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) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... |
/-- `-1 : ZMod n` lifts to `n - 1 : R`. This avoids the characteristic assumption in `cast_neg`. -/
| Mathlib/Data/ZMod/Basic.lean | 529 | 530 |
/-
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
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | rw [hl₁.diff_eq_filter, mem_filter, decide_eq_true_iff]
protected theorem Nodup.set :
| Mathlib/Data/List/Nodup.lean | 346 | 348 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,645 | 1,647 | |
/-
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.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.Basis.Defs
import Mathlib.LinearAlgebra.BilinearForm.Basic
import Mathlib.Linear... |
namespace LinearMap
| Mathlib/LinearAlgebra/BilinearForm/Hom.lean | 90 | 92 |
/-
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.Module.Equiv.Defs
import Mathlib.Data.DFinsupp.Module
import Mathlib.Data.Finsupp.SMul
/-!
# Conversion between `Finsupp` and homogeneous `DFinsupp`... | { sigmaFinsuppEquivDFinsupp with
toFun := sigmaFinsuppEquivDFinsupp
invFun := sigmaFinsuppEquivDFinsupp.symm
map_add' := sigmaFinsuppEquivDFinsupp_add }
attribute [-instance] Finsupp.instAddZeroClass
| Mathlib/Data/Finsupp/ToDFinsupp.lean | 313 | 318 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | _ ≤ ‖x‖ ^ n * (n.succ * (n.factorial * n : ℝ)⁻¹) := by
gcongr
| Mathlib/Data/Complex/Exponential.lean | 394 | 395 |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | calc
degree (p %ₘ q) < degree q := degree_modByMonic_lt _ hq
_ ≤ _ := by
rw [degree_mul' hlc, degree_eq_natDegree hq.ne_zero, degree_eq_natDegree hdiv0, ←
Nat.cast_add, Nat.cast_le]
exact Nat.le_add_right _ _
calc
degree q + degree (p /ₘ q) = degree (q * (p /ₘ q)) := Eq... | Mathlib/Algebra/Polynomial/Div.lean | 276 | 292 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 899 | 901 | |
/-
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 | 866 | 867 | |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov, Winston Yin
-/
import Mathlib.Algebra.Group.End
import Mathlib.Topology.EMetricSpace.Diam
/-!
# Lipschitz co... |
/-- A Lipschitz function is continuous. -/
| Mathlib/Topology/EMetricSpace/Lipschitz.lean | 181 | 182 |
/-
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.Order.WellFounded
import Mathlib.Tactic.Common
/-!
# Lexicographic order on Pi types
This file defines the lexicographic order for Pi types. `a` is less ... |
instance [LinearOrder ι] [WellFoundedLT ι] [∀ a, PartialOrder (β a)]
| Mathlib/Order/PiLex.lean | 165 | 166 |
/-
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... | noncomputable def toRightMovesNim {o : Ordinal} : Set.Iio o ≃ (nim o).RightMoves :=
(enumIsoToType o).toEquiv.trans (Equiv.cast (rightMoves_nim o).symm)
| Mathlib/SetTheory/Game/Nim.lean | 73 | 75 |
/-
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.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
/-!
# Half braidings and the Drinfeld center of a monoidal c... | section
variable (C)
| Mathlib/CategoryTheory/Monoidal/Center.lean | 301 | 303 |
/-
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, Sander Dahmen,
Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimensi... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 342 | 344 | |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | Mathlib/Topology/Semicontinuous.lean | 169 | 170 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... |
@[measurability]
protected theorem Homeomorph.measurable (h : α ≃ₜ γ) : Measurable h :=
h.continuous.measurable
/-- A homeomorphism between two Borel spaces is a measurable equivalence. -/
def Homeomorph.toMeasurableEquiv (h : γ ≃ₜ γ₂) : γ ≃ᵐ γ₂ where
measurable_toFun := h.measurable
measurable_invFun := h.symm... | Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 521 | 532 |
/-
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.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.Pointwise
import Mat... |
@[to_additive IsUpperSet.cthickening]
protected theorem IsUpperSet.cthickening' (hs : IsUpperSet s) (ε : ℝ) :
IsUpperSet (cthickening ε s) := by
| Mathlib/Analysis/Normed/Order/UpperLower.lean | 50 | 53 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | P3 x₁ x₂ y₁ y₂ := by
obtain (hl|he) := lt_or_equiv_of_le hl
· exact (h.2.2.2 hl).trans h.2.2.1
· rw [P3, h.1 he, h.2.1 he]
exact h.2.2.1
| Mathlib/SetTheory/Surreal/Multiplication.lean | 403 | 408 |
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Set.Finite.Range
import Mathlib.Logic.Equiv.Embedding
/-!
