Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 451 | 457 | theorem MapsTo.iterate_restrict {f : α → α} {s : Set α} (h : MapsTo f s s) (n : ℕ) :
(h.restrict f s s)^[n] = (h.iterate n).restrict _ _ _ := by |
funext x
rw [Subtype.ext_iff, MapsTo.val_restrict_apply]
induction' n with n ihn generalizing x
· rfl
· simp [Nat.iterate, ihn]
|
/-
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 | 1,364 | 1,366 | theorem t2Space_iff_nhds :
T2Space X ↔ Pairwise fun x y : X => ∃ U ∈ 𝓝 x, ∃ V ∈ 𝓝 y, Disjoint U V := by |
simp only [t2Space_iff_disjoint_nhds, Filter.disjoint_iff, Pairwise]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,466 | 1,466 | theorem ker_incl : N.incl.ker = ⊥ := by | simp [← LieSubmodule.coe_toSubmodule_eq_iff]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,529 | 1,532 | theorem setIntegral_withDensity_eq_setIntegral_smul {f : X → ℝ≥0} (f_meas : Measurable f)
(g : X → E) {s : Set X} (hs : MeasurableSet s) :
∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ := by |
rw [restrict_withDensity hs, integral_withDensity_eq_integral_smul f_meas]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 110 | 113 | theorem comp_differentiableOn_iff {f : G → E} {s : Set G} :
DifferentiableOn 𝕜 (iso ∘ f) s ↔ DifferentiableOn 𝕜 f s := by |
rw [DifferentiableOn, DifferentiableOn]
simp only [iso.comp_differentiableWithinAt_iff]
|
/-
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 | 1,833 | 1,835 | theorem exists_nat_eq_of_le_nat {c : Cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m := by |
lift c to ℕ using h.trans_lt (nat_lt_aleph0 _)
exact ⟨c, mod_cast h, rfl⟩
|
/-
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.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-communit... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 62 | 64 | theorem charmatrix_apply_ne (h : i ≠ j) : charmatrix M i j = -C (M i j) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, diagonal_apply_ne _ h,
map_apply, sub_eq_neg_self]
|
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.D... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 350 | 351 | theorem isClosed_iff_zeroLocus (Z : Set (ProjectiveSpectrum 𝒜)) :
IsClosed Z ↔ ∃ s, Z = zeroLocus 𝒜 s := by | rw [← isOpen_compl_iff, isOpen_iff, compl_compl]
|
/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathl... | Mathlib/Combinatorics/SetFamily/LYM.lean | 92 | 110 | theorem card_div_choose_le_card_shadow_div_choose (hr : r ≠ 0)
(h𝒜 : (𝒜 : Set (Finset α)).Sized r) : (𝒜.card : 𝕜) / (Fintype.card α).choose r
≤ (∂ 𝒜).card / (Fintype.card α).choose (r - 1) := by |
obtain hr' | hr' := lt_or_le (Fintype.card α) r
· rw [choose_eq_zero_of_lt hr', cast_zero, div_zero]
exact div_nonneg (cast_nonneg _) (cast_nonneg _)
replace h𝒜 := card_mul_le_card_shadow_mul h𝒜
rw [div_le_div_iff] <;> norm_cast
· cases' r with r
· exact (hr rfl).elim
rw [tsub_add_eq_add_tsub h... |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.EffectiveEpi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.Tactic.ApplyFun
/-!
# Effective epimorphic ... | Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean | 61 | 93 | theorem effectiveEpiFamilyStructOfEffectiveEpiDesc_aux {B : C} {α : Type*} {X : α → C}
{π : (a : α) → X a ⟶ B} [HasCoproduct X]
[∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)]
[∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a ↦ pullback g (Sigma.ι X a)]
[∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.des... |
apply_fun ((Sigma.desc fun a ↦ pullback.fst (f := g₁) (g := (Sigma.ι X a))) ≫ ·) using
(fun a b ↦ (cancel_epi _).mp)
ext a
simp only [colimit.ι_desc_assoc, Discrete.functor_obj, Cofan.mk_pt, Cofan.mk_ι_app]
rw [← Category.assoc, pullback.condition]
simp only [Category.assoc, colimit.ι_desc, Cofan.mk_pt, ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 995 | 1,000 | theorem Inducing.isCompact_iff {f : X → Y} (hf : Inducing f) :
IsCompact s ↔ IsCompact (f '' s) := by |
refine ⟨fun hs => hs.image hf.continuous, fun hs F F_ne_bot F_le => ?_⟩
obtain ⟨_, ⟨x, x_in : x ∈ s, rfl⟩, hx : ClusterPt (f x) (map f F)⟩ :=
hs ((map_mono F_le).trans_eq map_principal)
exact ⟨x, x_in, hf.mapClusterPt_iff.1 hx⟩
|
/-
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 | 2,095 | 2,099 | theorem snorm_one_le_of_le' {r : ℝ} {f : α → ℝ} (hfint : Integrable f μ) (hfint' : 0 ≤ ∫ x, f x ∂μ)
(hf : ∀ᵐ ω ∂μ, f ω ≤ r) : snorm f 1 μ ≤ 2 * μ Set.univ * ENNReal.ofReal r := by |
refine snorm_one_le_of_le hfint hfint' ?_
simp only [Real.coe_toNNReal', le_max_iff]
filter_upwards [hf] with ω hω using Or.inl hω
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import... | Mathlib/Data/Real/Irrational.lean | 651 | 652 | theorem irrational_div_nat_iff : Irrational (x / n) ↔ n ≠ 0 ∧ Irrational x := by |
