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) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,337 | 1,341 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 476 | 483 | |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... | (hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by
refine lhopital_zero_right_on_Ioo hab hff' hgg' hg' ?_ ?_ hdiv
· rw [← hfa, ← nhdsWithin_Ioo_eq_nhdsGT hab]
exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto
· rw [← hga, ← nhdsW... | Mathlib/Analysis/Calculus/LHopital.lean | 95 | 104 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,038 | 1,038 | |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... | Absorbent 𝕜 s ↔ ∀ x : E, ∀ᶠ c : 𝕜 in 𝓝[≠] 0, c • x ∈ s :=
forall_congr' fun x ↦ by simp only [absorbs_iff_eventually_nhdsNE_zero, mapsTo_singleton]
| Mathlib/Analysis/LocallyConvex/Basic.lean | 166 | 168 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Minchao Wu
-/
import Mathlib.Data.Prod.Basic
import Mathlib.Order.Lattice
import Mathlib.Order.BoundedOrder.Basic
import Mathlib.Tactic.Tauto
/-!
# Lexicographic order
Th... | section PartialOrderPreorder
variable [PartialOrder α] [Preorder β] {x y : α × β}
/-- Variant of `Prod.Lex.toLex_le_toLex` for partial orders. -/
| Mathlib/Data/Prod/Lex.lean | 115 | 119 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Equiv.Opposite
import Mathlib.Algebra.... | @[simp]
protected theorem smul_def (f : Module.End R M) (a : M) : f • a = f a :=
rfl
| Mathlib/Algebra/Module/LinearMap/End.lean | 199 | 202 |
/-
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.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... |
namespace ModEq
| Mathlib/Data/Nat/ModEq.lean | 89 | 91 |
/-
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.Algebra.Polynomial.Module.Basic
import Mathlib.RingTheory.Finiteness.Nakayama
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheor... |
variable (F F')
/-- The `R[IX]`-submodule of `M[X]` associated with an `I`-filtration. -/
protected def submodule : Submodule (reesAlgebra I) (PolynomialModule R M) where
carrier := { f | ∀ i, f i ∈ F.N i }
| Mathlib/RingTheory/Filtration.lean | 227 | 232 |
/-
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... | variable [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α]
instance (priority := 100) : IsAtomistic α where
isLUB_atoms b := (eq_bot_or_eq_top b).elim (fun h ↦ ⟨∅, by simp [h]⟩) (fun h ↦ ⟨{⊤}, by simp [h]⟩)
instance (priority := 100) : IsCoatomistic α :=
isAtomistic_dual_iff_isCoatomistic.1 (by infer_instance)
... | Mathlib/Order/Atoms.lean | 852 | 862 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,763 | 2,764 | |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
/-!
# LawfulTraversable instan... | -- Porting note: added `ApplicativeTransformation` theorems
induction x <;> simp! [*, functor_norm, ApplicativeTransformation.preserves_map,
ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]
| Mathlib/Control/Traversable/Instances.lean | 84 | 86 |
/-
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.ChosenFiniteProducts
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
/-!
# Functor categories have chosen finite products
If `C` is a... | change (f ⊗ g).app j ≫ (fst F₁' F₂').app j = _
rw [← NatTrans.comp_app, tensorHom_fst, NatTrans.comp_app]
rfl
| Mathlib/CategoryTheory/ChosenFiniteProducts/FunctorCategory.lean | 111 | 114 |
/-
Copyright (c) 2023 Mohanad Ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Eigenvalues are characteristic polynomial roots.
In fields we show that:
*... | · rw [Matrix.trace, Fintype.sum_empty, Matrix.charpoly,
det_eq_one_of_card_eq_zero (Fintype.card_eq_zero_iff.2 h), Polynomial.roots_one,
Multiset.empty_eq_zero, Multiset.sum_zero]
· rw [trace_eq_neg_charpoly_coeff, neg_eq_iff_eq_neg,
← Polynomial.sum_roots_eq_nextCoeff_of_monic_of_split A.charpoly... | Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean | 67 | 75 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... | hfc.leftDeriv_eq_sSup_slope_of_mem_interior hxs]
refine csSup_le ?_ ?_
· rw [image_nonempty]
obtain ⟨z, haz, hzx⟩ := exists_between hxab.1
exact ⟨z, habs ⟨haz, hzx.trans hxab.2⟩, hzx⟩
rintro _ ⟨z, ⟨hzs, hzx⟩, rfl⟩
| Mathlib/Analysis/Convex/Deriv.lean | 531 | 536 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by
intro _ _ h
-- Use above extensionality lemma to prove injectivity by showing that `h_add` and `h_mul` hold.
