Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#ali... | Mathlib/SetTheory/Ordinal/Notation.lean | 780 | 781 | theorem repr_scale (x) [NF x] (o) [NF o] : repr (scale x o) = ω ^ repr x * repr o := by |
simp only [scale_eq_mul, repr_mul, repr, PNat.one_coe, Nat.cast_one, mul_one, add_zero]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 288 | 290 | theorem coeff_zero_X : coeff R 0 (X : R⟦X⟧) = 0 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [coeff, Finsupp.single_zero, X, MvPowerSeries.coeff_zero_X]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,315 | 1,315 | theorem mk_fin (n : ℕ) : #(Fin n) = n := by | simp
|
/-
Copyright (c) 2023 Mantas Bakšys, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Rearrangement
i... | Mathlib/Algebra/Order/Chebyshev.lean | 109 | 112 | theorem MonovaryOn.sum_mul_sum_le_card_mul_sum (hfg : MonovaryOn f g s) :
((∑ i ∈ s, f i) * ∑ i ∈ s, g i) ≤ s.card * ∑ i ∈ s, f i * g i := by |
rw [← nsmul_eq_mul]
exact hfg.sum_smul_sum_le_card_smul_sum
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.r... | Mathlib/Data/ENNReal/Operations.lean | 382 | 383 | theorem natCast_sub (m n : ℕ) : ↑(m - n) = (m - n : ℝ≥0∞) := by |
rw [← coe_natCast, Nat.cast_tsub, coe_sub, coe_natCast, coe_natCast]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 265 | 271 | theorem IsPrime.mem_of_pow_mem {I : Ideal α} (hI : I.IsPrime) {r : α} (n : ℕ) (H : r ^ n ∈ I) :
r ∈ I := by |
induction' n with n ih
· rw [pow_zero] at H
exact (mt (eq_top_iff_one _).2 hI.1).elim H
· rw [pow_succ] at H
exact Or.casesOn (hI.mem_or_mem H) ih id
|
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Logic.Function.Iterate
#align_import dynamics.flow from "leanprover-community/mathlib"@"717c073262cd9d59b1a1dcda7e8ab570... | Mathlib/Dynamics/Flow.lean | 178 | 181 | theorem continuous_toFun (t : τ) : Continuous (ϕ.toFun t) := by |
rw [← curry_uncurry ϕ.toFun]
apply continuous_curry
exact ϕ.cont'
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureThe... | Mathlib/MeasureTheory/Integral/Periodic.lean | 287 | 306 | theorem intervalIntegral_add_zsmul_eq (hf : Periodic f T) (n : ℤ) (t : ℝ)
(h_int : ∀ t₁ t₂, IntervalIntegrable f MeasureSpace.volume t₁ t₂) :
∫ x in t..t + n • T, f x = n • ∫ x in t..t + T, f x := by |
-- Reduce to the case `b = 0`
suffices (∫ x in (0)..(n • T), f x) = n • ∫ x in (0)..T, f x by
simp only [hf.intervalIntegral_add_eq t 0, (hf.zsmul n).intervalIntegral_add_eq t 0, zero_add,
this]
-- First prove it for natural numbers
have : ∀ m : ℕ, (∫ x in (0)..m • T, f x) = m • ∫ x in (0)..T, f x :=... |
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 195 | 197 | theorem finSuccEquiv'_last_apply_castSucc (i : Fin n) :
finSuccEquiv' (Fin.last n) (Fin.castSucc i) = i := by |
rw [← Fin.succAbove_last, finSuccEquiv'_succAbove]
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Antoine Labelle
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.Tens... | Mathlib/Algebra/Module/Projective.lean | 98 | 116 | theorem projective_lifting_property [h : Projective R P] (f : M →ₗ[R] N) (g : P →ₗ[R] N)
(hf : Function.Surjective f) : ∃ h : P →ₗ[R] M, f.comp h = g := by |
/-
Here's the first step of the proof.
Recall that `X →₀ R` is Lean's way of talking about the free `R`-module
on a type `X`. The universal property `Finsupp.total` says that to a map
`X → N` from a type to an `R`-module, we get an associated R-module map
`(X →₀ R) →ₗ N`. Apply this to a (noncomp... |
/-
Copyright (c) 2023 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Etienne Marion
-/
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.StoneCech
import Mathlib.Topology.Filter
import Mathlib.Order.Filter.Cofinite
/-!
# P... | Mathlib/Topology/ProperMap.lean | 313 | 329 | theorem isProperMap_iff_isCompact_preimage [T2Space Y] [WeaklyLocallyCompactSpace Y] :
IsProperMap f ↔ Continuous f ∧ ∀ ⦃K⦄, IsCompact K → IsCompact (f ⁻¹' K) := by |
constructor <;> intro H
-- The direct implication follows from the previous results
· exact ⟨H.continuous, fun K hK ↦ H.isCompact_preimage hK⟩
· rw [isProperMap_iff_ultrafilter_of_t2]
-- Let `𝒰 : Ultrafilter X`, and assume that `f` tends to some `y` along `𝒰`.
refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩
-- Pi... |
/-
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.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic
#align_import algebra.algebraic_card from "leanprover-communit... | Mathlib/Algebra/AlgebraicCard.lean | 93 | 95 | theorem cardinal_mk_le_mul : #{ x : A // IsAlgebraic R x } ≤ #R[X] * ℵ₀ := by |
rw [← lift_id #_, ← lift_id #R[X]]
exact cardinal_mk_lift_le_mul R A
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 555 | 561 | theorem Int.ModEq.pow_card_sub_one_eq_one {p : ℕ} (hp : Nat.Prime p) {n : ℤ} (hpn : IsCoprime n p) :
n ^ (p - 1) ≡ 1 [ZMOD p] := by |
haveI : Fact p.Prime := ⟨hp⟩
have : ¬(n : ZMod p) = 0 := by
rw [CharP.intCast_eq_zero_iff _ p, ← (Nat.prime_iff_prime_int.mp hp).coprime_iff_not_dvd]
· exact hpn.symm
simpa [← ZMod.intCast_eq_intCast_iff] using ZMod.pow_card_sub_one_eq_one this
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptoti... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 124 | 134 | theorem isEquivalent_atTop_div :
(fun x => eval x P / eval x Q) ~[atTop] fun x =>
P.leadingCoeff / Q.leadingCoeff * x ^ (P.natDegree - Q.natDegree : ℤ) := by |
by_cases hP : P = 0
· simp [hP, IsEquivalent.refl]
by_cases hQ : Q = 0
· simp [hQ, IsEquivalent.refl]
refine
(P.isEquivalent_atTop_lead.symm.div Q.isEquivalent_atTop_lead.symm).symm.trans
(EventuallyEq.isEquivalent ((eventually_gt_atTop 0).mono fun x hx => ?_))
simp [← div_mul_div_comm, hP, hQ, z... |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.Algebra.Homology.ImageToKernel
#align_import algebra.homology.exact from "leanprover-community/mathlib"@"3feb151caefe53df080ca6ca67a0c6685cfd1b82"
/-!
... | Mathlib/Algebra/Homology/Exact.lean | 211 | 216 | theorem exact_comp_mono_iff [Mono h] : Exact f (g ≫ h) ↔ Exact f g := by |
refine
⟨fun hfg => ⟨zero_of_comp_mono h (by rw [Category.assoc, hfg.1]), ?_⟩, fun h =>
exact_comp_mono h⟩
rw [← (Iso.eq_comp_inv _).1 (imageToKernel_comp_mono _ _ h hfg.1)]
haveI := hfg.2; infer_instance
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,348 | 1,352 | theorem setToFun_zero_left' (hT : DominatedFinMeasAdditive μ T C)
(h_zero : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0) : setToFun μ T hT f = 0 := by |
by_cases hf : Integrable f μ
· rw [setToFun_eq hT hf]; exact L1.setToL1_zero_left' hT h_zero _
· exact setToFun_undef hT hf
|
/-
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 | 374 | 375 | theorem smul_tmul_smul (r s : R) (m : M) (n : N) : (r • m) ⊗ₜ[R] (s • n) = (r * s) • m ⊗ₜ[R] n := by |
simp_rw [smul_tmul, tmul_smul, mul_smul]
|
/-
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 | 898 | 898 | theorem div_pos {b c : Ordinal} (h : c ≠ 0) : 0 < b / c ↔ c ≤ b := by | simp [lt_div h]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Real sign fu... | Mathlib/Data/Real/Sign.lean | 92 | 98 | theorem sign_mul_nonneg (r : ℝ) : 0 ≤ sign r * r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn]
exact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le
· rw [mul_zero]
· rw [sign_of_pos hp, one_mul]
exact hp.le
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
/-!
# N-ary images of s... | Mathlib/Data/Set/NAry.lean | 177 | 177 | theorem image2_singleton : image2 f {a} {b} = {f a b} := by | simp
|
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
imp... | Mathlib/Algebra/Module/Defs.lean | 365 | 368 | theorem nsmul_eq_smul_cast (n : ℕ) (b : M) : n • b = (n : R) • b := by |
induction' n with n ih
· rw [Nat.cast_zero, zero_smul, zero_smul]
· rw [Nat.cast_succ, add_smul, add_smul, one_smul, ih, one_smul]
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 109 | 113 | theorem traverse_comp (g : α → F β) (h : β → G γ) :
traverse (Comp.mk ∘ map h ∘ g) =
(Comp.mk ∘ map (traverse h) ∘ traverse g : t α → Comp F G (t γ)) := by |
ext
exact comp_traverse _ _ _
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomi... | Mathlib/Algebra/Polynomial/Taylor.lean | 98 | 102 | theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p := by |
refine map_natDegree_eq_natDegree _ ?_
nontriviality R
intro n c c0
simp [taylor_monomial, natDegree_C_mul_eq_of_mul_ne_zero, natDegree_pow_X_add_C, c0]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 350 | 355 | theorem sub_zero {X Y : C} (a : X ⟶ Y) : a - 0 = a := by |
rw [sub_def]
conv_lhs =>
congr; congr; rw [← Category.comp_id a]
case a.g => rw [show 0 = a ≫ (0 : Y ⟶ Y) by simp]
rw [← prod.comp_lift, Category.assoc, lift_σ, Category.comp_id]
|
/-
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 | 760 | 768 | theorem prod_primeFactors_gcd_mul_prod_primeFactors_mul {β : Type*} [CommMonoid β] (m n : ℕ)
(f : ℕ → β) :
(m.gcd n).primeFactors.prod f * (m * n).primeFactors.prod f =