# Numb... | theorem card_embedding_eq {α β : Type*} [Fintype α] [Fintype β] [emb : Fintype (α ↪ β)] :
‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖ := by
rw [Subsingleton.elim emb Embedding.fintype]
refine Fintype.induction_empty_option (P := fun t ↦ ‖t ↪ β‖ = ‖β‖.descFactorial ‖t‖)
(fun α₁ α₂ h₂ e ih ↦ ?_) (?_) (fun γ h ih ↦ ?_... | Mathlib/Data/Fintype/CardEmbedding.lean | 36 | 50 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... |
theorem neg_one_ne_one {n : ℕ} [Fact (2 < n)] : (-1 : ZMod n) ≠ 1 :=
CharP.neg_one_ne_one (ZMod n) n
@[simp]
theorem neg_eq_self_mod_two (a : ZMod 2) : -a = a := by
fin_cases a <;> apply Fin.ext <;> simp [Fin.coe_neg, Int.natMod]; rfl
@[simp]
| Mathlib/Data/ZMod/Basic.lean | 911 | 919 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | exists_measurable_superset_forall_eq_trim
(Option.elim' (⨆ i, μ (PLift.down i)) (μ ∘ PLift.down)) s
simp only [Option.forall, Option.elim'] at hμt
simp only [iSup_apply, ← hμt.1]
exact iSup_congr hμt.2
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 445 | 449 |
/-
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.Basic
/-!
# Multivariate quotients of polynomial functors.
Basic definition of multivariate QPF. QPFs form a co... | rcases this with ⟨a, f, xeq, h'⟩
refine ⟨a, f, xeq.symm, ?_⟩
intro a' f' h''
rintro hu u ⟨j, _h₂, hfi⟩
have hh : u ∈ supp x a' := by rw [← hfi]; apply h'
exact (mem_supp x _ u).mp hh _ _ hu
rintro ⟨a, f, xeq, h⟩ p; rw [liftP_iff]; constructor
· rintro ⟨a', f', xeq', h'⟩ i u usuppx
rcases... | Mathlib/Data/QPF/Multivariate/Basic.lean | 180 | 203 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
/-!
# Egorov theorem
This file contains the Egorov theorem which states that an almost everywhere convergen... | simp only [Set.mem_inter_iff, Set.mem_iInter, mem_notConvergentSeq_iff]
push_neg
rintro ⟨hmem, hx⟩
refine ⟨hmem, 1 / (n + 1 : ℝ), Nat.one_div_pos_of_nat, fun N => ?_⟩
obtain ⟨n, hn₁, hn₂⟩ := hx N
exact ⟨n, hn₁, hn₂.le⟩
theorem notConvergentSeq_measurableSet [Preorder ι] [Countable ι]
(hf : ∀ n, Strongl... | Mathlib/MeasureTheory/Function/Egorov.lean | 59 | 70 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Multiset.Bind
import Mathlib.Order.SetNotation
/-!
