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) 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 | 1,535 | 1,536 | theorem lt_one_iff_zero {c : Cardinal} : c < 1 ↔ c = 0 := by |
simpa using lt_succ_bot_iff (a := c)
|
/-
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 | 649 | 653 | theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree := by |
apply le_antisymm
· exact Finset.sup_mono (normalizeScaleRoots_support p)
· rw [← degree_scaleRoots, ← leadingCoeff_smul_normalizeScaleRoots]
exact degree_smul_le _ _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,481 | 2,490 | theorem exists_smul_of_dvd_count (s : Multiset α) {k : ℕ}
(h : ∀ a : α, a ∈ s → k ∣ Multiset.count a s) : ∃ u : Multiset α, s = k • u := by |
use ∑ a ∈ s.toFinset, (s.count a / k) • {a}
have h₂ :
(∑ x ∈ s.toFinset, k • (count x s / k) • ({x} : Multiset α)) =
∑ x ∈ s.toFinset, count x s • {x} := by
apply Finset.sum_congr rfl
intro x hx
rw [← mul_nsmul', Nat.mul_div_cancel' (h x (mem_toFinset.mp hx))]
rw [← Finset.sum_nsmul, h₂, to... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 315 | 326 | theorem choose_le_middle (r n : ℕ) : choose n r ≤ choose n (n / 2) := by |
cases' le_or_gt r n with b b
· rcases le_or_lt r (n / 2) with a | h
· apply choose_le_middle_of_le_half_left a
· rw [← choose_symm b]
apply choose_le_middle_of_le_half_left
rw [div_lt_iff_lt_mul' Nat.zero_lt_two] at h
rw [le_div_iff_mul_le' Nat.zero_lt_two, Nat.mul_sub_right_distrib, Nat.... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 1,392 | 1,395 | theorem toSeq_ofSeq (s : Seq α) : toSeq (ofSeq s) = s := by |
apply Subtype.eq; funext n
dsimp [toSeq]; apply get_eq_of_mem
rw [get?_ofSeq]; apply ret_mem
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 316 | 318 | theorem polar_self (x : M) : polar Q x x = 2 * Q x := by |
rw [polar, map_add_self, sub_sub, sub_eq_iff_eq_add, ← two_mul, ← two_mul, ← mul_assoc]
norm_num
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 708 | 723 | theorem ae_tendsto_rnDeriv :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x)) := by |
let t := μ.withDensity (ρ.rnDeriv μ)
have eq_add : ρ = ρ.singularPart μ + t := haveLebesgueDecomposition_add _ _
have A : ∀ᵐ x ∂μ, Tendsto (fun a => ρ.singularPart μ a / μ a) (v.filterAt x) (𝓝 0) :=
v.ae_eventually_measure_zero_of_singular (mutuallySingular_singularPart ρ μ)
have B : ∀ᵐ x ∂μ, t.rnDeriv μ ... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Nor... | Mathlib/Analysis/Convex/Normed.lean | 92 | 97 | theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by |
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e... | Mathlib/Analysis/Analytic/Inverse.lean | 202 | 228 | theorem comp_rightInv_aux1 {n : ℕ} (hn : 0 < n) (p : FormalMultilinearSeries 𝕜 E F)
(q : FormalMultilinearSeries 𝕜 F E) (v : Fin n → F) :
p.comp q n v =
∑ c ∈ {c : Composition n | 1 < c.length}.toFinset,
p c.length (q.applyComposition c v) +
p 1 fun _ => q n v := by |
have A :
(Finset.univ : Finset (Composition n)) =
{c | 1 < Composition.length c}.toFinset ∪ {Composition.single n hn} := by
refine Subset.antisymm (fun c _ => ?_) (subset_univ _)
by_cases h : 1 < c.length
· simp [h, Set.mem_toFinset (s := {c | 1 < Composition.length c})]
· have : c.length =... |
/-
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.VectorMeasure
import Mathlib.Order.SymmDiff
#align_import measure_theory.decomposition.signed_hahn from "leanprover-community/mathlib"... | Mathlib/MeasureTheory/Decomposition/SignedHahn.lean | 266 | 328 | theorem exists_subset_restrict_nonpos (hi : s i < 0) :
∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by |
have hi₁ : MeasurableSet i := by_contradiction fun h => ne_of_lt hi <| s.not_measurable h
by_cases h : s ≤[i] 0; · exact ⟨i, hi₁, Set.Subset.refl _, h, hi⟩
by_cases hn : ∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0
swap; · exact exists_subset_restrict_nonpos' hi₁ hi hn
set A := i \ ⋃ l, restrictNonp... |
/-
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 | 513 | 515 | theorem all_node4R {P l x m y r} :
@All α P (node4R l x m y r) ↔ All P l ∧ P x ∧ All P m ∧ P y ∧ All P r := by |
cases m <;> simp [node4R, all_node', All, all_node3R, and_assoc]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,386 | 2,395 | theorem natCast_div (m n : ℕ) : ((m / n : ℕ) : Ordinal) = m / n := by |
