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) 2020. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Massot
-/
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FailIfNoProgress
import Mathlib.Algebra.Group.Commutator
/-!
# `group` tactic
Normalizes expressions in the langu... | theorem zpow_trick_one' {G : Type*} [Group G] (a b : G) (n : ℤ) :
a * b ^ n * b = a * b ^ (n + 1) := by rw [mul_assoc, mul_zpow_self]
| Mathlib/Tactic/Group.lean | 43 | 44 |
/-
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.GroupTheory.Torsion
import Mathlib.Data.ENat.Lattice
/-!
# Minimum order of an element
This file defines the minimum order of an element of a monoid.
##... |
@[simp]
protected lemma minOrder {n : ℕ} (hn : n ≠ 0) (hn₁ : n ≠ 1) : minOrder (ZMod n) = n.minFac := by
have : Fact (1 < n) := ⟨one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn, hn₁⟩⟩
classical
have : (↑(n / n.minFac) : ZMod n) ≠ 0 := by
rw [Ne, ringChar.spec, ringChar.eq (ZMod n) n]
exact
not_dvd_of_pos_of_... | Mathlib/GroupTheory/Order/Min.lean | 75 | 90 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import... | theorem antideriv_bernoulliFun (k : ℕ) (x : ℝ) :
HasDerivAt (fun x => bernoulliFun (k + 1) x / (k + 1)) (bernoulliFun k x) x := by
convert (hasDerivAt_bernoulliFun (k + 1) x).div_const _ using 1
field_simp [Nat.cast_add_one_ne_zero k]
| Mathlib/NumberTheory/ZetaValues.lean | 67 | 71 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.LinearAlgebra.BilinearForm.Hom
import Mathlib.LinearAlgebra.Dual.Lemmas
/-!
# Bilinear form
This file defines various properties of ... | Mathlib/LinearAlgebra/BilinearForm/Properties.lean | 521 | 526 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | theorem zpow_add {A : M} (ha : IsUnit A.det) (m n : ℤ) : A ^ (m + n) = A ^ m * A ^ n := by
induction n with
| hz => simp
| hp n ihn => simp only [← add_assoc, zpow_add_one ha, ihn, mul_assoc]
| hn n ihn => rw [zpow_sub_one ha, ← mul_assoc, ← ihn, ← zpow_sub_one ha, add_sub_assoc]
theorem zpow_add_of_nonpos {A ... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 149 | 158 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... |
lemma map'_inv_eq_inv_map' {n m : ℕ} (h : n+1 ≤ m) {f g : ComposableArrows C m}
(app : f.obj' n ≅ g.obj' n) (app' : f.obj' (n+1) ≅ g.obj' (n+1))
| Mathlib/CategoryTheory/ComposableArrows.lean | 707 | 709 |
/-
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.RingTheory.FinitePresentation
import Mathlib.RingTheory.FiniteStability
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.Ideal.Quotient.Ni... | variable {R : Type u} [CommRing R]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (f : P →ₐ[R] A)
theorem of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
in... | Mathlib/RingTheory/Smooth/Basic.lean | 188 | 196 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
/-!
# Well-approximable numbers and Gallagher's ergodic theorem
Gallagher's e... | /-- In a seminormed group `A`, given a sequence of distances `δ₁, δ₂, ...`, `wellApproximable A δ`
is the limsup as `n → ∞` of the sets `approxOrderOf A n δₙ`. Thus, it is the set of points that
lie in infinitely many of the sets `approxOrderOf A n δₙ`. -/
| Mathlib/NumberTheory/WellApproximable.lean | 77 | 79 |
/-
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.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | simp [mul_comm c, hc]
| Mathlib/Algebra/ModEq.lean | 290 | 291 |
/-
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.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | ring_nf
rfl
· have ID : ‖D / (1 + D) - 1‖ = 1 / (1 + D) := by
rw [Real.norm_of_nonpos]
· simp only [B.ne', Ne, not_false_iff, mul_one, neg_sub, add_tsub_cancel_right,
field_simps]
· simp only [B.ne', Ne, not_false_iff, mul_one, field_simps]
a... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 406 | 418 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Algebra.Module... | Mathlib/Algebra/Module/Submodule/Basic.lean | 482 | 485 | |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Order.ScottContinuity
import Mathlib.Topology.Order.UpperLowerSetTopology
/-!
