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) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | /-- Entourages are neighborhoods of the diagonal. -/
theorem iSup_nhds_le_uniformity : ⨆ x : α, 𝓝 (x, x) ≤ 𝓤 α :=
| Mathlib/Topology/UniformSpace/Basic.lean | 140 | 141 |
/-
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.Order.Monovary
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Product of co... | ← smul_smul_smul_comm b b, smul_eq_mul, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_comm b a,
add_comm ((a * b) • f x • g y), add_comm ((a * b) • f x • g y), add_add_add_comm]
lemma ConvexOn.smul'' [OrderedSMul 𝕜 E] (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g)
(hf₀ : ∀ ⦃x⦄, x ∈ s → f x ≤ 0) (hg₀ : ∀ ⦃x⦄, ... | Mathlib/Analysis/Convex/Mul.lean | 69 | 73 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probability measures
This file i... | lemma map_prod_map {α' : Type*} [MeasurableSpace α'] {β' : Type*} [MeasurableSpace β']
{f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) :
(μ.map f_mble.aemeasurable).prod (ν.map g_mble.aemeasurable)
= (μ.prod ν).map (f_mble.prodMap g_mble).aemeasurable := by
apply Subtype.ext
... | Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 125 | 131 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | theorem oangle_add_cyc3 {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle x y + o.oangle y z + o.oangle z x = 0 := by simp [hx, hy, hz]
/-- Given three nonzero vectors, adding the angles between them in cyclic order, with the first
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 471 | 474 |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Spectrum mapping theorem
This file develops proves the spectral mappi... | theorem nonempty_of_isAlgClosed_of_finiteDimensional [IsAlgClosed 𝕜] [Nontrivial A]
[I : FiniteDimensional 𝕜 A] (a : A) : (σ a).Nonempty := by
obtain ⟨p, ⟨h_mon, h_eval_p⟩⟩ := isIntegral_of_noetherian (IsNoetherian.iff_fg.2 I) a
| Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 143 | 145 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | contrapose! hvz with hw1
rw [hw1, neg_sq, one_pow, add_eq_right] at hq
exact pow_eq_zero hq
have hQ : ∀ x : ℚ, 1 + x ^ 2 ≠ 0 := by
intro q
apply ne_of_gt
exact lt_add_of_pos_of_le zero_lt_one (sq_nonneg q)
have hp : (⟨v, w⟩ : ℚ × ℚ) ∈ { p : ℚ × ℚ | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1 } := ⟨hq, ... | Mathlib/NumberTheory/PythagoreanTriples.lean | 445 | 460 |
/-
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.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | · rintro ⟨hx, h'x⟩
exact exists_mem_Ico_zpow hx.ne' h'x.ne hy h'y
· rintro ⟨n, hn, h'n⟩
constructor
· apply lt_of_lt_of_le _ hn
exact ENNReal.zpow_pos (zero_lt_one.trans hy).ne' h'y _
· apply lt_trans h'n _
exact ENNReal.zpow_lt_top (zero_lt_one.trans hy).ne' h'y _
@[gcongr]
theorem zpo... | Mathlib/Data/ENNReal/Inv.lean | 666 | 677 |
/-
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.Complex.Basic
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Gaussian intege... |
theorem norm_le_norm_mul_left (x : ℤ[i]) {y : ℤ[i]} (hy : y ≠ 0) :
(norm x).natAbs ≤ (norm (x * y)).natAbs := by
| Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 216 | 218 |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | rw [pow_succ', mul_comm, pow_mul, Int.sq_mod_four_eq_one_of_odd hx.pow]
theorem Int.two_pow_two_pow_sub_pow_two_pow {x y : ℤ} (n : ℕ) (hxy : 4 ∣ x - y) (hx : ¬2 ∣ x) :
emultiplicity 2 (x ^ 2 ^ n - y ^ 2 ^ n) = emultiplicity 2 (x - y) + n := by
simp only [pow_two_pow_sub_pow_two_pow n, emultiplicity_mul Int.pri... | Mathlib/NumberTheory/Multiplicity.lean | 305 | 326 |
/-
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.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Pi
import Mathlib.Algebra.O... | (collapse_nonneg h₄) (collapse_modular hu h₁ h₂ h₃ h₄ h 𝒜 ℬ) Subset.rfl Subset.rfl
