Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Sphere.Power
import Mathlib.Geometry.Euclidean.Triangle
/-!
# Ptolemy's theorem
This file proves Ptolemy's th... | dist a b * dist c d + dist b c * dist d a = dist a c * dist b d := by
have h' : Cospherical ({a, c, b, d} : Set P) := by rwa [Set.insert_comm c b {d}]
have hmul := mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi h' hapc hbpd
have hbp := left_dist_ne_zero_of_angle_eq_pi hbpd
have h₁ : dist c d = dist c p ... | Mathlib/Geometry/Euclidean/Sphere/Ptolemy.lean | 53 | 70 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | MDifferentiableWithinAt I I' f s x ↔
ContinuousWithinAt f s x ∧
DifferentiableWithinAt 𝕜 (e'.extend I' ∘ f ∘ (e.extend I).symm)
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 292 | 294 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | dsimp [dNext]
rw [shape _ _ _ hi, zero_comp]
-- This is not a simp lemma; the LHS already simplifies.
| Mathlib/Algebra/Homology/Homotopy.lean | 51 | 54 |
/-
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
import Mathlib.Algebra.Star.SelfAdjoint
/-... | theorem iSupLift_mk {i : ι} (x : K i) (hx : (x : A) ∈ T) :
iSupLift K dir f hf T hT ⟨x, hx⟩ = f i x := by
| Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 1,088 | 1,089 |
/-
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.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
/-!
# Left and right limits
We define the (strict) left and right limits of a f... | (hf.countable_not_continuousWithinAt_Ioi.union hf.countable_not_continuousWithinAt_Iio).mono
_
refine compl_subset_compl.1 ?_
simp only [compl_union]
rintro x ⟨hx, h'x⟩
simp only [mem_setOf_eq, Classical.not_not, mem_compl_iff] at hx h'x ⊢
exact continuousAt_iff_continuous_left'_right'.2 ⟨h'x, hx⟩
... | Mathlib/Topology/Order/LeftRightLim.lean | 240 | 252 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... |
@[simp]
theorem ae_restrict_union_eq (s t : Set α) :
ae (μ.restrict (s ∪ t)) = ae (μ.restrict s) ⊔ ae (μ.restrict t) := by
simp [union_eq_iUnion, iSup_bool_eq]
theorem ae_restrict_biUnion_eq (s : ι → Set α) {t : Set ι} (ht : t.Countable) :
ae (μ.restrict (⋃ i ∈ t, s i)) = ⨆ i ∈ t, ae (μ.restrict (s i)) := b... | Mathlib/MeasureTheory/Measure/Restrict.lean | 500 | 507 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Ring.NonZeroDivisors
import Mathlib.RingTheory.Localization.FractionRing
/-!
# The field of rational... | The value of `f p 0` for any `p` is never used and in principle this may be anything,
although many usages of `lift_on'` assume `f p 0 = f 0 1`.
-/
protected irreducible_def liftOn' {P : Sort v} (x : RatFunc K) (f : K[X] → K[X] → P)
| Mathlib/FieldTheory/RatFunc/Defs.lean | 189 | 192 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 644 | 644 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
/-... | [IsLocallySurjective J f₁] :
IsLocallySurjective J f₃ ↔ IsLocallySurjective J f₂ := by
constructor
· intro
exact isLocallySurjective_of_isLocallySurjective_fac J fac
· intro
rw [← fac]
infer_instance
lemma comp_isLocallySurjective_iff
| Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 168 | 177 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,098 | 1,105 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | (ofNat(n) : Matrix (Fin 3) (Fin 3) α) =
!![ofNat(n), 0, 0; 0, ofNat(n), 0; 0, 0, ofNat(n)] :=
natCast_fin_three _
| Mathlib/Data/Matrix/Notation.lean | 443 | 446 |
/-
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.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... |
theorem lt_sqrt (hx : 0 ≤ x) : x < √y ↔ x ^ 2 < y := by
| Mathlib/Data/Real/Sqrt.lean | 332 | 333 |
/-
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 ... | k.Coprime n → (m * k).gcd n = m.gcd n := by rw [mul_comm]; apply Coprime.gcd_mul_left_cancel
theorem Coprime.gcd_mul_left_cancel_right (m : ℕ+) {n k : ℕ+} :
k.Coprime m → m.gcd (k * n) = m.gcd n := by
| Mathlib/Data/PNat/Prime.lean | 192 | 195 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | -- This happens even if we make `inv` irreducible!