# Scott topology
This file introduces the Scott topology on a p... | `d` lies in `u` then there is a tail of `d` which is a subset of `u`. -/
class IsScottHausdorff : Prop where
topology_eq_scottHausdorff : ‹TopologicalSpace α› = scottHausdorff α D
| Mathlib/Topology/Order/ScottTopology.lean | 164 | 166 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Uniqueness
import Mathlib.Analysis.Calculus.DiffContOnCl
import Mathlib.Analysis.Calculus.DSlope
import Mathlib.Analysis.Calculus.F... | · intro fa
filter_upwards [fa.eventually_analyticAt]
apply AnalyticAt.differentiableAt
· intro d
rcases _root_.eventually_nhds_iff.mp d with ⟨s, d, o, m⟩
have h : AnalyticOnNhd ℂ f s := by
refine DifferentiableOn.analyticOnNhd ?_ o
intro z m
exact (d z m).differentiableWithinAt
... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 639 | 651 |
/-
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.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 722 | 730 | |
/-
Copyright (c) 2024 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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | (⨆ i, f i).card ≤ Cardinal.sum (fun i ↦ (f i).card) := by
have := lift_card_iSup_le_sum_card f
rwa [Cardinal.lift_id'] at this
theorem card_iSup_Iio_le_sum_card {o : Ordinal.{u}} (f : Iio o → Ordinal.{max u v}) :
(⨆ a : Iio o, f a).card ≤ Cardinal.sum fun i ↦ (f ((enumIsoToType o).symm i)).card := by
app... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 61 | 70 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,815 | 1,817 | |
/-
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... | lemma InvOn.prodMap (h₁ : InvOn g₁ f₁ s₁ t₁) (h₂ : InvOn g₂ f₂ s₂ t₂) :
InvOn (fun x ↦ (g₁ x.1, g₂ x.2)) (fun x ↦ (f₁ x.1, f₂ x.2)) (s₁ ×ˢ s₂) (t₁ ×ˢ t₂) :=
| Mathlib/Data/Set/Function.lean | 1,227 | 1,228 |
/-
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.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | rw [← sub_le_iff_le_add']
exact le_of_abs_le (dist_eq a b).ge
| Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 105 | 107 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... |
nonrec theorem ContinuousWithinAt.div (hf : ContinuousWithinAt f s a)
(hg : ContinuousWithinAt g s a) (h₀ : g a ≠ 0) : ContinuousWithinAt (f / g) s a :=
| Mathlib/Topology/Algebra/GroupWithZero.lean | 177 | 179 |
/-
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... | zero := 0
add := (· + ·)
| Mathlib/Data/Rat/Defs.lean | 289 | 290 |
/-
Copyright (c) 2016 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Control.Basic
import Mathlib.Data.Set.Defs
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Data.Set.Notation
/-!