rw [← cast_natCast, irrational_div_rat_iff, Nat.cast_ne_zero]
|
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 589 | 604 | theorem isSheaf_iff_multiequalizer [∀ (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)] :
IsSheaf J P ↔ ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P) := by |
rw [isSheaf_iff_multifork]
refine forall₂_congr fun X S => ⟨?_, ?_⟩
· rintro ⟨h⟩
let e : P.obj (op X) ≅ multiequalizer (S.index P) :=
h.conePointUniqueUpToIso (limit.isLimit _)
exact (inferInstance : IsIso e.hom)
· intro h
refine ⟨IsLimit.ofIsoLimit (limit.isLimit _) (Cones.ext ?_ ?_)⟩
· ... |
/-
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.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Locus of une... | Mathlib/Data/Finsupp/NeLocus.lean | 159 | 160 | theorem neLocus_self_add_right : neLocus f (f + g) = g.support := by |
rw [← neLocus_zero_left, ← neLocus_add_left f 0 g, add_zero]
|
/-
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.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 355 | 359 | theorem exists_iUnion_eq_diff (π : Prepartition I) :
∃ π' : Prepartition I, π'.iUnion = ↑I \ π.iUnion := by |
rcases π.eventually_splitMany_inf_eq_filter.exists with ⟨s, hs⟩
use (splitMany I s).filter fun J => ¬(J : Set (ι → ℝ)) ⊆ π.iUnion
simp [← hs]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomia... | Mathlib/Algebra/Polynomial/Reverse.lean | 272 | 277 | theorem reverse_natDegree_le (f : R[X]) : f.reverse.natDegree ≤ f.natDegree := by |
rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero]
intro n hn
rw [Nat.cast_lt] at hn
rw [coeff_reverse, revAt, Function.Embedding.coeFn_mk, if_neg (not_le_of_gt hn),
coeff_eq_zero_of_natDegree_lt hn]
|
/-
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.LinearAlgebra.Dual
/-!
# Perfect pairings of modules
A perfect pairing of two (left) modules may be defined either as:
1. A bilinear map `M × N → R` such ... | Mathlib/LinearAlgebra/PerfectPairing.lean | 85 | 89 | theorem apply_apply_toDualRight_symm (x : M) (f : Dual R M) :
(p x) (p.toDualRight.symm f) = f x := by |
have h := LinearEquiv.apply_symm_apply p.toDualRight f
rw [toDualRight_apply] at h
exact congrFun (congrArg DFunLike.coe h) x
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 235 | 237 | theorem fromBlocks_zero [Zero α] : fromBlocks (0 : Matrix n l α) 0 0 (0 : Matrix o m α) = 0 := by |
ext i j
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.Fin... | Mathlib/Analysis/Analytic/Polynomial.lean | 47 | 52 | theorem AnalyticAt.aeval_mvPolynomial (hf : ∀ i, AnalyticAt 𝕜 (f · i) z) (p : MvPolynomial σ A) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by |
apply p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_ -- `refine` doesn't work
· simp_rw [aeval_C]; apply analyticAt_const
· simp_rw [map_add]; exact hp.add hq
· simp_rw [map_mul, aeval_X]; exact hp.mul (hf i)
|
/-
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 | 354 | 367 | theorem integral_piecewise_zero {m : MeasurableSpace α} (f : α →ₛ F) (μ : Measure α) {s : Set α}
(hs : MeasurableSet s) : (piecewise s hs f 0).integral μ = f.integral (μ.restrict s) := by |
classical
refine (integral_eq_sum_of_subset ?_).trans
((sum_congr rfl fun y hy => ?_).trans (integral_eq_sum_filter _ _).symm)
· intro y hy
simp only [mem_filter, mem_range, coe_piecewise, coe_zero, piecewise_eq_indicator,
mem_range_indicator] at *
rcases hy with ⟨⟨rfl, -⟩ | ⟨x, -, rfl⟩, h₀⟩
... |
/-
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.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 281 | 294 | theorem ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring [IsDedekindDomain R] {I J : (Ideal R)⁰} :
ClassGroup.mk0 I =
ClassGroup.mk0 J ↔ ∃ (x : _) (_ : x ≠ (0 : K)), spanSingleton R⁰ x * I = J := by |
refine (ClassGroup.equiv K).injective.eq_iff.symm.trans ?_
simp only [ClassGroup.equiv_mk0, QuotientGroup.mk'_eq_mk', mem_principal_ideals_iff,
Units.ext_iff, Units.val_mul, FractionalIdeal.coe_mk0, exists_prop]
constructor
· rintro ⟨X, ⟨x, hX⟩, hx⟩
refine ⟨x, ?_, ?_⟩
· rintro rfl; simp [X.ne_zero.... |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe... | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 170 | 173 | theorem IsLocalMinOn.hasFDerivWithinAt_eq_zero {s : Set E} (h : IsLocalMinOn f s a)
(hf : HasFDerivWithinAt f f' s a) {y} (hy : y ∈ posTangentConeAt s a)
(hy' : -y ∈ posTangentConeAt s a) : f' y = 0 := by |
simpa using h.neg.hasFDerivWithinAt_eq_zero hf.neg hy hy'
|
/-
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.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
/-!