rcases eq_or_ne n 0 with (rfl | hn)
· simp
· have hn' := natCast_ne_zero.2 hn
apply le_antisymm
· rw [le_div hn', ← natCast_mul, natCast_le, mul_comm]
apply Nat.div_mul_le_self
· rw [div_le hn', ← add_one_eq_succ, ← Nat.cast_succ, ← natCast_mul, natCast_lt, mul_comm, ←
Nat.div_lt_iff_lt... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 129 | 129 | theorem measure_union_null (hs : μ s = 0) (ht : μ t = 0) : μ (s ∪ t) = 0 := by | simp [*]
|
/-
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.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import d... | Mathlib/Data/PNat/Factors.lean | 121 | 123 | theorem coePNat_prime (v : PrimeMultiset) (p : ℕ+) (h : p ∈ (v : Multiset ℕ+)) : p.Prime := by |
rcases Multiset.mem_map.mp h with ⟨⟨_, hp'⟩, ⟨_, h_eq⟩⟩
exact h_eq ▸ hp'
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 1,508 | 1,537 | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by |
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
simp [Complex.cos, sub_div, add_div, neg_div, div_self (two_ne_zero' ℂ)]
_ = abs
(((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) +
(Complex... |
/-
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.Calculus.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpac... | Mathlib/Geometry/Manifold/BumpFunction.lean | 246 | 248 | theorem support_updateRIn {r : ℝ} (hr : r ∈ Ioo 0 f.rOut) :
support (f.updateRIn r hr) = support f := by |
simp only [support_eq_inter_preimage, updateRIn_rOut]
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.cha... | Mathlib/RepresentationTheory/Character.lean | 54 | 55 | theorem char_mul_comm (V : FdRep k G) (g : G) (h : G) :
V.character (h * g) = V.character (g * h) := by | simp only [trace_mul_comm, character, map_mul]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 707 | 715 | theorem mem_comp_comp {V W M : Set (β × β)} (hW' : SymmetricRel W) {p : β × β} :
p ∈ V ○ M ○ W ↔ (ball p.1 V ×ˢ ball p.2 W ∩ M).Nonempty := by |
cases' p with x y
constructor
· rintro ⟨z, ⟨w, hpw, hwz⟩, hzy⟩
exact ⟨(w, z), ⟨hpw, by rwa [mem_ball_symmetry hW']⟩, hwz⟩
· rintro ⟨⟨w, z⟩, ⟨w_in, z_in⟩, hwz⟩
rw [mem_ball_symmetry hW'] at z_in
exact ⟨z, ⟨w, w_in, hwz⟩, z_in⟩
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 374 | 384 | theorem AffineIndependent.map' {p : ι → P} (hai : AffineIndependent k p) (f : P →ᵃ[k] P₂)
(hf : Function.Injective f) : AffineIndependent k (f ∘ p) := by |
cases' isEmpty_or_nonempty ι with h h
· haveI := h
apply affineIndependent_of_subsingleton
obtain ⟨i⟩ := h
rw [affineIndependent_iff_linearIndependent_vsub k p i] at hai
simp_rw [affineIndependent_iff_linearIndependent_vsub k (f ∘ p) i, Function.comp_apply, ←
f.linearMap_vsub]
have hf' : LinearMap.... |
/-
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 | 690 | 690 | theorem mem_ball_zero : y ∈ ball p 0 r ↔ p y < r := by | rw [mem_ball, sub_zero]
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 299 | 303 | theorem natSepDegree_ne_zero (h : f.natDegree ≠ 0) : f.natSepDegree ≠ 0 := by |
rw [natSepDegree, ne_eq, Finset.card_eq_zero, ← ne_eq, ← Finset.nonempty_iff_ne_empty]
use rootOfSplits _ (SplittingField.splits f) (ne_of_apply_ne _ h)
rw [Multiset.mem_toFinset, mem_aroots]
exact ⟨ne_of_apply_ne _ h, map_rootOfSplits _ (SplittingField.splits f) (ne_of_apply_ne _ h)⟩
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import ... | Mathlib/Algebra/Polynomial/Inductions.lean | 207 | 228 | theorem natDegree_ne_zero_induction_on {M : R[X] → Prop} {f : R[X]} (f0 : f.natDegree ≠ 0)
(h_C_add : ∀ {a p}, M p → M (C a + p)) (h_add : ∀ {p q}, M p → M q → M (p + q))
(h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (monomial n a)) : M f := by |
suffices f.natDegree = 0 ∨ M f from Or.recOn this (fun h => (f0 h).elim) id
refine Polynomial.induction_on f ?_ ?_ ?_
· exact fun a => Or.inl (natDegree_C _)
· rintro p q (hp | hp) (hq | hq)
· refine Or.inl ?_
rw [eq_C_of_natDegree_eq_zero hp, eq_C_of_natDegree_eq_zero hq, ← C_add, natDegree_C]
·... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 350 | 350 | theorem sinh_ofReal_im (x : ℝ) : (sinh x).im = 0 := by | rw [← ofReal_sinh_ofReal_re, ofReal_im]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.UnitInterval