have : 𝒜 ⊼ ℬ ⊆ powerset (insert a u) := by simpa using infs_subset h𝒜 hℬ
have : 𝒜 ⊻ ℬ ⊆ powerset (insert a u) := by simpa using sups_subset h𝒜 hℬ
simpa only [powerset_sups_powerset_self, powerset_infs_powerset_self... | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | 256 | 259 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | · rfl
· exact ((Germ.coe_eq.mp <| congr_arg Germ.ofFun <| condExp_of_stronglyMeasurable (𝒢.le _)
(hfadp _) <| hfmgle.integrable _).symm.trans_le (hfmgle.2.1 k (k + 1) k.le_succ)).trans ih
/-- A predictable martingale is a.e. equal to its initial state. -/
theorem Martingale.eq_zero_of_predictable [SigmaFini... | Mathlib/Probability/Martingale/Basic.lean | 454 | 460 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,026 | 1,028 | |
/-
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... | have h₁ : (n.succ : α) ≠ 1 :=
@Nat.cast_one α _ ▸ mt Nat.cast_inj.1 (mt Nat.succ.inj (pos_iff_ne_zero.1 hn))
have h₂ : (n.succ : α) ≠ 0 := by positivity
have h₃ : (n.factorial * n : α) ≠ 0 := by positivity
| Mathlib/Data/Complex/Exponential.lean | 358 | 361 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Nat.Basic
/-!
# Cast of integers (additional theorems)
This ... | -- `HasLiftT` appeared in the type signature
| Mathlib/Data/Int/Cast/Basic.lean | 33 | 34 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | @[simp]
theorem orthogonal_eq_bot_iff [K.HasOrthogonalProjection] : Kᗮ = ⊥ ↔ K = ⊤ := by
refine ⟨?_, fun h => by rw [h, Submodule.top_orthogonal_eq_bot]⟩
intro h
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 804 | 807 |
/-
Copyright (c) 2022 Joanna Choules. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joanna Choules
-/
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Homomorphisms from finite subgraphs
This file defines the typ... | @[simp, norm_cast] lemma coe_hnot (G' : G.Finsubgraph) : ↑(￢G') = (￢G' : G.Subgraph) := rfl
@[simp, norm_cast]
| Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 93 | 95 |
/-
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.Real
/-!
# Power function... | · rcases le_iff_eq_or_lt.1 h with (H | H)
· simp [hx, H.symm]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 498 | 499 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... |
theorem WithSeminorms.isVonNBounded_iff_finset_seminorm_bounded {s : Set E} (hp : WithSeminorms p) :
Bornology.IsVonNBounded 𝕜 s ↔ ∀ I : Finset ι, ∃ r > 0, ∀ x ∈ s, I.sup p x < r := by
rw [hp.hasBasis.isVonNBounded_iff]
constructor
· intro h I
simp only [id] at h
specialize h ((I.sup p).ball 0 1) (p... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 478 | 496 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | split_ifs at hba
· rw [← hba]
exact disjoint_sdiff_self_right
· cases ne_of_mem_of_not_mem hb ha hba
| Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 214 | 217 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | theorem disjoint_cylinder_iff [Nonempty (∀ i, α i)] {s t : Finset ι} {S : Set (∀ i : s, α i)}
{T : Set (∀ i : t, α i)} [DecidableEq ι] :
Disjoint (cylinder s S) (cylinder t T) ↔
Disjoint
(Finset.restrict₂ Finset.subset_union_left ⁻¹' S)
(Finset.restrict₂ Finset.subset_union_right ⁻¹' T) :=... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 237 | 244 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... | (hu : IsUniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ := by
have := (UniformFun.postcomp_isUniformInducing (α := ι) hu).isInducing
simp only [equicontinuousWithinAt_iff_continuousWithinAt, this.continuousWithinAt_iff]
rfl
/-- Given `u : α → β` a uniform ... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 717 | 725 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 480 | 485 | |
/-
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.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
/-!