# Associ... | Mathlib/Data/List/AList.lean | 421 | 422 | theorem extract_eq_lookup_erase (a : α) (s : AList β) : extract a s = (lookup a s, erase a s) := by |
simp [extract]; constructor <;> rfl
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Ring.Ideal
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed_space.units from "leanprover-community/mathl... | Mathlib/Analysis/NormedSpace/Units.lean | 189 | 196 | theorem inverse_add_norm_diff_nth_order (x : Rˣ) (n : ℕ) :
(fun t : R => inverse (↑x + t) - (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹) =O[𝓝 (0 : R)]
fun t => ‖t‖ ^ n := by |
refine EventuallyEq.trans_isBigO (.sub (inverse_add_nth_order x n) (.refl _ _)) ?_
simp only [add_sub_cancel_left]
refine ((isBigO_refl _ _).norm_right.mul (inverse_add_norm x)).trans ?_
simp only [mul_one, isBigO_norm_left]
exact ((isBigO_refl _ _).norm_right.const_mul_left _).pow _
|
/-
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 | 237 | 239 | theorem mk_preimage_prod_left_eq_empty (hb : b ∉ t) : (fun a => (a, b)) ⁻¹' s ×ˢ t = ∅ := by |
ext a
simp [hb]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071a... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 165 | 169 | theorem jacobiSymNat.even_odd₅ (a b c : ℕ) (r : ℤ) (ha : a % 2 = 0) (hb : b % 8 = 5)
(hc : a / 2 = c) (hr : jacobiSymNat c b = r) : jacobiSymNat a b = -r := by |
simp only [jacobiSymNat, ← hr, ← hc, Int.ofNat_ediv, Nat.cast_ofNat]
rw [← jacobiSym.even_odd (mod_cast ha), if_pos (by simp [hb])]
rw [← Nat.mod_mod_of_dvd, hb]; norm_num
|
/-
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 | 2,069 | 2,071 | theorem filter_singleton {a : α} (p : α → Prop) [DecidablePred p] :
filter p {a} = if p a then {a} else ∅ := by |
simp only [singleton, filter_cons, filter_zero, add_zero, empty_eq_zero]
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 77 | 77 | theorem isBounded_bot : IsBounded r ⊥ ↔ Nonempty α := by | simp [IsBounded, exists_true_iff_nonempty]
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 162 | 162 | theorem nonneg_iff {a : SignType} : 0 ≤ a ↔ a = 0 ∨ a = 1 := by | cases a <;> decide
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 199 | 202 | theorem pow_isRoot_minpoly {m : ℕ} (hcop : Nat.Coprime m n) :
IsRoot (map (Int.castRingHom K) (minpoly ℤ μ)) (μ ^ m) := by |
simp only [minpoly_eq_pow_coprime h hcop, IsRoot.def, eval_map]
exact minpoly.aeval ℤ (μ ^ m)
|
/-
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
-/
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"... | Mathlib/LinearAlgebra/Isomorphisms.lean | 67 | 70 | theorem comap_leq_ker_subToSupQuotient (p p' : Submodule R M) :
comap (Submodule.subtype p) (p ⊓ p') ≤ ker (subToSupQuotient p p') := by |
rw [LinearMap.ker_comp, Submodule.inclusion, comap_codRestrict, ker_mkQ, map_comap_subtype]
exact comap_mono (inf_le_inf_right _ le_sup_left)
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theory.group_action.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-... | Mathlib/GroupTheory/GroupAction/Support.lean | 58 | 63 | theorem Supports.smul (g : H) (h : Supports G s b) : Supports G (g • s) (g • b) := by |
rintro g' hg'
rw [smul_comm, h]
rintro a ha
have := Set.forall_mem_image.1 hg' ha
rwa [smul_comm, smul_left_cancel_iff] at this
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 433 | 438 | theorem iInf_ker_truncate : ⨅ i : ℕ, RingHom.ker (@WittVector.truncate p _ i R _) = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
ext
simp only [WittVector.mem_ker_truncate, Ideal.mem_iInf, WittVector.zero_coeff] at hx ⊢
exact hx _ _ (Nat.lt_succ_self _)
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 818 | 824 | theorem _root_.HasCompactSupport.integral_Ioi_deriv_eq (hf : ContDiff ℝ 1 f)
(h2f : HasCompactSupport f) (b : ℝ) : ∫ x in Ioi b, deriv f x = - f b := by |
have := fun x (_ : x ∈ Ioi b) ↦ hf.differentiable le_rfl x |>.hasDerivAt
rw [integral_Ioi_of_hasDerivAt_of_tendsto hf.continuous.continuousWithinAt this, zero_sub]
· refine hf.continuous_deriv le_rfl |>.integrable_of_hasCompactSupport h2f.deriv |>.integrableOn
rw [hasCompactSupport_iff_eventuallyEq, Filter.coc... |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,043 | 1,050 | theorem integral_of_hasDerivAt_of_tendsto [CompleteSpace E]
(hderiv : ∀ x, HasDerivAt f (f' x) x) (hf' : Integrable f')
(hbot : Tendsto f atBot (𝓝 m)) (htop : Tendsto f atTop (𝓝 n)) : ∫ x, f' x = n - m := by |
rw [← integral_univ, ← Set.Iic_union_Ioi (a := 0),
integral_union (Iic_disjoint_Ioi le_rfl) measurableSet_Ioi hf'.integrableOn hf'.integrableOn,
integral_Iic_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn hbot,
integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ ↦ hderiv x) hf'.integrableOn ... |
/-
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.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,373 | 1,373 | theorem iSup_univ {f : β → α} : ⨆ x ∈ (univ : Set β), f x = ⨆ x, f x := by | simp