import Mathlib.Algebra.Star.Subalge... | Mathlib/Topology/ContinuousFunction/Polynomial.lean | 177 | 215 | theorem polynomialFunctions.comap_compRightAlgHom_iccHomeoI (a b : ℝ) (h : a < b) :
(polynomialFunctions I).comap (compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap) =
polynomialFunctions (Set.Icc a b) := by |
ext f
fconstructor
· rintro ⟨p, ⟨-, w⟩⟩
rw [DFunLike.ext_iff] at w
dsimp at w
let q := p.comp ((b - a)⁻¹ • Polynomial.X + Polynomial.C (-a * (b - a)⁻¹))
refine ⟨q, ⟨?_, ?_⟩⟩
· simp
· ext x
simp only [q, neg_mul, RingHom.map_neg, RingHom.map_mul, AlgHom.coe_toRingHom,
Polynom... |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 382 | 383 | theorem sqrt_mul' (x) {y : ℝ} (hy : 0 ≤ y) : √(x * y) = √x * √y := by |
rw [mul_comm, sqrt_mul hy, mul_comm]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 378 | 378 | theorem multichoose_zero_right (n : ℕ) : multichoose n 0 = 1 := by | cases n <;> simp [multichoose]
|
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 226 | 248 | theorem _root_.IsOpen.exists_iUnion_isClosed {U : Set α} (hU : IsOpen U) :
∃ F : ℕ → Set α, (∀ n, IsClosed (F n)) ∧ (∀ n, F n ⊆ U) ∧ ⋃ n, F n = U ∧ Monotone F := by |
obtain ⟨a, a_pos, a_lt_one⟩ : ∃ a : ℝ≥0∞, 0 < a ∧ a < 1 := exists_between zero_lt_one
let F := fun n : ℕ => (fun x => infEdist x Uᶜ) ⁻¹' Ici (a ^ n)
have F_subset : ∀ n, F n ⊆ U := fun n x hx ↦ by
by_contra h
have : infEdist x Uᶜ ≠ 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'
exact this (infEdist... |
/-
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, Patrick Massot
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
/-!
# The dominated convergence t... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 506 | 513 | theorem continuous_primitive (h_int : ∀ a b, IntervalIntegrable f μ a b) (a : ℝ) :
Continuous fun b => ∫ x in a..b, f x ∂μ := by |
rw [continuous_iff_continuousAt]
intro b₀
cases' exists_lt b₀ with b₁ hb₁
cases' exists_gt b₀ with b₂ hb₂
apply ContinuousWithinAt.continuousAt _ (Icc_mem_nhds hb₁ hb₂)
exact continuousWithinAt_primitive (measure_singleton b₀) (h_int _ _)
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.Subsemigroup.Center
import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
/-!
# `NonUnitalSubring... | Mathlib/RingTheory/NonUnitalSubring/Basic.lean | 486 | 486 | theorem range_eq_map (f : R →ₙ+* S) : f.range = NonUnitalSubring.map f ⊤ := by | ext; simp
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 507 | 510 | theorem index_ne_zero_of_finite [hH : Finite (G ⧸ H)] : H.index ≠ 0 := by |
cases nonempty_fintype (G ⧸ H)
rw [index_eq_card]
exact Fintype.card_ne_zero
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 1,396 | 1,406 | theorem mem_of_max {s : Finset α} : ∀ {a : α}, s.max = a → a ∈ s := by |
induction' s using Finset.induction_on with b s _ ih
· intro _ H; cases H
· intro a h
by_cases p : b = a
· induction p
exact mem_insert_self b s
· cases' max_choice (↑b) s.max with q q <;> rw [max_insert, q] at h
· cases h
cases p rfl
· exact mem_insert_of_mem (ih h)
|
/-
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 | 78 | 80 | theorem mem_graph_iff {c : α × M} {f : α →₀ M} : c ∈ f.graph ↔ f c.1 = c.2 ∧ c.2 ≠ 0 := by |
cases c
exact mk_mem_graph_iff
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
/-!... | Mathlib/Algebra/MvPolynomial/Comap.lean | 83 | 87 | theorem comap_eq_id_of_eq_id (f : MvPolynomial σ R →ₐ[R] MvPolynomial σ R) (hf : ∀ φ, f φ = φ)
(x : σ → R) : comap f x = x := by |
convert comap_id_apply x
ext1 φ
simp [hf, AlgHom.id_apply]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Covering.DensityTheorem
#align_import measure_theory.covering.liminf_limsup from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ce... | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | 41 | 150 | theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
(hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
{M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop... |
/- Sketch of proof:
Assume that `p` is identically true for simplicity. Let `Y₁ i = cthickening (r₁ i) (s i)`, define
`Y₂` similarly except using `r₂`, and let `(Z i) = ⋃_{j ≥ i} (Y₂ j)`. Our goal is equivalent to
showing that `μ ((limsup Y₁) \ (Z i)) = 0` for all `i`.
