Context stringlengths 295 65.3k | file_name stringlengths 21 74 | start int64 14 1.41k | end int64 20 1.41k | theorem stringlengths 27 1.42k | proof stringlengths 0 4.57k |
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/-
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.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
/-!
# Algebraic order homomorphism classes
This file defines hom classes for common properties at the ... | Mathlib/Algebra/Order/Hom/Basic.lean | 241 | 243 | theorem abs_sub_map_le_div [Group α] [AddCommGroup β] [LinearOrder β] [IsOrderedAddMonoid β]
[GroupSeminormClass F α β]
(f : F) (x y : α) : |f x - f y| ≤ f (x / y) := by | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 65 | 65 | theorem left_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₂ := by | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 743 | 751 | theorem oangle_sign_smul_right (x y : V) (r : ℝ) :
(o.oangle x (r • y)).sign = SignType.sign r * (o.oangle x y).sign := by | rcases lt_trichotomy r 0 with (h | h | h) <;> simp [h]
/-- Auxiliary lemma for the proof of `oangle_sign_smul_add_right`; not intended to be used
outside of that proof. -/
theorem oangle_smul_add_right_eq_zero_or_eq_pi_iff {x y : V} (r : ℝ) :
o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = π ↔
o.oangle x... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 377 | 380 | theorem contMDiffWithinAt_iff_target_of_mem_source
[IsManifold I' n M'] {x : M} {y : M'} (hy : f x ∈ (chartAt H' y).source) :
ContMDiffWithinAt I I' n f s x ↔
ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (extChartAt I' y ∘ f) s x := by | |
/-
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.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | Mathlib/Data/Nat/Choose/Basic.lean | 192 | 194 | theorem choose_symm_of_eq_add {n a b : ℕ} (h : n = a + b) : Nat.choose n a = Nat.choose n b := by | suffices choose n (n - b) = choose n b by
rw [h, Nat.add_sub_cancel_right] at this; rwa [h] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | Mathlib/Analysis/SpecificLimits/Basic.lean | 329 | 335 | theorem tsum_geometric_inv_two : (∑' n : ℕ, (2 : ℝ)⁻¹ ^ n) = 2 :=
(inv_eq_one_div (2 : ℝ)).symm ▸ tsum_geometric_two
/-- The sum of `2⁻¹ ^ i` for `n ≤ i` equals `2 * 2⁻¹ ^ n`. -/
theorem tsum_geometric_inv_two_ge (n : ℕ) :
(∑' i, ite (n ≤ i) ((2 : ℝ)⁻¹ ^ i) 0) = 2 * 2⁻¹ ^ n := by | have A : Summable fun i : ℕ ↦ ite (n ≤ i) ((2⁻¹ : ℝ) ^ i) 0 := by |
/-
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.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 301 | 303 | theorem surjective_of_injective {f : α → α} (hinj : Injective f) : Surjective f := by | intro x
have := Classical.propDecidable |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 49 | 49 | theorem logb_one : logb b 1 = 0 := by | simp [logb] |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | Mathlib/Data/Set/Card.lean | 1,042 | 1,051 | theorem ncard_le_one_iff_subset_singleton [Nonempty α]
(hs : s.Finite := by | toFinite_tac) :
s.ncard ≤ 1 ↔ ∃ x : α, s ⊆ {x} := by
simp_rw [ncard_eq_toFinset_card _ hs, Finset.card_le_one_iff_subset_singleton,
Finite.toFinset_subset, Finset.coe_singleton]
/-- A `Set` of a subsingleton type has cardinality at most one. -/
theorem ncard_le_one_of_subsingleton [Subsingleton α] (s : Set α... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 969 | 981 | theorem const_lintegral_restrict (c : ℝ≥0∞) (s : Set α) :
(const α c).lintegral (μ.restrict s) = c * μ s := by | rw [const_lintegral, Measure.restrict_apply MeasurableSet.univ, univ_inter]
theorem restrict_const_lintegral (c : ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
((const α c).restrict s).lintegral μ = c * μ s := by
rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict]
@[gcongr]
theorem lintegr... |
/-
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
/-!
# Locus of unequal values of finitely supported functions
Let `α N` be two Types, assume that `N` has a `0` and let `f g : α →₀ N... | Mathlib/Data/Finsupp/NeLocus.lean | 149 | 150 | theorem neLocus_self_sub_left : neLocus (f - g) f = g.support := by | rw [neLocus_comm, neLocus_self_sub_right] |
/-
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
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | Mathlib/LinearAlgebra/Determinant.lean | 643 | 654 | theorem Basis.det_unitsSMul (e : Basis ι R M) (w : ι → Rˣ) :
(e.unitsSMul w).det = (↑(∏ i, w i)⁻¹ : R) • e.det := by | ext f
change
(Matrix.det fun i j => (e.unitsSMul w).repr (f j) i) =
(↑(∏ i, w i)⁻¹ : R) • Matrix.det fun i j => e.repr (f j) i
simp only [e.repr_unitsSMul]
convert Matrix.det_mul_column (fun i => (↑(w i)⁻¹ : R)) fun i j => e.repr (f j) i
simp [← Finset.prod_inv_distrib]
/-- The determinant of a basis... |
/-
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.GroupTheory.Coset.Basic
/-!