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_im... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 265 | 270 | theorem dist_coe_le (z w : ℍ) : dist (z : ℂ) w ≤ w.im * (Real.exp (dist z w) - 1) :=
calc
dist (z : ℂ) w ≤ dist (z : ℂ) (w.center (dist z w)) + dist (w : ℂ) (w.center (dist z w)) :=
dist_triangle_right _ _ _
_ = w.im * (Real.exp (dist z w) - 1) := by |
rw [dist_center_dist, dist_self_center, ← mul_add, ← add_sub_assoc, Real.sinh_add_cosh]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sebastien Gouezel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Ba... | Mathlib/Topology/VectorBundle/Basic.lean | 354 | 357 | theorem coordChangeL_apply' (e e' : Trivialization F (π F E)) [e.IsLinear R] [e'.IsLinear R] {b : B}
(hb : b ∈ e.baseSet ∩ e'.baseSet) (y : F) :
coordChangeL R e e' b y = (e' (e.toPartialHomeomorph.symm (b, y))).2 := by |
rw [e.coordChangeL_apply e' hb, e.mk_symm hb.1]
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Mo... | Mathlib/Data/Nat/Nth.lean | 314 | 322 | theorem filter_range_nth_eq_insert {k : ℕ}
(hlt : ∀ hf : (setOf p).Finite, k + 1 < hf.toFinset.card) :
(range (nth p (k + 1))).filter p = insert (nth p k) ((range (nth p k)).filter p) := by |
refine (filter_range_nth_subset_insert p k).antisymm fun a ha => ?_
simp only [mem_insert, mem_filter, mem_range] at ha ⊢
have : nth p k < nth p (k + 1) := nth_lt_nth' k.lt_succ_self hlt
rcases ha with (rfl | ⟨hlt, hpa⟩)
· exact ⟨this, nth_mem _ fun hf => k.lt_succ_self.trans (hlt hf)⟩
· exact ⟨hlt.trans t... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,264 | 1,265 | theorem vectorSpan_pair_rev (p₁ p₂ : P) : vectorSpan k ({p₁, p₂} : Set P) = k ∙ p₂ -ᵥ p₁ := by |
rw [pair_comm, vectorSpan_pair]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-comm... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 88 | 104 | theorem hasFiniteIntegral_compProd_iff ⦃f : β × γ → E⦄ (h1f : StronglyMeasurable f) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by |
simp only [HasFiniteIntegral]
rw [kernel.lintegral_compProd _ _ _ h1f.ennnorm]
have : ∀ x, ∀ᵐ y ∂η (a, x), 0 ≤ ‖f (x, y)‖ := fun x => eventually_of_forall fun y => norm_nonneg _
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
... |
/-
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.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Completion
import Mathlib.Topology.Metri... | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 576 | 648 | theorem ghDist_le_of_approx_subsets {s : Set X} (Φ : s → Y) {ε₁ ε₂ ε₃ : ℝ}
(hs : ∀ x : X, ∃ y ∈ s, dist x y ≤ ε₁) (hs' : ∀ x : Y, ∃ y : s, dist x (Φ y) ≤ ε₃)
(H : ∀ x y : s, |dist x y - dist (Φ x) (Φ y)| ≤ ε₂) : ghDist X Y ≤ ε₁ + ε₂ / 2 + ε₃ := by |
refine le_of_forall_pos_le_add fun δ δ0 => ?_
rcases exists_mem_of_nonempty X with ⟨xX, _⟩
rcases hs xX with ⟨xs, hxs, Dxs⟩
have sne : s.Nonempty := ⟨xs, hxs⟩
letI : Nonempty s := sne.to_subtype
have : 0 ≤ ε₂ := le_trans (abs_nonneg _) (H ⟨xs, hxs⟩ ⟨xs, hxs⟩)
have : ∀ p q : s, |dist p q - dist (Φ p) (Φ q... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Michael Stoll
-/
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import number_theory.l_series from "leanprover-community/ma... | Mathlib/NumberTheory/LSeries/Basic.lean | 166 | 168 | theorem LSeriesSummable_zero {s : ℂ} : LSeriesSummable 0 s := by |
simp only [LSeriesSummable, funext (term_def 0 s), Pi.zero_apply, zero_div, ite_self,
summable_zero]
|
/-
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 | 1,117 | 1,130 | theorem mul_finprod_cond_ne (a : α) (hf : (mulSupport f).Finite) :
(f a * ∏ᶠ (i) (_ : i ≠ a), f i) = ∏ᶠ i, f i := by |
classical
rw [finprod_eq_prod _ hf]
have h : ∀ x : α, f x ≠ 1 → (x ≠ a ↔ x ∈ hf.toFinset \ {a}) := by
intro x hx
rw [Finset.mem_sdiff, Finset.mem_singleton, Finite.mem_toFinset, mem_mulSupport]
exact ⟨fun h => And.intro hx h, fun h => h.2⟩
rw [finprod_cond_eq_prod_of_cond_iff f (fun hx ... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 619 | 621 | theorem inner_neg_left (x y : E) : ⟪-x, y⟫ = -⟪x, y⟫ := by |
rw [← neg_one_smul 𝕜 x, inner_smul_left]
simp
|
/-
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 | 986 | 997 | theorem ball_norm_mul_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r := by |
rcases eq_or_ne k 0 with (rfl | hk)
· rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl]
exact empty_subset _
· intro x
rw [Set.mem_smul_set, Seminorm.mem_ball_zero]
refine fun hx => ⟨k⁻¹ • x, ?_, ?_⟩
· rwa [Seminorm.mem_ball_zero, map_smul_eq_mul, norm_inv, ←
mul_lt_mul_left <| norm_pos... |
/-
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.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,421 | 1,429 | theorem ContDiffWithinAt.sum {ι : Type*} {f : ι → E → F} {s : Finset ι} {t : Set E} {x : E}