| Mathlib/Algebra/Ring/Ext.lean | 171 | 174 |
/-
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.CategoryTheory.FinCategory.AsType
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import ... | (@ULiftHom.category (ULift J) (@uliftCategory J hJ)) C _ J hJ
(@Equivalence.symm J hJ (ULiftHom (ULift J))
(@ULiftHom.category (ULift J) (@uliftCategory J hJ)) l) _
section
open WalkingParallelPair WalkingParallelPairHom
instance fintypeWalkingParallelPair : Fintype WalkingParallelPair where
elem... | Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean | 123 | 134 |
/-
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.Normed.Group.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Sums over residue classes
We consider... | if and only if it is summable. -/
lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : Antitone f) (k : ZMod m) :
Summable ({n : ℕ | (n : ZMod m) = k}.indicator f) ↔ Summable f := by
refine ⟨fun H ↦ ?_, fun H ↦ Summable.indicator H _⟩
rw [Finset.sum_indicator_mod m f]
convert summable_sum (s ... | Mathlib/Analysis/SumOverResidueClass.lean | 87 | 95 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | S.leftHomologyData.hπ
@[reassoc (attr := simp)]
lemma liftCycles_comp_cyclesMap (φ : S ⟶ S₁) [S₁.HasLeftHomology] :
S.liftCycles k hk ≫ cyclesMap φ =
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 978 | 982 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | kernelSubobject_factors f h⟩
/-- A factorisation of `h : W ⟶ X` through `kernelSubobject f`, assuming `h ≫ f = 0`. -/
| Mathlib/CategoryTheory/Subobject/Limits.lean | 110 | 112 |
/-
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.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Tactic.AdaptationNote
/-!
# Inversion in an affine sp... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 73 | 73 | |
/-
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... | rfl
theorem rothNumberNat_le (N : ℕ) : rothNumberNat N ≤ N :=
(addRothNumber_le _).trans (card_range _).le
theorem rothNumberNat_spec (n : ℕ) :
∃ t ⊆ range n, #t = rothNumberNat n ∧ ThreeAPFree (t : Set ℕ) :=
addRothNumber_spec _
/-- A verbose specialization of `threeAPFree.le_addRothNumber`, sometimes con... | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 407 | 420 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
/-!
# Adapted and progressively measurable processes
This file defi... | (progMeasurable_const _ 1) h
@[to_additive]
protected theorem finset_prod {γ} [CommMonoid β] [ContinuousMul β] {U : γ → ι → Ω → β}
{s : Finset γ} (h : ∀ c ∈ s, ProgMeasurable f (U c)) :
ProgMeasurable f fun i a => ∏ c ∈ s, U c i a := by
convert ProgMeasurable.finset_prod' h using 1; ext (i a); simp only ... | Mathlib/Probability/Process/Adapted.lean | 135 | 143 |
/-
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... | theorem det_units_conj' (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
det (M⁻¹.val * N * ↑M.val) = det N :=
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 194 | 195 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
import Mathlib.Algebra.Star.SelfAdjoint
/-... | open scoped Pointwise
open NonUnitalStarSubalgebra
variable (R)
/-- The minimal non-unital subalgebra that includes `s`. -/
def adjoin (s : Set A) : NonUnitalStarSubalgebra R A where
toNonUnitalSubalgebra := NonUnitalAlgebra.adjoin R (s ∪ star s)
star_mem' _ := by
rwa [NonUnitalSubalgebra.mem_carrier, ← NonU... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 646 | 657 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | ### `IsCycle`
-/
section IsCycle
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 216 | 219 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,364 | 2,365 | |
/-
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... | { bicone := _
isBilimit := binaryBiconeIsBilimitOfLimitConeOfIsLimit (limit.isLimit _) }
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 367 | 369 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... | Mathlib/ModelTheory/Substructures.lean | 1,039 | 1,044 | |
/-
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... | theorem IsUniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
(hu : IsUniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) := by
have := UniformFun.postcomp_isUniformInducing (α := ι) hu
simp only [uniformEquicontinuous_iff_uniformContinuous, this.uniformContinuou... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 741 | 745 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... |
@[simp]
theorem reflect_add (f g : R[X]) (N : ℕ) : reflect N (f + g) = reflect N f + reflect N g := by
ext
simp only [coeff_add, coeff_reflect]
@[simp]
| Mathlib/Algebra/Polynomial/Reverse.lean | 113 | 119 |
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | simpa only [List.drop, _root_.zero_le, ite_true] using H 0 (zero_le _)
intro k hk
induction hk using Nat.decreasingInduction with
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 384 | 386 |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
/-!