# Functoriality of `Set`
... | eq_univ_of_forall fun ⟨_, ha⟩ => mem_of_mem_coe <| hγ.symm ▸ ha
| Mathlib/Data/Set/Functor.lean | 93 | 94 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | ∃ (φ : α → β) (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v :=
isEquivalent_iff_exists_eq_mul.mp huv
theorem isEquivalent_of_tendsto_one (hz : ∀ᶠ x in l, v x = 0 → u x = 0)
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 187 | 190 |
/-
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.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | rw [find?_eq_some_iff_getElem]
exact ⟨fun ⟨hpb, i, hi, hfb, h⟩ ↦
⟨hpb, ⟨⟨i, length_ofFn (f := f) ▸ hi⟩, by simpa using hfb, fun j hj ↦ by simpa using h j hj⟩⟩,
fun ⟨hpb, i, hfb, h⟩ ↦
⟨hpb, ⟨i, (length_ofFn (f := f)).symm ▸ i.isLt, by simpa using hfb,
fun j hj ↦ by simpa using h ⟨j, by omega⟩... | Mathlib/Data/List/OfFn.lean | 162 | 169 |
/-
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... | @[deprecated nmul_nadd_le₃ (since := "2024-11-19")]
theorem nmul_nadd_le₃' {a' b' c' : Ordinal} (ha : a' ≤ a) (hb : b' ≤ b) (hc : c' ≤ c) :
a' ⨳ (b ⨳ c) ♯ a ⨳ (b' ⨳ c) ♯ a ⨳ (b ⨳ c') ♯ a' ⨳ (b' ⨳ c') ≤
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 607 | 609 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | Mathlib/Data/Nat/Bits.lean | 479 | 479 | |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | Mathlib/Algebra/BigOperators/Finprod.lean | 1,183 | 1,198 | |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | · rw [lowerCentralSeries_succ]
exact (LieSubmodule.mono_lie_right ⊤ (ih hk)).trans (LieSubmodule.lie_le_right _ _)
· exact hk.symm ▸ le_rfl
theorem eventually_iInf_lowerCentralSeries_eq [IsArtinian R M] :
∀ᶠ l in Filter.atTop, ⨅ k, lowerCentralSeries R L M k = lowerCentralSeries R L M l := by
have ... | Mathlib/Algebra/Lie/Nilpotent.lean | 174 | 183 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 687 | 708 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 897 | 899 | |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | lemma ext_r (s s' : S.Splitting) (h : s.r = s'.r) : s = s' := by
have := s.epi_g
have eq := s.id
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 518 | 520 |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
import Mathlib.MeasureTheory.Function.LpSpace.Complete
import Mathlib.MeasureThe... | have zero_mem : (0 : G) ∈ (range (g : α → G) ∪ {0} : Set G) ∩ { y | 0 ≤ y } := by
simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or,
mem_setOf_eq, le_refl, and_self_iff]
have : SeparableSpace ((range (g : α → G) ∪ {0}) ∩ { y | 0 ≤ y } : Set G) := by
apply IsSeparable... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 750 | 766 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Option.Basic
import Batteries.Tactic.Congr
import Mathlib.Data.Set.Basic
import Mathlib.Tactic.Contrapose
/-!
# Partial Equivalences
In this file, ... | section Single
| Mathlib/Data/PEquiv.lean | 285 | 286 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | @[to_additive const_sub]
theorem Filter.Tendsto.const_div' (b : G) {c : G} {f : α → G} {l : Filter α}
(h : Tendsto f l (𝓝 c)) : Tendsto (fun k : α => b / f k) l (𝓝 (b / c)) :=
tendsto_const_nhds.div' h
| Mathlib/Topology/Algebra/Group/Basic.lean | 902 | 905 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | rcases hx.eq_or_lt with (rfl | hlt)
· rcases eq_or_ne y 0 with (rfl | hy) <;> simp [*]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 257 | 258 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | have B : ContinuousAt F 0 := by apply Continuous.continuousAt; fun_prop
have : f₀ =ᶠ[𝓝 (F 0)] f₁ := by convert h; simp [F]
exact B.preimage_mem_nhds this
theorem Filter.EventuallyEq.lineDifferentiableAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
LineDifferentiableAt 𝕜 f₀ x v ↔ LineDifferentiableAt 𝕜 f₁ x v :=
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 321 | 326 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.Normed.Operator.LinearIsometry
import Mathlib.Analysi... | map and existence of a bound on the norm of the image. The linear map can be constructed using
`LinearMap.mk₂`.