(h : ∀ i ∈ s, ContDiffWithinAt 𝕜 n (fun x => f i x) t x) :
ContDiffWithinAt 𝕜 n (fun x => ∑ i ∈ s, f i x) t x := by |
classical
induction' s using Finset.induction_on with i s is IH
· simp [contDiffWithinAt_const]
· simp only [is, Finset.sum_insert, not_false_iff]
exact (h _ (Finset.mem_insert_self i s)).add
(IH fun j hj => h _ (Finset.mem_insert_of_mem hj))
|
/-
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
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_impor... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,444 | 1,447 | theorem card_typein_lt (r : α → α → Prop) [IsWellOrder α r] (x : α) (h : ord #α = type r) :
card (typein r x) < #α := by |
rw [← lt_ord, h]
apply typein_lt_type
|
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 41 | 43 | theorem coeff_derivativeFun (f : R⟦X⟧) (n : ℕ) :
coeff R n f.derivativeFun = coeff R (n + 1) f * (n + 1) := by |
rw [derivativeFun, coeff_mk]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
import Mathli... | Mathlib/RingTheory/GradedAlgebra/Basic.lean | 142 | 145 | theorem coe_decompose_mul_add_of_right_mem [AddRightCancelMonoid ι] [GradedRing 𝒜] {a b : A}
(b_mem : b ∈ 𝒜 j) : (decompose 𝒜 (a * b) (i + j) : A) = decompose 𝒜 a i * b := by |
lift b to 𝒜 j using b_mem
rw [decompose_mul, decompose_coe, coe_mul_of_apply_add]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprove... | Mathlib/Analysis/Complex/ReImTopology.lean | 188 | 190 | theorem frontier_setOf_le_re_and_le_im (a b : ℝ) :
frontier { z | a ≤ re z ∧ b ≤ im z } = { z | a ≤ re z ∧ im z = b ∨ re z = a ∧ b ≤ im z } := by |
simpa only [closure_Ici, frontier_Ici] using frontier_reProdIm (Ici a) (Ici b)
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 86 | 88 | theorem cons_one {α : Fin (n + 2) → Type*} (x : α 0) (p : ∀ i : Fin n.succ, α i.succ) :
cons x p 1 = p 0 := by |
rw [← cons_succ x p]; rfl
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 990 | 993 | theorem div_right_injective : Function.Injective fun a ↦ b / a := by |
-- FIXME see above
simp only [div_eq_mul_inv]
exact fun a a' h ↦ inv_injective (mul_right_injective b h)
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 423 | 447 | theorem id_comp (p : FormalMultilinearSeries 𝕜 E F) (h : p 0 = 0) : (id 𝕜 F).comp p = p := by |
ext1 n
by_cases hn : n = 0
· rw [hn, h]
ext v
rw [comp_coeff_zero', id_apply_ne_one _ _ zero_ne_one]
rfl
· dsimp [FormalMultilinearSeries.comp]
have n_pos : 0 < n := bot_lt_iff_ne_bot.mpr hn
rw [Finset.sum_eq_single (Composition.single n n_pos)]
· show compAlongComposition (id 𝕜 F) p (... |
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib... | Mathlib/Topology/UrysohnsLemma.lean | 386 | 403 | theorem exists_continuous_one_zero_of_isCompact [RegularSpace X] [LocallyCompactSpace X]
{s t : Set X} (hs : IsCompact s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C(X, ℝ), EqOn f 1 s ∧ EqOn f 0 t ∧ HasCompactSupport f ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 := by |
obtain ⟨k, k_comp, k_closed, sk, kt⟩ : ∃ k, IsCompact k ∧ IsClosed k ∧ s ⊆ interior k ∧ k ⊆ tᶜ :=
exists_compact_closed_between hs ht.isOpen_compl hd.symm.subset_compl_left
rcases exists_continuous_zero_one_of_isCompact hs isOpen_interior.isClosed_compl
(disjoint_compl_right_iff_subset.mpr sk) with ⟨⟨f, hf... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,134 | 1,137 | theorem re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four (x y : E) :
re ⟪x, y⟫ = (‖x + y‖ * ‖x + y‖ - ‖x - y‖ * ‖x - y‖) / 4 := by |
rw [@norm_add_mul_self 𝕜, @norm_sub_mul_self 𝕜]
ring
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 1,106 | 1,108 | theorem convex_closedBall : Convex ℝ (closedBall p x r) := by |
rw [closedBall_eq_biInter_ball]
exact convex_iInter₂ fun _ _ => convex_ball _ _ _
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 450 | 477 | theorem bound (hN : 4096 ≤ N) : (N : ℝ) ^ (nValue N : ℝ)⁻¹ / exp 1 < dValue N := by |
apply div_lt_floor _
rw [← log_le_log_iff, log_rpow, mul_comm, ← div_eq_mul_inv]
· apply le_trans _ (div_le_div_of_nonneg_left _ _ (ceil_lt_mul _).le)
· rw [mul_comm, ← div_div, div_sqrt, le_div_iff]
· set_option tactic.skipAssignedInstances false in norm_num; exact le_sqrt_log hN
· norm_num1
... |
/-
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 | 3,254 | 3,254 | theorem map₂Right'_nil_left : map₂Right' f [] bs = (bs.map (f none), []) := by | cases bs <;> rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 239 | 240 | theorem measure_diff (h : s₂ ⊆ s₁) (h₂ : MeasurableSet s₂) (h_fin : μ s₂ ≠ ∞) :
μ (s₁ \ s₂) = μ s₁ - μ s₂ := by | rw [measure_diff' _ h₂ h_fin, union_eq_self_of_subset_right h]
|
/-
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.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 116 | 120 | theorem tendsto_toNNReal {a : ℝ≥0∞} (ha : a ≠ ∞) :
Tendsto ENNReal.toNNReal (𝓝 a) (𝓝 a.toNNReal) := by |
lift a to ℝ≥0 using ha
