/Frobenius.lean | 218 | 220 | theorem coeff_frobeniusFun (x : 𝕎 R) (n : ℕ) :
coeff (frobeniusFun x) n = MvPolynomial.aeval x.coeff (frobeniusPoly p n) := by |
rw [frobeniusFun, coeff_mk]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,216 | 1,218 | theorem isIdealMorphism_of_surjective (h : Function.Surjective f) : f.IsIdealMorphism := by |
rw [isIdealMorphism_def, f.idealRange_eq_top_of_surjective h, f.range_eq_top.mpr h,
LieIdeal.top_coe_lieSubalgebra]
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Kalle Kytölä
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analys... | Mathlib/Analysis/LocallyConvex/Polar.lean | 106 | 109 | theorem polar_zero : B.polar ({0} : Set E) = Set.univ := by |
refine Set.eq_univ_iff_forall.mpr fun y x hx => ?_
rw [Set.mem_singleton_iff.mp hx, map_zero, LinearMap.zero_apply, norm_zero]
exact zero_le_one
|
/-
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.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 184 | 187 | theorem encard_lt_encard_iff_encard_diff_lt_encard_diff (h : (s ∩ t).Finite) :
s.encard < t.encard ↔ (s \ t).encard < (t \ s).encard := by |
rw [← encard_diff_add_encard_inter s t, ← encard_diff_add_encard_inter t s, inter_comm t s,
WithTop.add_lt_add_iff_right h.encard_lt_top.ne]
|
/-
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 | 96 | 99 | theorem HasSeparableContraction.eq_degree {f : F[X]} (hf : HasSeparableContraction 1 f) :
hf.degree = f.natDegree := by |
let ⟨a, ha⟩ := hf.dvd_degree'
rw [← ha, one_pow a, mul_one]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... | Mathlib/Algebra/Divisibility/Units.lean | 81 | 83 | theorem dvd : u ∣ a := by |
rcases hu with ⟨u, rfl⟩
apply Units.coe_dvd
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 590 | 594 | theorem snorm'_mono_measure (f : α → F) (hμν : ν ≤ μ) (hq : 0 ≤ q) :
snorm' f q ν ≤ snorm' f q μ := by |
simp_rw [snorm']
gcongr
exact lintegral_mono' hμν le_rfl
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 117 | 119 | theorem mem_maximals_iff_forall_lt_not_mem' (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt x y → y ∉ s := by |
simp [maximals, right_iff_left_not_left_of r rlt, not_imp_not, imp.swap (a := _ ∈ _)]
|
/-
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.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 225 | 226 | theorem tsub_encard_le_encard_diff (s t : Set α) : s.encard - t.encard ≤ (s \ t).encard := by |
rw [tsub_le_iff_left, add_comm]; apply encard_le_encard_diff_add_encard
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd0... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 307 | 310 | theorem mem_iff_lt_rowLen {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ ↔ j < μ.rowLen i := by |
rw [rowLen, Nat.lt_find_iff]
push_neg
exact ⟨fun h _ hmj => μ.up_left_mem (by rfl) hmj h, fun h => h _ (by rfl)⟩
|
/-
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.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,667 | 1,686 | theorem exists_prime_dvd_of_not_inf_one {a b : α} (ha : a ≠ 0) (hb : b ≠ 0)
(h : Associates.mk a ⊓ Associates.mk b ≠ 1) : ∃ p : α, Prime p ∧ p ∣ a ∧ p ∣ b := by |
have hz : factors (Associates.mk a) ⊓ factors (Associates.mk b) ≠ 0 := by
contrapose! h with hf
change (factors (Associates.mk a) ⊓ factors (Associates.mk b)).prod = 1
rw [hf]
exact Multiset.prod_zero
rw [factors_mk a ha, factors_mk b hb, ← WithTop.coe_inf] at hz
obtain ⟨⟨p0, p0_irr⟩, p0_mem⟩ := ... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 907 | 908 | theorem reindex_reindex_symm {m n : ℕ} (s : Simplex k P m) (e : Fin (m + 1) ≃ Fin (n + 1)) :
(s.reindex e).reindex e.symm = s := by | rw [← reindex_trans, Equiv.self_trans_symm, reindex_refl]
|
/-
Copyright (c) 2020 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 159 | 164 | theorem orderOf_sr (i : ZMod n) : orderOf (sr i) = 2 := by |
apply orderOf_eq_prime
· rw [sq, sr_mul_self]
· -- Porting note: Previous proof was `decide`
revert n
simp_rw [one_def, ne_eq, forall_const, not_false_eq_true]
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#... | Mathlib/Data/Complex/Basic.lean | 766 | 766 | theorem I_pow_four : I ^ 4 = 1 := by | rw [(by norm_num : 4 = 2 * 2), pow_mul, I_sq, neg_one_sq]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monoto... | Mathlib/Data/Nat/Factorial/Basic.lean | 408 | 411 | theorem descFactorial_eq_div {n k : ℕ} (h : k ≤ n) : n.descFactorial k = n ! / (n - k)! := by |
apply Nat.mul_left_cancel (n - k).factorial_pos
rw [factorial_mul_descFactorial h]
exact (Nat.mul_div_cancel' <| factorial_dvd_factorial <| Nat.sub_le n k).symm
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,452 | 1,455 | theorem _root_.List.aestronglyMeasurable_prod