If you have an explicit bound, use `LinearMap.mkContinuous₂` instead, as a norm estimate will
follow automatically in `LinearMap.mkContinuous₂_norm_le`. -/
| Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean | 92 | 96 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | exact mod_cast h
/-- `q` is the minimum factor of `mersenne p`, so `M p = 0` in `X q`. -/
| Mathlib/NumberTheory/LucasLehmer.lean | 397 | 399 |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite.Basic
/-!
# Coverings and sieves... |
theorem compatiblePreserving_opens_map :
CompatiblePreserving (Opens.grothendieckTopology X) (Opens.map f) :=
compatiblePreservingOfFlat _ _
| Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 187 | 190 |
/-
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.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Formal (multivariate) ... | /-- The `n`th truncation of a multivariate formal power series to a multivariate polynomial
If `f : MvPowerSeries σ R` and `n : σ →₀ ℕ` is a (finitely-supported) function from `σ`
to the naturals, then `trunc' R n f` is the multivariable power series obtained from `f`
by keeping only the monomials $c\prod_i X_i^{a_i}$... | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 77 | 92 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... |
variable {_mα : MeasurableSpace α}
@[symm]
| Mathlib/Probability/Independence/Kernel.lean | 275 | 278 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
deprecated_module (since := "2025-04-13")
| Mathlib/MeasureTheory/Integral/Periodic.lean | 158 | 173 | |
/-
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... | lemma RightHomologyData.rightHomologyIso_hom_comp_homologyIso_inv (h : S.RightHomologyData) :
h.rightHomologyIso.hom ≫ h.homologyIso.inv = S.rightHomologyIso.hom := by
dsimp only [homologyIso]
simp only [Iso.trans_inv, Iso.symm_inv, Iso.hom_inv_id_assoc]
@[reassoc]
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 983 | 988 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra... | section Ring
| Mathlib/Algebra/Ring/Units.lean | 50 | 50 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Data.Set.Order
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.Interval.Set.LinearOrder
import Mathlib.Tac... | lemma uIoo_self : uIoo a a = ∅ := by simp [uIoo]
lemma Ioo_subset_uIoo : Ioo a b ⊆ uIoo a b := Ioo_subset_Ioo inf_le_left le_sup_right
/-- Same as `Ioo_subset_uIoo` but with `Ioo a b` replaced by `Ioo b a`. -/
lemma Ioo_subset_uIoo' : Ioo b a ⊆ uIoo a b := Ioo_subset_Ioo inf_le_right le_sup_left
variable {x : α}
le... | Mathlib/Order/Interval/Set/UnorderedInterval.lean | 349 | 361 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,284 | 2,286 | |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... | /-- An implementation of the extended GCD algorithm.
At each step we are computing a triple `(r, s, t)`, where `r` is the next value of the GCD
algorithm, to compute the greatest common divisor of the input (say `x` and `y`), and `s` and `t`
are the coefficients in front of `x` and `y` to obtain `r` (i.e. `r = s * x + ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 179 | 189 |
/-
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... | InnerProductGeometry.norm_div_cos_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))]
/-- A side of a right-angled triangle divided by the cosine of the adjacent angle equals the
hypotenuse, version subtracting vectors. -/
the... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 397 | 403 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... | star_ratCast _
| Mathlib/Algebra/Star/SelfAdjoint.lean | 269 | 270 |
/-
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.Logic.Function.Iterate
import Mathlib.Order.Monotone.Basic
/-!