# Double cosets
This file defines double cosets for two subgroups `H K` o... | Mathlib/GroupTheory/DoubleCoset.lean | 52 | 57 | theorem mem_doset_of_not_disjoint {H K : Subgroup G} {a b : G}
(h : ¬Disjoint (doset a H K) (doset b H K)) : b ∈ doset a H K := by | rw [Set.not_disjoint_iff] at h
simp only [mem_doset] at *
obtain ⟨x, ⟨l, hl, r, hr, hrx⟩, y, hy, ⟨r', hr', rfl⟩⟩ := h
refine ⟨y⁻¹ * l, H.mul_mem (H.inv_mem hy) hl, r * r'⁻¹, K.mul_mem hr (K.inv_mem hr'), ?_⟩ |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Topology.Algebra.Module.Equiv
import Mathlib.Topology.Algebra.Module.UniformConvergence
import Mathlib.Topology.Algebra.Separation... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 168 | 177 | theorem t2Space [TopologicalSpace F] [IsTopologicalAddGroup F] [T2Space F]
(𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = univ) : T2Space (UniformConvergenceCLM σ F 𝔖) := by | letI : UniformSpace F := IsTopologicalAddGroup.toUniformSpace F
haveI : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
haveI : T2Space (E →ᵤ[𝔖] F) := UniformOnFun.t2Space_of_covering h𝔖
exact (isEmbedding_coeFn σ F 𝔖).t2Space
instance instDistribMulAction (M : Type*) [Monoid M] [DistribMulAction M F... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 682 | 686 | theorem dist_div_cos_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₁ p₂ / Real.Angle.cos (∡ p₃ p₁ p₂) = dist p₁ p₃ := by | have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.cos_coe, dist_comm p₁ p₃,
dist_div_cos_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h) |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 894 | 900 | theorem commute_of_normal_of_disjoint (H₁ H₂ : Subgroup G) (hH₁ : H₁.Normal) (hH₂ : H₂.Normal)
(hdis : Disjoint H₁ H₂) (x y : G) (hx : x ∈ H₁) (hy : y ∈ H₂) : Commute x y := by | suffices x * y * x⁻¹ * y⁻¹ = 1 by
show x * y = y * x
· rw [mul_assoc, mul_eq_one_iff_eq_inv] at this
simpa
apply hdis.le_bot |
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Defs
import Mathlib.Data.Set.Function
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variab... | Mathlib/Data/List/Iterate.lean | 25 | 26 | theorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f a n = [] ↔ n = 0 := by | rw [← length_eq_zero_iff, length_iterate] |
/-
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
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | Mathlib/Algebra/Polynomial/EraseLead.lean | 245 | 248 | theorem leadingCoeff_eraseLead_eq_nextCoeff (h : f.nextCoeff ≠ 0) :
f.eraseLead.leadingCoeff = f.nextCoeff := by | have := natDegree_pos_of_nextCoeff_ne_zero h
rw [leadingCoeff, nextCoeff, natDegree_eraseLead h, if_neg, |
/-
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.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | Mathlib/RingTheory/PowerSeries/Basic.lean | 438 | 448 | theorem X_pow_mul_cancel {k : ℕ} {φ ψ : R⟦X⟧} (h : X ^ k * φ = X ^ k * ψ) :
φ = ψ := by | rw [PowerSeries.ext_iff] at h ⊢
intro n
simpa using h (n + k)
theorem X_pow_mul_injective {k : ℕ} : Function.Injective (X ^ k * · : R⟦X⟧ → R⟦X⟧) :=
fun _ _ ↦ X_pow_mul_cancel
theorem X_pow_mul_inj {k : ℕ} {φ ψ : R⟦X⟧} :
X ^ k * φ = X ^ k * ψ ↔ φ = ψ := |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | Mathlib/GroupTheory/Coxeter/Length.lean | 314 | 321 | theorem not_isLeftDescent_iff {w : W} {i : B} :
¬cs.IsLeftDescent w i ↔ ℓ (s i * w) = ℓ w + 1 := by | unfold IsLeftDescent
constructor
· intro _
exact (cs.length_simple_mul w i).resolve_right (by omega)
· omega |
/-
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.Data.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... | Mathlib/NumberTheory/Dioph.lean | 85 | 86 | theorem IsPoly.add {f g : (α → ℕ) → ℤ} (hf : IsPoly f) (hg : IsPoly g) : IsPoly (f + g) := by | rw [← sub_neg_eq_add]; exact hf.sub hg.neg |
/-
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.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 518 | 520 | theorem iterate_pos_eq_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
f^[n] x = x + n * m ↔ f x = x + m := by | simpa only [nsmul_eq_mul, add_right_iterate] using |
/-
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.Complex
/-!