Assume for contradiction that `μ ((li... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.Nonem... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 583 | 586 | theorem len_le_of_epi {x y : SimplexCategory} {f : x ⟶ y} : Epi f → y.len ≤ x.len := by |
intro hyp_f_epi
have f_surj : Function.Surjective f.toOrderHom.toFun := epi_iff_surjective.1 hyp_f_epi
simpa using Fintype.card_le_of_surjective f.toOrderHom.toFun f_surj
|
/-
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.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 264 | 265 | theorem range_smul (f : V →ₗ[K] V₂) (a : K) (h : a ≠ 0) : range (a • f) = range f := by |
simpa only [range_eq_map] using Submodule.map_smul f _ a h
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 354 | 383 | theorem GammaSeq_tendsto_Gamma (s : ℂ) : Tendsto (GammaSeq s) atTop (𝓝 <| Gamma s) := by |
suffices ∀ m : ℕ, -↑m < re s → Tendsto (GammaSeq s) atTop (𝓝 <| GammaAux m s) by
rw [Gamma]
apply this
rw [neg_lt]
rcases lt_or_le 0 (re s) with (hs | hs)
· exact (neg_neg_of_pos hs).trans_le (Nat.cast_nonneg _)
· refine (Nat.lt_floor_add_one _).trans_le ?_
rw [sub_eq_neg_add, Nat.floo... |
/-
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.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra
#align_impo... | Mathlib/Algebra/DirectSum/Internal.lean | 232 | 241 | theorem coe_of_mul_apply_of_not_le {i : ι} (r : A i) (r' : ⨁ i, A i) (n : ι) (h : ¬i ≤ n) :
((of (fun i => A i) i r * r') n : R) = 0 := by |
classical
rw [coe_mul_apply_eq_dfinsupp_sum]
apply (DFinsupp.sum_single_index _).trans
swap
· simp_rw [ZeroMemClass.coe_zero, zero_mul, ite_self]
exact DFinsupp.sum_zero
· rw [DFinsupp.sum, Finset.sum_ite_of_false _ _ fun x _ H => _, Finset.sum_const_zero]
exact fun x _ H => h ((self_... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,977 | 1,983 | theorem isNormal_iff_lt_succ_and_blsub_eq {f : Ordinal.{u} → Ordinal.{max u v}} :
IsNormal f ↔ (∀ a, f a < f (succ a)) ∧
∀ o, IsLimit o → (blsub.{_, v} o fun x _ => f x) = f o := by |
rw [isNormal_iff_lt_succ_and_bsup_eq.{u, v}, and_congr_right_iff]
intro h
constructor <;> intro H o ho <;> have := H o ho <;>
rwa [← bsup_eq_blsub_of_lt_succ_limit ho fun a _ => h a] at *
|
/-
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 | 544 | 545 | theorem eq_ones_iff_le_length {c : Composition n} : c = ones n ↔ n ≤ c.length := by |
simp [eq_ones_iff_length, le_antisymm_iff, c.length_le]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 376 | 379 | theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ := by |
simp only [coeff, Finsupp.single_add, add_comm n 1]
convert φ.coeff_add_monomial_mul (single () 1) (single () n) _
rw [one_mul]; rfl
|
/-
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.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 219 | 223 | theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
(h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
(U : Opens G) :
μ.innerContent (Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) = μ.innerContent U := by |
convert μ.innerContent_comap (Homeomorph.mulLeft g) (fun K => h g) U
|
/-
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 | 435 | 437 | theorem iterate_le_nfp (f a n) : f^[n] a ≤ nfp f a := by |
rw [← sup_iterate_eq_nfp]
exact le_sup _ n
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 811 | 812 | theorem sum_elim_dotProduct_sum_elim : Sum.elim u x ⬝ᵥ Sum.elim v y = u ⬝ᵥ v + x ⬝ᵥ y := by |
simp [dotProduct]
|
/-
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 | 719 | 721 | theorem localTriv_symm_apply {b : B} (hb : b ∈ Z.baseSet i) (v : F) :
(Z.localTriv i).symm b v = Z.coordChange i (Z.indexAt b) b v := by |
apply (Z.localTriv i).symm_apply hb v
|
/-
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.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Math... | Mathlib/Combinatorics/Quiver/Covering.lean | 307 | 309 | theorem Prefunctor.costar_conj_star (u : U) :
φ.costar u = Quiver.starEquivCostar (φ.obj u) ∘ φ.star u ∘ (Quiver.starEquivCostar u).symm := by |
ext ⟨v, f⟩ <;> simp
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 113 | 115 | theorem mem_minimals_iff_forall_lt_not_mem' (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt y x → y ∉ s := by |