# Unions of finite sets
This file defines the union of a fami... | section BUnion
variable [DecidableEq β]
| Mathlib/Data/Finset/Union.lean | 129 | 130 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | dist x₂ y₂ ≤ dist x₁ y₁ + (dist x₁ x₂ + dist y₁ y₂) := by
rw [add_left_comm, dist_comm x₁, ← add_assoc]
apply dist_triangle4
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 203 | 206 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | exact g.mono hxy
· rw [sub_nonneg]
exact f.mono hxy
@[simp]
lemma measure_zero : StieltjesFunction.measure 0 = 0 :=
| Mathlib/MeasureTheory/Measure/Stieltjes.lean | 539 | 544 |
/-
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.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | theorem affineCombination_affineCombinationSingleWeights [DecidableEq ι] (p : ι → P) {i : ι}
(hi : i ∈ s) : s.affineCombination k p (affineCombinationSingleWeights k i) = p i := by
refine s.affineCombination_of_eq_one_of_eq_zero _ _ hi (by simp) ?_
rintro j - hj
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 690 | 693 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | (p + q) • s = p • s + q • s := (add_smul_subset _ _ _).antisymm <| by
rintro _ ⟨_, ⟨v₁, h₁, rfl⟩, _, ⟨v₂, h₂, rfl⟩, rfl⟩
exact h_conv.exists_mem_add_smul_eq h₁ h₂ hp hq
| Mathlib/Analysis/Convex/Basic.lean | 538 | 540 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | ∀ x y z, glueDist Φ Ψ ε x z ≤ glueDist Φ Ψ ε x y + glueDist Φ Ψ ε y z
| .inl _, .inl _, .inl _ => dist_triangle _ _ _
| .inr _, .inr _, .inr _ => dist_triangle _ _ _
| .inr x, .inl y, .inl z => by
simp only [← glueDist_swap Φ]
apply glueDist_triangle_inl_inr_inr
| .inr x, .inr y, .inl z => by
si... | Mathlib/Topology/MetricSpace/Gluing.lean | 132 | 150 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | def smulg {α} [AddCommGroup α] (n : ℤ) (x : α) : α := n • x
| Mathlib/Tactic/Abel.lean | 242 | 242 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.... |
See also `center_equiv_rootsOfUnity'`. -/
noncomputable def center_equiv_rootsOfUnity :
center (SpecialLinearGroup n R) ≃* rootsOfUnity (max (Fintype.card n) 1) R :=
(isEmpty_or_nonempty n).by_cases
(fun hn ↦ by
rw [center_eq_bot_of_subsingleton, Fintype.card_eq_zero, max_eq_right_of_lt zero_lt_one,
... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 285 | 293 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | (fun bc => by rwa [sup_eq_left.2 bc] at h)
| Mathlib/Order/Lattice.lean | 675 | 675 |
/-
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.Order.Group.Int
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Int.Parity
im... | Mathlib/Algebra/Order/Ring/Int.lean | 103 | 104 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | /-! ### Properties of the iterated derivative on the whole space -/
| Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 206 | 208 |
/-
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.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | · rw [pow_succ, pow_succ, ← hm, Nat.succ_mul, coeff_mul_of_natDegree_le _ pn]
refine natDegree_pow_le.trans (le_trans ?_ (le_refl _))
exact mul_le_mul_of_nonneg_left pn m.zero_le
theorem coeff_pow_eq_ite_of_natDegree_le_of_le {o : ℕ}
(pn : natDegree p ≤ n) (mno : m * n ≤ o) :
coeff (p ^ m) o = if o =... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 163 | 169 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | constructor
· rintro ⟨y, hy, rfl⟩
exact
⟨(pullback.fst f g) y, hy, CategoryTheory.congr_fun pullback.condition y⟩
· rintro ⟨y, hy, eq⟩
-- next 5 lines were
-- `exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq⟩, by simpa, by simp⟩` before https://github.com/leanprover-community/mathlib4/pul... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 367 | 373 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | theorem cos_pi_sub (x : ℂ) : cos (π - x) = -cos x :=
cos_neg x ▸ cos_antiperiodic.sub_eq'
theorem cos_two_pi_sub (x : ℂ) : cos (2 * π - x) = cos x :=
cos_neg x ▸ cos_periodic.sub_eq'
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,079 | 1,084 |
/-
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... | RatFunc.mk p q =
ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p
| Mathlib/FieldTheory/RatFunc/Defs.lean | 154 | 155 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | rw [sub_eq_add_neg 1 x]
simpa only [v.map_one, v.map_neg] using v.map_add_eq_of_lt_left h
| Mathlib/RingTheory/Valuation/Basic.lean | 327 | 328 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | `compacts G → ℝ`, with the topology of pointwise convergence.