# Order-connected sets
We say that a set `s :... |
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
| Mathlib/Order/Interval/Set/OrdConnected.lean | 93 | 96 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | `Subgroup.closure_induction_left`. -/
| Mathlib/Algebra/Group/Basic.lean | 885 | 885 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
import Mathlib.CategoryTheory.Limits.Shapes.Grothendieck
import Mathlib.CategoryTheory.Comma.StructuredArrow.Functo... | colimit.ι (proj L X ⋙ F) f :=
LeftExtension.IsPointwiseLeftKanExtensionAt.ι_isoColimit_hom (F := F)
(isPointwiseLeftKanExtensionLeftKanExtensionUnit L F X) f
| Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.lean | 87 | 90 |
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | variable {s t}
theorem le_chainHeight_iff {n : ℕ} : ↑n ≤ s.chainHeight ↔ ∃ l ∈ s.subchain, length l = n :=
(le_chainHeight_TFAE s n).out 0 1
theorem length_le_chainHeight_of_mem_subchain (hl : l ∈ s.subchain) : ↑l.length ≤ s.chainHeight :=
| Mathlib/Order/Height.lean | 109 | 114 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | simp [fderivWithin, hasFDerivWithinAt_inter ht, DifferentiableWithinAt]
theorem fderivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : fderivWithin 𝕜 f s x = fderiv 𝕜 f x := by
rw [← fderivWithin_univ, ← univ_inter s, fderivWithin_inter h]
theorem fderivWithin_of_isOpen (hs : IsOpen s) (hx : x ∈ s) : fderivWithin 𝕜 f s x =... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 675 | 680 |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... |
variable {s : Set R}
-- attribute [local reducible] closure -- Porting note: not available in Lean4
@[elab_as_elim]
protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s) (h1 : C 1)
(hneg1 : C (-1)) (hs : ∀ z ∈ s, ∀ n, C n → C (z * n)) (ha : ∀ {x y}, C x → C y → C (x + y)) :
C x := by
... | Mathlib/Algebra/Ring/Subring/Basic.lean | 917 | 944 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | /-- There are exactly `⌊N/n⌋` positive multiples of `n` that are `≤ N`.
See `Nat.card_multiples` for a "shifted-by-one" version. -/
lemma card_multiples' (N n : ℕ) : #{k ∈ range N.succ | k ≠ 0 ∧ n ∣ k} = N / n := by
induction N with
| zero => simp [Finset.filter_false_of_mem]
| succ N ih =>
rw [Finset... | Mathlib/Data/Nat/Factorization/Basic.lean | 593 | 604 |
/-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Algebra.ZMod
import Mathlib.Algebra.Field.ZMod
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mat... | (hv : IsTranscendenceBasis R v') (hR : #R ≤ ℵ₀) (hK : ℵ₀ < #K') : #K' = #ι' := by
simpa using cardinal_eq_cardinal_transcendence_basis_of_aleph0_lt v' hv hR hK
end Cardinal
variable {K : Type u} {L : Type v} [Field K] [Field L] [IsAlgClosed K] [IsAlgClosed L]
/-- Two uncountable algebraically closed fields of ... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 146 | 159 |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | and_congr_left_iff]
tauto
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 153 | 155 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | ⟨?_, fun h y hy a b ha hb hab =>
h ha hb hab (add_mem_add (smul_mem_smul_set <| mem_singleton _) ⟨_, hy, rfl⟩)⟩
rintro hA a b ha hb hab w ⟨au, ⟨u, rfl : u = x, rfl⟩, bv, ⟨v, hv, rfl⟩, rfl⟩
exact hA hv ha hb hab
theorem starConvex_empty (x : E) : StarConvex 𝕜 x ∅ := fun _ hy => hy.elim
| Mathlib/Analysis/Convex/Star.lean | 94 | 100 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,752 | 1,766 | |
/-
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.FieldTheory.Finite.Basic
/-!