# Inequalities on iterates
In this file we prove some inequalities comparing `f^[n] ... | induction n with
| zero => exact hn.false.elim
| succ n ihn =>
suffices x n ≤ y n from (hx n n.lt_succ_self).trans_le ((hf this).trans <| hy n n.lt_succ_self)
cases n with
| zero => exact h₀
| succ n =>
refine (ihn n.zero_lt_succ (fun k hk => hx _ ?_) fun k hk => hy _ ?_).le <;>
exact hk.trans n.s... | Mathlib/Order/Iterate.lean | 51 | 60 |
/-
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... | from
| Mathlib/NumberTheory/Dioph.lean | 302 | 302 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.... | end Pullback
section Map
| Mathlib/CategoryTheory/Subobject/Basic.lean | 561 | 564 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Typeclasses... | · apply restrict_eq_self_of_ae_mem
rw [ae_dirac_iff] <;> assumption
· rw [restrict_eq_zero, dirac_apply' _ hs, indicator_of_not_mem has]
| Mathlib/MeasureTheory/Measure/Dirac.lean | 166 | 168 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... |
theorem integrable_stoppedProcess [LocallyFiniteOrderBot ι] (hτ : IsStoppingTime ℱ τ)
(hu : ∀ n, Integrable (u n) μ) (n : ι) : Integrable (stoppedProcess u τ n) μ :=
integrable_stoppedProcess_of_mem_finset hτ hu n fun _ h => Finset.mem_Iio.mpr h
end StoppedProcess
end StoppedValueOfMemFinset
section AdaptedSt... | Mathlib/Probability/Process/Stopping.lean | 908 | 928 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 685 | 689 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,641 | 1,641 | |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,561 | 1,564 | |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | _ ≤ b i * ⌈(b i)⁻¹⌉₊ := by gcongr; exact Nat.le_ceil _
_ < b i * n := by gcongr
case le => calc b i * n ≤ 1 * n := by have := R.b_lt_one i; gcongr
_ = n := by rw [one_mul]
positivity
filter_upwards [h_mai... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 1,150 | 1,157 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... |
theorem eq_iff_card_ge_of_superset (hst : s ⊆ t) : #t ≤ #s ↔ t = s :=
(eq_iff_card_le_of_subset hst).trans eq_comm
theorem subset_iff_eq_of_card_le (h : #t ≤ #s) : s ⊆ t ↔ s = t :=
⟨fun hst => eq_of_subset_of_card_le hst h, Eq.subset'⟩
theorem map_eq_of_subset {f : α ↪ α} (hs : s.map f ⊆ s) : s.map f = s :=
| Mathlib/Data/Finset/Card.lean | 272 | 279 |
/-
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... | rw [hs.cthickening_eq_biUnion_closedBall hδ]
| Mathlib/Analysis/Normed/Group/Pointwise.lean | 196 | 196 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Compactness.LocallyCompact
import Mathlib.Topology.Compactness.LocallyFinite
/... | (hcomp : ∀ s : S, IsSigmaCompact s (X := X)) : IsSigmaCompact (⋃₀ S) := by
have : Countable S := countable_coe_iff.mpr hc
apply sUnion_eq_iUnion.symm ▸ isSigmaCompact_iUnion _ hcomp
/-- Countable unions of σ-compact sets are σ-compact. -/
lemma isSigmaCompact_biUnion {s : Set ι} {S : ι → Set X} (hc : Set.Count... | Mathlib/Topology/Compactness/SigmaCompact.lean | 72 | 78 |
/-
Copyright (c) 2024 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.PiTensorProduct
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Data.Finset.NoncommProd
/-!
# T... | | add _ _ h1 h2 => simp [map_add, h1, h2]
protected lemma mul_one (x : ⨂[R] i, A i) : mul x (tprod R 1) = x := by
induction x using PiTensorProduct.induction_on with
| Mathlib/RingTheory/PiTensorProduct.lean | 98 | 101 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.SetTheory.Game.Short
/-!