rw [nhds_coe, tendsto_map'_iff]
exact tendsto_id
|
/-
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, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 327 | 328 | theorem finite_int_prime_iff {a : ℤ} : Finite (p : ℤ) a ↔ a ≠ 0 := by |
simp [finite_int_iff, hp.1.ne_one]
|
/-
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.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 501 | 516 | theorem integral_sub_zpow_of_ne {n : ℤ} (hn : n ≠ -1) (c w : ℂ) (R : ℝ) :
(∮ z in C(c, R), (z - w) ^ n) = 0 := by |
rcases em (w ∈ sphere c |R| ∧ n < -1) with (⟨hw, hn⟩ | H)
· exact integral_sub_zpow_of_undef (hn.trans (by decide)) hw
push_neg at H
have hd : ∀ z, z ≠ w ∨ -1 ≤ n →
HasDerivAt (fun z => (z - w) ^ (n + 1) / (n + 1)) ((z - w) ^ n) z := by
intro z hne
convert ((hasDerivAt_zpow (n + 1) _ (hne.imp _ _... |
/-
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 | 749 | 752 | theorem splitWrtComposition_join (L : List (List α)) (c : Composition L.join.length)
(h : map length L = c.blocks) : splitWrtComposition (join L) c = L := by |
simp only [eq_self_iff_true, and_self_iff, eq_iff_join_eq, join_splitWrtComposition,
map_length_splitWrtComposition, h]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 330 | 331 | theorem preimage_const_sub_Ico : (fun x => a - x) ⁻¹' Ico b c = Ioc (a - c) (a - b) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, inter_comm]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 202 | 203 | theorem IsLittleO.eventuallyLE (h : f =o[l] g) : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖ := by |
simpa using h.def zero_lt_one
|
/-
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 | 1,211 | 1,213 | theorem castPred_le_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ last n) :
castPred i hi ≤ j ↔ i ≤ castSucc j := by |
rw [← castSucc_le_castSucc_iff, castSucc_castPred]
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 140 | 140 | theorem zero'_eval : zero'.eval = fun v => pure (0 :: v) := by | simp [eval]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
/-!
# Cycles of a li... | Mathlib/Data/List/Cycle.lean | 605 | 610 | theorem Subsingleton.congr {s : Cycle α} (h : Subsingleton s) :
∀ ⦃x⦄ (_hx : x ∈ s) ⦃y⦄ (_hy : y ∈ s), x = y := by |
induction' s using Quot.inductionOn with l
simp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff,
length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or_iff] at h
rcases h with (rfl | ⟨z, rfl⟩) <;> simp
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 153 | 155 | theorem spanSingleton_div_self {x : K} (hx : x ≠ 0) :
spanSingleton R₁⁰ x / spanSingleton R₁⁰ x = 1 := by |
rw [spanSingleton_div_spanSingleton, div_self hx, spanSingleton_one]
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.Separation
import... | Mathlib/Topology/UniformSpace/Compact.lean | 159 | 164 | theorem CompactSpace.uniformContinuous_of_continuous [CompactSpace α] {f : α → β}
(h : Continuous f) : UniformContinuous f :=
calc map (Prod.map f f) (𝓤 α)
= map (Prod.map f f) (𝓝ˢ (diagonal α)) := by | rw [nhdsSet_diagonal_eq_uniformity]
_ ≤ 𝓝ˢ (diagonal β) := (h.prod_map h).tendsto_nhdsSet mapsTo_prod_map_diagonal
_ ≤ 𝓤 β := nhdsSet_diagonal_le_uniformity
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Init.Data.Int.Order
import Mathlib.Order.Compare
import Mathlib.Order.Max
import Math... | Mathlib/Order/Monotone/Basic.lean | 587 | 593 | theorem injective_of_lt_imp_ne [LinearOrder α] {f : α → β} (h : ∀ x y, x < y → f x ≠ f y) :
Injective f := by |
intro x y hf
rcases lt_trichotomy x y with (hxy | rfl | hxy)
· exact absurd hf <| h _ _ hxy
· rfl
· exact absurd hf.symm <| h _ _ hxy
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 2,317 | 2,321 | theorem ContinuousLinearMap.reApplyInnerSelf_smul (T : E →L[𝕜] E) (x : E) {c : 𝕜} :
T.reApplyInnerSelf (c • x) = ‖c‖ ^ 2 * T.reApplyInnerSelf x := by |
simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.reApplyInnerSelf_apply,
inner_smul_left, inner_smul_right, ← mul_assoc, mul_conj, ← ofReal_pow, ← smul_re,
Algebra.smul_def (‖c‖ ^ 2) ⟪T x, x⟫, algebraMap_eq_ofReal]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-com... | Mathlib/Tactic/NormNum/Prime.lean | 67 | 82 | theorem minFacHelper_1 {n k k' : ℕ} (e : k + 2 = k') (h : MinFacHelper n k)
(np : minFac n ≠ k) : MinFacHelper n k' := by |
rw [← e]
refine ⟨Nat.lt_add_right _ h.1, ?_, ?_⟩
· rw [add_mod, mod_self, add_zero, mod_mod]
exact h.2.1
rcases h.2.2.eq_or_lt with rfl|h2
· exact (np rfl).elim
rcases (succ_le_of_lt h2).eq_or_lt with h2|h2
· refine ((h.1.trans_le h.2.2).ne ?_).elim
have h3 : 2 ∣ minFac n := by
rw [Nat.dvd_... |
/-
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.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 642 | 643 | theorem Perm.subtypeCongr.right_apply {a : ε} (h : ¬p a) : ep.subtypeCongr en a = en ⟨a, h⟩ := by |
simp [Perm.subtypeCongr.apply, h]
