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) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
/-!
# Number field discriminant
This file defines the discrimi... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 443 | 455 | theorem numberField_discr : discr ℚ = 1 := by |
let b : Basis (Fin 1) ℤ (𝓞 ℚ) :=
Basis.map (Basis.singleton (Fin 1) ℤ) ringOfIntegersEquiv.toAddEquiv.toIntLinearEquiv.symm
calc NumberField.discr ℚ
_ = Algebra.discr ℤ b := by convert (discr_eq_discr ℚ b).symm
_ = Algebra.trace ℤ (𝓞 ℚ) (b default * b default) := by
rw [Algebra.discr_def, Matri... |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 1,006 | 1,011 | theorem convolution_precompR_apply {g : G → E'' →L[𝕜] E'} (hf : LocallyIntegrable f μ)
(hcg : HasCompactSupport g) (hg : Continuous g) (x₀ : G) (x : E'') :
(f ⋆[L.precompR E'', μ] g) x₀ x = (f ⋆[L, μ] fun a => g a x) x₀ := by |
have := hcg.convolutionExists_right (L.precompR E'' : _) hf hg x₀
simp_rw [convolution_def, ContinuousLinearMap.integral_apply this]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 544 | 556 | theorem mapDomain_apply' (S : Set α) {f : α → β} (x : α →₀ M) (hS : (x.support : Set α) ⊆ S)
(hf : Set.InjOn f S) {a : α} (ha : a ∈ S) : mapDomain f x (f a) = x a := by |
classical
rw [mapDomain, sum_apply, sum]
simp_rw [single_apply]
by_cases hax : a ∈ x.support
· rw [← Finset.add_sum_erase _ _ hax, if_pos rfl]
convert add_zero (x a)
refine Finset.sum_eq_zero fun i hi => if_neg ?_
exact (hf.mono hS).ne (Finset.mem_of_mem_erase hi) hax (Finset.ne_of_... |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 276 | 285 | theorem hasStrictFDerivAt_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.1) :
HasStrictFDerivAt (fun x : ℝ × ℝ => x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ) p := by |
have : (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) :=
(continuousAt_fst.eventually (lt_mem_nhds hp)).mono fun p hp => rpow_def_of_pos hp _
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
convert ((hasStrictFDerivAt_fst.log hp.ne').mul hasStrictFDerivAt_snd).exp using 1
rw... |
/-
Copyright (c) 2023 Hanneke Wiersema. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Hanneke Wiersema
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
/-!
# The cyclotomic character
Let `L` be an integral domain and let `n : ℕ+` be a positive integer... | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 120 | 125 | theorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
(hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by |
apply IsCyclic.ext rfl (fun ζ ↦ ?_)
specialize hc ζ
suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this
rw [← toFun_spec g ζ, hc]
|
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 690 | 692 | theorem smul_comp [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p q : R[X]) :
(s • p).comp q = s • p.comp q := by |
rw [← smul_one_smul R s p, comp, comp, eval₂_smul, ← smul_eq_C_mul, smul_assoc, one_smul]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 940 | 945 | theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by |
have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp
rw [show (k : ℚ_[p]) = ((k : ℚ) : ℚ_[p]) by norm_cast, eq_padicNorm, this]
norm_cast
rw [← padicNorm.dvd_iff_norm_le]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 256 | 256 | theorem top_bihimp : ⊤ ⇔ a = a := by | rw [bihimp_comm, bihimp_top]
|
/-
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, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 434 | 436 | theorem smul_mk' (x y : R) (m : M) : x • mk' S y m = mk' S (x * y) m := by |
nth_rw 2 [← one_mul m]
rw [mk'_mul, mk'_one, Algebra.smul_def]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 303 | 306 | theorem mdifferentiableWithinAt_inter' (ht : t ∈ 𝓝[s] x) :
MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x := by |
rw [MDifferentiableWithinAt, MDifferentiableWithinAt,
(differentiable_within_at_localInvariantProp I I').liftPropWithinAt_inter' ht]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 840 | 842 | theorem oangle_sign_smul_left (x y : V) (r : ℝ) :
(o.oangle (r • x) y).sign = SignType.sign r * (o.oangle x y).sign := by |
rcases lt_trichotomy r 0 with (h | h | h) <;> simp [h]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 96 | 99 | theorem integral_comp_neg_Ioi {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(c : ℝ) (f : ℝ → E) : (∫ x in Ioi c, f (-x)) = ∫ x in Iic (-c), f x := by |