simp [minimals, right_iff_left_not_left_of r rlt, not_imp_not, imp.swap (a := _ ∈ _)]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 124 | 127 | theorem sum_ite_self_eq_aux [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) :
(if a ∈ f.support then f a else 0) = f a := by |
simp only [mem_support_iff, ne_eq, ite_eq_left_iff, not_not]
exact fun h ↦ h.symm
|
/-
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 | 552 | 555 | theorem rpow_le_rpow {x y z : ℝ} (h : 0 ≤ x) (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z := by |
rcases eq_or_lt_of_le h₁ with (rfl | h₁'); · rfl
rcases eq_or_lt_of_le h₂ with (rfl | h₂'); · simp
exact le_of_lt (rpow_lt_rpow h h₁' h₂')
|
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 85 | 89 | theorem bottom_row_coprime {R : Type*} [CommRing R] (g : SL(2, R)) :
IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) R) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) R) 1 1) := by |
use -(↑g : Matrix (Fin 2) (Fin 2) R) 0 1, (↑g : Matrix (Fin 2) (Fin 2) R) 0 0
rw [add_comm, neg_mul, ← sub_eq_add_neg, ← det_fin_two]
exact g.det_coe
|
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 675 | 678 | theorem Coplanar.finiteDimensional_vectorSpan {s : Set P} (h : Coplanar k s) :
FiniteDimensional k (vectorSpan k s) := by |
refine IsNoetherian.iff_fg.1 (IsNoetherian.iff_rank_lt_aleph0.2 (lt_of_le_of_lt h ?_))
exact Cardinal.lt_aleph0.2 ⟨2, rfl⟩
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Sets.Opens
#align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589b... | Mathlib/Topology/PartialHomeomorph.lean | 938 | 941 | theorem ofSet_trans_ofSet {s : Set X} (hs : IsOpen s) {s' : Set X} (hs' : IsOpen s') :
(ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') (IsOpen.inter hs hs') := by |
rw [(ofSet s hs).trans_ofSet hs']
ext <;> simp [hs'.interior_eq]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.Quot... | Mathlib/Analysis/Normed/Group/Quotient.lean | 147 | 148 | theorem quotient_norm_sub_rev {S : AddSubgroup M} (x y : M ⧸ S) : ‖x - y‖ = ‖y - x‖ := by |
rw [← neg_sub, quotient_norm_neg]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,006 | 1,007 | theorem lift_spec_mul (z w v) : f.lift hg z * w = v ↔ g (f.sec z).1 * w = g (f.sec z).2 * v := by |
erw [mul_comm, ← mul_assoc, mul_inv_left hg, mul_comm]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#... | Mathlib/Data/Rat/Defs.lean | 537 | 540 | theorem coe_int_num_of_den_eq_one {q : ℚ} (hq : q.den = 1) : (q.num : ℚ) = q := by |
conv_rhs => rw [← num_divInt_den q, hq]
rw [intCast_eq_divInt]
rfl
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_seri... | Mathlib/RingTheory/PowerSeries/Order.lean | 99 | 101 | theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by |
contrapose! h
exact order_le _ h
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-communi... | Mathlib/MeasureTheory/Group/Prod.lean | 108 | 116 | theorem measurable_measure_mul_right (hs : MeasurableSet s) :
Measurable fun x => μ ((fun y => y * x) ⁻¹' s) := by |
suffices
Measurable fun y =>
μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s))
by convert this using 1; ext1 x; congr 1 with y : 1; simp
apply measurable_measure_prod_mk_right
apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs)
infer_instance... |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algeb... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 274 | 286 | theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
(ι R <| f i) * (List.ofFn fun i => ι R <| f i).prod = 0 := by |
induction' n with n hn
· exact i.elim0
· rw [List.ofFn_succ, List.prod_cons, ← mul_assoc]
by_cases h : i = 0
· rw [h, ι_sq_zero, zero_mul]
· replace hn :=
congr_arg (ι R (f 0) * ·) <| hn (fun i => f <| Fin.succ i) (i.pred h)
simp only at hn
rw [Fin.succ_pred, ← mul_assoc, mul_zero... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsup... | Mathlib/LinearAlgebra/DFinsupp.lean | 206 | 209 | theorem mapRange.linearMap_id :
(mapRange.linearMap fun i => (LinearMap.id : β₂ i →ₗ[R] _)) = LinearMap.id := by |
ext
simp [linearMap]
|
/-
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.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
/-!