# The `arctan` function.
Inequalit... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 193 | 194 | theorem arctan_inv_of_pos {x : ℝ} (h : 0 < x) : arctan x⁻¹ = π / 2 - arctan x := by | rw [← arctan_tan (x := _ - _), tan_pi_div_two_sub, tan_arctan] |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 286 | 296 | theorem convexBodySumFun_eq_zero_iff (x : mixedSpace K) :
convexBodySumFun x = 0 ↔ x = 0 := by | rw [← forall_normAtPlace_eq_zero_iff, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg
fun _ _ ↦ mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
conv =>
enter [1, w, hw]
rw [mul_left_mem_nonZeroDivisors_eq_zero_iff
(mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mu... |
/-
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
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 135 | 138 | theorem jacobiSymNat.even_odd₇ (a b c : ℕ) (r : ℤ) (ha : a % 2 = 0) (hb : b % 8 = 7)
(hc : a / 2 = c) (hr : jacobiSymNat c b = r) : jacobiSymNat a b = r := by | simp only [jacobiSymNat, ← hr, ← hc, Int.natCast_ediv, Nat.cast_ofNat]
rw [← jacobiSym.even_odd (mod_cast ha), if_neg (by simp [hb])] |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 90 | 91 | theorem memLp_zero_iff_aestronglyMeasurable [TopologicalSpace ε] {f : α → ε} :
MemLp f 0 μ ↔ AEStronglyMeasurable f μ := by | simp [MemLp, eLpNorm_exponent_zero] |
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Anthony DeRossi
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespac... | Mathlib/Data/List/ReduceOption.lean | 19 | 21 | theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by | simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 286 | 286 | theorem natDegree_intCast (n : ℤ) : natDegree (n : R[X]) = 0 := by | |
/-
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
import Mathlib.... | Mathlib/LinearAlgebra/Trace.lean | 289 | 291 | theorem IsProj.trace {p : Submodule R M} {f : M →ₗ[R] M} (h : IsProj p f) [Module.Free R p]
[Module.Finite R p] [Module.Free R (ker f)] [Module.Finite R (ker f)] :
trace R M f = (finrank R p : R) := by | |
/-
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.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 106 | 116 | theorem integral_exp_mul_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) :
∫ x : ℝ in Set.Iic c, Real.exp (a * x) = Real.exp (a * c) / a := by | simpa [neg_mul, ← mul_neg, integral_comp_neg_Ioi (f := fun x : ℝ ↦ Real.exp (a * x))]
using integral_exp_mul_Ioi (a := -a) (by simpa) (-c)
/-- If `0 < c`, then `(fun t : ℝ ↦ t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. -/
theorem integrableOn_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
I... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
/-!
# Angles between vectors
This fil... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 106 | 108 | theorem angle_zero_left (x : V) : angle 0 x = π / 2 := by | unfold angle
rw [inner_zero_left, zero_div, Real.arccos_zero] |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 370 | 381 | theorem UniformCauchySeqOn.uniformCauchySeqOnFilter (hF : UniformCauchySeqOn F p s) :
UniformCauchySeqOnFilter F p (𝓟 s) := by | rwa [← uniformCauchySeqOn_iff_uniformCauchySeqOnFilter]
/-- A sequence that converges uniformly is also uniformly Cauchy -/
theorem TendstoUniformlyOnFilter.uniformCauchySeqOnFilter (hF : TendstoUniformlyOnFilter F f p p') :
UniformCauchySeqOnFilter F p p' := by
intro u hu
rcases comp_symm_of_uniformity hu wit... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | Mathlib/Tactic/NormNum/Basic.lean | 119 | 120 | theorem isintCast {R} [Ring R] (n m : ℤ) :
IsInt n m → IsInt (n : R) m := by | rintro ⟨⟨⟩⟩; exact ⟨rfl⟩ |
/-
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, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 678 | 681 | theorem separatingLeft_iff_det_ne_zero {B : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁} (b : Basis ι R₁ M₁) :
B.SeparatingLeft ↔ (toMatrix₂ b b B).det ≠ 0 := by | rw [← Matrix.nondegenerate_iff_det_ne_zero, nondegenerate_toMatrix_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.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | Mathlib/GroupTheory/Nilpotent.lean | 504 | 509 | theorem isNilpotent_of_ker_le_center {H : Type*} [Group H] (f : G →* H) (hf1 : f.ker ≤ center G)
(hH : IsNilpotent H) : IsNilpotent G := by | rw [nilpotent_iff_lowerCentralSeries] at *
rcases hH with ⟨n, hn⟩
use n + 1
refine lowerCentralSeries_succ_eq_bot (le_trans ((Subgroup.map_eq_bot_iff _).mp ?_) hf1) |
/-