# Lagrange's four square theorem
The main result in this file is `sum_four_squares`,
a proof that every natural number is th... | 2 * 2 * m = (x - y) ^ 2 + (x + y) ^ 2 := by rw [mul_assoc, h]; ring
_ = (2 * ((x - y) / 2)) ^ 2 + (2 * ((x + y) / 2)) ^ 2 := by
rw [Int.mul_ediv_cancel' _, Int.mul_ediv_cancel' _] <;>
simpa [sq, parity_simps, ← even_iff_two_dvd]
_ = 2 * 2 * (((x - y) / 2) ^ 2 + ((x + y) / 2) ^ 2) := ... | Mathlib/NumberTheory/SumFourSquares.lean | 46 | 59 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.Categor... | simp [Set.image_preimage_eq_range_inter]
lemma image_injective : Function.Injective (f ''ᵁ ·) := by
intro U V hUV
| Mathlib/AlgebraicGeometry/OpenImmersion.lean | 129 | 132 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | mul_add, add_mul])
(fun m i c x ↦ by
have : (List.finRange n).idxOf i < n := by simp
simp [List.ofFn_eq_map, (List.nodup_finRange n).map_update, List.prod_set, this])
variable {R A n}
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,157 | 1,163 |
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | theorem le_degrees_add_left (h : Disjoint p.degrees q.degrees) : p.degrees ≤ (p + q).degrees := by
classical
apply Finset.sup_le
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 144 | 146 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | theorem ContinuousWithinAt.finSnoc
{f : α → ∀ j : Fin n, π (Fin.castSucc j)} {g : α → π (Fin.last _)} {a : α} {s : Set α}
(hf : ContinuousWithinAt f s a) (hg : ContinuousWithinAt g s a) :
ContinuousWithinAt (fun a => Fin.snoc (f a) (g a)) s a :=
hf.tendsto.finSnoc hg
theorem ContinuousOn.finSnoc
{f :... | Mathlib/Topology/ContinuousOn.lean | 1,239 | 1,257 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | theorem factorial_mul_descFactorial : ∀ {n k : ℕ}, k ≤ n → (n - k)! * n.descFactorial k = n !
| n, 0 => fun _ => by rw [descFactorial_zero, Nat.mul_one, Nat.sub_zero]
| 0, succ k => fun h => by
exfalso
exact not_succ_le_zero k h
| succ n, succ k => fun h => by
| Mathlib/Data/Nat/Factorial/Basic.lean | 368 | 373 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,695 | 1,697 | |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | ae_eq_set_pi fun _ _ => Iio_ae_eq_Iic
theorem pi_Ioi_ae_eq_pi_Ici {s : Set ι} {f : ∀ i, α i} :
(pi s fun i => Ioi (f i)) =ᵐ[Measure.pi μ] pi s fun i => Ici (f i) :=
ae_eq_set_pi fun _ _ => Ioi_ae_eq_Ici
theorem univ_pi_Iio_ae_eq_Iic {f : ∀ i, α i} :
| Mathlib/MeasureTheory/Constructions/Pi.lean | 439 | 445 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | Nat.find ((nilpotent_iff_finite_descending_central_series G).mp hG) =
Group.nilpotencyClass G := by
rw [← least_ascending_central_series_length_eq_nilpotencyClass]
refine le_antisymm (Nat.find_mono ?_) (Nat.find_mono ?_)
· rintro n ⟨H, ⟨hH, hn⟩⟩
refine ⟨fun m => H (n - m), is_descending_rev_series_of_... | Mathlib/GroupTheory/Nilpotent.lean | 397 | 411 |
/-
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 | 1,402 | 1,404 | |
/-
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 | 1,291 | 1,293 | |
/-
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.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Nat
/-!