rw [← neg_neg c, ← integral_comp_neg_Iic]
simp only [neg_neg]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 735 | 736 | theorem prod.lift_fst_snd {X Y : C} [HasBinaryProduct X Y] :
prod.lift prod.fst prod.snd = 𝟙 (X ⨯ Y) := by | ext <;> simp
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 339 | 341 | theorem nhdsWithin_pi_neBot {ι : Type*} {α : ι → Type*} [∀ i, TopologicalSpace (α i)] {I : Set ι}
{s : ∀ i, Set (α i)} {x : ∀ i, α i} : (𝓝[pi I s] x).NeBot ↔ ∀ i ∈ I, (𝓝[s i] x i).NeBot := by |
simp [neBot_iff, nhdsWithin_pi_eq_bot]
|
/-
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 | 1,841 | 1,844 | theorem lintegral_trim_ae {μ : Measure α} (hm : m ≤ m0) {f : α → ℝ≥0∞}
(hf : AEMeasurable f (μ.trim hm)) : ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ := by |
rw [lintegral_congr_ae (ae_eq_of_ae_eq_trim hf.ae_eq_mk), lintegral_congr_ae hf.ae_eq_mk,
lintegral_trim hm hf.measurable_mk]
|
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 446 | 455 | theorem ContMDiffWithinAt.change_section_trivialization {f : M → TotalSpace F E}
(hp : ContMDiffWithinAt IM IB n (π F E ∘ f) s x)
(hf : ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e (f y)).2) s x)
(he : f x ∈ e.source) (he' : f x ∈ e'.source) :
ContMDiffWithinAt IM 𝓘(𝕜, F) n (fun y ↦ (e' (f y)).2) s x ... |
rw [Trivialization.mem_source] at he he'
refine (hp.coordChange hf he he').congr_of_eventuallyEq ?_ ?_
· filter_upwards [hp.continuousWithinAt (e.open_baseSet.mem_nhds he)] with y hy
rw [Function.comp_apply, e.coordChange_apply_snd _ hy]
· rw [Function.comp_apply, e.coordChange_apply_snd _ he]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 472 | 476 | theorem restrict_biInf {ι} {I : Set ι} (hI : I.Nonempty) (s : Set α) (m : ι → OuterMeasure α) :
restrict s (⨅ i ∈ I, m i) = ⨅ i ∈ I, restrict s (m i) := by |
haveI := hI.to_subtype
rw [← iInf_subtype'', ← iInf_subtype'']
exact restrict_iInf _ _
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 824 | 826 | theorem inv_smul_integral_comp_add_div (c d) :
(c⁻¹ • ∫ x in a..b, f (d + x / c)) = ∫ x in d + a / c..d + b / c, f x := by |
by_cases hc : c = 0 <;> simp [hc, integral_comp_add_div]
|
/-
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 | 404 | 409 | theorem finset_sup_smul (p : ι → Seminorm 𝕜 E) (s : Finset ι) (C : ℝ≥0) :
s.sup (C • p) = C • s.sup p := by |
ext x
rw [smul_apply, finset_sup_apply, finset_sup_apply]
symm
exact congr_arg ((↑) : ℝ≥0 → ℝ) (NNReal.mul_finset_sup C s (fun i ↦ ⟨p i x, apply_nonneg _ _⟩))
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 306 | 310 | theorem convexBodySumFun_add_le (x y : E K) :
convexBodySumFun (x + y) ≤ convexBodySumFun x + convexBodySumFun y := by |
simp_rw [convexBodySumFun, ← Finset.sum_add_distrib, ← mul_add]
exact Finset.sum_le_sum
fun _ _ ↦ mul_le_mul_of_nonneg_left (normAtPlace_add_le _ x y) (Nat.cast_pos.mpr mult_pos).le
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_spa... | Mathlib/Analysis/NormedSpace/Real.lean | 76 | 78 | theorem frontier_ball (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (ball x r) = sphere x r := by |
rw [frontier, closure_ball x hr, isOpen_ball.interior_eq, closedBall_diff_ball]
|
/-
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 | 672 | 675 | theorem ext_fourfold {g h : ((M ⊗[R] N) ⊗[R] P) ⊗[R] Q →ₗ[R] S}
(H : ∀ w x y z, g (w ⊗ₜ x ⊗ₜ y ⊗ₜ z) = h (w ⊗ₜ x ⊗ₜ y ⊗ₜ z)) : g = h := by |
ext w x y z
exact H w x y z
|
/-
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.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 214 | 216 | theorem lift_aux (x : L) : ∃ y : ℕ × K, i y.2 = x ^ p ^ y.1 := by |
obtain ⟨n, y, h⟩ := IsPRadical.pow_mem i p x
exact ⟨(n, y), h⟩
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2... | Mathlib/Order/Filter/Lift.lean | 45 | 53 | theorem HasBasis.mem_lift_iff {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α}
(hf : f.HasBasis p s) {β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ}
{g : Set α → Filter γ} (hg : ∀ i, (g <| s i).HasBasis (pg i) (sg i)) (gm : Monotone g)
{s : Set γ} : s ∈ f.lift g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg... |
refine (mem_biInf_of_directed ?_ ⟨univ, univ_sets _⟩).trans ?_
· intro t₁ ht₁ t₂ ht₂
exact ⟨t₁ ∩ t₂, inter_mem ht₁ ht₂, gm inter_subset_left, gm inter_subset_right⟩
· simp only [← (hg _).mem_iff]
exact hf.exists_iff fun t₁ t₂ ht H => gm ht H