m.primeFactors.prod f * n.primeFactors.prod f := by |
obtain rfl | hm₀ := eq_or_ne m 0
· simp
obtain rfl | hn₀ := eq_or_ne n 0
· simp
· rw [primeFactors_mul hm₀ hn₀, primeFactors_gcd hm₀ hn₀, mul_comm, Finset.prod_union_inter]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 425 | 428 | theorem whiskerLeft_dite {P : Prop} [Decidable P]
(X : C) {Y Z : C} (f : P → (Y ⟶ Z)) (f' : ¬P → (Y ⟶ Z)) :
X ◁ (if h : P then f h else f' h) = if h : P then X ◁ f h else X ◁ f' h := by |
split_ifs <;> rfl
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 621 | 624 | theorem lintegral_finset_sum_measure {ι} {m : MeasurableSpace α} (s : Finset ι) (f : α → ℝ≥0∞)
(μ : ι → Measure α) : ∫⁻ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫⁻ a, f a ∂μ i := by |
rw [← Measure.sum_coe_finset, lintegral_sum_measure, ← Finset.tsum_subtype']
simp only [Finset.coe_sort_coe]
|
/-
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.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 352 | 376 | theorem affine_openCover_TFAE {X Y : Scheme.{u}} [IsAffine Y] (f : X ⟶ Y) :
List.TFAE
[sourceAffineLocally (@P) f,
∃ (𝒰 : Scheme.OpenCover.{u} X) (_ : ∀ i, IsAffine (𝒰.obj i)),
∀ i : 𝒰.J, P (Scheme.Γ.map (𝒰.map i ≫ f).op),
∀ (𝒰 : Scheme.OpenCover.{u} X) [∀ i, IsAffine (𝒰.obj i)... |
tfae_have 1 → 4
· intro H U g _ hg
specialize H ⟨⟨_, hg.base_open.isOpen_range⟩, rangeIsAffineOpenOfOpenImmersion g⟩
rw [← hP.respectsIso.cancel_right_isIso _ (Scheme.Γ.map (IsOpenImmersion.isoOfRangeEq g
(X.ofRestrict (Opens.openEmbedding ⟨_, hg.base_open.isOpen_range⟩))
Subtype.range_coe.symm... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#a... | Mathlib/Data/Real/Archimedean.lean | 332 | 353 | theorem iInf_Ioi_eq_iInf_rat_gt {f : ℝ → ℝ} (x : ℝ) (hf : BddBelow (f '' Ioi x))
(hf_mono : Monotone f) : ⨅ r : Ioi x, f r = ⨅ q : { q' : ℚ // x < q' }, f q := by |
refine le_antisymm ?_ ?_
· have : Nonempty { r' : ℚ // x < ↑r' } := by
obtain ⟨r, hrx⟩ := exists_rat_gt x
exact ⟨⟨r, hrx⟩⟩
refine le_ciInf fun r => ?_
obtain ⟨y, hxy, hyr⟩ := exists_rat_btwn r.prop
refine ciInf_set_le hf (hxy.trans ?_)
exact_mod_cast hyr
· refine le_ciInf fun q => ?_
... |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
/-!
# N-ary images of s... | Mathlib/Data/Set/NAry.lean | 374 | 379 | theorem image2_inter_union_subset_union :
image2 f (s ∩ s') (t ∪ t') ⊆ image2 f s t ∪ image2 f s' t' := by |
rw [image2_union_right]
exact
union_subset_union (image2_subset_right inter_subset_left)
(image2_subset_right inter_subset_right)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 609 | 619 | theorem single_mul_apply_of_not_exists_mul [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G)
(h : ¬∃ d, g' = g * d) : (single g r * x) g' = 0 := by |
classical
rw [mul_apply, Finsupp.sum_single_index]
swap
· simp_rw [Finsupp.sum, zero_mul, ite_self, Finset.sum_const_zero]
· apply Finset.sum_eq_zero
simp_rw [ite_eq_right_iff]
rintro g'' _hg'' rfl
exfalso
exact h ⟨_, rfl⟩
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import... | Mathlib/NumberTheory/DiophantineApproximation.lean | 232 | 258 | theorem den_le_and_le_num_le_of_sub_lt_one_div_den_sq {ξ q : ℚ}
(h : |ξ - q| < 1 / (q.den : ℚ) ^ 2) :
q.den ≤ ξ.den ∧ ⌈ξ * q.den⌉ - 1 ≤ q.num ∧ q.num ≤ ⌊ξ * q.den⌋ + 1 := by |
have hq₀ : (0 : ℚ) < q.den := Nat.cast_pos.mpr q.pos
replace h : |ξ * q.den - q.num| < 1 / q.den := by
rw [← mul_lt_mul_right hq₀] at h
conv_lhs at h => rw [← abs_of_pos hq₀, ← abs_mul, sub_mul, mul_den_eq_num]
rwa [sq, div_mul, mul_div_cancel_left₀ _ hq₀.ne'] at h
constructor
· rcases eq_or_ne ξ q... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 334 | 364 | theorem Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card
{t : Finset M} (h : finrank R M + 1 < t.card) :
∃ f : M → R, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by |
-- Pick an element x₀ ∈ t,
obtain ⟨x₀, x₀_mem⟩ := card_pos.1 ((Nat.succ_pos _).trans h)
-- and apply the previous lemma to the {xᵢ - x₀}
let shift : M ↪ M := ⟨(· - x₀), sub_left_injective⟩
classical
let t' := (t.erase x₀).map shift
have h' : finrank R M < t'.card := by
rw [card_map, card_erase_of_mem... |
/-
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 | 235 | 237 | theorem map_exp_comap_re_atTop : map exp (comap re atTop) = cobounded ℂ := by |
rw [← comap_exp_cobounded, map_comap, range_exp, inf_eq_left, le_principal_iff]
exact eventually_ne_cobounded _
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 412 | 415 | theorem X_pow_eq (s : σ) (n : ℕ) : (X s : MvPowerSeries σ R) ^ n = monomial R (single s n) 1 := by |
induction' n with n ih
· simp
· rw [pow_succ, ih, Finsupp.single_add, X, monomial_mul_monomial, one_mul]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
/-!