# Double countings
This... | _ < ∑ _b ∈ t, n := sum_lt_sum_of_nonempty ht hn
_ = _ := sum_const _
| Mathlib/Combinatorics/Enumerative/DoubleCounting.lean | 143 | 145 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Combinatorics.Additive.FreimanHom
/-!
# Corners
This file defines corners, namely triples of the form `(x, y), (x, y + d), (x + d, y)`, and the
property ... | hf' (hAs hxy.fst_fst_mem).1 (hAs hxy.snd_fst_mem).1 <| hA <| hxy.image hf hAs
lemma isCorner_image (hf : IsAddFreimanIso 2 s t f) (hAs : A ⊆ s ×ˢ s)
(hx₁ : x₁ ∈ s) (hy₁ : y₁ ∈ s) (hx₂ : x₂ ∈ s) (hy₂ : y₂ ∈ s) :
IsCorner (Prod.map f f '' A) (f x₁) (f y₁) (f x₂) (f y₂) ↔ IsCorner A x₁ y₁ x₂ y₂ := by
have h... | Mathlib/Combinatorics/Additive/Corner/Defs.lean | 81 | 90 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,225 | 2,226 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | variable (E)
theorem isVonNBounded_ball (r : ℝ) : Bornology.IsVonNBounded 𝕜 (Metric.ball (0 : E) r) :=
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 446 | 448 |
/-
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 | 1,470 | 1,472 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... | lemma gaussianPDFReal_nonneg (μ : ℝ) (v : ℝ≥0) (x : ℝ) : 0 ≤ gaussianPDFReal μ v x := by
rw [gaussianPDFReal]
positivity
| Mathlib/Probability/Distributions/Gaussian.lean | 61 | 64 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | theorem polar_neg (f : M → N) (x y : M) : polar (-f) x y = -polar f x y := by
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 107 | 108 |
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Cofinal
import Mathlib.Order.UpperLower.Principal... | Iff.rfl
lemma mem_principal_self : x ∈ principal x :=
| Mathlib/Order/Ideal.lean | 268 | 270 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | apply le_antisymm
· simp_rw [map_le_iff_le_comap]
refine mul_le.2 fun m hm n hn => ?_
| Mathlib/Algebra/Algebra/Operations.lean | 436 | 438 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem IsLindelof.disjoint_nhdsSet_left {l : Filter X} [CountableInterFilter l]
(hs : IsLindelof s) :
Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l := by
refine ⟨fun h x hx ↦ h.mono_left <| nhds_le_nhdsSet hx, fun H ↦ ?_⟩
choose! U hxU hUl using fun x hx ↦ (nhds_basis_opens x).disjoint_iff_left.1 (H x hx... | Mathlib/Topology/Compactness/Lindelof.lean | 196 | 206 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
/-!