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.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 291 | 292 | theorem ext_inner_right {x y : E} (h : ∀ v, ⟪x, v⟫ = ⟪y, v⟫) : x = y := by | rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜, inner_sub_left, sub_eq_zero, h (x - y)] |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.AlgebraicGeometry.StructureSheaf
import Mathlib.RingTheory.Localization.Localizatio... | Mathlib/AlgebraicGeometry/Spec.lean | 292 | 295 | theorem Spec_Γ_naturality {R S : CommRingCat.{u}} (f : R ⟶ S) :
f ≫ toSpecΓ S = toSpecΓ R ≫ Γ.map (Spec.toLocallyRingedSpace.map f.op).op := by | -- Porting note (https://github.com/leanprover-community/mathlib4/issues/11041): `ext` failed to pick up one of the three lemmas
ext : 2 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 152 | 202 | theorem norm_eval_le_injectiveSeminorm (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
‖lift f.toMultilinearMap x‖ ≤ ‖f‖ * injectiveSeminorm x := by | /- If `F` were in `Type (max uι u𝕜 uE)` (which is the type of `⨂[𝕜] i, E i`), then the
property that we want to prove would hold by definition of `injectiveSeminorm`. This is
not necessarily true, but we will show that there exists a normed vector space `G` in
`Type (max uι u𝕜 uE)` and an injective isome... |
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson
-/
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Defs
import Mathlib.Logic.Equiv.Option
import Mathlib.Tactic... | Mathlib/Combinatorics/Derangements/Basic.lean | 129 | 134 | theorem RemoveNone.fiber_some (a : α) :
RemoveNone.fiber (some a) = { f : Perm α | fixedPoints f ⊆ {a} } := by | ext f
constructor
· rw [RemoveNone.mem_fiber]
rintro ⟨F, F_derangement, F_none, rfl⟩ x x_fixed |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | Mathlib/NumberTheory/Pell.lean | 256 | 260 | theorem y_pow_succ_pos {a : Solution₁ d} (hax : 0 < a.x) (hay : 0 < a.y) (n : ℕ) :
0 < (a ^ n.succ).y := by | induction n with
| zero => simp only [pow_one, hay]
| succ n ih => rw [pow_succ']; exact y_mul_pos hax hay (x_pow_pos hax _) ih |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | Mathlib/NumberTheory/Bernoulli.lean | 216 | 221 | theorem bernoulli_spec' (n : ℕ) :
(∑ k ∈ antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli k.1) =
if n = 0 then 1 else 0 := by | cases n with | zero => simp | succ n =>
rw [if_neg (succ_ne_zero _)]
-- algebra facts |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | Mathlib/Data/List/DropRight.lean | 51 | 51 | theorem rdrop_eq_reverse_drop_reverse : l.rdrop n = reverse (l.reverse.drop n) := by | |
/-
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.Complex
/-!
# The `arctan` function.
Inequalit... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 178 | 179 | theorem arctan_eq_arccos {x : ℝ} (h : 0 ≤ x) : arctan x = arccos (√(1 + x ^ 2))⁻¹ := by | rw [arctan_eq_arcsin, arccos_eq_arcsin]; swap; · exact inv_nonneg.2 (sqrt_nonneg _) |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | Mathlib/Topology/ContinuousOn.lean | 104 | 107 | theorem nhds_of_nhdsWithin_of_nhds {s t : Set α} {a : α} (h1 : s ∈ 𝓝 a) (h2 : t ∈ 𝓝[s] a) :
t ∈ 𝓝 a := by | rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.mp h2 with ⟨_, Hw, hw⟩
exact (𝓝 a).sets_of_superset ((𝓝 a).inter_sets Hw h1) hw |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 283 | 286 | theorem liftIsometry_apply_apply (f : ContinuousMultilinearMap 𝕜 E F) (x : ⨂[𝕜] i, E i) :
liftIsometry 𝕜 E F f x = lift f.toMultilinearMap x := by | simp only [liftIsometry, LinearIsometryEquiv.coe_mk, liftEquiv_apply,
LinearMap.mkContinuous_apply] |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | Mathlib/Algebra/Order/ToIntervalMod.lean | 377 | 379 | theorem toIcoMod_sub_zsmul (a b : α) (m : ℤ) : toIcoMod hp a (b - m • p) = toIcoMod hp a b := by | rw [sub_eq_add_neg, ← neg_smul, toIcoMod_add_zsmul] |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 149 | 150 | theorem singleton_div_closedBall_one : {x} / closedBall 1 δ = closedBall x δ := by | rw [singleton_div_closedBall, div_one] |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 158 | 158 | theorem sdiff_symmDiff_eq_sup : (a \ b) ∆ b = a ⊔ b := by | |
/-
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
/-!
# Rewriting arrows and paths along vertex... | Mathlib/Combinatorics/Quiver/Cast.lean | 118 | 121 | theorem cast_eq_of_cons_eq_cons {u v v' w : U} {p : Path u v} {p' : Path u v'} {e : v ⟶ w}
{e' : v' ⟶ w} (h : p.cons e = p'.cons e') : p.cast rfl (obj_eq_of_cons_eq_cons h) = p' := by | rw [Path.cast_eq_iff_heq]
exact heq_of_cons_eq_cons h |
/-
Copyright (c) 2023 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
/-!