-- I don't understand how this is happening: it is likely a bug.
| Mathlib/CategoryTheory/Iso.lean | 248 | 250 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | rw [diff_singleton_eq_self hfa, diff_singleton_eq_self] at this
· exact ⟨by simp [← this, mapsTo'], h.injOn.mono (subset_insert ..),
by simp [← this, surjOn_image]⟩
simp only [mem_image, not_exists, not_and]
intro x hx
rw [h.injOn.eq_iff (by simp [hx]) (by simp)]
| Mathlib/Data/Set/Function.lean | 718 | 723 |
/-
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.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | Mathlib/Order/WithBot.lean | 1,246 | 1,247 | |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 897 | 906 | |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | def HasColimit.isoOfEquivalence {F : J ⥤ C} [HasColimit F] {G : K ⥤ C} [HasColimit G] (e : J ≌ K)
(w : e.functor ⋙ G ≅ F) : colimit F ≅ colimit G :=
IsColimit.coconePointsIsoOfEquivalence (colimit.isColimit F) (colimit.isColimit G) e w
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 853 | 856 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... |
lemma neg_of_smul_neg_right [SMulPosReflectLT α β] (h : a • b < 0) (hb : 0 ≤ b) : a < 0 :=
| Mathlib/Algebra/Order/Module/Defs.lean | 511 | 512 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | · rw [angle_eq_iff_two_pi_dvd_sub, ← coe_neg, angle_eq_iff_two_pi_dvd_sub]
rintro (⟨k, H⟩ | ⟨k, H⟩)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 210 | 211 |
/-
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.Multiset.Dedup
import Mathlib.Data.List.Infix
/-!
# Preparations for defining operations on `Finset`.
The operations here ignore multiplicities,... | by_cases h : a ∈ s <;> simp [h]
| Mathlib/Data/Multiset/FinsetOps.lean | 74 | 75 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | ⟨fun s t ⟨a⟩ => Quot.inductionOn a fun a' => by simp only [RingQuot.smul_quot, smul_comm]⟩
instance instAddCommMonoid (r : R → R → Prop) : AddCommMonoid (RingQuot r) where
add := (· + ·)
zero := 0
| Mathlib/Algebra/RingQuot.lean | 255 | 259 |
/-
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.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathl... | simp only [ContinuousLinearMap.ext_iff, ContinuousLinearMap.comp_apply] at hg'inv hg'invf ⊢
simp [ContinuousLinearEquiv.eq_symm_apply, *]
| Mathlib/Analysis/Calculus/Implicit.lean | 196 | 198 |
/-
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 | 169 | 170 | |
/-
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.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | intro x
rcases hf x with ⟨U, hU, h'U⟩
exact ⟨U, hU, h'U.indicator hs⟩
theorem locallyIntegrable_map_homeomorph [BorelSpace X] [BorelSpace Y] (e : X ≃ₜ Y) {f : Y → E}
{μ : Measure X} : LocallyIntegrable f (Measure.map e μ) ↔ LocallyIntegrable (f ∘ e) μ := by
refine ⟨fun h x => ?_, fun h x => ?_⟩
· rcases ... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 276 | 283 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | /-- Casting out of `SignType` respects composition with functions preserving `0, 1, -1`. -/
| Mathlib/Data/Sign.lean | 223 | 223 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... |