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 433 | 434 | theorem Left.mul_nonneg [PosMulMono α] (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a * b := by |
simpa only [mul_zero] using mul_le_mul_of_nonneg_left hb ha
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 498 | 501 | theorem hasDerivAt_fourier_neg (n : ℤ) (x : ℝ) :
HasDerivAt (fun y : ℝ => fourier (-n) (y : AddCircle T))
(-2 * π * I * n / T * fourier (-n) (x : AddCircle T)) x := by |
simpa using hasDerivAt_fourier T (-n) x
|
/-
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.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 543 | 544 | theorem coe_succ_le_iff {n : ℕ} {e : PartENat} : ↑n.succ ≤ e ↔ ↑n < e := by |
rw [Nat.succ_eq_add_one n, Nat.cast_add, Nat.cast_one, add_one_le_iff_lt (natCast_ne_top n)]
|
/-
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.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 760 | 766 | theorem affineCombination_affineCombinationLineMapWeights [DecidableEq ι] (p : ι → P) {i j : ι}
(hi : i ∈ s) (hj : j ∈ s) (c : k) :
s.affineCombination k p (affineCombinationLineMapWeights i j c) =
AffineMap.lineMap (p i) (p j) c := by |
rw [affineCombinationLineMapWeights, ← weightedVSub_vadd_affineCombination,
weightedVSub_const_smul, s.affineCombination_affineCombinationSingleWeights k p hi,
s.weightedVSub_weightedVSubVSubWeights k p hj hi, AffineMap.lineMap_apply]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 568 | 575 | theorem Ordered.lowerBound?_lt [@TransCmp α cmp] [IsStrictCut cmp cut] (ht : Ordered cmp t)
(H : t.lowerBound? cut = some x) (hy : y ∈ t) : cmp x y = .lt ↔ cut y = .lt := by |
refine ⟨fun h => ?_, fun h => OrientedCmp.cmp_eq_gt.1 ?_⟩
· cases e : cut x
· cases lowerBound?_le H e
· exact IsStrictCut.exact e |>.symm.trans h
· exact ht.lowerBound?_greatest H hy h e
· by_contra h'; exact lowerBound?_le H <| IsCut.le_lt_trans (cmp := cmp) (cut := cut) h' h
|
/-
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.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 400 | 401 | theorem cancel_counitInv_right {X Y : D} (f f' : X ⟶ Y) :
f ≫ e.counitInv.app Y = f' ≫ e.counitInv.app Y ↔ f = f' := by | simp only [cancel_mono]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 185 | 187 | theorem _root_.NumberField.mixedEmbedding_injective [NumberField K] :
Function.Injective (NumberField.mixedEmbedding K) := by |
exact RingHom.injective _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,869 | 1,871 | theorem set_eventuallyEq_iff_inf_principal {s t : Set α} {l : Filter α} :
s =ᶠ[l] t ↔ l ⊓ 𝓟 s = l ⊓ 𝓟 t := by |
simp only [eventuallyLE_antisymm_iff, le_antisymm_iff, set_eventuallyLE_iff_inf_principal_le]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
#align_import analysis.convex.quasiconvex from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Q... | Mathlib/Analysis/Convex/Quasiconvex.lean | 138 | 143 | theorem quasilinearOn_iff_mem_uIcc : QuasilinearOn 𝕜 s f ↔ Convex 𝕜 s ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄,
y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ∈ uIcc (f x) (f y) := by |
rw [QuasilinearOn, quasiconvexOn_iff_le_max, quasiconcaveOn_iff_min_le, and_and_and_comm,
and_self_iff]
apply and_congr_right'
simp_rw [← forall_and, ← Icc_min_max, mem_Icc, and_comm]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,321 | 2,322 | theorem mem_conjugatesOfSet_iff {x : G} : x ∈ conjugatesOfSet s ↔ ∃ a ∈ s, IsConj a x := by |
erw [Set.mem_iUnion₂]; simp only [conjugatesOf, isConj_iff, Set.mem_setOf_eq, exists_prop]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Anne Baanen
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-c... | Mathlib/Order/Hom/Order.lean | 117 | 119 | theorem coe_iSup {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) :
((⨆ i, f i : α →o β) : α → β) = ⨆ i, (f i : α → β) := by |
funext x; simp [iSup_apply]
|
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheo... | Mathlib/Algebra/BigOperators/Fin.lean | 136 | 138 | theorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by |
rw [prod_univ_castSucc, prod_univ_three]
rfl
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 293 | 296 | theorem dotProduct_mulVec_adjMatrix [NonAssocSemiring α] (x y : V → α) :
x ⬝ᵥ (G.adjMatrix α).mulVec y = ∑ i : V, ∑ j : V, if G.Adj i j then x i * y j else 0 := by |
simp only [dotProduct, mulVec, adjMatrix_apply, ite_mul, one_mul, zero_mul, mul_sum, mul_ite,