# Lemmas about the divisibility relation in product (sem... | Mathlib/Algebra/Divisibility/Prod.lean | 35 | 36 | theorem pi_dvd_iff {x y : ∀ i, G i} : x ∣ y ↔ ∀ i, x i ∣ y i := by | simp_rw [dvd_def, funext_iff, Classical.skolem]; 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.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,160 | 1,176 | theorem contDiff_iff_ftaylorSeries {n : ℕ∞} :
ContDiff 𝕜 n f ↔ HasFTaylorSeriesUpTo n f (ftaylorSeries 𝕜 f) := by | constructor
· rw [← contDiffOn_univ, ← hasFTaylorSeriesUpToOn_univ_iff, ← ftaylorSeriesWithin_univ]
exact fun h ↦ ContDiffOn.ftaylorSeriesWithin h uniqueDiffOn_univ
· exact fun h ↦ ⟨ftaylorSeries 𝕜 f, h⟩
theorem contDiff_iff_continuous_differentiable {n : ℕ∞} :
ContDiff 𝕜 n f ↔
(∀ m : ℕ, m ≤ n → Co... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | Mathlib/MeasureTheory/Integral/Average.lean | 122 | 123 | theorem setLAverage_eq (f : α → ℝ≥0∞) (s : Set α) :
⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s := by | rw [laverage_eq, restrict_apply_univ] |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 305 | 310 | theorem borel_eq_generateFrom_Ico (α : Type*) [TopologicalSpace α] [SecondCountableTopology α]
[LinearOrder α] [OrderTopology α] :
borel α = .generateFrom { S : Set α | ∃ (l u : α), l < u ∧ Ico l u = S } := by | simpa only [exists_prop, mem_univ, true_and] using
(@dense_univ α _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>
mem_univ _ |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 468 | 480 | theorem adjugate_pow (A : Matrix n n α) (k : ℕ) : adjugate (A ^ k) = adjugate A ^ k := by | induction k with
| zero => simp
| succ k IH => rw [pow_succ', adjugate_mul_distrib, IH, pow_succ]
theorem det_smul_adjugate_adjugate (A : Matrix n n α) :
det A • adjugate (adjugate A) = det A ^ (Fintype.card n - 1) • A := by
have : A * (A.adjugate * A.adjugate.adjugate) =
A * (A.det ^ (Fintype.card n -... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 102 | 102 | theorem content_zero : content (0 : R[X]) = 0 := by | rw [← C_0, content_C, normalize_zero] |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Group.Defs
/-!
# Flows and invariant sets
This file defines a flow on... | Mathlib/Dynamics/Flow.lean | 158 | 162 | theorem continuous_toFun (t : τ) : Continuous (ϕ.toFun t) := by | fun_prop
/-- The map `ϕ t` as a homeomorphism. -/
def toHomeomorph (t : τ) : (α ≃ₜ α) where |
/-
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.SetTheory.Ordinal.Exponential
import Mathlib.SetTheory.Ordinal.Family
/-!
# Cantor Normal Form
The Cantor normal form of an ordinal is generally defi... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 114 | 116 | theorem CNF_snd_lt {b o : Ordinal.{u}} (hb : 1 < b) {x : Ordinal × Ordinal} :
x ∈ CNF b o → x.2 < b := by | refine CNFRec b ?_ (fun o ho IH ↦ ?_) o |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | Mathlib/Algebra/Polynomial/Mirror.lean | 151 | 153 | theorem natDegree_mul_mirror : (p * p.mirror).natDegree = 2 * p.natDegree := by | by_cases hp : p = 0
· rw [hp, zero_mul, natDegree_zero, mul_zero] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 206 | 209 | theorem HasDerivWithinAt.cpow (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x)
(h0 : f x ∈ slitPlane) : HasDerivWithinAt (fun x => f x ^ g x)
(g x * f x ^ (g x - 1) * f' + f x ^ g x * Complex.log (f x) * g') s x := by | simpa only [aux] using (hf.hasFDerivWithinAt.cpow hg h0).hasDerivWithinAt |
/-
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.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | Mathlib/Algebra/Algebra/Operations.lean | 462 | 466 | theorem comap_op_mul (M N : Submodule R Aᵐᵒᵖ) :
comap (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) (M * N) =
comap (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) N *
comap (↑(opLinearEquiv R : A ≃ₗ[R] Aᵐᵒᵖ) : A →ₗ[R] Aᵐᵒᵖ) M := by | simp_rw [comap_equiv_eq_map_symm, map_unop_mul] |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | Mathlib/GroupTheory/OrderOfElement.lean | 586 | 601 | theorem isOfFinOrder_inv_iff {x : G} : IsOfFinOrder x⁻¹ ↔ IsOfFinOrder x := by | simp [isOfFinOrder_iff_pow_eq_one]
@[to_additive] alias ⟨IsOfFinOrder.of_inv, IsOfFinOrder.inv⟩ := isOfFinOrder_inv_iff
@[to_additive]
theorem orderOf_dvd_iff_zpow_eq_one : (orderOf x : ℤ) ∣ i ↔ x ^ i = 1 := by
rcases Int.eq_nat_or_neg i with ⟨i, rfl | rfl⟩
· rw [Int.natCast_dvd_natCast, orderOf_dvd_iff_pow_eq_on... |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.Group.Prod
/-!