@[simp]
theorem testBit_lxor (m n k) : testBit (Int.xor m n) k = xor (testBit m k) (testBit n k) := by
rw [← bitwise_xor, testBit_bitwise]
@[simp]
theorem testBit_lnot : ∀ n k, testBit (lnot n) k = not (testBit n k)
| (n : ℕ), k => by simp [lnot, testBit]
| -[n+1], k => by simp [lnot, testBit]
@[simp]
theorem ... | Mathlib/Data/Int/Bitwise.lean | 336 | 348 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... | traverse (m := Id) (pure ∘ f) = pure ∘ (map f : t β → t γ) := by
ext
exact traverse_eq_map_id _ _
| Mathlib/Control/Traversable/Lemmas.lean | 103 | 105 |
/-
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... | fun _ x1 _ y1 xy => (one_div_lt_one_div (Set.mem_Ioi.mp y1) (Set.mem_Ioi.mp x1)).mpr xy
| Mathlib/Algebra/Order/Field/Basic.lean | 239 | 240 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | rw [Finsupp.sum_single_index (zero_smul R (x ^ n).val)]
| Mathlib/Algebra/Polynomial/Laurent.lean | 636 | 636 |
/-
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 ... |
theorem lintegral_eq_of_subset' (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0∞} (hs : f.range \ {0} ⊆ s) :
f.lintegral μ = ∑ x ∈ s, x * μ (f ⁻¹' {x}) :=
f.lintegral_eq_of_subset fun x hfx _ =>
hs <| Finset.mem_sdiff.2 ⟨f.mem_range_self x, mt Finset.mem_singleton.1 hfx⟩
/-- Calculate the integral of `(g ∘ f)`, where `g : ... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 857 | 870 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
/-!
# Traversing collections
This file proves basic properties of traversable and applicative ... | end Traversable
| Mathlib/Control/Traversable/Lemmas.lean | 123 | 126 |
/-
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... |
theorem iteratedFDerivWithin_eq_iteratedFDeriv {n : ℕ}
(hs : UniqueDiffOn 𝕜 s) (h : ContDiffAt 𝕜 n f x) (hx : x ∈ s) :
iteratedFDerivWithin 𝕜 n f s x = iteratedFDeriv 𝕜 n f x := by
rw [← iteratedFDerivWithin_univ]
rcases h.contDiffOn' le_rfl (by simp) with ⟨u, u_open, xu, hu⟩
rw [← iteratedFDerivWith... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,006 | 1,018 |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | Adj v w := if v = t then if w = t then False else G.Adj s w
else if w = t then G.Adj v s else G.Adj v w
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 52 | 53 |
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... |
/-- Pythagorean theorem, subtracting vectors, vector angle form. -/
theorem norm_sub_sq_eq_norm_sq_add_norm_sq' (x y : V) (h : angle x y = π / 2) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ :=
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 56 | 59 |
/-
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.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | obtain ⟨I, hI⟩ := M.exists_isBasis X (hXY.trans hY)
obtain ⟨J, hJ, hIJ⟩ := hI.indep.subset_isBasis_of_subset (hI.subset.trans hXY)
exact ⟨_, _, hI, hJ, hIJ⟩
| Mathlib/Data/Matroid/Basic.lean | 959 | 962 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | theorem inv_mul' : (a * b)⁻¹ = a⁻¹ / b := by simp
| Mathlib/Algebra/Group/Basic.lean | 554 | 554 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
/-!