# Preservation of zero objects and zero mo... | _ = F.map 0 ≫ (i.hom ≫ i.inv) ≫ F.map 0 := by rw [Iso.hom_inv_id, Category.id_comp]
_ = 0 := by simp only [zero_of_to_zero i.hom, zero_comp, comp_zero]
instance (priority := 100) preservesZeroMorphisms_of_preserves_initial_object
[PreservesColimit (Functor.empty.{0} C) F] : PreservesZeroMorphisms F :=
... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean | 142 | 147 |
/-
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 ... | (a ⊔ v) \ u ∈ s := by
rw [mem_compression] at h
obtain h | ⟨-, b, hb, hba⟩ := h
· cases ha h.1
unfold compress at hba
split_ifs at hba with h
· rwa [← hba, sdiff_sup_cancel (le_sup_of_le_left h.2), sup_sdiff_right_self,
h.1.symm.sdiff_eq_left]
· cases ne_of_mem_of_not_mem hb ha hba
| Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 220 | 228 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | IsLimit s :=
IsLimit.ofIsoLimit hs (Cones.extendIso s (asIso i))
| Mathlib/CategoryTheory/Limits/IsLimit.lean | 315 | 317 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
/-!
# Lemmas about `extendFrom` in an order topology... | · rw [closure_Ioo hab.ne]
exact Ico_subset_Icc_self
· intro x x_in
rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)
· use la
simpa [hab]
· exact ⟨f x, hf x h⟩
theorem continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [Topolog... | Mathlib/Topology/Order/ExtendFrom.lean | 50 | 61 |
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... |
@[simp]
theorem Monic.degree_map [Semiring S] [Nontrivial S] {P : R[X]} (hmo : P.Monic) (f : R →+* S) :
| Mathlib/Algebra/Polynomial/Monic.lean | 371 | 373 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | funext a
rwa [List.getD_eq_getElem?_getD, List.getElem?_reverse, ← List.getD_eq_getElem?_getD]
/-- An empty list can always be decidably checked for the presence of an element.
Not an instance because it would clash with `DecidableEq α`. -/
def decidableGetDNilNe (a : α) : DecidablePred fun i : ℕ => getD ([] : Lis... | Mathlib/Data/List/GetD.lean | 50 | 56 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... | lemma stableUnderSpecialization_iUnion {ι : Sort*} (S : ι → Set X)
(H : ∀ i, StableUnderSpecialization (S i)) : StableUnderSpecialization (⋃ i, S i) :=
isLowerSet_iUnion H
| Mathlib/Topology/Inseparable.lean | 266 | 268 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic... | theorem isClosed_iff {X : Compactum} (S : Set X) :
IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
rw [← isOpen_compl_iff]
constructor
· intro cond F h
| Mathlib/Topology/Category/Compactum.lean | 157 | 161 |
/-
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... | coeff_monomial _ _ _
theorem coeff_zero_one : coeff R (0 : σ →₀ ℕ) 1 = 1 :=
coeff_monomial_same 0 1
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 199 | 202 |
/-
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.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | convexJoin_subset_convexHull _ _
theorem convexHull_union (hs : s.Nonempty) (ht : t.Nonempty) :
| Mathlib/Analysis/Convex/Join.lean | 165 | 167 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... | (h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by
apply toMeasure_injective
| Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 204 | 205 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
/-!