# Typeclasses for power-associative structures
In this... | Mathlib/Algebra/Group/NatPowAssoc.lean | 117 | 121 | theorem Nat.cast_npow (R : Type*) [NonAssocSemiring R] [Pow R ℕ] [NatPowAssoc R] (n m : ℕ) :
(↑(n ^ m) : R) = (↑n : R) ^ m := by | induction m with
| zero => simp only [pow_zero, Nat.cast_one, npow_zero]
| succ m ih => rw [npow_add, npow_add, Nat.cast_mul, ih, npow_one, npow_one] |
/-
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.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 319 | 323 | theorem exp_eq_one_iff : exp x = 1 ↔ x = 0 :=
exp_injective.eq_iff' exp_zero
@[simp]
theorem one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x := by | rw [← exp_zero, exp_lt_exp] |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 243 | 243 | theorem oangle_add_oangle_rev_neg_right (x y : V) : o.oangle x (-y) + o.oangle y (-x) = 0 := by | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | Mathlib/Data/Nat/Factorial/Basic.lean | 272 | 274 | theorem pow_lt_ascFactorial' (n k : ℕ) : (n + 1) ^ (k + 2) < (n + 1).ascFactorial (k + 2) := by | rw [Nat.pow_succ, ascFactorial, Nat.mul_comm]
exact Nat.mul_lt_mul_of_lt_of_le' (Nat.lt_add_of_pos_right k.succ_pos) |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 143 | 147 | theorem tendstoInMeasure_of_tendsto_ae [IsFiniteMeasure μ] (hf : ∀ n, AEStronglyMeasurable (f n) μ)
(hfg : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))) : TendstoInMeasure μ f atTop g := by | have hg : AEStronglyMeasurable g μ := aestronglyMeasurable_of_tendsto_ae _ hf hfg
refine TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm ?_
refine tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathl... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 108 | 113 | theorem locally_ne_zero (hp : HasFPowerSeriesAt f p z₀) (h : p ≠ 0) : ∀ᶠ z in 𝓝[≠] z₀, f z ≠ 0 := by | rw [eventually_nhdsWithin_iff]
have h2 := (has_fpower_series_iterate_dslope_fslope p.order hp).continuousAt
have h3 := h2.eventually_ne (iterate_dslope_fslope_ne_zero hp h)
filter_upwards [eq_pow_order_mul_iterate_dslope hp, h3] with z e1 e2 e3
simpa [e1, e2, e3] using pow_ne_zero p.order (sub_ne_zero.mpr e3) |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 254 | 257 | theorem boundedBy_eq_ofFunction (m_empty : m ∅ = 0) (s : Set α) :
boundedBy m s = OuterMeasure.ofFunction m m_empty s := by | have : (fun s : Set α => ⨆ _ : s.Nonempty, m s) = m := by
ext1 t |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.Algebra.Module.Submodule.Ker
/-!
# Iterate maps and comaps of submodules
Some preliminary work for establishing the strong rank condition for noetherian rings.
Giv... | Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 88 | 92 | theorem ker_le_of_iterateMapComap_eq_succ (K : Submodule R N)
(m : ℕ) (heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K)
(hf : Surjective f) (hi : Injective i) : LinearMap.ker f ≤ K := by | rw [show K = _ from f.iterateMapComap_eq_succ i K m heq hf hi 0]
exact f.ker_le_comap |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | Mathlib/GroupTheory/Coxeter/Length.lean | 277 | 279 | theorem not_isLeftDescent_one (i : B) : ¬cs.IsLeftDescent 1 i := by | simp [IsLeftDescent]
theorem not_isRightDescent_one (i : B) : ¬cs.IsRightDescent 1 i := by simp [IsRightDescent] |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | Mathlib/Data/Nat/Digits.lean | 199 | 199 | theorem coe_int_ofDigits (b : ℕ) (L : List ℕ) : ((ofDigits b L : ℕ) : ℤ) = ofDigits (b : ℤ) L := by | |
/-
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.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 555 | 556 | theorem real_inner_add_sub_eq_zero_iff (x y : F) : ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ = ‖y‖ := by | conv_rhs => rw [← mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)] |
/-
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.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 106 | 109 | theorem mellin_div_const (f : ℝ → ℂ) (s a : ℂ) : mellin (fun t => f t / a) s = mellin f s / a := by | simp_rw [mellin, smul_eq_mul, ← mul_div_assoc, integral_div]
theorem mellin_comp_rpow (f : ℝ → E) (s : ℂ) (a : ℝ) : |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 108 | 109 | theorem volume_real_Ioc {a b : ℝ} : volume.real (Ioc a b) = max (b - a) 0 := by | simp [measureReal_def, ENNReal.toReal_ofReal'] |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 451 | 455 | theorem norm_withSeminorms (𝕜 E) [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] :
WithSeminorms fun _ : Fin 1 => normSeminorm 𝕜 E := by | let p : SeminormFamily 𝕜 E (Fin 1) := fun _ => normSeminorm 𝕜 E
refine
⟨SeminormedAddCommGroup.toIsTopologicalAddGroup.ext |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 457 | 458 | theorem symmDiff_eq_right : a ∆ b = b ↔ a = ⊥ := by | rw [symmDiff_comm, symmDiff_eq_left] |
/-
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.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 63 | 64 | theorem ordConnectedComponent_inter (s t : Set α) (x : α) :
ordConnectedComponent (s ∩ t) x = ordConnectedComponent s x ∩ ordConnectedComponent t x := by | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 61 | 73 | theorem div_lt_div' (hac : a ≤ c) (hbd : d < b) (c0 : 0 < c) (d0 : 0 < d) : a / b < c / d :=
div_lt_div₀' hac hbd c0 d0
/-!