# Noetheri... | Mathlib/Topology/NoetherianSpace.lean | 53 | 56 | theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) := by |
rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact,
CompleteLattice.isSupFiniteCompact_iff_all_elements_compact]
exact forall_congr' Opens.isCompactElement_iff
|
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Nilpotent.lean | 384 | 392 | theorem least_ascending_central_series_length_eq_nilpotencyClass :
Nat.find ((nilpotent_iff_finite_ascending_central_series G).mp hG) =
Group.nilpotencyClass G := by |
refine le_antisymm (Nat.find_mono ?_) (Nat.find_mono ?_)
· intro n hn
exact ⟨upperCentralSeries G, upperCentralSeries_isAscendingCentralSeries G, hn⟩
· rintro n ⟨H, ⟨hH, hn⟩⟩
rw [← top_le_iff, ← hn]
exact ascending_central_series_le_upper H hH n
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e... | Mathlib/CategoryTheory/Subobject/Limits.lean | 98 | 100 | theorem kernelSubobject_arrow :
(kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow := by |
simp [kernelSubobjectIso]
|
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.specia... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 251 | 255 | theorem tendsto_stirlingSeq_sqrt_pi : Tendsto stirlingSeq atTop (𝓝 (√π)) := by |
obtain ⟨a, hapos, halimit⟩ := stirlingSeq_has_pos_limit_a
have hπ : π / 2 = a ^ 2 / 2 :=
tendsto_nhds_unique Wallis.tendsto_W_nhds_pi_div_two (second_wallis_limit a hapos.ne' halimit)
rwa [(div_left_inj' (two_ne_zero' ℝ)).mp hπ, sqrt_sq hapos.le]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 439 | 440 | theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by |
by_cases h : arg x = π <;> simp [arg_inv, 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.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib... | Mathlib/Order/Chain.lean | 107 | 110 | theorem Monotone.isChain_range [LinearOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
IsChain (· ≤ ·) (range f) := by |
rw [← image_univ]
exact (isChain_of_trichotomous _).image (· ≤ ·) _ _ hf
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import ... | Mathlib/LinearAlgebra/Trace.lean | 186 | 191 | theorem trace_one : trace R M 1 = (finrank R M : R) := by |
cases subsingleton_or_nontrivial R
· simp [eq_iff_true_of_subsingleton]
have b := Module.Free.chooseBasis R M
rw [trace_eq_matrix_trace R b, toMatrix_one, finrank_eq_card_chooseBasisIndex]
simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.deriv.zpow from "leanpro... | Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 138 | 140 | theorem iter_deriv_inv (k : ℕ) (x : 𝕜) :
deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - i : 𝕜)) * x ^ (-1 - k : ℤ) := by |
simpa only [zpow_neg_one, Int.cast_neg, Int.cast_one] using iter_deriv_zpow (-1) x k
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
# Basic... | Mathlib/Topology/Bornology/Basic.lean | 294 | 295 | theorem isBounded_sUnion {S : Set (Set α)} (hs : S.Finite) :
IsBounded (⋃₀ S) ↔ ∀ s ∈ S, IsBounded s := by | rw [sUnion_eq_biUnion, isBounded_biUnion hs]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Normalized
#align_import algebraic_topology.dold_kan.homotopy_equivalence from "leanprover-community/mathlib"@"f951e201d416fb50cc78261... | Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean | 52 | 58 | theorem homotopyPToId_eventually_constant {q n : ℕ} (hqn : n < q) :
((homotopyPToId X (q + 1)).hom n (n + 1) : X _[n] ⟶ X _[n + 1]) =
(homotopyPToId X q).hom n (n + 1) := by |
simp only [homotopyHσToZero, AlternatingFaceMapComplex.obj_X, Nat.add_eq, Homotopy.trans_hom,
Homotopy.ofEq_hom, Pi.zero_apply, Homotopy.add_hom, Homotopy.compLeft_hom, add_zero,
Homotopy.nullHomotopy'_hom, ComplexShape.down_Rel, hσ'_eq_zero hqn (c_mk (n + 1) n rfl),
dite_eq_ite, ite_self, comp_zero, zer... |
/-
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.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 1,546 | 1,547 | theorem mk_quaternionAlgebra_of_infinite [Infinite R] : #(ℍ[R,c₁,c₂]) = #R := by |
rw [mk_quaternionAlgebra, pow_four]
|
/-
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 | 742 | 750 | theorem comapDomain_single (f : α → β) (a : α) (m : M)
(hif : Set.InjOn f (f ⁻¹' (single (f a) m).support)) :
comapDomain f (Finsupp.single (f a) m) hif = Finsupp.single a m := by |
rcases eq_or_ne m 0 with (rfl | hm)
· simp only [single_zero, comapDomain_zero]
· rw [eq_single_iff, comapDomain_apply, comapDomain_support, ← Finset.coe_subset, coe_preimage,
support_single_ne_zero _ hm, coe_singleton, coe_singleton, single_eq_same]
rw [support_single_ne_zero _ hm, coe_singleton] at h... |
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Scott Morrison
-/
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 382 | 382 | theorem one_minus_pos (x : Ioo (0 : β) 1) : 0 < 1 - (x : β) := by | simpa using x.2.2
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Star.Spectrum
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Algebra
import Mathlib.Topology.Cont... | Mathlib/Analysis/NormedSpace/Star/GelfandDuality.lean | 145 | 158 | theorem gelfandTransform_isometry : Isometry (gelfandTransform ℂ A) := by |
nontriviality A
refine AddMonoidHomClass.isometry_of_norm (gelfandTransform ℂ A) fun a => ?_