# D... | Mathlib/SetTheory/Game/Domineering.lean | 93 | 98 | theorem card_of_mem_left {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) : 2 ≤ Finset.card b := by |
have w₁ : m ∈ b := (Finset.mem_inter.1 h).1
have w₂ : (m.1, m.2 - 1) ∈ b.erase m := snd_pred_mem_erase_of_mem_left h
have i₁ := Finset.card_erase_lt_of_mem w₁
have i₂ := Nat.lt_of_le_of_lt (Nat.zero_le _) (Finset.card_erase_lt_of_mem w₂)
exact Nat.lt_of_le_of_lt i₂ i₁
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 693 | 693 | theorem le_add_left (s t : Multiset α) : s ≤ t + s := by | simpa using add_le_add_right (zero_le t) s
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Yury Kudryashov
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Na... | Mathlib/Combinatorics/Pigeonhole.lean | 134 | 141 | theorem exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum
(ht : ∀ y ∉ t, ∑ x ∈ s.filter fun x => f x = y, w x ≤ 0)
(hb : t.card • b < ∑ x ∈ s, w x) : ∃ y ∈ t, b < ∑ x ∈ s.filter fun x => f x = y, w x :=
exists_lt_of_sum_lt <|
calc
∑ _y ∈ t, b < ∑ x ∈ s, w x := by | simpa
_ ≤ ∑ y ∈ t, ∑ x ∈ s.filter fun x => f x = y, w x :=
sum_le_sum_fiberwise_of_sum_fiber_nonpos ht
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-commun... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 68 | 72 | theorem integral_circleTransform (f : ℂ → E) :
(∫ θ : ℝ in (0)..2 * π, circleTransform R z w f θ) =
(2 * ↑π * I)⁻¹ • ∮ z in C(z, R), (z - w)⁻¹ • f z := by |
simp_rw [circleTransform, circleIntegral, deriv_circleMap, circleMap]
simp
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprove... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 45 | 46 | theorem terminatedAt_iff_part_num_none : g.TerminatedAt n ↔ g.partialNumerators.get? n = none := by |
rw [terminatedAt_iff_s_none, part_num_none_iff_s_none]
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
#alig... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 212 | 216 | theorem frobeniusRotation_nonzero {a₁ a₂ : 𝕎 k} (ha₁ : a₁.coeff 0 ≠ 0) (ha₂ : a₂.coeff 0 ≠ 0) :
frobeniusRotation p ha₁ ha₂ ≠ 0 := by |
intro h
apply solution_nonzero p ha₁ ha₂
simpa [← h, frobeniusRotation, frobeniusRotationCoeff] using WittVector.zero_coeff p k 0
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 94 | 96 | theorem val_unit' {n : ZMod 0} : IsUnit n ↔ n.val = 1 := by |
simp only [val]
rw [Int.isUnit_iff, Int.natAbs_eq_iff, Nat.cast_one]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 382 | 384 | theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by |
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 144 | 145 | theorem Sized.pos {s l x r} (h : Sized (@node α s l x r)) : 0 < s := by |
rw [h.1]; apply Nat.le_add_left
|
/-
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, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Gr... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 149 | 156 | theorem measure_ne_zero [MeasurableSpace G] [Countable G] [MeasurableSMul G α]
[SMulInvariantMeasure G α μ] (hμ : μ ≠ 0) (h : IsFundamentalDomain G s μ) :
μ s ≠ 0 := by |
have hc := measure_univ_pos.mpr hμ
contrapose! hc
rw [← measure_congr h.iUnion_smul_ae_eq]
refine le_trans (measure_iUnion_le _) ?_
simp_rw [measure_smul, hc, tsum_zero, le_refl]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
/-!
# Measure with a given density with respect to another measure
For a measure `μ` on `α` and a fun... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 150 | 154 | theorem withDensity_absolutelyContinuous {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥0∞) :
μ.withDensity f ≪ μ := by |
refine AbsolutelyContinuous.mk fun s hs₁ hs₂ => ?_
rw [withDensity_apply _ hs₁]
exact set_lintegral_measure_zero _ _ hs₂
|
/-
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.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 474 | 480 | theorem hasFDerivWithinAt_apply (i : ι) (f : ∀ i, F' i) (s' : Set (∀ i, F' i)) :
HasFDerivWithinAt (𝕜:=𝕜) (fun f : ∀ i, F' i => f i) (proj i) s' f := by |
let id' := ContinuousLinearMap.id 𝕜 (∀ i, F' i)
have h := ((hasFDerivWithinAt_pi'
(Φ := fun (f : ∀ i, F' i) (i' : ι) => f i') (Φ':=id') (x:=f) (s:=s'))).1
have h' : comp (proj i) id' = proj i := by rfl
rw [← h']; apply h; apply hasFDerivWithinAt_id
|
/-
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 | 1,359 | 1,360 | theorem indexOf_get? [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) :
get? l (indexOf a l) = some a := by | rw [get?_eq_get, indexOf_get (indexOf_lt_length.2 h)]
|
/-
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 | 256 | 260 | theorem equicontinuousWithinAt_finite [Finite ι] {F : ι → X → α} {S : Set X} {x₀ : X} :
EquicontinuousWithinAt F S x₀ ↔ ∀ i, ContinuousWithinAt (F i) S x₀ := by |
simp [EquicontinuousWithinAt, ContinuousWithinAt,
(nhds_basis_uniformity' (𝓤 α).basis_sets).tendsto_right_iff, UniformSpace.ball,