### Relating one division and involving `1`
-/
@[bound]
theorem div_le_self (ha : 0 ≤ a) (hb : 1 ≤ b) : a / b ≤ a := by | simpa only [div_one] using div_le_div_of_nonneg_left ha zero_lt_one hb
@[bound] |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 604 | 609 | theorem Topology.IsInducing.isLindelof_iff {f : X → Y} (hf : IsInducing f) :
IsLindelof s ↔ IsLindelof (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) 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 | 127 | 133 | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by | cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative] |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | Mathlib/Data/Set/Card.lean | 402 | 408 | theorem InjOn.encard_image (h : InjOn f s) : (f '' s).encard = s.encard := by | rw [encard, ENat.card_image_of_injOn h, encard]
theorem encard_congr (e : s ≃ t) : s.encard = t.encard := by
rw [← encard_univ_coe, ← encard_univ_coe t, encard_univ, encard_univ, ENat.card_congr e]
theorem _root_.Function.Injective.encard_image (hf : f.Injective) (s : Set α) : |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
/-!
# Action of regular elements on a module
We introduce `M`-regular elements, in the... | Mathlib/Algebra/Regular/SMul.lean | 225 | 227 | theorem IsUnit.isSMulRegular (ua : IsUnit a) : IsSMulRegular M a := by | rcases ua with ⟨a, rfl⟩
exact a.isSMulRegular M |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | Mathlib/Tactic/Ring/Basic.lean | 453 | 454 | theorem mul_zero (a : R) : a * 0 = 0 := by | simp |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Renaming variables of polynomials
This file establishes the `rename` operation on mul... | Mathlib/Algebra/MvPolynomial/Rename.lean | 228 | 247 | theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by | classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q, rfl⟩
refine ⟨s ∪ t, ⟨?_, ?_⟩⟩
· refine rename (Subtype.map id ?_) p + rename (Subtype.map id ?_) q <;>
simp +contextual only [id, true_or, or_true,
Finset.mem_union, forall_true_... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | Mathlib/RingTheory/Localization/InvSubmonoid.lean | 77 | 81 | theorem surj'' (z : S) : ∃ (r : R) (m : M), z = r • (toInvSubmonoid M S m : S) := by | rcases IsLocalization.surj M z with ⟨⟨r, m⟩, e : z * _ = algebraMap R S r⟩
refine ⟨r, m, ?_⟩
rw [Algebra.smul_def, ← e, mul_assoc]
simp |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 699 | 700 | theorem separatingRight_iff_flip_ker_eq_bot {B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M} :
B.SeparatingRight ↔ LinearMap.ker B.flip = ⊥ := by | |
/-
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.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | Mathlib/Topology/MetricSpace/Gluing.lean | 159 | 171 | theorem Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s := by | simp only [Sum.uniformity, Filter.mem_sup, Filter.mem_map, mem_uniformity_dist, mem_preimage]
constructor
· rintro ⟨⟨δX, δX0, hX⟩, δY, δY0, hY⟩
refine ⟨min (min δX δY) ε, lt_min (lt_min δX0 δY0) hε, ?_⟩
rintro (a | a) (b | b) h <;> simp only [lt_min_iff] at h
· exact hX h.1.1
· exact absurd h.2 (le_... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | Mathlib/Data/ZMod/Basic.lean | 636 | 641 | theorem val_add_val_of_le {n : ℕ} [NeZero n] {a b : ZMod n} (h : n ≤ a.val + b.val) :
a.val + b.val = (a + b).val + n := by | rw [val_add, Nat.add_mod_add_of_le_add_mod, Nat.mod_eq_of_lt (val_lt _),
Nat.mod_eq_of_lt (val_lt _)]
rwa [Nat.mod_eq_of_lt (val_lt _), Nat.mod_eq_of_lt (val_lt _)] |
/-
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.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 291 | 292 | theorem polar_sub_right (x y y' : M) : polar Q x (y - y') = polar Q x y - polar Q x y' := by | rw [sub_eq_add_neg, sub_eq_add_neg, polar_add_right, polar_neg_right] |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# A lemma about `Approxi... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FiniteDimensional.lean | 27 | 47 | theorem exists_homeomorph_extension {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] {s : Set E}
{f : E → F} {f' : E ≃L[ℝ] F} {c : ℝ≥0} (hf : ApproximatesLinearOn f (f' : E →L[ℝ] F) s c)
(hc : Subsingleton E ∨ lipschitzExtensio... | -- the difference `f - f'` is Lipschitz on `s`. It can be extended to a Lipschitz function `u`
-- on the whole space, with a slightly worse Lipschitz constant. Then `f' + u` will be the
-- desired homeomorphism.