(l : List (α → M)) (hl : ∀ f ∈ l, AEStronglyMeasurable f μ) :
AEStronglyMeasurable (fun x => (l.map fun f : α → M => f x).prod) μ := by |
simpa only [← Pi.list_prod_apply] using l.aestronglyMeasurable_prod' hl
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic fr... | Mathlib/Topology/MetricSpace/Basic.lean | 87 | 88 | theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by |
simpa only [not_le] using not_congr dist_le_zero
|
/-
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 | 1,496 | 1,501 | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ := by |
subst er
refine exp_approx_succ _ en _ _ ?_ h
field_simp [show (m : ℝ) ≠ 0 by norm_cast; omega]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Real
#align_import probability.process.filtration from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/Process/Filtration.lean | 281 | 305 | theorem filtrationOfSet_eq_natural [MulZeroOneClass β] [Nontrivial β] {s : ι → Set Ω}
(hsm : ∀ i, MeasurableSet[m] (s i)) :
filtrationOfSet hsm = natural (fun i => (s i).indicator (fun _ => 1 : Ω → β)) fun i =>
stronglyMeasurable_one.indicator (hsm i) := by |
simp only [filtrationOfSet, natural, measurableSpace_iSup_eq, exists_prop, mk.injEq]
ext1 i
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_)
· rintro _ ⟨j, hij, rfl⟩
refine measurableSet_generateFrom ⟨j, measurableSet_generateFrom ⟨hij, ?_⟩⟩
rw [comap_eq_generateFrom]
refine measurableS... |
/-
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 | 1,486 | 1,487 | theorem subtypeRestr_source : (e.subtypeRestr hs).source = (↑) ⁻¹' e.source := by |
simp only [subtypeRestr_def, mfld_simps]
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 239 | 241 | theorem mem_support_bindOnSupport_iff (b : β) :
b ∈ (p.bindOnSupport f).support ↔ ∃ (a : α) (h : a ∈ p.support), b ∈ (f a h).support := by |
simp only [support_bindOnSupport, Set.mem_setOf_eq, Set.mem_iUnion]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 451 | 470 | theorem mvPolynomial_aeval_of_surjective_of_closure [AddCommMonoid M] [CommSemiring R] {S : Set M}
(hS : closure S = ⊤) :
Function.Surjective
(MvPolynomial.aeval fun s : S => of' R M ↑s : MvPolynomial S R → R[M]) := by |
intro f
induction' f using induction_on with m f g ihf ihg r f ih
· have : m ∈ closure S := hS.symm ▸ mem_top _
refine AddSubmonoid.closure_induction this (fun m hm => ?_) ?_ ?_
· exact ⟨MvPolynomial.X ⟨m, hm⟩, MvPolynomial.aeval_X _ _⟩
· exact ⟨1, AlgHom.map_one _⟩
· rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, ... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Loca... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 465 | 467 | theorem isLocalization_of_eq_basicOpen {V : Opens X} (i : V ⟶ U) (e : V = X.basicOpen f) :
@IsLocalization.Away _ _ f (X.presheaf.obj (op V)) _ (X.presheaf.map i.op).toAlgebra := by |
subst e; convert isLocalization_basicOpen hU f using 3
|
/-
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 | 340 | 341 | theorem preimage_const_sub_Ioo : (fun x => a - x) ⁻¹' Ioo b c = Ioo (a - c) (a - b) := by |
simp [← Ioi_inter_Iio, inter_comm]
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.of_norm from "leanprover-communi... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 305 | 309 | theorem innerProp (r : 𝕜) : innerProp' E r := by |
intro x y
rw [← re_add_im r, add_smul, add_left, real_prop _ x, ← smul_smul, real_prop _ _ y, I_prop,
map_add, map_mul, conj_ofReal, conj_ofReal, conj_I]
ring
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sh... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 103 | 107 | theorem mem_grothendieckTopology :
Sieve.generate (presieveOfCovering U) ∈ Opens.grothendieckTopology X (iSup U) := by |
intro x hx
obtain ⟨i, hxi⟩ := Opens.mem_iSup.mp hx
exact ⟨U i, Opens.leSupr U i, ⟨U i, 𝟙 _, Opens.leSupr U i, ⟨i, rfl⟩, Category.id_comp _⟩, hxi⟩
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 363 | 364 | theorem apply_mem_support {x : α} : f x ∈ f.support ↔ x ∈ f.support := by |
rw [mem_support, mem_support, Ne, Ne, apply_eq_iff_eq]
|
/-
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
-/
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathl... | Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | 88 | 93 | theorem toBilin'Aux_toMatrixAux [DecidableEq n] (B₂ : BilinForm R₂ (n → R₂)) :
-- Porting note: had to hint the base ring even though it should be clear from context...
Matrix.toBilin'Aux (BilinForm.toMatrixAux (R₂ := R₂)
(fun j => stdBasis R₂ (fun _ => R₂) j 1) B₂) = B₂ := by |
rw [BilinForm.toMatrixAux, Matrix.toBilin'Aux,
toLinearMap₂'Aux_toMatrix₂Aux]
|
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