# Basic facts about real ... |
theorem interior_closedBall (x : E) {r : ℝ} (hr : r ≠ 0) :
interior (closedBall x r) = ball x r := by
| Mathlib/Analysis/NormedSpace/Real.lean | 76 | 78 |
/-
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.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... |
theorem mul_self_le_self {I : FractionalIdeal S P} (hI : I ≤ 1) : I * I ≤ I := by
convert mul_left_mono I hI
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 612 | 614 |
/-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Data.Set.... | section Unital
variable [CommRing 𝕜] [NoZeroDivisors 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
[ContinuousConstSMul 𝕜 𝕜] [TopologicalSpace A] [Semiring A] [Algebra 𝕜 A]
/-- In a unital algebra, elements of the character space are algebra homomorphisms. -/
instance instAlgHomClass : AlgHomClass (characterSpac... | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 128 | 134 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
/-- Let `s, t ⊆ X` be two subsets of a topological space `X`. If `t ⊆ s` and the topology induced
by `X`on `s` is discrete, then also the topology induces on `t` is discrete. -/
theorem DiscreteTopology.of_subset {X : Type*} [TopologicalSpace X] {s t : Set X}
(_ : DiscreteTopology s) (ts : t ⊆ s) : DiscreteTopolo... | Mathlib/Topology/Constructions.lean | 470 | 477 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... |
variable [TopologicalSpace H] [CommGroup H] [PartialOrder H] [IsOrderedMonoid H] [ContinuousInv H]
@[to_additive]
theorem tendsto_inv_nhdsGT {a : H} : Tendsto Inv.inv (𝓝[>] a) (𝓝[<] a⁻¹) :=
| Mathlib/Topology/Algebra/Group/Basic.lean | 440 | 444 |
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Geometry.Euclidean.Sphere.Basic
/-!
# Power of a point (intersecting ch... | simp_all only [Classical.not_not, one_smul]
exacts [hab (vsub_left_cancel hab₁).symm, hcd (vsub_left_cancel hcd₁).symm]
end EuclideanGeometry
| Mathlib/Geometry/Euclidean/Sphere/Power.lean | 122 | 129 |
/-
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.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... | (hasFDerivAt_finset_prod.comp x <| hasFDerivAt_pi.mpr fun i ↦ hg (Subtype.val i) i.prop :)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach])
theorem HasFDerivWithinAt.finset_prod [DecidableEq ι] {x : E}
(hg : ∀ i ∈ u, HasFDerivWithinAt (g i) (g' i) s x) :
HasFDerivWithinAt (∏ i ∈ u, g i... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 775 | 782 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
/-!
# Sesquilinear maps
... | simp [x', hx']
lemma IsSymm.nondegenerate_restrict_of_isCompl_ker {B : M →ₗ[R] M →ₗ[R] R} (hB : B.IsSymm)
| Mathlib/LinearAlgebra/SesquilinearForm.lean | 731 | 733 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | Fintype.card_ofSubsingleton, rank]
@[simp]
theorem unitLattice_rank :
| Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 367 | 370 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
i... |
lemma sinZeta_neg (a : UnitAddCircle) (s : ℂ) :
sinZeta (-a) s = -sinZeta a s := by
| Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 467 | 469 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... |
theorem Sbtw.trans_wbtw_left_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Sbtw R w y z)
| Mathlib/Analysis/Convex/Between.lean | 458 | 459 |
/-
Copyright (c) 2017 Mario Carneiro. 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.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... |
@[simp]
theorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a := by
| Mathlib/SetTheory/Ordinal/Exponential.lean | 72 | 74 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | infer_instance
theorem unop {W : MorphismProperty C} {L : Cᵒᵖ ⥤ Dᵒᵖ} (h : W.op.IsInvertedBy L) :
W.IsInvertedBy L.unop := fun X Y f hf => by
haveI := h f.op hf
| Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 63 | 67 |
/-
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.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | homGroup X Y :=
| Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 399 | 399 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.RingTheory.TensorProduct.Basic
/-!