obtain ⟨u, hu, uf⟩ :
∃ u : E → F, LipschitzWith (lipschitzExtensionConstant F * c) u ∧ EqOn (f - ⇑... |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | Mathlib/Data/Rat/Lemmas.lean | 154 | 166 | theorem add_num_den' (q r : ℚ) :
(q + r).num * q.den * r.den = (q.num * r.den + r.num * q.den) * (q + r).den := by | let s := divInt (q.num * r.den + r.num * q.den) (q.den * r.den : ℤ)
have hs : (q.den * r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.mpr (Nat.mul_pos q.pos r.pos)
obtain ⟨c, ⟨c_mul_num, c_mul_den⟩⟩ :=
exists_eq_mul_div_num_and_eq_mul_div_den (q.num * r.den + r.num * q.den) hs
rw [c_mul_num, mul_assoc, mul_com... |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | Mathlib/Data/PNat/Prime.lean | 210 | 214 | theorem gcd_one {n : ℕ+} : gcd n 1 = 1 := by | rw [gcd_comm]
apply one_gcd
@[symm] |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | Mathlib/Data/Nat/Totient.lean | 134 | 144 | theorem totient_div_of_dvd {n d : ℕ} (hnd : d ∣ n) :
φ (n / d) = #{k ∈ range n | n.gcd k = d} := by | rcases d.eq_zero_or_pos with (rfl | hd0); · simp [eq_zero_of_zero_dvd hnd]
rcases hnd with ⟨x, rfl⟩
rw [Nat.mul_div_cancel_left x hd0]
apply Finset.card_bij fun k _ => d * k
· simp only [mem_filter, mem_range, and_imp, Coprime]
refine fun a ha1 ha2 => ⟨(mul_lt_mul_left hd0).2 ha1, ?_⟩
rw [gcd_mul_left, ... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.SuccPred
/-!
# Successors and predecessors of integers
In this file, we show that `ℤ` is both an archimede... | Mathlib/Data/Int/SuccPred.lean | 55 | 59 | theorem natCast_covBy {a b : ℕ} : (a : ℤ) ⋖ b ↔ a ⋖ b := by | rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]
exact Int.natCast_inj
end Int |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | Mathlib/Topology/MetricSpace/Infsep.lean | 176 | 179 | theorem einfsep_pair (hxy : x ≠ y) : ({x, y} : Set α).einfsep = edist x y := by | nth_rw 1 [← min_self (edist x y)]
convert einfsep_pair_eq_inf hxy using 2
rw [edist_comm] |
/-
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, Kim Morrison, Jens Wagemaker, Andrew Yang, Yuyang Zhao
-/
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# The... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 71 | 73 | theorem integralNormalization_coeff_degree_ne {i : ℕ} (hi : p.degree ≠ i) :
coeff (integralNormalization p) i = coeff p i * p.leadingCoeff ^ (p.natDegree - 1 - i) := by | rw [integralNormalization_coeff, if_neg hi] |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 139 | 142 | theorem mem_span_latticeBasis [NumberField K] {x : (K →+* ℂ) → ℂ} :
x ∈ Submodule.span ℤ (Set.range (latticeBasis K)) ↔
x ∈ ((canonicalEmbedding K).comp (algebraMap (𝓞 K) K)).range := by | rw [show Set.range (latticeBasis K) = |
/-
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.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L`... | Mathlib/Algebra/Lie/IdealOperations.lean | 111 | 116 | theorem lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N' := by | rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩
rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩
exact h x hx m hm |
/-
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.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
/-!
# The Karoubi envelope ... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 97 | 98 | theorem hom_ext_iff {P Q : Karoubi C} {f g : P ⟶ Q} : f = g ↔ f.f = g.f := by | constructor |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 154 | 157 | theorem image2_insert_left : image2 f (insert a s) t = (fun b => f a b) '' t ∪ image2 f s t := by | rw [insert_eq, image2_union_left, image2_singleton_left]
@[simp] |
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