/- By `spectrum.gelfandTransform_eq`, the spectra of `star a * a` and its
`gelfandTransform` coincide. Therefore, so do their spectral radii, and since they are
self-adjoint, so also do their norms. Applying the ... |
/-
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.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-c... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 529 | 550 | theorem quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
(hB : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
(hre : ∀ x : ℝ, x ≤ 0 → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C) (hz_re : z.re ≤ 0)
(hz_im : z.im ≤ 0) : ‖f z‖ ≤ C := by |
obtain ⟨z, rfl⟩ : ∃ z', -z' = z := ⟨-z, neg_neg z⟩
simp only [neg_re, neg_im, neg_nonpos] at hz_re hz_im
change ‖(f ∘ Neg.neg) z‖ ≤ C
have H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0) := by
intro w hw
simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, mem_Iio] using hw
refine
quadrant... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import ... | Mathlib/Data/List/Chain.lean | 223 | 225 | theorem chain'_append_cons_cons {b c : α} {l₁ l₂ : List α} :
Chain' R (l₁ ++ b :: c :: l₂) ↔ Chain' R (l₁ ++ [b]) ∧ R b c ∧ Chain' R (c :: l₂) := by |
rw [chain'_split, chain'_cons]
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 191 | 194 | theorem Convex.sum_mem (hs : Convex R s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1)
(hz : ∀ i ∈ t, z i ∈ s) : (∑ i ∈ t, w i • z i) ∈ s := by |
simpa only [h₁, centerMass, inv_one, one_smul] using
hs.centerMass_mem h₀ (h₁.symm ▸ zero_lt_one) hz
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-communi... | Mathlib/Algebra/DirectSum/Decomposition.lean | 145 | 147 | theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) :
i = j := by |
contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx]
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 325 | 328 | theorem NonUnitalStarSubalgebra.toStarSubalgebra_toNonUnitalStarSubalgebra
(S : NonUnitalStarSubalgebra R A) (h1 : (1 : A) ∈ S) :
(S.toStarSubalgebra h1).toNonUnitalStarSubalgebra = S := by |
cases S; rfl
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | Mathlib/Analysis/MeanInequalitiesPow.lean | 332 | 339 | theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0∞) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
(a + b) ^ p ≤ a ^ p + b ^ p := by |
rcases hp.eq_or_lt with (rfl | hp_pos)
· simp
have h := rpow_add_rpow_le a b hp_pos hp1
rw [one_div_one] at h
repeat' rw [ENNReal.rpow_one] at h
exact (ENNReal.le_rpow_one_div_iff hp_pos).mp h
|
/-
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 | 837 | 838 | theorem mem_iSup₂_iff {f : ∀ i, κ i → LowerSet α} : (a ∈ ⨆ (i) (j), f i j) ↔ ∃ i j, a ∈ f i j := by |
simp_rw [mem_iSup_iff]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
#align_import topology.sheaves.skyscrape... | Mathlib/Topology/Sheaves/Skyscraper.lean | 68 | 74 | theorem skyscraperPresheaf_eq_pushforward
[hd : ∀ U : Opens (TopCat.of PUnit.{u + 1}), Decidable (PUnit.unit ∈ U)] :
skyscraperPresheaf p₀ A =
ContinuousMap.const (TopCat.of PUnit) p₀ _*
skyscraperPresheaf (X := TopCat.of PUnit) PUnit.unit A := by |
convert_to @skyscraperPresheaf X p₀ (fun U => hd <| (Opens.map <| ContinuousMap.const _ p₀).obj U)
C _ _ A = _ <;> congr
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 284 | 289 | theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by |
rw [mem_emetric_ball_zero_iff] at hx
refine .of_nonneg_of_le
(fun _ ↦ norm_nonneg _) (fun n ↦ ((p n).le_opNorm _).trans_eq ?_) (p.summable_norm_mul_pow hx)
simp
|
/-
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.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathl... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 348 | 352 | theorem sum_le' {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α}
(hF : ∀ i ∈ s, padicNorm p (F i) ≤ t) (ht : 0 ≤ t) : padicNorm p (∑ i ∈ s, F i) ≤ t := by |
obtain rfl | hs := Finset.eq_empty_or_nonempty s
· simp [ht]
· exact sum_le hs hF
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7... | Mathlib/Algebra/RingQuot.lean | 616 | 618 | theorem mkAlgHom_coe (s : A → A → Prop) : (mkAlgHom S s : A →+* RingQuot s) = mkRingHom s := by |
simp_rw [mkAlgHom_def, mkRingHom_def]
rfl
|
/-
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.MeasureTheory.Group.Measure
/-!
# Lebesgue Integration on Groups
We develop properties of integrals with a group as domain.
This file contains pr... | Mathlib/MeasureTheory/Group/LIntegral.lean | 46 | 49 | theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (x * g) ∂μ) = ∫⁻ x, f x ∂μ := by |
convert (lintegral_map_equiv f <| MeasurableEquiv.mulRight g).symm using 1
simp [map_mul_right_eq_self μ g]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "lean... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 136 | 138 | theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 := by |
refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)
split_ifs with h <;> simp only [h, zero_le', le_rfl]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Chris Hughes
-/
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
/-!