@forall_swap _ ι]
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.RingTheory.Valuation.Integers
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localizat... | Mathlib/RingTheory/Valuation/ValuationRing.lean | 312 | 333 | theorem iff_isInteger_or_isInteger :
ValuationRing R ↔ ∀ x : K, IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹ := by |
constructor
· intro H x
obtain ⟨x : R, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := R) x
have := (map_ne_zero_iff _ (IsFractionRing.injective R K)).mpr (nonZeroDivisors.ne_zero hy)
obtain ⟨s, rfl | rfl⟩ := ValuationRing.cond x y
· exact Or.inr
⟨s, eq_inv_of_mul_eq_one_left <| by rwa [... |
/-
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.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 204 | 213 | theorem Subpresheaf.eq_sheafify (h : Presieve.IsSheaf J F) (hG : Presieve.IsSheaf J G.toPresheaf) :
G = G.sheafify J := by |
apply (G.le_sheafify J).antisymm
intro U s hs
suffices ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).1 = s by
rw [← this]
exact ((hG _ hs).amalgamate _ (G.family_of_elements_compatible s)).2
apply (h _ hs).isSeparatedFor.ext
intro V i hi
exact (congr_arg Subtype.val ((hG _ hs).valid_... |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne... | Mathlib/Tactic/Linarith/Lemmas.lean | 39 | 40 | theorem lt_of_lt_of_eq {α} [OrderedSemiring α] {a b : α} (ha : a < 0) (hb : b = 0) : a + b < 0 := by |
simp [*]
|
/-
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.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 506 | 510 | theorem tendsto_const_mul_rpow_nhds_zero_of_pos {c : ℝ≥0∞} (hc : c ≠ ∞) {y : ℝ} (hy : 0 < y) :
Tendsto (fun x : ℝ≥0∞ => c * x ^ y) (𝓝 0) (𝓝 0) := by |
convert ENNReal.Tendsto.const_mul (ENNReal.continuous_rpow_const.tendsto 0) _
· simp [hy]
· exact Or.inr hc
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 82 | 84 | theorem star_eigenvectorUnitary_mulVec (j : n) :
(star (eigenvectorUnitary hA : Matrix n n 𝕜)) *ᵥ ⇑(hA.eigenvectorBasis j) = Pi.single j 1 := by |
rw [← eigenvectorUnitary_mulVec, mulVec_mulVec, unitary.coe_star_mul_self, one_mulVec]
|
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
/-!
# Hada... | Mathlib/Data/Matrix/Hadamard.lean | 121 | 123 | theorem one_hadamard : (1 : Matrix n n α) ⊙ M = diagonal fun i => M i i := by |
ext i j
by_cases h : i = j <;> simp [h]
|
/-
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.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/... | Mathlib/Algebra/Lie/Solvable.lean | 136 | 138 | theorem abelian_iff_derived_succ_eq_bot (I : LieIdeal R L) (k : ℕ) :
IsLieAbelian (derivedSeriesOfIdeal R L k I) ↔ derivedSeriesOfIdeal R L (k + 1) I = ⊥ := by |
rw [add_comm, derivedSeriesOfIdeal_add I 1 k, abelian_iff_derived_one_eq_bot]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 261 | 262 | theorem le_mul_inv_iff_le : 1 ≤ a * b⁻¹ ↔ b ≤ a := by |
rw [← mul_le_mul_iff_right b, one_mul, inv_mul_cancel_right]
|
/-
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 | 345 | 349 | theorem Finite.eq_insert_of_subset_of_encard_eq_succ (hs : s.Finite) (h : s ⊆ t)
(hst : t.encard = s.encard + 1) : ∃ a, t = insert a s := by |
rw [← encard_diff_add_encard_of_subset h, add_comm,
WithTop.add_left_cancel_iff hs.encard_lt_top.ne, encard_eq_one] at hst
obtain ⟨x, hx⟩ := hst; use x; rw [← diff_union_of_subset h, hx, singleton_union]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
import Mathlib.CategoryTheory.MorphismProperty.Compositio... | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | 95 | 112 | theorem StableUnderBaseChange.pullback_map [HasPullbacks C] {P : MorphismProperty C}
(hP : StableUnderBaseChange P) [P.IsStableUnderComposition] {S X X' Y Y' : C} {f : X ⟶ S}
{g : Y ⟶ S} {f' : X' ⟶ S} {g' : Y' ⟶ S} {i₁ : X ⟶ X'} {i₂ : Y ⟶ Y'} (h₁ : P i₁) (h₂ : P i₂)
(e₁ : f = i₁ ≫ f') (e₂ : g = i₂ ≫ g') :
... |
have :
pullback.map f g f' g' i₁ i₂ (𝟙 _) ((Category.comp_id _).trans e₁)
((Category.comp_id _).trans e₂) =
((pullbackSymmetry _ _).hom ≫
((Over.baseChange _).map (Over.homMk _ e₂.symm : Over.mk g ⟶ Over.mk g')).left) ≫
(pullbackSymmetry _ _).hom ≫
((Over.baseChange g')... |
/-
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 | 742 | 744 | theorem closedBall_sup (p : Seminorm 𝕜 E) (q : Seminorm 𝕜 E) (e : E) (r : ℝ) :
closedBall (p ⊔ q) e r = closedBall p e r ∩ closedBall q e r := by |
simp_rw [closedBall, ← Set.setOf_and, coe_sup, Pi.sup_apply, sup_le_iff]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
#align_import category_theory.natural... | Mathlib/CategoryTheory/NatIso.lean | 172 | 173 | theorem naturality_2 (α : F ≅ G) (f : X ⟶ Y) : α.hom.app X ≫ G.map f ≫ α.inv.app Y = F.map f := by |
simp
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 76 | 79 | theorem conj_eigenvalue_eq_self {μ : 𝕜} (hμ : HasEigenvalue T μ) : conj μ = μ := by |