# Bimodules
One frequently encounters situations in which several sets of scalars act on a single space, subject
to comp... |
theorem smul_mem' (p : Submodule (A ⊗[R] B) M) (b : B) {m : M} (hm : m ∈ p) : b • m ∈ p := by
suffices b • m = (1 : A) ⊗ₜ[R] b • m by exact this.symm ▸ p.smul_mem _ hm
| Mathlib/Algebra/Module/Bimodule.lean | 90 | 92 |
/-
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 prod_apply_symm (s : Set (α × β)) (s_mble : MeasurableSet s) :
μ.prod ν s = ENNReal.toNNReal (∫⁻ y, μ.toMeasure ((fun x ↦ ⟨x, y⟩) ⁻¹' s) ∂ν) := by
simp [coeFn_def, Measure.prod_apply_symm s_mble]
| Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 53 | 55 |
/-
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.Functor.Currying
import Mathlib.CategoryTheory.Localization.Predicate
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
# Localizati... | intro _ _ f₂ hf₂
haveI : ∀ (X₁ : W₁.Localization),
IsIso (((Functor.flip (prodLift₁ F hF)).map f₂).app X₁) := fun X₁ => by
obtain ⟨X₁, rfl⟩ := (Construction.objEquiv W₁).surjective X₁
| Mathlib/CategoryTheory/Localization/Prod.lean | 76 | 79 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | exact ⟨h.2.2.1, h.2.2.2, H.2⟩
| Mathlib/Computability/TMToPartrec.lean | 1,183 | 1,183 |
/-
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.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,381 | 1,382 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | Mathlib/Order/Monotone/Basic.lean | 850 | 852 | |
/-
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)] [∀ a, OrderTop (β a)] :
| Mathlib/Order/PiLex.lean | 160 | 161 |
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Tactic.FieldSimp
import Mathli... | rw [← mul_div_right_comm, ← mul_assoc, le_div_iff₀ (cast_pos.2 hU.1.card_pos), mul_comm _ C,
← mul_assoc, mul_comm _ C]
exact mod_cast pluennecke_petridis_inequality_mul C (mul_aux hU.1 hU.2 hUA)
| Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 184 | 187 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | suffices setToL1 hT = setToL1 hT' by rw [this]
refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ hT) _ _ _ _ ?_
ext1 f
suffices setToL1 hT' f = setToL1SCLM α E μ hT f by rw [← this]; simp [coeToLp]
rw [setToL1_eq_setToL1SCLM]
exact setToL1SCLM_congr_left hT' hT h.symm f
theorem setToL1_congr_left... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 432 | 445 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | rw [← Nat.cast_two, ΨSq_ofNat, preΨ'_two, one_pow, one_mul, if_pos even_two]
@[simp]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 271 | 273 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
/-!
# Trivial Lie modules and Abelian Lie algebras
The action of a Lie algebra `L` on ... | @[simp]
theorem toLinearMap_maxTrivLinearMapEquivLieModuleHom_symm (f : M →ₗ⁅R,L⁆ N) :
| Mathlib/Algebra/Lie/Abelian.lean | 228 | 229 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Alg... | have hp_sup : p s.prod := prod_induction_nonempty p hp_mul hs_empty hsa_restrict
have hp_a : p a := hsa_prop a (mem_cons_self a s)
exact (h_mul a _ hp_a hp_sup).trans (mul_le_mul_left' (hs hs_empty hsa_restrict) _)
@[to_additive le_sum_nonempty_of_subadditive]
| Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean | 111 | 115 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | (v : m → α) : replicateCol ι (v ᵥ* M) = (replicateRow ι v * M)ᵀ := by
ext
rfl
| Mathlib/Data/Matrix/RowCol.lean | 168 | 171 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | rfl
@[simp]
theorem get_inits : ∀ (n : ℕ) (s : Stream' α), get (inits s) n = take (succ n) s := by
intro n; induction' n with n' ih
· intros
| Mathlib/Data/Stream/Init.lean | 674 | 679 |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... | @[simp]
theorem toList_comp (p : Path a b) : ∀ {c} (q : Path b c), (p.comp q).toList = q.toList ++ p.toList
| _, nil => by simp
| Mathlib/Combinatorics/Quiver/Path.lean | 171 | 173 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... | lemma mem_nonZeroDivisorsLeft_iff : x ∈ nonZeroDivisorsLeft M₀ ↔ ∀ y, y * x = 0 → y = 0 := .rfl
lemma nmem_nonZeroDivisorsLeft_iff :
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 45 | 47 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 257 | 262 | |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 800 | 804 | |
/-
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.Fintype.EquivFin
import Mathlib.Data.Finset.Option
/-!