mul_zero]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 593 | 598 | theorem exists_not_mem_null_le_average (hμ : μ ≠ 0) (hf : Integrable f μ) (hN : μ N = 0) :
∃ x, x ∉ N ∧ f x ≤ ⨍ a, f a ∂μ := by |
have := measure_le_average_pos hμ hf
rw [← measure_diff_null hN] at this
obtain ⟨x, hx, hxN⟩ := nonempty_of_measure_ne_zero this.ne'
exact ⟨x, hxN, hx⟩
|
/-
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.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mat... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 539 | 541 | theorem nextCoeffUp_C_eq_zero (c : R) : nextCoeffUp (C c) = 0 := by |
rw [nextCoeffUp]
simp
|
/-
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.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theor... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 89 | 102 | theorem closure_induction_left {p : (x : G) → x ∈ closure s → Prop} (one : p 1 (one_mem _))
(mul_left : ∀ x (hx : x ∈ s), ∀ (y) hy, p y hy → p (x * y) (mul_mem (subset_closure hx) hy))
(mul_left_inv : ∀ x (hx : x ∈ s), ∀ (y) hy, p y hy →
p (x⁻¹ * y) (mul_mem (inv_mem (subset_closure hx)) hy))
{x : G} ... |
revert h
simp_rw [← mem_toSubmonoid, closure_toSubmonoid] at *
intro h
induction h using Submonoid.closure_induction_left with
| one => exact one
| mul_left x hx y hy ih =>
cases hx with
| inl hx => exact mul_left _ hx _ hy ih
| inr hx => simpa only [inv_inv] using mul_left_inv _ hx _ hy ih
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,449 | 2,450 | theorem comap_inf_principal_range : comap m (g ⊓ 𝓟 (range m)) = comap m g := by |
simp
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-communit... | Mathlib/AlgebraicGeometry/Gluing.lean | 256 | 262 | theorem isOpen_iff (U : Set D.glued.carrier) : IsOpen U ↔ ∀ i, IsOpen ((D.ι i).1.base ⁻¹' U) := by |
rw [← (TopCat.homeoOfIso D.isoCarrier.symm).isOpen_preimage]
rw [TopCat.GlueData.isOpen_iff]
apply forall_congr'
intro i
erw [← Set.preimage_comp, ← ι_isoCarrier_inv]
rfl
|
/-
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 | 319 | 326 | 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 ... |
simp only [← Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *
exact ⟨lhopital_zero_nhds_left hff'.1 hgg'.1 hg'.1 hfa.1 hga.1 hdiv.1,
lhopital_zero_nhds_right hff'.2 hgg'.2 hg'.2 hfa.2 hga.2 hdiv.2⟩
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Floris van Doorn
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.GroupTheory.GroupAc... | Mathlib/Data/Set/Pointwise/SMul.lean | 982 | 983 | theorem smul_set_compl : a • sᶜ = (a • s)ᶜ := by |
simp_rw [Set.compl_eq_univ_diff, smul_set_sdiff, smul_set_univ]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
#align_import analysi... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 74 | 83 | theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) :
∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c := by |
cases' hc with hE hc
· refine ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => ?_⟩
simp [@Subsingleton.elim E hE x y]
have := hf.def hc
rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
rcases this with ⟨s, has, hs⟩
exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
|
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 151 | 153 | theorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by |
rw [mul_comm]
exact mul_left_not_lt b h
|
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanpro... | Mathlib/Algebra/Star/SelfAdjoint.lean | 169 | 170 | theorem conjugate' {x : R} (hx : IsSelfAdjoint x) (z : R) : IsSelfAdjoint (star z * x * z) := by |
simp only [isSelfAdjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
imp... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 153 | 172 | theorem cardPowDegree_anti_archimedean {x y z : Fq[X]} {a : ℤ} (hxy : cardPowDegree (x - y) < a)
(hyz : cardPowDegree (y - z) < a) : cardPowDegree (x - z) < a := by |
have ha : 0 < a := lt_of_le_of_lt (AbsoluteValue.nonneg _ _) hxy
by_cases hxy' : x = y
· rwa [hxy']
by_cases hyz' : y = z
· rwa [← hyz']
by_cases hxz' : x = z
· rwa [hxz', sub_self, map_zero]
rw [← Ne, ← sub_ne_zero] at hxy' hyz' hxz'
refine lt_of_le_of_lt ?_ (max_lt hxy hyz)
rw [cardPowDegree_nonz... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 735 | 738 | theorem pi_eq_empty (hs : i ∈ s) (ht : t i = ∅) : s.pi t = ∅ := by |
ext f
simp only [mem_empty_iff_false, not_forall, iff_false_iff, mem_pi, Classical.not_imp]
exact ⟨i, hs, by simp [ht]⟩
|
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Jacobson