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_hasEigenvector
rw [mem_eigenspace_iff] at hv₁
simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
|
/-
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 | 771 | 773 | theorem restr_source_inter (s : Set X) : e.restr (e.source ∩ s) = e.restr s := by |
refine PartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_
simp [e.open_source.interior_eq, ← inter_assoc]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Compl... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 139 | 141 | theorem neg (hf : IsBoundedLinearMap 𝕜 f) : IsBoundedLinearMap 𝕜 fun e => -f e := by |
rw [show (fun e => -f e) = fun e => (-1 : 𝕜) • f e by funext; simp]
exact smul (-1) hf
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa... | Mathlib/Data/Matrix/PEquiv.lean | 109 | 114 | theorem toMatrix_trans [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring α] (f : l ≃. m)
(g : m ≃. n) : ((f.trans g).toMatrix : Matrix l n α) = f.toMatrix * g.toMatrix := by |
ext i j
rw [mul_matrix_apply]
dsimp [toMatrix, PEquiv.trans]
cases f i <;> simp
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 532 | 536 | theorem finprod_mem_eq_one_of_infinite {f : α → M} {s : Set α} (hs : (s ∩ mulSupport f).Infinite) :
∏ᶠ i ∈ s, f i = 1 := by |
rw [finprod_mem_def]
apply finprod_of_infinite_mulSupport
rwa [← mulSupport_mulIndicator] at hs
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 736 | 741 | theorem eq_zero_of_constant_mem_of_maximal (hR : IsField R) (I : Ideal R[X]) [hI : I.IsMaximal]
(x : R) (hx : C x ∈ I) : x = 0 := by |
refine Classical.by_contradiction fun hx0 => hI.ne_top ((eq_top_iff_one I).2 ?_)
obtain ⟨y, hy⟩ := hR.mul_inv_cancel hx0
convert I.mul_mem_left (C y) hx
rw [← C.map_mul, hR.mul_comm y x, hy, RingHom.map_one]
|
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 246 | 251 | theorem zeroRootSubalgebra_eq_of_is_cartan (H : LieSubalgebra R L) [H.IsCartanSubalgebra]
[IsNoetherian R L] : zeroRootSubalgebra R L H = H := by |
refine le_antisymm ?_ (le_zeroRootSubalgebra R L H)
suffices rootSpace H 0 ≤ H.toLieSubmodule by exact fun x hx => this hx
obtain ⟨k, hk⟩ := (rootSpace H 0).isNilpotent_iff_exists_self_le_ucs.mp (by infer_instance)
exact hk.trans (LieSubmodule.ucs_le_of_normalizer_eq_self (by simp) k)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Gluing
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
#align_import algebraic... | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | 156 | 179 | theorem f_invApp_f_app (i j k : D.J) (U : Opens (D.V (i, j)).carrier) :
(D.f_open i j).invApp U ≫ (D.f i k).c.app _ =
(π₁ i, j, k).c.app (op U) ≫
(π₂⁻¹ i, j, k) (unop _) ≫
(D.V _).presheaf.map
(eqToHom
(by
delta IsOpenImmersion.openFunctor
... |
have := PresheafedSpace.congr_app (@pullback.condition _ _ _ _ _ (D.f i j) (D.f i k) _)
dsimp only [comp_c_app] at this
rw [← cancel_epi (inv ((D.f_open i j).invApp U)), IsIso.inv_hom_id_assoc,
IsOpenImmersion.inv_invApp]
simp_rw [Category.assoc]
erw [(π₁ i, j, k).c.naturality_assoc, reassoc_of% this, ← ... |
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functorial constructions
Lemmas on the interaction... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 284 | 289 | theorem tsum_op [T2Space α] :
∑' x, op (f x) = op (∑' x, f x) := by |
by_cases h : Summable f
· exact h.hasSum.op.tsum_eq
· have ho := summable_op.not.mpr h
rw [tsum_eq_zero_of_not_summable h, tsum_eq_zero_of_not_summable ho, op_zero]
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Init.Order.Defs
set_option autoImplicit true
structure UFModel (n) where
parent : Fin n → Fin n
rank : Nat ... | Mathlib/Data/UnionFind.lean | 82 | 84 | theorem get_eq {arr : Array α} {n} {m : Fin n → β} (H : Agrees arr f m) :
∀ i h₁ h₂, f (arr.get ⟨i, h₁⟩) = m ⟨i, h₂⟩ := by |
cases H; exact fun i h _ ↦ rfl
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 480 | 489 | theorem isPurelyInseparable_of_finSepDegree_eq_one [Algebra.IsAlgebraic F E]
(hdeg : finSepDegree F E = 1) : IsPurelyInseparable F E := by |
rw [isPurelyInseparable_iff]
refine fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, fun hsep ↦ ?_⟩
have : Algebra.IsAlgebraic F⟮x⟯ E := Algebra.IsAlgebraic.tower_top (K := F) F⟮x⟯
have := finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E
rw [hdeg, mul_eq_one, (finSepDegree_adjoin_simple_eq_finrank_iff F E x
... |
/-
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.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 870 | 871 | theorem tendsto_const_nhds_iff [T1Space X] {l : Filter Y} [NeBot l] {c d : X} :
Tendsto (fun _ => c) l (𝓝 d) ↔ c = d := by | simp_rw [Tendsto, Filter.map_const, pure_le_nhds_iff]
|
/-
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.ContMDiff.Defs
/-!