# Bounded and unbounded sets
We prove miscellaneous lemmas about ... |
theorem bounded_lt_inter_le [LinearOrder α] (a : α) :
Bounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· < ·) s := by
convert @bounded_lt_inter_not_lt _ s _ a
exact not_lt.symm
| Mathlib/Order/Bounded.lean | 301 | 306 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.Ca... | Function.Bijective (fun f : Aut A ↦ F.map f.hom a) :=
⟨evaluation_aut_injective_of_isConnected F A a, evaluation_aut_surjective_of_isGalois F A a⟩
/-- For Galois `A` and a point `a` of the fiber of `A`, the evaluation at `A` as an equivalence. -/
noncomputable def evaluationEquivOfIsGalois (A : C) [IsGalois A] (... | Mathlib/CategoryTheory/Galois/GaloisObjects.lean | 115 | 119 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | exact hVU <| hVclosed.mem_of_tendsto ((h₁ x hxS).prodMk_nhds (h₁ y hyS)) <|
Eventually.of_forall hxy
/-- If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. -/
theorem Filter.Tendsto.uniform... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 942 | 951 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | rw [IsRoot.def, eq_natCast, h, ← coeff_zero_eq_eval_zero] at hroot
by_cases hone : n = 1
· simp only [hone, cyclotomic_one, zero_sub, coeff_one_zero, coeff_X_zero, neg_eq_zero,
one_ne_zero, coeff_sub] at hroot
rw [cyclotomic_coeff_zero (ZMod p) (Nat.succ_le_of_lt
(lt_of_le_of_ne (Nat.succ_le_of_lt hpo... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 559 | 575 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | end ΨSq
section Ψ
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 305 | 307 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Discrete.Basic
/-!
# Categorical (co)products
This file defines (co)products ... | apply Category.id_comp
uniq := fun s m w => by
specialize w default
erw [Category.id_comp] at w
exact w }
instance (priority := 100) hasCoproduct_unique [Nonempty β] [Subsingleton β] (f : β → C) :
HasCoproduct f :=
let ⟨_⟩ := nonempty_unique β; HasColimit.mk (colimitCoconeOf... | Mathlib/CategoryTheory/Limits/Shapes/Products.lean | 760 | 769 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.LinearAlgebra.AffineSpace.Slope
/-!
# Derivative as the limit of the slope
In this file we relate the ... | ContinuousAt (Function.update (fun x ↦ (f x - f c) / (x - c)) c a) c := by
rw [← slope_fun_def_field]
exact continuousAt_update_same.mpr <| hasDerivAt_iff_tendsto_slope.mp hf
end NormedField
/-! ### Upper estimates on liminf and limsup -/
| Mathlib/Analysis/Calculus/Deriv/Slope.lean | 145 | 151 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,286 | 1,291 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 776 | 781 | |
/-
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.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | rw [lintegral_indicator ht, setLIntegral_one] at h1 h2
rw [← h1, mul_left_comm, h2]
/-- Left invariant Borel measures on a measurable group are unique (up to a scalar). -/
@[to_additive
" Left invariant Borel measures on an additive measurable group are unique (up to a scalar). "]
theorem measure_eq_div_smul (h2s ... | Mathlib/MeasureTheory/Group/Prod.lean | 325 | 334 |
/-
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.NumberTheory.Padics.PadicVal.Basic
/-!
# p-adic norm
This file defines the `p`-adic norm on `ℚ`.
The `p`-adic valuation on `ℚ` is the difference o... | padicNorm.nonarchimedean.trans
(max_le (ht a (Finset.mem_insert_self a S))
| Mathlib/NumberTheory/Padics/PadicNorm.lean | 298 | 299 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | instance : Add NatOrdinal := ⟨nadd⟩
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 327 | 327 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
/-!
# The initial algebra of a multivariate qpf is again a qpf.
Fo... |
theorem recF_eq' {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (x : q.P.W α) :
recF g x = g (abs (appendFun id (recF g) <$$> q.P.wDest' x)) := by
apply q.P.w_cases _ x
| Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 64 | 67 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | rcases h𝔖 with ⟨V, hV, hVx⟩
filter_upwards [prod_mem_nhds (UniformOnFun.gen_mem_nhds _ _ _ hV hU)
(inter_mem hVx <| hc <| UniformSpace.ball_mem_nhds _ hU)]
with ⟨g, y⟩ ⟨hg, hyV, hy⟩ using ⟨toFun 𝔖 f y, hy, hg y hyV⟩
/-- If each point of `α` admits a neighbourhood `V ∈ 𝔖`,
then the evaluation of `f : α →... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 951 | 958 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `α` is given by `Nat.card α`. This function has
the "junk val... | Mathlib/Data/Finite/Card.lean | 232 | 234 | |
/-
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.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | the predicate is true everywhere in the chain.