# Games described via "the state of the board".
We provide a simple mechanism for constructing combinatorial (pre-)games, by des... | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| Mathlib/SetTheory/Game/State.lean | 50 | 54 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | Mathlib/SetTheory/Game/Nim.lean | 67 | 67 | |
/-
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 ... | theorem adjointAux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) :
⟪x, adjointAux A y⟫ = ⟪A x, y⟫ := by
rw [← inner_conj_symm, adjointAux_inner_left, inner_conj_symm]
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 80 | 82 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | · lift x to t using (h trivial).resolve_left hxs
rw [extend_apply', Subtype.coe_injective.extend_apply]
exact fun ⟨y, hy⟩ ↦ hxs <| hy ▸ y.2
theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableSpace α}
[TopologicalSpace β] {f : α → β} {s : Set α} (hs : MeasurableSet s)
(h₁ :... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 777 | 793 |
/-
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 | 640 | 644 | |
/-
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
... | exact fun h_diff ↦ hv <| this ▸ h_diff.mul_const _
theorem deriv_mul_const (hc : DifferentiableAt 𝕜 c x) (d : 𝔸) :
deriv (fun y => c y * d) x = deriv c x * d :=
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 258 | 261 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
/-!
# The mo... | · simp
· apply IsLimit.hom_ext (ℬ _ _).isLimit
rintro ⟨⟨⟩⟩
· simp
· simp
end MonoidalOfChosenFiniteProducts
open MonoidalOfChosenFiniteProducts
| Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 297 | 306 |
/-
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... | (cyclotomic n R).coeff 0 = 1 := by
induction' n using Nat.strong_induction_on with n hi
have hprod : (∏ i ∈ Nat.properDivisors n, (Polynomial.cyclotomic i R).coeff 0) = -1 := by
rw [← Finset.insert_erase (Nat.one_mem_properDivisors_iff_one_lt.2
(lt_of_lt_of_le one_lt_two hn)), Finset.prod_insert (Fins... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 523 | 535 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | · intro h
have : Mono φ.τ₂ := by
rw [← S₂.liftCycles_i φ.τ₂ w]
apply mono_comp
refine ⟨?_, this⟩
apply exact_of_f_is_kernel
exact IsLimit.ofIsoLimit S₂.cyclesIsKernel
(Fork.ext (asIso (S₂.liftCycles φ.τ₂ w)).symm (by simp))
· rintro ⟨h₁, h₂⟩
refine ⟨⟨h₁.lift S₂.iCycles (by simp... | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 843 | 865 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | · refine (le_div_iff₀ ?_).2 ht₂
exact mod_cast h₂.card_pos
/-- If `s₂ ⊆ s₁`, `t₂ ⊆ t₁` and they take up all but a `δ`-proportion, then the difference in edge
densities is at most `2 * δ`. -/
theorem abs_edgeDensity_sub_edgeDensity_le_two_mul (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁) (hδ : 0 ≤ δ)
(hscard : (1 - δ) * #s₁ ... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 219 | 246 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | theorem map_inv [Group G] [DivisionMonoid H] [MonoidHomClass F G H]
(f : F) (a : G) : f a⁻¹ = (f a)⁻¹ :=
eq_inv_of_mul_eq_one_left <| map_mul_eq_one f <| inv_mul_cancel _
| Mathlib/Algebra/Group/Hom/Defs.lean | 425 | 427 |
/-
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.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
/-!
# Lie algeb... |
theorem coe_map_toEnd_le :
(N : Submodule R M).map (LieModule.toEnd R L M x) ≤ N := by
| Mathlib/Algebra/Lie/OfAssociative.lean | 321 | 323 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | theorem mkReal_coe (φ : { φ : K →+* ℂ // ComplexEmbedding.IsReal φ }) :
(mkReal φ : InfinitePlace K) = mk (φ : K →+* ℂ) := rfl
@[simp]
theorem mkComplex_coe (φ : { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ }) :
(mkComplex φ : InfinitePlace K) = mk (φ : K →+* ℂ) := rfl
section NumberField
variable [NumberFiel... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 523 | 535 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | ext ⟨v, hv⟩ ⟨w, hw⟩
simp only [SimpleGraph.deleteEdges_adj, coe_adj, deleteEdges_adj, Set.mem_image, not_exists,
not_and, and_congr_right_iff]
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,067 | 1,069 |
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Yaël Dillies, Jineon Baek
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebr... | In particular, it holds trivially for `[Field R]`.