# Closure, interior, and frontier of preimages under `re`... | theorem frontier_setOf_le_re_and_im_le (a b : ℝ) :
frontier { z | a ≤ re z ∧ im z ≤ b } = { z | a ≤ re z ∧ im z = b ∨ re z = a ∧ im z ≤ b } := by
| Mathlib/Analysis/Complex/ReImTopology.lean | 164 | 165 |
/-
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.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | · have : n - 1 - (ν - 1) = n - ν := by omega
rw [this, ascPochhammer_eval_succ]
rw_mod_cast [tsub_add_cancel_of_le (h'.trans n.pred_le)]
· simp only [not_le] at h
rw [tsub_eq_zero_iff_le.mpr (Nat.le_sub_one_of_lt h), eq_zero_of_lt R h]
simp [pos_iff_ne_zero.mp (pos_of_gt h)]
theorem ite... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 173 | 193 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,055 | 1,058 | |
/-
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
/-... | isGLB_Ioi.lowerBounds_eq
end
| Mathlib/Order/Bounds/Basic.lean | 470 | 472 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | @[simp, rclike_simps]
theorem I_im (z : K) : im z * im (I : K) = im z :=
| Mathlib/Analysis/RCLike/Basic.lean | 263 | 264 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
theorem Ico_subset_Ico_union_Ico {a b c : α} : Ico a c ⊆ Ico a b ∪ Ico b c := by
rw [← coe_subset, coe_union, coe_Ico, coe_Ico, coe_Ico]
exact Set.Ico_subset_Ico_union_Ico
| Mathlib/Order/Interval/Finset/Basic.lean | 829 | 832 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 732 | 737 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | (hf : ∀ᶠ x in 𝓝 b, ∃ c, Tendsto f (comap i <| 𝓝 x) (𝓝 c)) : ContinuousAt (di.extend f) b := by
set φ := di.extend f
haveI := di.comap_nhds_neBot
suffices ∀ V' ∈ 𝓝 (φ b), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 b by
simpa [ContinuousAt, (closed_nhds_basis (φ b)).tendsto_right_iff]
intro V' V'_in V'_closed
set ... | Mathlib/Topology/DenseEmbedding.lean | 176 | 184 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... |
end RpowCpow
open Nat
/-! ### Integral of `sin x ^ n` -/
theorem integral_sin_pow_aux :
(∫ x in a..b, sin x ^ (n + 2)) =
(sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ x in a..b, sin x ^ n) -
(↑n + 1) * ∫ x in a..b, sin x ^ (n + 2) := by
let C := sin a ^ (n + 1) * cos a - sin ... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 618 | 634 |
/-
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 | 1,537 | 1,540 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,636 | 1,650 | |
/-
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.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.AffineSpace.Pointwise
import Mathlib.LinearAlgebra.Basis.SMul
/-!
# Affine bases and bary... | linear := -(b.basisOf i).sumCoords
map_vadd' q v := by
| Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 134 | 135 |
/-
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... | · intro _ _
refine ⟨i, fun j hj => ?_⟩
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 793 | 794 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | rw [← inv_div b c, div_inv_eq_mul, mul_comm, div_mul_div_cancel]
| Mathlib/Algebra/Group/Basic.lean | 998 | 999 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Junyan Xu
-/
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
/-!
# Skyscraper (pre)sheaves
A skyscra... | fromStalk p₀ (toSkyscraperPresheaf _ f) = f := by
refine 𝓕.stalk_hom_ext fun U hxU ↦ ?_
rw [germ_fromStalk, toSkyscraperPresheaf_app, dif_pos hxU, Category.assoc, Category.assoc,
eqToHom_trans, eqToHom_refl, Category.comp_id, Presheaf.germ]
/-- The unit in `Presheaf.stalkFunctor ⊣ skyscraperPresheafFuncto... | Mathlib/Topology/Sheaves/Skyscraper.lean | 300 | 308 |
/-
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 | 784 | 785 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... |
@[simp]
theorem cancel_iso_inv_right {X Y Z : C} (f f' : X ⟶ Y) (g : Z ≅ Y) :
f ≫ g.inv = f' ≫ g.inv ↔ f = f' := by
simp only [cancel_mono]
| Mathlib/CategoryTheory/Iso.lean | 479 | 483 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Initial and terminal objects in a category.
##... | Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | 743 | 747 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | classical
simp only [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_C_mul _ hx]
@[simp]
theorem natSepDegree_smul_nonzero {x : F} (hx : x ≠ 0) :
(x • f).natSepDegree = f.natSepDegree := by
classical
simp only [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_smul_nonzero _ hx]
@[si... | Mathlib/FieldTheory/SeparableDegree.lean | 389 | 419 |
/-
Copyright (c) 2021 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.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... | rw [coe_image]
simpa using (hf.subset hsA hfsA.bijOn_subset.mapsTo).threeGPFree (hf'.injOn.mono hsA) hs
| Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 343 | 346 |
/-
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.Topology.MetricSpace.Lipschitz
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
/-!