That is, we can propagate the predicate down the chain.
-/
theorem Chain.induction (p : α → Prop) (l : List α) (h : Chain r a l)
(carries : ∀ ⦃x y : α⦄, r x y → p x → p y) (initial : p a) : ∀ i ∈ l, p i := by
induction h with
| nil => simp
| @cons a b t hab _ h_i... | Mathlib/Data/List/Chain.lean | 370 | 380 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... |
theorem pow_mod_card_support_cycleOf_self_apply [DecidableEq α] [Fintype α]
(f : Perm α) (n : ℕ) (x : α) : (f ^ (n % #(f.cycleOf x).support)) x = (f ^ n) x := by
by_cases hx : f x = x
· rw [pow_apply_eq_self_of_apply_eq_self hx, pow_apply_eq_self_of_apply_eq_self hx]
· rw [← cycleOf_pow_apply_self, ← cycleOf... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 306 | 322 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "covering pres... | lemma eq_top_pullback {X Y : C} {S T : Sieve X} (h : S ≤ T) (f : Y ⟶ X) (hf : S f) :
T.pullback f = ⊤ := by
ext Z g
simp only [Sieve.pullback_apply, Sieve.top_apply, iff_true]
apply h
apply S.downward_closed
exact hf
| Mathlib/CategoryTheory/Sites/Coverage.lean | 189 | 195 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... | field_simp [linePolynomial, sub_ne_zero_of_ne hy]
· field_simp [linePolynomial, slope_of_X_ne hx, sub_ne_zero_of_ne hx]
ring1
nth_rw 1 [hy₂]
simp only [XYIdeal, XClass, YClass, linePolynomial]
rw [← span_pair_add_mul_right <| mk W <| C <| C <| -W.slope x₁ x₂ y₁ y₂, ← map_mul,
| Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 288 | 293 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | Mathlib/GroupTheory/GroupAction/Basic.lean | 537 | 541 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Locally convex topological modules
A `LocallyConvexSpace` is a ... | variable {ι : Sort*} {𝕜 E F : Type*} [Semiring 𝕜] [PartialOrder 𝕜] [IsOrderedRing 𝕜]
[AddCommMonoid E] [Module 𝕜 E]
[AddCommMonoid F] [Module 𝕜 F]
theorem locallyConvexSpace_sInf {ts : Set (TopologicalSpace E)}
(h : ∀ t ∈ ts, @LocallyConvexSpace 𝕜 E _ _ _ _ _ t) :
@LocallyConvexSpace 𝕜 E _ _ _ _ _ ... | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | 152 | 161 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Discriminant
/-!