2. This variant is equivalent to the original `FermatLastTheoremWith R` for `R = ℕ` or `ℤ`.
In general, they are equivalent if there is no solutions of units to the Fermat equation.
3. For a polynomial ring `R = k[X]`, the original `FermatLastTheoremWith R` is fa... | Mathlib/NumberTheory/FLT/Basic.lean | 155 | 169 |
/-
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.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Prod
/-!
# The multiplicative and additive convolution of measures
In this file w... | unfold mconv
simp
| Mathlib/MeasureTheory/Group/Convolution.lean | 69 | 71 |
/-
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.Order.Monotone.Odd
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.Spe... | @[simp]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 707 | 707 |
/-
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.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | theorem coeff_zero (m : σ →₀ ℕ) : coeff m (0 : MvPolynomial σ R) = 0 :=
rfl
@[simp]
| Mathlib/Algebra/MvPolynomial/Basic.lean | 559 | 562 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Hom.Bounded
import Mathlib.Topology.Order.Hom.Basic
/-!
# Esakia morphisms
This file defines pseudo-epimorphisms and Esakia morphisms.
We use the ... | map_continuous f := f.continuous_toFun
exists_map_eq_of_map_le f := f.exists_map_eq_of_map_le'
| Mathlib/Topology/Order/Hom/Esakia.lean | 223 | 225 |
/-
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.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | exact h₁₂ (by simpa only [← h, ← h₁] using ComplexShape.rel_π₂ c₁ c₁₂ i₁ h₂)
· exact d₂_eq_zero _ _ _ _ _ h₂
@[reassoc (attr := simp)]
lemma D₁_D₁ (i₁₂ i₁₂' i₁₂'' : I₁₂) : K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0 := by
by_cases h₁ : c₁₂.Rel i₁₂ i₁₂'
· by_cases h₂ : c₁₂.Rel i₁₂' i₁₂''
· ext ⟨i₁, i₂⟩ h
... | Mathlib/Algebra/Homology/TotalComplex.lean | 170 | 188 |
/-
Copyright (c) 2022 Alex J. Best, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Ring.Equiv
/-!
# Ordered ring homomorphisms
Homomorphisms between ordered (se... | coe_injective' f g h₁ h₂ := by
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := f
obtain ⟨⟨⟨_, _⟩, _⟩, _⟩ := g
| Mathlib/Algebra/Order/Hom/Ring.lean | 308 | 310 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.Analysis.NormedSpace.Multiline... |
theorem exists_separating_of_ne {x y : V} (h : x ≠ y) :
∃ f : V →L[R] R, f x ≠ f y := by
rcases exists_ne_zero (R := R) (sub_ne_zero_of_ne h) with ⟨f, hf⟩
| Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 57 | 60 |
/-
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... | theorem leadingCoeff_neg (p : R[X]) : (-p).leadingCoeff = -p.leadingCoeff := by
rw [leadingCoeff, leadingCoeff, natDegree_neg, coeff_neg]
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 292 | 293 |
/-
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.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... | simp only [Cofan.mk_pt, cofan_mk_inj]
split
· subst ‹j = i›; simp
· rw [Category.assoc, ← IsIso.eq_inv_comp]
exact initialIsInitial.hom_ext _ _
· intro t m hm
simp [← hm i]
theorem isPullback_initial_to_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {F : Discrete ι ⥤ C}
{... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 729 | 755 |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 783 | 784 | |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheo... | · simp
-- now `n ≠ 0`
rw [term_of_ne_zero hn, convolution_def, Finset.sum_div]
refine Finset.sum_congr rfl fun p hp ↦ ?_
have ⟨hp₁, hp₂⟩ := ne_zero_of_mem_divisorsAntidiagonal hp
rw [term_of_ne_zero hp₁, term_of_ne_zero hp₂, mul_comm_div, div_div, ← mul_div_assoc,