We show that the intersection of all these sets is nonempty, and the Haar measure
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 128 | 129 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | Subtype.range_coe_subtype, mem_affineSpan_singleton, setOf_eq_eq_singleton]
| Mathlib/Analysis/Convex/Intrinsic.lean | 116 | 116 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | ⟨h.1.weak, h.2, h.3⟩
theorem valid'_nil {o₁ o₂} (h : Bounded nil o₁ o₂) : Valid' o₁ (@nil α) o₂ :=
| Mathlib/Data/Ordmap/Ordset.lean | 83 | 85 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... |
instance {X Y : C} (h : X = Y) : IsIso (eqToHom h) :=
(eqToIso h).isIso_hom
@[simp]
| Mathlib/CategoryTheory/EqToHom.lean | 223 | 227 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | theorem compl_compl_image [BooleanAlgebra α] (S : Set α) :
HasCompl.compl '' (HasCompl.compl '' S) = S := by
rw [← image_comp, compl_comp_compl, image_id]
theorem image_insert_eq {f : α → β} {a : α} {s : Set α} :
f '' insert a s = insert (f a) (f '' s) := by
| Mathlib/Data/Set/Image.lean | 318 | 323 |
/-
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 | 1,064 | 1,073 | |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | ← lTensor.inverse_comp_lTensor Q hfg hg]
apply symm
apply LinearMap.ker_comp_of_ker_eq_bot
rw [LinearMap.ker_eq_bot]
exact (lTensor.equiv Q hfg hg).symm.injective
| Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 284 | 288 |
/-
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... | isUniformEmbedding_comap Subtype.val_injective
theorem isUniformInducing : IsUniformInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
simpleFunc.isUniformEmbedding.isUniformInducing
lemma isDenseEmbedding (hp_ne_top : p ≠ ∞) :
| Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 658 | 663 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathli... | apply congr_arg det
ext
symm
| Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 167 | 169 |
/-
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 | 164 | 165 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | exact KaehlerDifferential.one_smul_sub_smul_one_mem_ideal _ _
· rintro x (hx : _ = _)
have : x - TensorProduct.lmul' (S := S) R x ⊗ₜ[R] (1 : S) = x := by
rw [hx, TensorProduct.zero_tmul, sub_zero]
rw [← this]
clear this hx
refine TensorProduct.induction_on x ?_ ?_ ?_
· rw [map_zero, Tens... | Mathlib/RingTheory/Kaehler/Basic.lean | 109 | 132 |
/-
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... | rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_sub_eq_arcsin_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))]
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 259 | 263 |
/-
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... | ext X; dsimp; apply comp_zero
zero_comp := fun F {G H} η => by
ext X; dsimp; apply zero_comp
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 145 | 147 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | Mathlib/Order/Bounds/Basic.lean | 955 | 958 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
/-!
# Birkhoff average in a normed space
In this file we prove some lemmas about ... | theorem uniformEquicontinuous_birkhoffAverage (hf : LipschitzWith 1 f) (hg : UniformContinuous g) :
UniformEquicontinuous (birkhoffAverage 𝕜 f g) := by
refine Metric.uniformity_basis_dist_le.uniformEquicontinuous_iff_right.2 fun ε hε ↦ ?_
rcases (uniformity_basis_edist_le.uniformContinuous_iff Metric.uniformit... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 106 | 122 |
/-
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 | 1,780 | 1,782 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | @[simp]
theorem filter_add [DecidablePred p] {v v' : α →₀ M} :
(v + v').filter p = v.filter p + v'.filter p :=
(filterAddHom p).map_add v v'
| Mathlib/Data/Finsupp/Basic.lean | 978 | 981 |
/-
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.Logic.Denumerable
/-!