# Field/algebra norm / trace and localizatio... | · haveI : Subsingleton Rₘ := Module.subsingleton R Rₘ
simp [eq_iff_true_of_subsingleton]
let b := Module.Free.chooseBasis R S
letI := Classical.decEq (Module.Free.ChooseBasisIndex R S)
rw [Algebra.trace_eq_matrix_trace (b.localizationLocalization Rₘ M Sₘ),
Algebra.trace_eq_matrix_trace b, ← Algebra.map_... | Mathlib/RingTheory/Localization/NormTrace.lean | 83 | 92 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 70 | 70 | |
/-
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... | rw [← inv_le_inv_of_neg (inv_lt_zero.2 hb) ha, inv_inv]
| Mathlib/Algebra/Order/Field/Basic.lean | 375 | 375 |
/-
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.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 876 | 894 | |
/-
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... | theorem orderOf_units {y : Gˣ} : orderOf (y : G) = orderOf y :=
orderOf_injective (Units.coeHom G) Units.ext y
/-- If the order of `x` is finite, then `x` is a unit with inverse `x ^ (orderOf x - 1)`. -/
@[to_additive (attr := simps) "If the additive order of `x` is finite, then `x` is an additive
unit with inverse ... | Mathlib/GroupTheory/OrderOfElement.lean | 338 | 355 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,341 | 1,348 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | · exact geom_sum_pos' hx' hn
theorem geom_sum_ne_zero [Ring R] [LinearOrder R] [IsStrictOrderedRing R]
(hx : x ≠ -1) (hn : n ≠ 0) :
∑ i ∈ range n, x ^ i ≠ 0 := by
obtain _ | _ | n := n
· cases hn rfl
· simp only [zero_add, range_one, sum_singleton, pow_zero, ne_eq, one_ne_zero, not_false_eq_true]
r... | Mathlib/Algebra/GeomSum.lean | 554 | 565 |
/-
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.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | rcases v with ⟨l⟩
dsimp [PrimeMultiset.toNatMultiset]
let l' := l.map ((↑) : Nat.Primes → ℕ)
have (p : ℕ) (hp : p ∈ l') : p.Prime := by
| Mathlib/Data/PNat/Factors.lean | 225 | 228 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | Int.natAbs (Algebra.norm ℤ (seq K w₁ hB n : 𝓞 K)) ≤ B := by
cases n with
| zero =>
have : 1 ≤ B := by
contrapose! hB
simp only [Nat.lt_one_iff.mp hB, CharP.cast_eq_zero, mul_zero, zero_le]
simp only [ne_eq, seq, map_one, Int.natAbs_one, this]
| succ n =>
rw [← Nat.cast_le (α... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 265 | 275 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 782 | 793 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | lemma refl {X Y : C} (z : W.LeftFraction X Y) : LeftFractionRel z z :=
⟨z.Y', 𝟙 _, 𝟙 _, rfl, rfl, by simpa only [Category.comp_id] using z.hs⟩
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 234 | 235 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
/-!
# Conserv... | rw [← map_add_atTop_eq_nat k, frequently_map]
refine (hx hk).mono fun n hn => ?_
rwa [add_comm, iterate_add_apply]
/-- If `f` is a conservative self-map and `s` is a measurable set of positive measure, then
`ae μ`-frequently we have `x ∈ s` and `s` returns to `s` under infinitely many iterations of `f`. -/
theor... | Mathlib/Dynamics/Ergodic/Conservative.lean | 176 | 183 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,120 | 3,121 | |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Lie algebras with non-degenerate invariant bilinear forms are s... | lemma restrict_orthogonal_nondegenerate (I : LieIdeal K L) (hI : IsAtom I) :
(Φ.restrict (orthogonal Φ hΦ_inv I)).Nondegenerate := by
rw [LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal hΦ_refl]
simp only [LieIdeal.toLieSubalgebra_toSubmodule, orthogonal_toSubmodule,
LinearMap.BilinForm.or... | Mathlib/Algebra/Lie/InvariantForm.lean | 150 | 177 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | Mathlib/CategoryTheory/Iso.lean | 605 | 608 | |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.DirichletCharacter.Basic
import Mathlib.NumberTheory.GaussSum
/-!
# Gauss sums for Dirichlet characters
-/
variable {N : ℕ} [NeZero N] {R... | /-- If `χ` is primitive, but `e` is not, then `gaussSum χ e = 0`. -/
lemma gaussSum_eq_zero_of_isPrimitive_of_not_isPrimitive [IsDomain R]
{χ : DirichletCharacter R N} (hχ : IsPrimitive χ) (he : ¬IsPrimitive e) :
gaussSum χ e = 0 := by
contrapose! hχ
rcases e.exists_divisor_of_not_isPrimitive he with ⟨d, hd... | Mathlib/NumberTheory/DirichletCharacter/GaussSum.lean | 44 | 51 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,523 | 3,531 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.RingHomProperties
/-!