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.MvPolynomial
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
... | Mathlib/RingTheory/Nullstellensatz.lean | 199 | 211 | theorem vanishingIdeal_zeroLocus_eq_radical (I : Ideal (MvPolynomial σ k)) :
vanishingIdeal (zeroLocus I) = I.radical := by |
rw [I.radical_eq_jacobson]
refine le_antisymm (le_sInf ?_) fun p hp x hx => ?_
· rintro J ⟨hJI, hJ⟩
obtain ⟨x, hx⟩ := (isMaximal_iff_eq_vanishingIdeal_singleton J).1 hJ
refine hx.symm ▸ vanishingIdeal_anti_mono fun y hy p hp => ?_
rw [← mem_vanishingIdeal_singleton_iff, Set.mem_singleton_iff.1 hy, ← ... |
/-
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 | 354 | 361 | theorem irreducible_iff_prime_of_exists_prime_factors {p : α} : Irreducible p ↔ Prime p := by |
by_cases hp0 : p = 0
· simp [hp0]
refine ⟨fun h => ?_, Prime.irreducible⟩
obtain ⟨f, hf⟩ := pf p hp0
obtain ⟨q, hq, rfl⟩ := prime_factors_irreducible h hf
rw [hq.prime_iff]
exact hf.1 q (Multiset.mem_singleton_self _)
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 183 | 187 | theorem isPrime_carrier : Ideal.IsPrime (carrier x) := by |
refine Ideal.IsPrime.comap _ (hK := ?_)
exact IsLocalization.isPrime_of_isPrime_disjoint
(Submonoid.powers f) _ _ inferInstance
((disjoint_powers_iff_not_mem _ (Ideal.IsPrime.isRadical inferInstance)).mpr x.2)
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 261 | 263 | theorem StrictConvex.add_left (hs : StrictConvex 𝕜 s) (z : E) :
StrictConvex 𝕜 ((fun x => z + x) '' s) := by |
simpa only [singleton_add] using (strictConvex_singleton z).add hs
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,078 | 3,081 | theorem map_injective_of_ker_le {H K : Subgroup G} (hH : f.ker ≤ H) (hK : f.ker ≤ K)
(hf : map f H = map f K) : H = K := by |
apply_fun comap f at hf
rwa [comap_map_eq, comap_map_eq, sup_of_le_left hH, sup_of_le_left hK] at hf
|
/-
Copyright (c) 2022 Siddhartha Prasad, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddhartha Prasad, Yaël Dillies
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Tactic.Monotonici... | Mathlib/Algebra/Order/Kleene.lean | 228 | 229 | theorem kstar_le_of_mul_le_left (hb : 1 ≤ b) : b * a ≤ b → a∗ ≤ b := by |
simpa using mul_kstar_le hb
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDe... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,303 | 1,304 | theorem mem_extChartAt_source (x : M) : x ∈ (extChartAt I x).source := by |
simp only [extChartAt_source, mem_chart_source]
|
/-
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 | 1,111 | 1,117 | theorem map [MeasureSpace β] (h : v ≪ᵥ w) (f : α → β) : v.map f ≪ᵥ w.map f := by |
by_cases hf : Measurable f
· refine mk fun s hs hws => ?_
rw [map_apply _ hf hs] at hws ⊢
exact h hws
· intro s _
rw [map_not_measurable v hf, zero_apply]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 823 | 824 | theorem inter_mem_inter [Inter α] (a b : Part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma ∩ mb ∈ a ∩ b := by | simp [inter_def]; aesop
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 44 | 45 | theorem intDegree_zero : intDegree (0 : RatFunc K) = 0 := by |
rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]
|
/-
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.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 854 | 860 | theorem exists_circumsphere_eq_of_cospherical_subset {s : AffineSubspace ℝ P} {ps : Set P}
(h : ps ⊆ s) [Nonempty s] {n : ℕ} [FiniteDimensional ℝ s.direction]
(hd : finrank ℝ s.direction = n) (hc : Cospherical ps) :
∃ c : Sphere P, ∀ sx : Simplex ℝ P n, Set.range sx.points ⊆ ps → sx.circumsphere = c := by |
obtain ⟨r, hr⟩ := exists_circumradius_eq_of_cospherical_subset h hd hc
obtain ⟨c, hc⟩ := exists_circumcenter_eq_of_cospherical_subset h hd hc
exact ⟨⟨c, r⟩, fun sx hsx => Sphere.ext _ _ (hc sx hsx) (hr sx hsx)⟩
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
#ali... | Mathlib/CategoryTheory/Subobject/Comma.lean | 172 | 179 | theorem unop_left_comp_underlyingIso_hom_unop {A : CostructuredArrow S T}
{P : (CostructuredArrow S T)ᵒᵖ} (f : P ⟶ op A) [Mono f.unop.left.op] :
f.unop.left ≫ (Subobject.underlyingIso f.unop.left.op).hom.unop =
(Subobject.mk f.unop.left.op).arrow.unop := by |
conv_lhs =>
congr
rw [← Quiver.Hom.unop_op f.unop.left]
rw [← unop_comp, Subobject.underlyingIso_hom_comp_eq_mk]
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 693 | 694 | theorem codisjoint_coe : Codisjoint (s : Set α) t ↔ Disjoint s t := by |