# Equivalences involving `List`-like types
This file defines some additional constructive equivalences using `Encodable` and th... | Mathlib/Logic/Equiv/List.lean | 357 | 359 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | exact coeff_eq_zero_of_natDegree_lt <| (Nat.le_of_lt_succ hdeg).trans_lt hi
obtain hFn | hFn := ne_or_eq ((finSuccEquiv R N F).coeff n) 0
· exact hF.exists_eval_ne_zero_of_coeff_finSuccEquiv_ne_zero_aux hFn
have hin : i < n := hin.lt_or_eq.elim id <| by aesop
obtain ⟨j, hj⟩ : ∃ j, i + (j + 1) = n ... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 368 | 407 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | simp only [CovBy, bot_lt_iff_ne_bot, IsAtom, not_imp_not]
| Mathlib/Order/Atoms.lean | 104 | 105 |
/-
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 | 1,614 | 1,618 | |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | · rw [zero_add, Gamma_zero, inv_zero, zero_mul]
/-- The reciprocal of the Gamma function is differentiable everywhere
(including the points where Gamma itself is not). -/
| Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 506 | 509 |
/-
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.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | obtain ⟨N, hN⟩ := LieAlgebra.nilpotent_ad_of_nilpotent_algebra R L
obtain ⟨m, rfl⟩ := hm (N + k)
let f₁ : Module.End R (L ⊗[R] M) := (LieAlgebra.ad R L x).rTensor M
let f₂ : Module.End R (L ⊗[R] M) := φ.lTensor L
replace hN : f₁ ^ N = 0 := by ext; simp [f₁, hN]
have h₁ : Commute f₁ f... | Mathlib/Algebra/Lie/Weights/Basic.lean | 407 | 417 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | Multiset.map_add, Finset.union_val, Multiset.add_eq_union_iff_disjoint.mpr _]
exact Finset.disjoint_val.2 h.disjoint_cycleFactorsFinset
@[simp]
theorem cycleType_inv (σ : Perm α) : σ⁻¹.cycleType = σ.cycleType :=
cycle_induction_on (P := fun τ : Perm α => τ⁻¹.cycleType = τ.cycleType) σ rfl
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 130 | 135 |
/-
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.Complex.LocallyUniformLimit
import Mathlib.NumberTheory.LSeries.Convergence
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.An... | the `m`th derivative of this L-series is `(-1)^m` times the L-series of `log^m * f`. -/
lemma LSeries_iteratedDeriv {f : ℕ → ℂ} (m : ℕ) {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
iteratedDeriv m (LSeries f) s = (-1) ^ m * LSeries (logMul^[m] f) s := by
induction m generalizing s with
| zero => simp
| succ m i... | Mathlib/NumberTheory/LSeries/Deriv.lean | 128 | 141 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | rw [rel_map]
exact hst.mono fun a _ b _ hr => h hr
| Mathlib/Data/Multiset/Bind.lean | 177 | 178 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | nth_rw 1 [hr]
rw [mul_comm, mul_div_cancel_right₀ _ pq0]
exact r0
theorem isCoprime_div_gcd_div_gcd (hq : q ≠ 0) :
IsCoprime (p / GCDMonoid.gcd p q) (q / GCDMonoid.gcd p q) :=
| Mathlib/RingTheory/EuclideanDomain.lean | 42 | 47 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... |
theorem inv_natCast_num_of_pos {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.num = 1 :=
inv_intCast_num_of_pos (mod_cast ha0 : 0 < (a : ℤ))
| Mathlib/Data/Rat/Lemmas.lean | 237 | 240 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | rw [sup_comm (c \ b), sup_sup_sup_comm, symmDiff, symmDiff]
| Mathlib/Order/SymmDiff.lean | 184 | 185 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... |
instance : IsOrderedAddMonoid (ℤ√d) :=
{ add_le_add_left := Zsqrtd.add_le_add_left }
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 851 | 853 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | namespace ENNReal
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
(hp : 1 ≤ p) : (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p := by
ha... | Mathlib/Analysis/MeanInequalitiesPow.lean | 217 | 224 |
/-
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.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | theorem pullbackDiagonalMapIso.hom_fst :
(pullbackDiagonalMapIso f i i₁ i₂).hom ≫ pullback.fst _ _ =
pullback.snd _ _ ≫ pullback.fst _ _ := by
delta pullbackDiagonalMapIso
| Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 158 | 161 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... | map_ωSup' c' := eq_of_forall_ge_iff fun a ↦ by simp [(c _).ωScottContinuous.map_ωSup]
@[simps ωSup]
instance : OmegaCompletePartialOrder (α →𝒄 β) :=
OmegaCompletePartialOrder.lift ContinuousHom.toMono ContinuousHom.ωSup
| Mathlib/Order/OmegaCompletePartialOrder.lean | 703 | 707 |
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Group.Torsion
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
/-!