# Constructors for properties of morphisms between schemes
This file pro... | letI := HasAffineProperty.of_isLocalAtTarget P
inferInstance
open MorphismProperty in
instance (P : MorphismProperty Scheme)
[P.HasOfPostcompProperty @IsOpenImmersion] [P.RespectsRight @IsOpenImmersion]
[IsLocalAtSource P] : IsLocalAtSource P.diagonal := by
let g {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens) ... | Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean | 151 | 166 |
/-
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.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | rw [sum_insert ns]
exact add _ _ A ih
rw [← sum_C_mul_X_pow_eq p, Polynomial.sum]
exact B (support p)
| Mathlib/Algebra/Polynomial/Basic.lean | 923 | 926 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | u.val.inner_map_map_of_mem_unitary u.property x y
/-- The unitary elements of continuous linear maps on a Hilbert space coincide with the linear
isometric equivalences on that Hilbert space. -/
noncomputable def linearIsometryEquiv : unitary (H →L[𝕜] H) ≃* (H ≃ₗᵢ[𝕜] H) where
toFun u :=
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 514 | 519 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... | rw [← mk_le_aleph0_iff, ← lift_le_aleph0]
exact lift_card_closure_le_card_term.trans mk_le_aleph0
variable {L} (S)
| Mathlib/ModelTheory/Substructures.lean | 319 | 322 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 3,576 | 3,579 | |
/-
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.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | noncomputable
def linearMap (b₁ : Basis ι₁ R M₁) (b₂ : Basis ι₂ R M₂) :
Basis (ι₂ × ι₁) R (M₁ →ₗ[R] M₂) :=
(Matrix.stdBasis R ι₂ ι₁).map (LinearMap.toMatrix b₁ b₂).symm
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 944 | 948 |
/-
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.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,354 | 1,356 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | /-- `Fact.out` contains the unwrapped witness for the fact represented by the instance of
`Fact p`. -/
| Mathlib/Logic/Basic.lean | 81 | 82 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | · norm_num
· simp only [differentiableAt_sin, implies_true]
· exact continuousOn_cos
theorem integral_cos_mul_complex {z : ℂ} (hz : z ≠ 0) (a b : ℝ) :
(∫ x in a..b, Complex.cos (z * x)) = Complex.sin (z * b) / z - Complex.sin (z * a) / z := by
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 531 | 536 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | theorem exists_list_of_mem_closure {s : Set M} {x : M} (hx : x ∈ closure s) :
∃ l : List M, (∀ y ∈ l, y ∈ s) ∧ l.prod = x := by
rwa [← SetLike.mem_coe, closure_eq_image_prod, Set.mem_image] at hx
| Mathlib/Algebra/Group/Submonoid/Membership.lean | 227 | 229 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | simp only [← cancel_mono S₂.leftHomologyIso.inv, assoc, ← leftHomologyIso_inv_naturality φ,
homologyπ_comp_leftHomologyIso_inv]
simp only [homologyπ, assoc, Iso.hom_inv_id_assoc, leftHomologyπ_naturality]
@[reassoc (attr := simp)]
lemma homologyι_naturality (φ : S₁ ⟶ S₂) [S₁.HasHomology] [S₂.HasHomology] :
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 907 | 912 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | (h_add : local_hAdd[R, inst₁] = local_hAdd[R, inst₂])
(h_one : (letI := inst₁; One.one : R) = (letI := inst₂; One.one)) :
inst₁ = inst₂ := by
have : inst₁.toAddMonoidWithOne = inst₂.toAddMonoidWithOne :=
AddMonoidWithOne.ext h_add h_one
have : inst₁.toNatCast = inst₂.toNatCast := congrArg (·.toNatCa... | Mathlib/Algebra/Ring/Ext.lean | 214 | 222 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,670 | 1,672 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 993 | 1,001 | |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | (hf : LocallyFinite fun i => mulSupport (f i)) :
Continuous fun x => ∏ᶠ (i) (_ : p i), f i x := by
simp only [← finprod_subtype_eq_finprod_cond]
exact continuous_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective)
| Mathlib/Topology/Algebra/Monoid.lean | 890 | 894 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.Order.Units
import Mathlib.RingTheory.Coprime.Basic
/-!