## Basic properties of smooth functions between manifolds
In this file, we show that sta... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 332 | 338 | theorem contMDiff_of_mulTSupport [One M'] {f : M → M'}
(hf : ∀ x ∈ mulTSupport f, ContMDiffAt I I' n f x) : ContMDiff I I' n f := by |
intro x
by_cases hx : x ∈ mulTSupport f
· exact hf x hx
· exact ContMDiffAt.congr_of_eventuallyEq contMDiffAt_const
(not_mem_mulTSupport_iff_eventuallyEq.1 hx)
|
/-
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 | 713 | 714 | theorem inv_le_of_neg (ha : a < 0) (hb : b < 0) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := by |
rw [← inv_le_inv_of_neg hb (inv_lt_zero.2 ha), inv_inv]
|
/-
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 | 151 | 157 | theorem mkSpanSingleton'_apply (x : E) (y : F) (H : ∀ c : R, c • x = 0 → c • y = 0) (c : R) (h) :
mkSpanSingleton' x y H ⟨c • x, h⟩ = c • y := by |
dsimp [mkSpanSingleton']
rw [← sub_eq_zero, ← sub_smul]
apply H
simp only [sub_smul, one_smul, sub_eq_zero]
apply Classical.choose_spec (mem_span_singleton.1 h)
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 165 | 165 | theorem refl_range {a : X} : range (Path.refl a) = {a} := by | simp [Path.refl, CoeFun.coe]
|
/-
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 | 40 | 47 | theorem revAtFun_invol {N i : ℕ} : revAtFun N (revAtFun N i) = i := by |
unfold revAtFun
split_ifs with h j
· exact tsub_tsub_cancel_of_le h
· exfalso
apply j
exact Nat.sub_le N i
· rfl
|
/-
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.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,586 | 1,623 | theorem ae_tendsto_of_cauchy_snorm' [CompleteSpace E] {f : ℕ → α → E} {p : ℝ}
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
(h_cau : ∀ N n m : ℕ, N ≤ n → N ≤ m → snorm' (f n - f m) p μ < B N) :
∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by |
have h_summable : ∀ᵐ x ∂μ, Summable fun i : ℕ => f (i + 1) x - f i x := by
have h1 :
∀ n, snorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' i, B i :=
snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' hf hp1 h_cau
have h2 :
∀ n,
(∫⁻ a, (∑ i ∈ Finset.range (n + 1... |
/-
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 | 493 | 496 | theorem val_add_eq_ite {n : ℕ} (a b : Fin n) :
(↑(a + b) : ℕ) = if n ≤ a + b then a + b - n else a + b := by |
rw [Fin.val_add, Nat.add_mod_eq_ite, Nat.mod_eq_of_lt (show ↑a < n from a.2),
Nat.mod_eq_of_lt (show ↑b < n from b.2)]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 526 | 528 | theorem edgeSet_inf : (G₁ ⊓ G₂).edgeSet = G₁.edgeSet ∩ G₂.edgeSet := by |
ext ⟨x, y⟩
rfl
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeMod... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 167 | 170 | theorem cardinal_mk_le_finrank [Module.Finite R M]
{ι : Type w} {b : ι → M} (h : LinearIndependent R b) : #ι ≤ finrank R M := by |
rw [← lift_le.{max v w}]
simpa only [← finrank_eq_rank, lift_natCast, lift_le_nat_iff] using h.cardinal_lift_le_rank
|
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6... | Mathlib/Algebra/ModEq.lean | 320 | 322 | theorem intCast_modEq_intCast {a b z : ℤ} : a ≡ b [PMOD (z : α)] ↔ a ≡ b [PMOD z] := by |
simp_rw [ModEq, ← Int.cast_mul_eq_zsmul_cast]
norm_cast
|
/-
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 | 461 | 463 | theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by |
rw [two_mul, sinh_add]
ring
|
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 505 | 515 | theorem eq_ones_iff {c : Composition n} : c = ones n ↔ ∀ i ∈ c.blocks, i = 1 := by |
constructor
· rintro rfl
exact fun i => eq_of_mem_replicate
· intro H
ext1
have A : c.blocks = replicate c.blocks.length 1 := eq_replicate_of_mem H
have : c.blocks.length = n := by
conv_rhs => rw [← c.blocks_sum, A]
simp
rw [A, this, ones_blocks]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | Mathlib/Algebra/Polynomial/Laurent.lean | 352 | 368 | theorem trunc_C_mul_T (n : ℤ) (r : R) : trunc (C r * T n) = ite (0 ≤ n) (monomial n.toNat r) 0 := by |
apply (toFinsuppIso R).injective
rw [← single_eq_C_mul_T, trunc, AddMonoidHom.coe_comp, Function.comp_apply]
-- Porting note (#10691): was `rw`
erw [comapDomain.addMonoidHom_apply Int.ofNat_injective]
rw [toFinsuppIso_apply]
-- Porting note: rewrote proof below relative to mathlib3.