# Lemmas about the ... | (zpow_right_strictMono ha).injective.eq_iff
@[to_additive zsmul_left_mono]
lemma zpow_right_mono (ha : 1 ≤ a) : Monotone fun n : ℤ ↦ a ^ n := by
refine monotone_int_of_le_succ fun n ↦ ?_
rw [zpow_add_one]
| Mathlib/Algebra/Order/Group/Basic.lean | 43 | 48 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | rw [abv_sub abv]; exact hi _ ik
theorem cauchy₃ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - f j) < ε :=
let ⟨i, H⟩ := hf.cauchy₂ ε0
⟨i, fun _ ij _ jk => H _ (le_trans ij jk) _ ij⟩
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 101 | 106 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 1,133 | 1,143 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-!
# Action of `Mᵈᵐᵃ` on `Lᵖ` spaces
In this file we... | theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 78 | 79 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 659 | 663 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingT... |
theorem dvd_map_of_isScalarTower' (R : Type*) {S : Type*} (K L : Type*) [CommRing R]
[CommRing S] [Field K] [Ring L] [Algebra R S] [Algebra R K] [Algebra S L] [Algebra K L]
[Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] (s : S) :
minpoly K (algebraMap S L s) ∣ map (algebraMap R K) (minpoly R s) ... | Mathlib/FieldTheory/Minpoly/Field.lean | 86 | 90 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | Nat.zero_add, zipWith_map_left, true_and]
rw [ih]
· suffices (fun i ↦ f (i + (n + 1))) = ((fun i ↦ f (i + n)) ∘ Nat.succ) by
rw [this]
rfl
funext n' a
simp only [comp, Nat.add_assoc, Nat.add_comm, Nat.add_succ]
simp only [length_cons, Nat.succ.injEq] at e; e... | Mathlib/Data/List/Indexes.lean | 59 | 69 |
/-
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.Data.Fintype.Option
import Mathlib.GroupTheory.Perm.Sign
/-!
# Permutations of `Option α`
-/
open Equiv
@[simp]
theorem Equiv.optionCongr_one {α : Type*}... | (Finset.univ : Finset <| Option α × Perm α).map Equiv.Perm.decomposeOption.symm.toEmbedding :=
(Finset.univ_map_equiv_to_embedding _).symm
| Mathlib/GroupTheory/Perm/Option.lean | 76 | 77 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | f.toMultilinearMap.exists_bound_of_continuous f.2
open Real
/-- The operator norm of a continuous multilinear map is the inf of all its bounds. -/
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 336 | 340 |
/-
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.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... |
@[simp, reassoc]
theorem braiding_inv_tensor_left (X Y Z : C) :
(β_ (X ⊗ Y) Z).inv =
(α_ Z X Y).inv ≫ (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫
X ◁ (β_ Y Z).inv ≫ (α_ X Y Z).inv :=
eq_of_inv_eq_inv (by simp)
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 102 | 108 |
/-
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.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | (g₂₂ : Y₂ ⟶ Z₂) :
Biprod.ofComponents f₁₁ f₁₂ f₂₁ f₂₂ ≫ Biprod.ofComponents g₁₁ g₁₂ g₂₁ g₂₂ =
Biprod.ofComponents (f₁₁ ≫ g₁₁ + f₁₂ ≫ g₂₁) (f₁₁ ≫ g₁₂ + f₁₂ ≫ g₂₂) (f₂₁ ≫ g₁₁ + f₂₂ ≫ g₂₁)
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 695 | 697 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | rw [derivWithin, fderivWithin_zero_of_nmem_closure h, ContinuousLinearMap.zero_apply]
theorem deriv_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : deriv f x = 0 := by
unfold deriv
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 246 | 249 |
/-
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.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | open MulOpposite
| Mathlib/Algebra/Quaternion.lean | 712 | 712 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 728 | 728 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | fixed : ∀ ϕ : A ≃+ A, H.comap ϕ.toAddMonoidHom = H
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 280 | 281 |
/-
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.Shift.CommShift
/-!