# fintype instances for option
-/
assert_not_exists MonoidWithZero MulAction
open Functi... | Mathlib/Data/Fintype/Option.lean | 114 | 119 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | Measurable (fun p : α × γ ↦ Real.toNNReal (rnDerivAux κ (κ + η) p.1 p.2)
- Real.toNNReal (1 - rnDerivAux κ (κ + η) p.1 p.2) * rnDeriv κ η p.1 p.2) := by fun_prop
lemma measurable_singularPart_fun_right (κ η : Kernel α γ) (a : α) :
| Mathlib/Probability/Kernel/RadonNikodym.lean | 264 | 267 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.RingHomProperties
/-!
# Constructors for properties of morphisms between schemes
This file pro... | variable (P : ∀ {α β : Type u} [TopologicalSpace α] [TopologicalSpace β] (_ : α → β), Prop)
/-- If a property of maps of topological spaces is stable under composition, the induced
morphism property of schemes is stable under composition. -/
lemma topologically_isStableUnderComposition
(hP : ∀ {α β γ : Type u} [To... | Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean | 221 | 241 |
/-
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... | nonrec theorem star_add_self : star a + a = 2 * a.re := by
simp [a.star_add_self, Quaternion.coe]
| Mathlib/Algebra/Quaternion.lean | 1,027 | 1,028 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Algebra.Lie.... | invertibleOfRightInverse _ _ (pso_inv p q R hi)
theorem indefiniteDiagonal_transform {i : R} (hi : i * i = -1) :
(Pso p q R i)ᵀ * indefiniteDiagonal p q R * Pso p q R i = 1 := by
ext (x y); rcases x with ⟨x⟩|⟨x⟩ <;> rcases y with ⟨y⟩|⟨y⟩
· -- x y : p
by_cases h : x = y <;>
simp [Pso, indefiniteDiagon... | Mathlib/Algebra/Lie/Classical.lean | 178 | 189 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,639 | 1,644 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | rw [← integral_comp_polarCoord_symm]
simp only [polarCoord_symm_apply, ofReal_mul, ofReal_cos, ofReal_sin]
_ = (∫ r in Ioi (0 : ℝ), r * cexp (-b * (r : ℂ) ^ 2)) * ∫ θ in Ioo (-π) π, 1 := by
rw [← setIntegral_prod_mul]
congr with p : 1
rw [mul_one]
congr
conv_rhs => rw [← on... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 205 | 235 |
/-
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... | have fin : x * x = d * y * y → (⟨x, y⟩ : ℤ√d) = 0 := fun e =>
match x, y, divides_sq_eq_zero_z e with
| _, _, ⟨rfl, rfl⟩ => rfl
exact
if z0 : z = 0 then
if w0 : w = 0 then
Or.inr
(match z, w, z0, w0 with
| _, _, rfl, rfl => rfl)
else
... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 813 | 848 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | theorem volume_real_ball {a r : ℝ} (hr : 0 ≤ r) : volume.real (Metric.ball a r) = 2 * r := by
simp [measureReal_def, hr]
@[simp]
theorem volume_closedBall (a r : ℝ) : volume (Metric.closedBall a r) = ofReal (2 * r) := by
rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel_left, two_mul]
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 127 | 132 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... |
theorem roots_expand_image_frobenius_subset [DecidableEq R] :
(expand R p f).roots.toFinset.image (frobenius R p) ⊆ f.roots.toFinset := by
rw [← iterateFrobenius_one]
convert ← roots_expand_pow_image_iterateFrobenius_subset p 1 f
apply pow_one
section PerfectRing
variable {p n f}
variable [PerfectRing R p]
... | Mathlib/FieldTheory/Perfect.lean | 267 | 278 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Batteries.Data.List.Basic
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two... |
theorem suffixLevenshtein_cons₁_aux {α} {x y : { l : List α // 0 < l.length }}
(w₀ : x.1[0]'x.2 = y.1[0]'y.2) (w : x.1.tail = y.1.tail) : x = y := by
match x, y with
| Mathlib/Data/List/EditDistance/Defs.lean | 221 | 224 |
/-
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.Group.Submonoid.Operations
import Mathlib.Data.DFinsupp.Sigma
import Mathlib.Data.DFinsupp.Submonoid
/-!