← natCast_mul_natCast_cpow, ← cast_mul,... | Mathlib/NumberTheory/LSeries/Convolution.lean | 97 | 108 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 901 | 906 | |
/-
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.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | rw [← this.integral_comp (MeasurableEquiv.measurableEmbedding _)]
simp only [neg_mul, Function.comp_def]
convert integral_cexp_neg_mul_sum_add hb (fun i ↦ c * w i) using 5 with _x y
· simp only [EuclideanSpace.measurableEquiv, MeasurableEquiv.symm_mk, MeasurableEquiv.coe_mk,
EuclideanSpace.norm_eq, WithLp... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 306 | 314 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
| p, hp =>
haveI := Classical.propDecidable (∃ x, IsRoot p x)
if h : ∃ x, IsRoot p x then
let ⟨x, hx⟩ := h
have hpd : 0 < degree p := degree_pos_of_root hp hx
have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
... | Mathlib/Algebra/Polynomial/RingDivision.lean | 245 | 256 |
/-
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... |
@[simp] theorem ncard_singleton (a : α) : ({a} : Set α).ncard = 1 := by
| Mathlib/Data/Set/Card.lean | 576 | 577 |
/-
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.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Types.Shapes
impo... | (isoWhiskerLeft _ (asIso adj.counit) ≪≫ Functor.rightUnitor _).hom).mp
have : ∀ (Z : C) (i : Discrete WalkingPair) (f : Z ⟶ (colimit.cocone (pair X Y ⋙ Gr)).pt),
PreservesLimit (cospan f ((colimit.cocone (pair X Y ⋙ Gr)).ι.app i)) Gl := by
have : pair X Y ⋙ Gr = pair (Gr.obj X) (Gr.obj Y) := by
... | Mathlib/CategoryTheory/Extensive.lean | 363 | 386 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,481 | 1,483 | |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sigma
import Mathlib.... | smash_castLE h i.succ
lemma smash_natAdd {p q : RelSeries r} (h : p.last = q.head) (i : Fin q.length) :
smash p q h (i.natAdd p.length).castSucc = q i.castSucc := by
dsimp only [smash, Fin.castSucc_natAdd]
apply Fin.addCases_right
| Mathlib/Order/RelSeries.lean | 553 | 558 |
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.IsPrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
import Mathlib.Data.Z... |
section
variable [UniqueFactorizationMonoid N] [UniqueFactorizationMonoid M]
open DivisorChain
theorem pow_image_of_prime_by_factor_orderIso_dvd
{m p : Associates M} {n : Associates N} (hn : n ≠ 0) (hp : p ∈ normalizedFactors m)
(d : Set.Iic m ≃o Set.Iic n) {s : ℕ} (hs' : p ^ s ≤ m) :
| Mathlib/RingTheory/ChainOfDivisors.lean | 234 | 243 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... |
theorem disjSups_inter_subset_left : (s₁ ∩ s₂) ○ t ⊆ s₁ ○ t ∩ s₂ ○ t := by
| Mathlib/Data/Finset/Sups.lean | 460 | 461 |
/-
Copyright (c) 2022 Justin Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justin Thomas
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Annihilating Ideal
Given ... | by_cases h : p = 0
· rwa [h, annIdealGenerator_eq_zero_iff, ← p_gen, Ideal.span_singleton_eq_bot.mpr]
· rw [← span_singleton_annIdealGenerator, Ideal.span_singleton_eq_span_singleton] at p_gen
rw [eq_comm]
apply eq_of_monic_of_associated p_monic _ p_gen
apply monic_annIdealGenerator _ _ ((Associated.n... | Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean | 155 | 164 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,020 | 2,021 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | inv := 0
hom_inv_id := by
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 403 | 404 |
/-
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... |
/-- The cosine of an angle in a right-angled triangle as a ratio of sides, version subtracting
vectors. -/
theorem cos_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 295 | 298 |
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