# Hölder continuous functions
In this file we define Höl... | theorem nndist_le (hf : HolderWith C r f) (x y : X) :
nndist (f x) (f y) ≤ C * nndist x y ^ (r : ℝ) :=
hf.nndist_le_of_le le_rfl
theorem dist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ} (hd : dist x y ≤ d) :
dist (f x) (f y) ≤ C * d ^ (r : ℝ) :=
| Mathlib/Topology/MetricSpace/Holder.lean | 250 | 255 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | convert not_bddAbove_fp_family fun _ : Unit => H
exact (Set.iInter_const _).symm
/-- The derivative of a normal function `f` is the sequence of fixed points of `f`.
This is defined as `Ordinal.derivFamily` applied to a trivial family consisting only of `f`. -/
| Mathlib/SetTheory/Ordinal/FixedPoint.lean | 299 | 304 |
/-
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... |
private theorem div_def (a) {b : Ordinal} (h : b ≠ 0) : a / b = sInf { o | a < b * succ o } :=
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 825 | 826 |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Principal Ideals
This file deals with the set of principal ideals of a `CommRing R`.
## Main definitions and res... |
theorem associatesEquivIsPrincipal_mul (x y : Associates R) :
(associatesEquivIsPrincipal R (x * y) : Ideal R) =
(associatesEquivIsPrincipal R x) * (associatesEquivIsPrincipal R y) := by
| Mathlib/RingTheory/Ideal/IsPrincipal.lean | 81 | 84 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.Analysis.Convex.Contractible
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.... | · change (Function.invFunOn UpperHalfPlane.coe Set.univ w) = _
simp only [invFunOn, dite_eq_right_iff, mem_univ, true_and]
| Mathlib/Analysis/Complex/UpperHalfPlane/Topology.lean | 137 | 138 |
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Equal and mixed characteristic
In commutative ... | Mathlib/Algebra/CharP/MixedCharZero.lean | 372 | 378 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | theorem Disjoint.symmetric : Symmetric (@Disjoint α) := fun _ _ => Disjoint.symm
| Mathlib/GroupTheory/Perm/Support.lean | 50 | 50 |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | (ae_restrict_mem hS.compl).mono fun x hx =>
ENNReal.ofReal_eq_zero.2 (not_lt.1 <| mt neg_pos.1 hx)
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 190 | 192 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Measurability of `⌊x⌋` etc
In this file we prove that `Int.floor`, `Int.ceil`, `Int.fract`, `Nat... | @[measurability, fun_prop]
theorem Measurable.fract [IsStrictOrderedRing R] [BorelSpace R] {f : α → R} (hf : Measurable f) :
Measurable fun x => Int.fract (f x) :=
measurable_fract.comp hf
| Mathlib/MeasureTheory/Function/Floor.lean | 47 | 50 |
/-
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.Altitude
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Monge point and orthocenter
This file defines the orthocenter of a trian... |
end Affine
namespace EuclideanGeometry
open Affine AffineSubspace Module
variable {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P]
/-- Four points form an orthocentric system if they consist of the
vertices of a triangle and its orthocenter. -/
def Orth... | Mathlib/Geometry/Euclidean/MongePoint.lean | 471 | 483 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | W'.Equation x y ↔ W'.Nonsingular x y := by
rw [equation_iff_variableChange, nonsingular_iff_variableChange,
equation_zero_iff_nonsingular_zero_of_Δ_ne_zero <| by
rwa [variableChange_Δ, inv_one, Units.val_one, one_pow, one_mul]]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 289 | 293 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Category.Grp.Basic
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.PUnit
import Mathlib.CategoryTheo... | /-- Build a `RingEquiv` from an isomorphism in the category `CommSemiRingCat`. -/
def commSemiRingCatIsoToRingEquiv {R S : CommSemiRingCat.{u}} (e : R ≅ S) : R ≃+* S :=
RingEquiv.ofHomInv e.hom.hom e.inv.hom (by ext; simp) (by ext; simp)
| Mathlib/Algebra/Category/Ring/Basic.lean | 662 | 665 |
/-