# Direct sum
This file defines the direct sum ... | def fromAddMonoid : (⨁ i, γ →+ β i) →+ γ →+ ⨁ i, β i :=
toAddMonoid fun i => AddMonoidHom.compHom (of β i)
| Mathlib/Algebra/DirectSum/Basic.lean | 243 | 245 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | · simp [ofFn_zero, Nat.zero_mul, ofFn_zero, flatten]
· simp_rw [ofFn_succ', succ_mul]
| Mathlib/Data/List/OfFn.lean | 80 | 81 |
/-
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.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | theorem birkhoffAverage_apply_sub_birkhoffAverage {α M : Type*} (R : Type*) [DivisionRing R]
[AddCommGroup M] [Module R M] (f : α → α) (g : α → M) (n : ℕ) (x : α) :
birkhoffAverage R f g n (f x) - birkhoffAverage R f g n x =
(n : R)⁻¹ • (g (f^[n] x) - g x) := by
simp only [birkhoffAverage, birkhoffSum_a... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 82 | 86 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,670 | 1,671 | |
/-
Copyright (c) 2023 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.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | rw [cliqueFinset_eq_map, card_map]
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 220 | 221 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-!
# Linear equivalences of tensor products as isometrie... | section tensorLId
theorem comp_tensorLId_eq (Q₂ : QuadraticForm R M₂) :
Q₂.comp (TensorProduct.lid R M₂) = QuadraticForm.tmul (sq (R := R)) Q₂ := by
refine (QuadraticMap.associated_rightInverse R).injective ?_
ext m₂ m₂'
dsimp [-associated_apply]
| Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 186 | 192 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl, Yuyang Zhao
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra... |
alias NE.ne.pos := pos_of_ne_zero
| Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 199 | 200 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Bhavik Mehta, Daniel Weber, Stefan Kebekus
-/
import Mathlib.Tactic.TautoSet
import Mathlib.Topology.Constructions
import Mathlib.Data.Set.Subset
import Mathlib.Topology.Sepa... |
section cofinite_cocompact
lemma tendsto_cofinite_cocompact_iff :
Tendsto f cofinite (cocompact _) ↔ ∀ K, IsCompact K → Set.Finite (f ⁻¹' K) := by
| Mathlib/Topology/DiscreteSubset.lean | 44 | 48 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | Real.cos_arccos (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
(div_le_one_of_le₀ _ (norm_nonneg _))]
rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),
norm_add_sq_eq_norm_sq_add_norm_sq_real h]
exact le_add_of_nonneg_right (mul_self_nonneg _)
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 123 | 128 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | (hm : ∀ i, NullMeasurableSet (s i) μ) {t : Set α} (ht : MeasurableSet t) :
μ.restrict (⋃ i, s i) t = ∑' i, μ.restrict (s i) t := by
simp only [restrict_apply, ht, inter_iUnion]
| Mathlib/MeasureTheory/Measure/Restrict.lean | 271 | 273 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | [TopologicalSpace ε] [ContinuousENorm ε]
[TopologicalSpace ε'] [ENormedAddMonoid ε']
theorem eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p) (hpq : p ≤ q)
(hf : AEStronglyMeasurable f μ) :
eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q) := by
have hq0_lt : 0 < q := lt... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 26 | 45 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | ¬G.CliqueFree (card α) := by
rw [not_cliqueFree_iff]
exact ⟨(Iso.completeGraph (Fintype.equivFin α)).symm.toEmbedding.trans f⟩
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 352 | 355 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... |
end CommRing
theorem sq_add_sq_ne_zero {R : Type*} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R]
| Mathlib/RingTheory/Coprime/Basic.lean | 393 | 396 |
/-
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 | 1,486 | 1,489 |
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