simp [disjoint_iff, codisjoint_iff, SetLike.ext'_iff]
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 479 | 481 | theorem unitors_equal : (λ_ (𝟙 a)).hom = (ρ_ (𝟙 a)).hom := by |
rw [← whiskerLeft_iff, ← cancel_epi (α_ _ _ _).hom, ← cancel_mono (ρ_ _).hom, triangle, ←
rightUnitor_comp, rightUnitor_naturality]
|
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanp... | Mathlib/RingTheory/Localization/NormTrace.lean | 115 | 119 | theorem Algebra.discr_localizationLocalization (b : Basis ι R S) :
Algebra.discr Rₘ (b.localizationLocalization Rₘ M Sₘ) =
algebraMap R Rₘ (Algebra.discr R b) := by |
rw [Algebra.discr_def, Algebra.discr_def, RingHom.map_det,
Algebra.traceMatrix_localizationLocalization]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebr... | Mathlib/Algebra/Ring/BooleanRing.lean | 211 | 213 | theorem sup_inf_self (a b : α) : a ⊔ a ⊓ b = a := by |
dsimp only [(· ⊔ ·), (· ⊓ ·)]
rw [← mul_assoc, mul_self, add_assoc, add_self, add_zero]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 1,265 | 1,267 | theorem LiftRelAux.ret_right (R : α → β → Prop) (C : Computation α → Computation β → Prop) (b ca) :
LiftRelAux R C (destruct ca) (Sum.inl b) ↔ ∃ a, a ∈ ca ∧ R a b := by |
rw [← LiftRelAux.swap, LiftRelAux.ret_left]
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 90 | 97 | theorem mul_mem_cancel_left {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hf : f ∈ S.arrows c d) :
f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e := by |
constructor
· rintro h
suffices Groupoid.inv f ≫ f ≫ g ∈ S.arrows d e by
simpa only [inv_eq_inv, IsIso.inv_hom_id_assoc] using this
apply S.mul (S.inv hf) h
· apply S.mul hf
|
/-
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.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_i... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 63 | 64 | theorem foldl_loop_eq (f : α → Fin n → α) (x) : foldl.loop n f x n = x := by |
rw [foldl.loop, dif_neg (Nat.lt_irrefl _)]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 390 | 393 | theorem mem_iSup_of_directed {ι} [Nonempty ι] (S : ι → Submodule R M) (H : Directed (· ≤ ·) S) {x} :
x ∈ iSup S ↔ ∃ i, x ∈ S i := by |
rw [← SetLike.mem_coe, coe_iSup_of_directed S H, mem_iUnion]
rfl
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topolog... | Mathlib/Analysis/Quaternion.lean | 173 | 178 | theorem norm_piLp_equiv_symm_equivTuple (x : ℍ) :
‖(WithLp.equiv 2 (Fin 4 → _)).symm (equivTuple ℝ x)‖ = ‖x‖ := by |
rw [norm_eq_sqrt_real_inner, norm_eq_sqrt_real_inner, inner_self, normSq_def', PiLp.inner_apply,
Fin.sum_univ_four]
simp_rw [RCLike.inner_apply, starRingEnd_apply, star_trivial, ← sq]
rfl
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv... | Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 39 | 40 | theorem natCast_mem [AddSubmonoidWithOneClass S R] (n : ℕ) : (n : R) ∈ s := by |
induction n <;> simp [zero_mem, add_mem, one_mem, *]
|
/-
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.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cf... | Mathlib/Data/Nat/Pairing.lean | 156 | 162 | theorem max_sq_add_min_le_pair (m n : ℕ) : max m n ^ 2 + min m n ≤ pair m n := by |
rw [pair]
cases' lt_or_le m n with h h
· rw [if_pos h, max_eq_right h.le, min_eq_left h.le, Nat.pow_two]
rw [if_neg h.not_lt, max_eq_left h, min_eq_right h, Nat.pow_two, Nat.add_assoc,
Nat.add_le_add_iff_left]
exact Nat.le_add_left _ _
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Scott Morrison
-/
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Multiset.Functor
#align_import data.finset.functor from "leanprove... | Mathlib/Data/Finset/Functor.lean | 200 | 202 | theorem id_traverse [DecidableEq α] (s : Finset α) : traverse (pure : α → Id α) s = s := by |
rw [traverse, Multiset.id_traverse]
exact s.val_toFinset
|
/-
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.Polynomial.Expand
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
import Mathlib.RingTheory.Adjoin.... | Mathlib/RingTheory/IntegralClosure.lean | 668 | 672 | theorem normalizeScaleRoots_monic (h : p ≠ 0) : (normalizeScaleRoots p).Monic := by |
delta Monic leadingCoeff
rw [natDegree_eq_of_degree_eq (normalizeScaleRoots_degree p)]
suffices p = 0 → (0 : R) = 1 by simpa [normalizeScaleRoots, coeff_monomial]
exact fun h' => (h h').elim
|
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.Probability.Kernel.Basic
/-!