|
/-
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, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Algebra.Module.Basic
import Mathlib.Topology.Separation
#align_import topology.supp... | Mathlib/Topology/Support.lean | 390 | 418 | theorem LocallyFinite.exists_finset_nhd_mulSupport_subset {U : ι → Set X} [One R] {f : ι → X → R}
(hlf : LocallyFinite fun i => mulSupport (f i)) (hso : ∀ i, mulTSupport (f i) ⊆ U i)
(ho : ∀ i, IsOpen (U i)) (x : X) :
∃ (is : Finset ι), ∃ n, n ∈ 𝓝 x ∧ (n ⊆ ⋂ i ∈ is, U i) ∧
∀ z ∈ n, (mulSupport fun i ... |
obtain ⟨n, hn, hnf⟩ := hlf x
classical
let is := hnf.toFinset.filter fun i => x ∈ U i
let js := hnf.toFinset.filter fun j => x ∉ U j
refine
⟨is, (n ∩ ⋂ j ∈ js, (mulTSupport (f j))ᶜ) ∩ ⋂ i ∈ is, U i, inter_mem (inter_mem hn ?_) ?_,
inter_subset_right, fun z hz => ?_⟩
· exact (biInter_f... |
/-
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 | 230 | 235 | theorem Surjective.lieAlgebra_isSolvable [h₁ : IsSolvable R L'] (h₂ : Surjective f) :
IsSolvable R L := by |
obtain ⟨k, hk⟩ := id h₁
use k
rw [← LieIdeal.derivedSeries_map_eq k h₂, hk]
simp only [LieIdeal.map_eq_bot_iff, bot_le]
|
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 57 | 60 | theorem Hom.cast_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
HEq (e.cast hu hv) e := by |
subst_vars
rfl
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import ... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 118 | 133 | theorem exists_immersion_euclidean [Finite ι] (f : SmoothBumpCovering ι I M) :
∃ (n : ℕ) (e : M → EuclideanSpace ℝ (Fin n)),
Smooth I (𝓡 n) e ∧ Injective e ∧ ∀ x : M, Injective (mfderiv I (𝓡 n) e x) := by |
cases nonempty_fintype ι
set F := EuclideanSpace ℝ (Fin <| finrank ℝ (ι → E × ℝ))
letI : IsNoetherian ℝ (E × ℝ) := IsNoetherian.iff_fg.2 inferInstance
letI : FiniteDimensional ℝ (ι → E × ℝ) := IsNoetherian.iff_fg.1 inferInstance
set eEF : (ι → E × ℝ) ≃L[ℝ] F :=
ContinuousLinearEquiv.ofFinrankEq finrank_e... |
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,088 | 1,091 | theorem tendsto_tan_pi_div_two : Tendsto tan (𝓝[<] (π / 2)) atTop := by |
convert tendsto_cos_pi_div_two.inv_tendsto_zero.atTop_mul zero_lt_one tendsto_sin_pi_div_two
using 1
simp only [Pi.inv_apply, ← div_eq_inv_mul, ← tan_eq_sin_div_cos]
|
/-
Copyright (c) 2021 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_t... | Mathlib/GroupTheory/DoubleCoset.lean | 44 | 45 | theorem mem_doset {s t : Set α} {a b : α} : b ∈ doset a s t ↔ ∃ x ∈ s, ∃ y ∈ t, b = x * a * y := by |
simp only [doset_eq_image2, Set.mem_image2, eq_comm]
|
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 330 | 333 | theorem toMeasure_bindOnSupport_apply [MeasurableSpace β] (hs : MeasurableSet s) :
(p.bindOnSupport f).toMeasure s =
∑' a, p a * if h : p a = 0 then 0 else (f a h).toMeasure s := by |
simp only [toMeasure_apply_eq_toOuterMeasure_apply _ _ hs, toOuterMeasure_bindOnSupport_apply]
|
/-
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 | 1,688 | 1,690 | theorem norm_setToFun_le (hT : DominatedFinMeasAdditive μ T C) (hf : Integrable f μ) (hC : 0 ≤ C) :
‖setToFun μ T hT f‖ ≤ C * ‖hf.toL1 f‖ := by |
rw [setToFun_eq hT hf]; exact L1.norm_setToL1_le_mul_norm hT hC _
|
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.... | Mathlib/Topology/Algebra/Order/Field.lean | 228 | 233 | theorem nhdsWithin_pos_comap_mul_left {x : 𝕜} (hx : 0 < x) :
comap (x * ·) (𝓝[>] 0) = 𝓝[>] 0 := by |
rw [nhdsWithin, comap_inf, comap_principal, preimage_const_mul_Ioi _ hx, zero_div]
congr 1
refine ((Homeomorph.mulLeft₀ x hx.ne').comap_nhds_eq _).trans ?_
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.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 451 | 452 | theorem rpow_sub_one {x : ℝ} (hx : x ≠ 0) (y : ℝ) : x ^ (y - 1) = x ^ y / x := by |
simpa using rpow_sub_nat hx y 1
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,420 | 1,430 | theorem copy_ok (q s a b c d) :
Reaches₁ (TM2.step tr) ⟨some (Λ'.copy q), s, K'.elim a b c d⟩
⟨some q, none, K'.elim (List.reverseAux b a) [] c (List.reverseAux b d)⟩ := by |
induction' b with x b IH generalizing a d s
· refine TransGen.single ?_
simp
refine TransGen.head rfl ?_
simp only [TM2.step, Option.mem_def, TM2.stepAux, elim_rev, List.head?_cons, Option.isSome_some,
List.tail_cons, elim_update_rev, ne_eq, Function.update_noteq, elim_main, elim_update_main,
elim_... |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Ring.Commute
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f... | Mathlib/Data/Nat/Cast/Commute.lean | 24 | 27 | theorem cast_commute (n : ℕ) (x : α) : Commute (n : α) x := by |
induction n with
| zero => rw [Nat.cast_zero]; exact Commute.zero_left x
| succ n ihn => rw [Nat.cast_succ]; exact ihn.add_left (Commute.one_left x)
|