# Unit... | rwa [add_comm, add_lt_add_iff_left, lt_add_iff_pos_left, tsub_pos_iff_lt]
· exact fun h => h.1.ne (add_comm k n)
· exact ⟨add_lt_add_right hkm n, add_lt_add_right hmn n⟩
· rw [← add_assoc, add_tsub_cancel_of_le hmn.le, add_comm]
exact fun h => h.1.ne rfl
· intro h
have := h.1
rw [add_comm, add_l... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 222 | 254 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic... | contradiction
-- If A ∈ F, then x ∈ cl A.
have claim2 : ∀ A : Set X, A ∈ F → x ∈ cl A := by
intro A hA
exact claim1 (cl A) (isClosed_cl A) (mem_of_superset hA (subset_cl A))
-- T0 is closed under intersections.
have claim3 : ∀ (S1) (_ : S1 ∈ T0) (S2) (_ : S2 ∈ T0), S1 ∩ S2 ∈ T0 := by
rintro S1 ⟨... | Mathlib/Topology/Category/Compactum.lean | 281 | 349 |
/-
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.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | · have h : √(1 - x ^ 2) = 0 := sqrt_eq_zero_of_nonpos (by nlinarith)
rw [h]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 280 | 281 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... |
theorem derivWithin_finset_prod
(hf : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x) :
derivWithin (∏ i ∈ u, f i ·) s x =
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 341 | 344 |
/-
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 Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... |
protected lemma add_comm : a + b = b + a := by
simp [add_def, Int.add_comm, Int.mul_comm, Nat.mul_comm]
protected theorem add_assoc : a + b + c = a + (b + c) :=
numDenCasesOn' a fun n₁ d₁ h₁ ↦ numDenCasesOn' b fun n₂ d₂ h₂ ↦ numDenCasesOn' c fun n₃ d₃ h₃ ↦ by
simp only [ne_eq, Int.natCast_eq_zero, h₁, not_fal... | Mathlib/Data/Rat/Defs.lean | 233 | 239 |
/-
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.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | have measurable supersets `u'` and `w'` of the same measure. We will show that these satisfy
the conclusion of the theorem, i.e., `μ (u' ∩ w') = 0`. For this, note that
`ρ (u' ∩ w') = ρ (u ∩ w')` (as `w'` is measurable, see `measure_toMeasurable_add_inter_left`).
The latter set is included in the set wh... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 277 | 416 |
/-
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 | 340 | 343 | |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | Mathlib/Data/Nat/Fib/Basic.lean | 312 | 318 | |
/-
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.Data.Nat.Prime.Factorial
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Ring.Lemmas
/-!
# Lemmas of Gauss and Eisenstein
... | ((∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p)) : ℕ) : ZMod 2) := by
simp only [mod_add_div]
_ = (∑ x ∈ Ico 1 (p / 2).succ, ((a * x : ℕ) : ZMod p).val : ℕ) +
(∑ x ∈ Ico 1 (p / 2).succ, a * x / p : ℕ) := by
simp only [val_natCast]
simp [sum_add_distrib, ← mul_sum, Nat.cast... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 118 | 144 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A c₁ c₂ c₃`: a basis... | simp only [mul_right_comm _ _ (c₁ * c₃), mul_comm _ (c₁ * c₃)]
simp only [mul_comm _ c₁, mul_right_comm _ _ c₁]
simp only [mul_comm _ c₂, mul_right_comm _ _ c₃]
simp only [← mul_comm c₁ c₂, ← mul_assoc]
simp only [mul_re, sub_eq_add_neg, add_smul, neg_smul, mul_imI, ← add_assoc, mul_imJ, mul_imK]
linear_com... | Mathlib/Algebra/QuaternionBasis.lean | 125 | 135 |
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