... | Mathlib/Data/List/Duplicate.lean | 98 | 99 | theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l := by |
simpa [duplicate_cons_iff, hx.symm] using 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.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Ring.Multiset
import Mathlib.Algebra.Field.Defs
import Mathlib.Data.Finty... | Mathlib/Algebra/BigOperators/Ring.lean | 232 | 237 | theorem sum_pow_mul_eq_add_pow (a b : α) (s : Finset ι) :
(∑ t ∈ s.powerset, a ^ t.card * b ^ (s.card - t.card)) = (a + b) ^ s.card := by |
classical
rw [← prod_const, prod_add]
refine Finset.sum_congr rfl fun t ht => ?_
rw [prod_const, prod_const, ← card_sdiff (mem_powerset.1 ht)]
|
/-
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 | 199 | 205 | theorem left_bot_eq_left_quot (H : Subgroup G) :
Quotient (⊥ : Subgroup G).1 (H : Set G) = (G ⧸ H) := by |
unfold Quotient
congr
ext
simp_rw [← bot_rel_eq_leftRel H]
rfl
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 414 | 415 | theorem cospanExt_hom_app_left :
(cospanExt iX iY iZ wf wg).hom.app WalkingCospan.left = iX.hom := by | dsimp [cospanExt]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 423 | 429 | theorem Cofix.dest_corec {α : Type u} (g : α → F α) (x : α) :
Cofix.dest (Cofix.corec g x) = Cofix.corec g <$> g x := by |
conv =>
lhs
rw [Cofix.dest, Cofix.corec];
dsimp
rw [corecF_eq, abs_map, abs_repr, ← comp_map]; rfl
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 761 | 762 | theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} :
(↑(⨅ i, S i) : Set A) = ⋂ i, S i := by | simp [iInf]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 451 | 455 | theorem setIntegral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
[MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
∫ x in s, f x ∂kernel.deterministic g hg a = if g a ∈ s then f (g a) else 0 := by |
rw [kernel.deterministic_apply, setIntegral_dirac f _ s]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.... | Mathlib/Analysis/Convex/Topology.lean | 228 | 230 | theorem Convex.add_smul_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s)
(hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • y ∈ interior s := by |
simpa only [add_sub_cancel_left] using hs.add_smul_sub_mem_interior' hx hy ht
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import... | Mathlib/Algebra/Algebra/Operations.lean | 653 | 655 | theorem prod_span_singleton {ι : Type*} (s : Finset ι) (x : ι → A) :
(∏ i ∈ s, span R ({x i} : Set A)) = span R {∏ i ∈ s, x i} := by |
rw [prod_span, Set.finset_prod_singleton]
|
/-
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.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 173 | 176 | theorem formPerm_eq_self_of_not_mem (s : Cycle α) (h : Nodup s) (x : α) (hx : x ∉ s) :
formPerm s h x = x := by |
induction s using Quot.inductionOn
simpa using List.formPerm_eq_self_of_not_mem _ _ hx
|
/-
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 | 94 | 97 | theorem Filter.Tendsto.atBot_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg
simpa [(· ∘ ·)] using tendsto_neg_atTop_atBot.comp this
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community... | Mathlib/Data/Nat/Choose/Basic.lean | 257 | 261 | theorem choose_eq_asc_factorial_div_factorial (n k : ℕ) :
(n + k).choose k = (n + 1).ascFactorial k / k ! := by |
apply Nat.mul_left_cancel k.factorial_pos
rw [← ascFactorial_eq_factorial_mul_choose]
exact (Nat.mul_div_cancel' <| factorial_dvd_ascFactorial _ _).symm
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
i... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 957 | 968 | theorem measurable_fderiv_with_param (hf : Continuous f.uncurry) :
Measurable (fun (p : α × E) ↦ fderiv 𝕜 (f p.1) p.2) := by |
refine measurable_of_isClosed (fun s hs ↦ ?_)
have :
(fun (p : α × E) ↦ fderiv 𝕜 (f p.1) p.2) ⁻¹' s =
{p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ s } ∪
{ p | ¬DifferentiableAt 𝕜 (f p.1) p.2} ∩ { _p | (0 : E →L[𝕜] F) ∈ s} :=
Set.ext (fun x ↦ mem_preimage.trans fderiv_mem_... |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 446 | 458 | theorem card_support_eq_three :
f.support.card = 3 ↔
∃ (k m n : ℕ) (hkm : k < m) (hmn : m < n) (x y z : R) (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0),
f = C x * X ^ k + C y * X ^ m + C z * X ^ n := by |
refine ⟨fun h => ?_, ?_⟩
· obtain ⟨k, x, hk, hx, rfl⟩ := card_support_eq.mp h
refine
⟨k 0, k 1, k 2, hk Nat.zero_lt_one, hk (Nat.lt_succ_self 1), x 0, x 1, x 2, hx 0, hx 1, hx 2,
?_⟩
rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc, Fin.sum_univ_one]
rfl
· rintro ⟨k, m, n, hkm, hmn, x, ... |
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