# Independence with respect to a kernel and a measure
A family of sets of sets... | Mathlib/Probability/Independence/Kernel.lean | 932 | 956 | theorem iIndepFun.indepFun_prod_mk [IsMarkovKernel κ] (hf_Indep : iIndepFun m f κ μ)
(hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
IndepFun (fun a => (f i a, f j a)) (f k) κ μ := by |
classical
have h_right : f k =
(fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘
fun a (j : ({k} : Finset ι)) => f j a := rfl
have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι),
β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _
let s : Finset... |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 578 | 583 | theorem geom_sum_neg_iff [LinearOrderedRing α] (hn : n ≠ 0) :
∑ i ∈ range n, x ^ i < 0 ↔ Even n ∧ x + 1 < 0 := by |
rw [← not_iff_not, not_lt, le_iff_lt_or_eq, eq_comm,
or_congr (geom_sum_pos_iff hn) (geom_sum_eq_zero_iff_neg_one hn), Nat.odd_iff_not_even, ←
add_eq_zero_iff_eq_neg, not_and, not_lt, le_iff_lt_or_eq, eq_comm, ← imp_iff_not_or, or_comm,
and_comm, Decidable.and_or_imp, or_comm]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.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 | 1,237 | 1,294 | theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
(H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by |
cases' r with rs rl rx rr; · cases H2
rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
rw [hr.2.size_eq] at H3
replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
... |
/-
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 | 98 | 100 | theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by |
rw [← mul_le_mul_iff_left a]
simp
|
/-
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 | 127 | 132 | theorem coe_rootSpaceWeightSpaceProduct_tmul (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃)
(x : rootSpace H χ₁) (m : weightSpace M χ₂) :
(rootSpaceWeightSpaceProduct R L H M χ₁ χ₂ χ₃ hχ (x ⊗ₜ m) : M) = ⁅(x : L), (m : M)⁆ := by |
simp only [rootSpaceWeightSpaceProduct, rootSpaceWeightSpaceProductAux, coe_liftLie_eq_lift_coe,
AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, lift_apply, LinearMap.coe_mk, AddHom.coe_mk,
Submodule.coe_mk]
|
/-
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.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-commu... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 88 | 94 | theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @UniversallyClosed := by |
rw [universallyClosed_eq]
apply universallyIsLocalAtTargetOfMorphismRestrict
· exact topologically_isClosedMap_respectsIso
· intro X Y f ι U hU H
simp_rw [topologically, morphismRestrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
|
/-
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 | 736 | 753 | theorem tendstoLocallyUniformlyOn_TFAE [LocallyCompactSpace α] (G : ι → α → β) (g : α → β)
(p : Filter ι) (hs : IsOpen s) :
List.TFAE [
TendstoLocallyUniformlyOn G g p s,
∀ K, K ⊆ s → IsCompact K → TendstoUniformlyOn G g p K,
∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v] := by |
tfae_have 1 → 2
· rintro h K hK1 hK2
exact (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK2).mp (h.mono hK1)
tfae_have 2 → 3
· rintro h x hx
obtain ⟨K, ⟨hK1, hK2⟩, hK3⟩ := (compact_basis_nhds x).mem_iff.mp (hs.mem_nhds hx)
exact ⟨K, nhdsWithin_le_nhds hK1, h K hK3 hK2⟩
tfae_have 3... |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
#align_import data.finmap from "leanprover-community/mathlib"@"c... | Mathlib/Data/Finmap.lean | 479 | 481 | theorem insert_toFinmap (a : α) (b : β a) (s : AList β) :
insert a b (AList.toFinmap s) = AList.toFinmap (s.insert a b) := by |
simp [insert]
|
/-
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 | 674 | 759 | theorem balance_eq_balance' {l x r} (hl : Balanced l) (hr : Balanced r) (sl : Sized l)
(sr : Sized r) : @balance α l x r = balance' l x r := by |
cases' l with ls ll lx lr
· cases' r with rs rl rx rr
· rfl
· rw [sr.eq_node'] at hr ⊢
cases' rl with rls rll rlx rlr <;> cases' rr with rrs rrl rrx rrr <;>
dsimp [balance, balance']
· rfl
· have : size rrl = 0 ∧ size rrr = 0 := by
have := balancedSz_zero.1 hr.1.symm
... |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 499 | 500 | theorem basis_one (h : IsAdjoinRootMonic S f) (hdeg : 1 < natDegree f) :
h.basis ⟨1, hdeg⟩ = h.root := by | rw [h.basis_apply, Fin.val_mk, pow_one]
|
/-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 546 | 554 | theorem Ideal.le_toIdeal_homogeneousHull : I ≤ (Ideal.homogeneousHull 𝒜 I).toIdeal := by |
intro r hr
classical
rw [← DirectSum.sum_support_decompose 𝒜 r]
refine Ideal.sum_mem _ ?_
intro j _
apply Ideal.subset_span
use j
use ⟨r, hr⟩
|
/-
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.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-communit... | Mathlib/SetTheory/Cardinal/Finite.lean | 97 | 98 | theorem card_eq_of_equiv_fin {α : Type*} {n : ℕ} (f : α ≃ Fin n) : Nat.card α = n := by |
simpa only [card_eq_fintype_card, Fintype.card_fin] using card_congr f
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
#align_import analysis.calculus.deriv.support from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff... | Mathlib/Analysis/Calculus/Deriv/Support.lean | 36 | 41 | theorem support_deriv_subset : support (deriv f) ⊆ tsupport f := by |
intro x
rw [← not_imp_not]
intro h2x
rw [not_mem_tsupport_iff_eventuallyEq] at h2x
exact nmem_support.mpr (h2x.deriv_eq.trans (deriv_const x 0))
|
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mat... | Mathlib/Data/Nat/Hyperoperation.lean | 49 | 50 | theorem hyperoperation_ge_three_eq_one (n m : ℕ) : hyperoperation (n + 3) m 0 = 1 := by |
rw [hyperoperation]
|
/-
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.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.m... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 665 | 666 | theorem haarMeasure_closure_self {K₀ : PositiveCompacts G} : haarMeasure K₀ (closure K₀) = 1 := by |
rw [K₀.isCompact.measure_closure, haarMeasure_self]
|
/-
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.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
/-!