# Shift induced from a category to another
In this file, we introduce a sufficient condition on a functor
`F : C ⥤ D` so tha... | @[simp]
lemma zero_hom_app_obj (X : C) :
(zero F s i).hom.app (F.obj X) =
(i 0).hom.app X ≫ F.map ((shiftFunctorZero C A).hom.app X) := by
have h : whiskerLeft F (zero F s i).hom = _ :=
((whiskeringLeft C D D).obj F).map_preimage _
exact (NatTrans.congr_app h X).trans (by simp)
| Mathlib/CategoryTheory/Shift/Induced.lean | 50 | 56 |
/-
Copyright (c) 2014 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.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 265 | 266 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | have hsvt := bsv.span_eq
rw [Basis.coe_extend] at hsvi hsvt
rw [linearIndependent_subtype_iff] at hsvi h
have hsv := h.subset_extend (Set.subset_univ _)
have h0 : ∀ v : V, v ∈ h.extend (Set.subset_univ _) → v ≠ 0 := by
intro v hv
simpa [bsv] using bsv.ne_zero ⟨v, hv⟩
rw [← linearInde... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 572 | 602 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | ⟨r, hf⟩
theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt 𝕜 f x :=
⟨p, hf⟩
theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
hf.hasFPowerSeriesAt.analyticAt
| Mathlib/Analysis/Analytic/Basic.lean | 467 | 473 |
/-
Copyright (c) 2020 Thomas Browning and Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
/-... | Mathlib/FieldTheory/AbelRuffini.lean | 49 | 49 | |
/-
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.Algebra.BigOperators.Field
import Mathlib.NumberTheory.LSeries.Basic
/-!
# Linearity of the L-series of `f` as a function of `f`
We show that the `LSer... |
@[simp]
lemma LSeries_sub {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s) :
LSeries (f - g) s = LSeries f s - LSeries g s := by
| Mathlib/NumberTheory/LSeries/Linearity.lean | 93 | 96 |
/-
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... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,086 | 1,090 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Aaron Anderson
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Pi
import Mathlib.Data.Fi... | section LinearOrder
variable [AddCommMonoid α] [LinearOrder α] [CanonicallyOrderedAdd α]
| Mathlib/Data/Finsupp/Order.lean | 285 | 288 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.H... | [HasLimit (parallelPair (prod.fst ≫ f) (prod.snd ≫ g))] : HasLimit (cospan f g) :=
let π₁ : X ⨯ Y ⟶ X := prod.fst
let π₂ : X ⨯ Y ⟶ Y := prod.snd
let e := equalizer.ι (π₁ ≫ f) (π₂ ≫ g)
HasLimit.mk
{ cone :=
PullbackCone.mk (e ≫ π₁) (e ≫ π₂) <| by
rw [Category.assoc, equalizer.condition]... | Mathlib/CategoryTheory/Limits/Constructions/Pullbacks.lean | 28 | 45 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... | rw [hom_bind₁, map_comp_C, ← eval₂Hom_map_hom]
rfl
@[simp]
| Mathlib/Algebra/MvPolynomial/Monad.lean | 239 | 242 |
/-
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.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | theorem ascPochhammer_ne_zero_eval_zero {n : ℕ} (h : n ≠ 0) : (ascPochhammer S n).eval 0 = 0 := by
simp [ascPochhammer_eval_zero, h]
theorem ascPochhammer_succ_right (n : ℕ) :
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 110 | 113 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | simp_rw [Function.comp_apply, cramer_apply, updateCol_submatrix_equiv,
det_submatrix_equiv_self e, Function.comp_def]
theorem cramer_reindex (e : m ≃ n) (A : Matrix m m α) (b : n → α) :
cramer (reindex e e A) b = cramer A (b ∘ e) ∘ e.symm :=
cramer_submatrix_equiv _ _ _
end Cramer
section Adjugate
| Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 160 | 170 |
/-
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.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | obtain ⟨r₂, p₂o, hp₂o, hp₂⟩ := hp₂
obtain rfl : ↑(s.orthogonalProjectionSpan p₁) = p₁o := by
subst hp₁
exact s.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection hp₁o
rw [h₁'] at hp₁
obtain rfl : ↑(s.orthogonalProjectionSpan p₂) = p₂o := by
subst hp₂
| Mathlib/Geometry/Euclidean/Circumcenter.lean | 781 | 787 |
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