by_cases n0 : 0 ≤ n
... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,648 | 1,657 | theorem set_lintegral_lt_top_of_bddAbove {s : Set α} (hs : μ s ≠ ∞) {f : α → ℝ≥0}
(hf : Measurable f) (hbdd : BddAbove (f '' s)) : ∫⁻ x in s, f x ∂μ < ∞ := by |
obtain ⟨M, hM⟩ := hbdd
rw [mem_upperBounds] at hM
refine
lt_of_le_of_lt (set_lintegral_mono hf.coe_nnreal_ennreal (@measurable_const _ _ _ _ ↑M) ?_) ?_
· simpa using hM
· rw [lintegral_const]
refine ENNReal.mul_lt_top ENNReal.coe_lt_top.ne ?_
simp [hs]
|
/-
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
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,707 | 1,709 | theorem uniformContinuous₂_curry (f : α × β → γ) :
UniformContinuous₂ (Function.curry f) ↔ UniformContinuous f := by |
rw [UniformContinuous₂, uncurry_curry]
|
/-
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 | 88 | 95 | theorem dart_edge_fiber_card [DecidableEq V] (e : Sym2 V) (h : e ∈ G.edgeSet) :
(univ.filter fun d : G.Dart => d.edge = e).card = 2 := by |
refine Sym2.ind (fun v w h => ?_) e h
let d : G.Dart := ⟨(v, w), h⟩
convert congr_arg card d.edge_fiber
rw [card_insert_of_not_mem, card_singleton]
rw [mem_singleton]
exact d.symm_ne.symm
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,045 | 1,051 | theorem degree_pow' : ∀ {n : ℕ}, leadingCoeff p ^ n ≠ 0 → degree (p ^ n) = n • degree p
| 0 => fun h => by rw [pow_zero, ← C_1] at *; rw [degree_C h, zero_nsmul]
| n + 1 => fun h => by
have h₁ : leadingCoeff p ^ n ≠ 0 := fun h₁ => h <| by rw [pow_succ, h₁, zero_mul]
have h₂ : leadingCoeff (p ^ n) * leadingC... |
rwa [pow_succ, ← leadingCoeff_pow' h₁] at h
rw [pow_succ, degree_mul' h₂, succ_nsmul, degree_pow' h₁]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 203 | 204 | theorem add_def' (a b : Rat) : a + b = mkRat (a.num * b.den + b.num * a.den) (a.den * b.den) := by |
rw [add_def, normalize_eq_mkRat]
|
/-
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.CharP.Invertible
import Mathlib.Data.Real.Sqrt
import Mathlib.Tactic.Polyrith
#align_import algebra.star.chsh from "leanprover-community/mathl... | Mathlib/Algebra/Star/CHSH.lean | 183 | 222 | theorem tsirelson_inequality [OrderedRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
[OrderedSMul ℝ R] [StarModule ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ √2 ^ 3 • (1 : R) := by |
-- abel will create `ℤ` multiplication. We will `simp` them away to `ℝ` multiplication.
have M : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = ((m : ℝ) * a) • x := fun m a x => by
rw [zsmul_eq_smul_cast ℝ, ← mul_smul]
let P := (√2)⁻¹ • (A₁ + A₀) - B₀
let Q := (√2)⁻¹ • (A₁ - A₀) + B₁
have w : √2 ^ 3 • (1 : R) - ... |
/-
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.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 516 | 519 | theorem prod_embDomain [Zero M] [CommMonoid N] {v : α →₀ M} {f : α ↪ β} {g : β → M → N} :
(v.embDomain f).prod g = v.prod fun a b => g (f a) b := by |
rw [prod, prod, support_embDomain, Finset.prod_map]
simp_rw [embDomain_apply]
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 211 | 214 | theorem IsBlock.subgroup {H : Subgroup G} {B : Set X} (hfB : IsBlock G B) :
IsBlock H B := by |
rw [IsBlock.def_one]; rintro ⟨g, _⟩
simpa only using hfB.smul_eq_or_disjoint g
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib... | Mathlib/Analysis/Calculus/LHopital.lean | 330 | 336 | theorem lhopital_zero_nhds (hff' : ∀ᶠ x in 𝓝 a, HasDerivAt f (f' x) x)
(hgg' : ∀ᶠ x in 𝓝 a, HasDerivAt g (g' x) x) (hg' : ∀ᶠ x in 𝓝 a, g' x ≠ 0)
(hfa : Tendsto f (𝓝 a) (𝓝 0)) (hga : Tendsto g (𝓝 a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝 a) l) : Tendsto (fun x => f x / g x) (𝓝[≠] a) l := b... |
apply @lhopital_zero_nhds' _ _ _ f' _ g' <;>
(first | apply eventually_nhdsWithin_of_eventually_nhds |
apply tendsto_nhdsWithin_of_tendsto_nhds) <;> assumption
|
/-
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.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.tagged from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97... | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | 83 | 85 | theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J := by |
convert Set.mem_iUnion₂
rw [Box.mem_coe, mem_toPrepartition, exists_prop]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 740 | 742 | theorem mk'_one (x) : f.mk' x (1 : S) = f.toMap x := by |
rw [mk', MonoidHom.map_one]
exact mul_one _
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.char_p.quotient from... | Mathlib/Algebra/CharP/Quotient.lean | 60 | 66 | theorem Ideal.Quotient.index_eq_zero {R : Type*} [CommRing R] (I : Ideal R) :
(↑I.toAddSubgroup.index : R ⧸ I) = 0 := by |
rw [AddSubgroup.index, Nat.card_eq]
split_ifs with hq; swap
· simp
letI : Fintype (R ⧸ I) := @Fintype.ofFinite _ hq
exact Nat.cast_card_eq_zero (R ⧸ I)
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Nor... | Mathlib/Analysis/Convex/Normed.lean | 39 | 44 | theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by |
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
|
/-
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, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 461 | 463 | theorem addHaar_ball_of_pos (x : E) {r : ℝ} (hr : 0 < r) :
μ (ball x r) = ENNReal.ofReal (r ^ finrank ℝ E) * μ (ball 0 1) := by |
rw [← addHaar_ball_mul_of_pos μ x hr, mul_one]
|
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