# Convex join
... | Mathlib/Analysis/Convex/Join.lean | 170 | 173 | theorem convexJoin_convexJoin_convexJoin_comm (s t u v : Set E) :
convexJoin 𝕜 (convexJoin 𝕜 s t) (convexJoin 𝕜 u v) =
convexJoin 𝕜 (convexJoin 𝕜 s u) (convexJoin 𝕜 t v) := by |
simp_rw [← convexJoin_assoc, convexJoin_right_comm]
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Tactic.IntervalCases
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import ... | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 174 | 176 | theorem card [NeZero n] : Fintype.card (QuaternionGroup n) = 4 * n := by |
rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul]
ring
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 827 | 829 | theorem L1.integral_eq_integral (f : α →₁[μ] E) : L1.integral f = ∫ a, f a ∂μ := by |
simp only [integral, L1.integral, integral_eq_setToFun]
exact (L1.setToFun_eq_setToL1 (dominatedFinMeasAdditive_weightedSMul μ) f).symm
|
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_dif... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 35 | 43 | theorem contDiffOn_clm_apply {n : ℕ∞} {f : E → F →L[𝕜] G} {s : Set E} [FiniteDimensional 𝕜 F] :
ContDiffOn 𝕜 n f s ↔ ∀ y, ContDiffOn 𝕜 n (fun x => f x y) s := by |
refine ⟨fun h y => h.clm_apply contDiffOn_const, fun h => ?_⟩
let d := finrank 𝕜 F
have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm
let e₁ := ContinuousLinearEquiv.ofFinrankEq hd
let e₂ := (e₁.arrowCongr (1 : G ≃L[𝕜] G)).trans (ContinuousLinearEquiv.piRing (Fin d))
rw [← id_comp f, ← e₂... |
/-
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.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 823 | 828 | theorem circumcenter_eq_of_cospherical_subset {s : AffineSubspace ℝ P} {ps : Set P} (h : ps ⊆ s)
[Nonempty s] {n : ℕ} [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = n)
(hc : Cospherical ps) {sx₁ sx₂ : Simplex ℝ P n} (hsx₁ : Set.range sx₁.points ⊆ ps)
(hsx₂ : Set.range sx₂.points ⊆ ps) : sx₁... |
rcases exists_circumcenter_eq_of_cospherical_subset h hd hc with ⟨r, hr⟩
rw [hr sx₁ hsx₁, hr sx₂ hsx₂]
|
/-
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.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 87 | 89 | theorem restrict_image_emb {f : α → β} (hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f)
(s : Set α) : MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s)) := by |
simpa only [Set.preimage_image_eq _ h₂.injective] using hf.restrict_preimage_emb h₂ (f '' s)
|
/-
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.Ring.Regular
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.modeq... | Mathlib/Data/Nat/ModEq.lean | 89 | 91 | theorem modEq_iff_dvd : a ≡ b [MOD n] ↔ (n : ℤ) ∣ b - a := by |
rw [ModEq, eq_comm, ← Int.natCast_inj, Int.natCast_mod, Int.natCast_mod,
Int.emod_eq_emod_iff_emod_sub_eq_zero, Int.dvd_iff_emod_eq_zero]
|
/-
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 | 316 | 321 | theorem rotate_injective (n : ℕ) : Function.Injective fun l : List α => l.rotate n := by |
rintro l l' (h : l.rotate n = l'.rotate n)
have hle : l.length = l'.length := (l.length_rotate n).symm.trans (h.symm ▸ l'.length_rotate n)
rw [rotate_eq_drop_append_take_mod, rotate_eq_drop_append_take_mod] at h
obtain ⟨hd, ht⟩ := append_inj h (by simp_all)
rw [← take_append_drop _ l, ht, hd, take_append_dro... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.angle.sphere from "leanprover... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 108 | 113 | theorem Cospherical.two_zsmul_oangle_eq {p₁ p₂ p₃ p₄ : P}
(h : Cospherical ({p₁, p₂, p₃, p₄} : Set P)) (hp₂p₁ : p₂ ≠ p₁) (hp₂p₄ : p₂ ≠ p₄)
(hp₃p₁ : p₃ ≠ p₁) (hp₃p₄ : p₃ ≠ p₄) : (2 : ℤ) • ∡ p₁ p₂ p₄ = (2 : ℤ) • ∡ p₁ p₃ p₄ := by |
obtain ⟨s, hs⟩ := cospherical_iff_exists_sphere.1 h
simp_rw [Set.insert_subset_iff, Set.singleton_subset_iff, Sphere.mem_coe] at hs
exact Sphere.two_zsmul_oangle_eq hs.1 hs.2.1 hs.2.2.1 hs.2.2.2 hp₂p₁ hp₂p₄ hp₃p₁ hp₃p₄
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_spa... | Mathlib/Analysis/NormedSpace/Real.lean | 158 | 159 | theorem interior_sphere' (x : E) (r : ℝ) : interior (sphere x r) = ∅ := by |
rw [← frontier_closedBall' x, interior_frontier isClosed_ball]
|
/-
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 | 969 | 974 | theorem AEEqFun.compMeasurePreserving_mem_Lp {β : Type*} [MeasurableSpace β]
{μb : MeasureTheory.Measure β} {g : β →ₘ[μb] E} (hg : g ∈ Lp E p μb) {f : α → β}
(hf : MeasurePreserving f μ μb) :
g.compMeasurePreserving f hf ∈ Lp E p μ := by |
rw [Lp.mem_Lp_iff_snorm_lt_top] at hg ⊢
rwa [snorm_compMeasurePreserving]
|
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