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_th... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 293 | 296 | theorem apply_lt_nfpBFamily (H : ∀ i hi, IsNormal (f i hi)) {a b} (hb : b < nfpBFamily.{u, v} o f a)
(i hi) : f i hi b < nfpBFamily.{u, v} o f a := by |
rw [← familyOfBFamily_enum o f]
apply apply_lt_nfpFamily (fun _ => H _ _) hb
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-c... | Mathlib/RingTheory/Discriminant.lean | 154 | 157 | theorem discr_eq_det_embeddingsMatrixReindex_pow_two [IsSeparable K L] (e : ι ≃ (L →ₐ[K] E)) :
algebraMap K E (discr K b) = (embeddingsMatrixReindex K E b e).det ^ 2 := by |
rw [discr_def, RingHom.map_det, RingHom.mapMatrix_apply,
traceMatrix_eq_embeddingsMatrixReindex_mul_trans, det_mul, det_transpose, pow_two]
|
/-
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.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,379 | 1,381 | theorem iSup_union {f : β → α} {s t : Set β} :
⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x := by |
simp_rw [mem_union, iSup_or, iSup_sup_eq]
|
/-
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.UniformConvergence
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 344 | 349 | theorem uniformContinuous_eval (x : α) :
UniformContinuous (Function.eval x ∘ toFun : (α →ᵤ β) → β) := by |
change _ ≤ _
rw [map_le_iff_le_comap,
(UniformFun.hasBasis_uniformity α β).le_basis_iff ((𝓤 _).basis_sets.comap _)]
exact fun U hU => ⟨U, hU, fun uv huv => huv x⟩
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprov... | Mathlib/ModelTheory/Ultraproducts.lean | 83 | 91 | theorem term_realize_cast {β : Type*} (x : β → ∀ a, M a) (t : L.Term β) :
(t.realize fun i => (x i : (u : Filter α).Product M)) =
(fun a => t.realize fun i => x i a : (u : Filter α).Product M) := by |
convert @Term.realize_quotient_mk' L _ ((u : Filter α).productSetoid M)
(Ultraproduct.setoidPrestructure M u) _ t x using 2
ext a
induction t with
| var => rfl
| func _ _ t_ih => simp only [Term.realize, t_ih]; rfl
|
/-
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.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 295 | 300 | theorem aeval_algHom_apply {F : Type*} [FunLike F A B] [AlgHomClass F R A B]
(f : F) (x : A) (p : R[X]) :
aeval (f x) p = f (aeval x p) := by |
refine Polynomial.induction_on p (by simp [AlgHomClass.commutes]) (fun p q hp hq => ?_)
(by simp [AlgHomClass.commutes])
rw [map_add, hp, hq, ← map_add, ← map_add]
|
/-
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.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,478 | 1,518 | theorem integral_comp_smul_deriv''' {f f' : ℝ → ℝ} {g : ℝ → G} (hf : ContinuousOn f [[a, b]])
(hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x)
(hg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))) (hg1 : IntegrableOn g (f '' [[a, b]]))
(hg2 : IntegrableOn (fun x => f' x • (g ∘... |
by_cases hG : CompleteSpace G; swap
· simp [intervalIntegral, integral, hG]
rw [hf.image_uIcc, ← intervalIntegrable_iff'] at hg1
have h_cont : ContinuousOn (fun u => ∫ t in f a..f u, g t) [[a, b]] := by
refine (continuousOn_primitive_interval' hg1 ?_).comp hf ?_
· rw [← hf.image_uIcc]; exact mem_image_... |
/-
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.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 323 | 326 | theorem ClassGroup.integralRep_mem_nonZeroDivisors
{I : FractionalIdeal R⁰ (FractionRing R)} (hI : I ≠ 0) :
I.num ∈ (Ideal R)⁰ := by |
rwa [mem_nonZeroDivisors_iff_ne_zero, ne_eq, FractionalIdeal.num_eq_zero_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 930 | 932 | theorem LocallyFinite.finite_nonempty_of_compact [CompactSpace X] {f : ι → Set X}
(hf : LocallyFinite f) : { i | (f i).Nonempty }.Finite := by |
simpa only [inter_univ] using hf.finite_nonempty_inter_compact isCompact_univ
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology... | Mathlib/Topology/TietzeExtension.lean | 96 | 100 | theorem ContinuousMap.exists_forall_mem_restrict_eq {Y : Type v} [TopologicalSpace Y] (f : C(s, Y))
{t : Set Y} (hf : ∀ x, f x ∈ t) [ht : TietzeExtension.{u, v} t] :
∃ (g : C(X, Y)), (∀ x, g x ∈ t) ∧ g.restrict s = f := by |
obtain ⟨g, hg⟩ := mk _ (map_continuous f |>.codRestrict hf) |>.exists_restrict_eq hs
exact ⟨comp ⟨Subtype.val, by continuity⟩ g, by simp, by ext x; congrm(($(hg) x : Y))⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,790 | 2,793 | theorem filter_insert (a : α) (s : Finset α) :
filter p (insert a s) = if p a then insert a (filter p s) else filter p s := by |
ext x
split_ifs with h <;> by_cases h' : x = a <;> simp [h, h']
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.extr from "leanprover-community/mathlib"@"1f00... | Mathlib/Order/Filter/Extr.lean | 510 | 512 | theorem IsMinOn.sub (hf : IsMinOn f s a) (hg : IsMaxOn g s a) :
IsMinOn (fun x => f x - g x) s a := by |
simpa only [sub_eq_add_neg] using hf.add hg.neg
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.