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, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | (lift k G A).symm F x = F (single x.toAdd 1) :=
rfl
theorem lift_of (F : Multiplicative G →* A) (x : Multiplicative G) :
lift k G A F (of k G x) = F x := MonoidAlgebra.lift_of F x
| Mathlib/Algebra/MonoidAlgebra/Basic.lean | 478 | 482 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Adjoin.Polynomial
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Ideal.Quotient.Operati... | exact MvPolynomial.adjoin_range_X⟩⟩
theorem of_restrictScalars_finiteType [Algebra S A] [IsScalarTower R S A] [hA : FiniteType R A] :
FiniteType S A := by
obtain ⟨s, hS⟩ := hA.out
refine ⟨⟨s, eq_top_iff.2 fun b => ?_⟩⟩
| Mathlib/RingTheory/FiniteType.lean | 93 | 98 |
/-
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.Set.Lattice
/-! # Semiquotients
A data type for semiquotients, which are classically equivalent to
nonempty sets, but are useful for programming... |
/-- If `f` is a constant on `q.s`, then `q.liftOn f` is the value of `f`
| Mathlib/Data/Semiquot.lean | 90 | 91 |
/-
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... | instance botCharacteristic : Characteristic (⊥ : Subgroup G) :=
characteristic_iff_le_map.mpr fun _ϕ => bot_le
@[to_additive]
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 327 | 330 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | theorem liminf_le_iff' [DenselyOrdered β] {x : β}
(h₁ : f.IsCoboundedUnder (· ≥ ·) u := by isBoundedDefault)
(h₂ : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) :
liminf u f ≤ x ↔ ∀ y > x, ∃ᶠ a in f, u a ≤ y := le_limsup_iff' (β := βᵒᵈ) h₁ h₂
lemma liminf_le_limsup_of_frequently_le {v : α → β} (h : ∃ᶠ... | Mathlib/Order/LiminfLimsup.lean | 911 | 921 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | · exact Nat.cast_pos.mpr Finite.card_pos
variable (G)
theorem inv_card_commutator_le_commProb : (↑(Nat.card (commutator G)))⁻¹ ≤ commProb G :=
(inv_le_iff_one_le_mul₀ (Nat.cast_pos.mpr Finite.card_pos)).mpr
(le_trans (ge_of_eq (commProb_eq_one_iff.mpr ⟨(Abelianization.commGroup G).mul_comm⟩))
(commutato... | Mathlib/GroupTheory/CommutingProbability.lean | 119 | 128 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | det A =
A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1
- A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0
+ A 0 2 * A 1 0 * A 2 1 - A 0 2 * A 1 1 * A 2 0 := by
simp only [det_succ_row_zero, submatrix_apply, Fin.succ_zero_eq_one, submatrix_submatrix,
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 819 | 823 |
/-
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.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... |
theorem leftDistributor_assoc {J : Type} [Finite J] (X Y : C) (f : J → C) :
(asIso (𝟙 X) ⊗ leftDistributor Y f) ≪≫ leftDistributor X _ =
| Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 188 | 190 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | rw [union_comm, union_ae_eq_left_iff_ae_subset]
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 315 | 316 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | @[ext]
theorem ext {φ ψ} (h : ∀ n : σ →₀ ℕ, coeff R n φ = coeff R n ψ) : φ = ψ :=
funext h
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 143 | 146 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 430 | 436 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | apply sum_congr rfl
intro i h
rw [mem_range] at h
rw [map_mul, map_pow (bind₁ _), algHom_C, H i h, algebraMap_eq]
| Mathlib/RingTheory/WittVector/WittPolynomial.lean | 285 | 289 |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, sSup_insert,
forall_eq_or_imp, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
constructor
· exact le_sup_of_le_left h
· intro i' h_i'
apply le_sup_of_le_right
apply le_sSup_of_le
· use i'
... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 137 | 146 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | simp
| Mathlib/Data/Seq/Computation.lean | 983 | 984 |
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