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 |
|---|---|---|---|---|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
/-! # Optional stopping theorem (fair game theorem)
The optional stopping theor... | (ε : ℝ) ≤ stoppedValue f (hitting f {y : ℝ | ↑ε ≤ y} 0 n) ω := by
intro x hx
simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx
refine stoppedValue_hitting_mem ?_
simp only [Set.mem_setOf_eq, exists_prop, hn]
exact
let ⟨j, hj₁, hj₂⟩ := hx
⟨j, hj₁, hj₂⟩
have h := setIntegral_... | Mathlib/Probability/Martingale/OptionalStopping.lean | 112 | 133 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | simp [h, map₂]
end Option
| Mathlib/Data/Option/NAry.lean | 195 | 197 |
/-
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.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | left_inv s := by
ext a; constructor <;> intro h <;>
| Mathlib/Data/Finset/Image.lean | 706 | 707 |
/-
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.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | lemma preNormEDS_even (m : ℤ) : preNormEDS b c d (2 * m) =
preNormEDS b c d (m - 1) ^ 2 * preNormEDS b c d m * preNormEDS b c d (m + 2) -
preNormEDS b c d (m - 2) * preNormEDS b c d m * preNormEDS b c d (m + 1) ^ 2 := by
induction m using Int.negInduction with
| nat m =>
rcases m with _ | _ | _ | m
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 250 | 255 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | attribute is now added locally where it is needed. Using this as the `@[ext]` lemma instead of
`TensorProduct.ext'` allows `ext` to apply lemmas specific to `M →ₗ _` and `N →ₗ _`.
See note [partially-applied ext lemmas]. -/
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 538 | 541 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Analysis.Specia... | theorem comp_symm_equiv' {π : P →+* R} (m : PerfectionMap p π) :
π.comp ↑m.equiv.symm = Perfection.coeff R p 0 :=
RingHom.ext m.comp_symm_equiv
variable (p R P)
/-- Given rings `R` and `S` of characteristic `p`, with `R` being perfect,
| Mathlib/RingTheory/Perfection.lean | 270 | 276 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | (p.support.sort (· ≤ ·))
match termPrecAndReprs with
| [] => "0"
| [(tprec, t)] => if prec ≥ tprec then Lean.Format.paren t else t
| Mathlib/Algebra/Polynomial/Basic.lean | 1,194 | 1,197 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
/-!
# Finite intervals in a disjoint union
This file provides the... | Mathlib/Data/Sum/Interval.lean | 416 | 418 | |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | ContinuousWithinAt (fun x => √(f x)) s x :=
h.sqrt
| Mathlib/Data/Real/Sqrt.lean | 397 | 398 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... | lemma addNSMul_eq_right [Archimedean α] (hδ : 0 < δ) :
∃ m, ∀ n ≥ m, addNSMul h δ n = b := by
obtain ⟨m, hm⟩ := Archimedean.arch (b - a) hδ
refine ⟨m, fun n hn ↦ ?_⟩
rw [addNSMul, coe_projIcc, add_comm, min_eq_left_iff.mpr, max_eq_right h]
exact sub_le_iff_le_add.mp (hm.trans <| nsmul_le_nsmul_left hδ.le hn... | Mathlib/Topology/UnitInterval.lean | 273 | 285 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
/-!
# Pi
This file contains lemmas which establish bounds on `Real.pi`.
Notably, t... |
theorem pi_lt_d4 : π < 3.1416 := by
| Mathlib/Data/Real/Pi/Bounds.lean | 171 | 172 |
/-
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
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | ⟨fun h => ⟨h.left_of_union, h.right_of_union⟩, fun h => h.1.union h.2⟩
@[simp]
theorem integrableOn_singleton_iff {x : α} [MeasurableSingletonClass α] :
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 165 | 168 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | Mathlib/Data/List/Sublists.lean | 428 | 432 | |
/-
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
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Set.Accumulate
import Mathlib.Data.Set.Pairwise.Lattice
import Mathlib.MeasureTheory.PiSystem
/-!... | refine hC.empty_nmem_disjointOfDiff hs ht ?_
rwa [← h]
rw [← hC.sUnion_disjointOfDiff hs ht]
exact subset_sUnion_of_mem hs_mem
| Mathlib/MeasureTheory/SetSemiring.lean | 114 | 117 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
/-!
# Non integrable functions
In this f... | ¬ IntegrableOn (·⁻¹) (Ioi a) := by
simpa only [IntegrableOn, restrict_Ioi_eq_restrict_Ici] using not_IntegrableOn_Ici_inv
| Mathlib/Analysis/SpecialFunctions/NonIntegrable.lean | 216 | 224 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Applications of the Hausdorf... | have Rpos : 0 < R := (norm_nonneg _).trans_lt hR
have : ‖c‖ / R < 1 := by
rw [div_lt_iff₀ Rpos]
simpa using hR
rcases riesz_lemma hFc hF this with ⟨x, xF, hx⟩
have x0 : x ≠ 0 := fun H => by simp [H] at xF
obtain ⟨d, d0, dxlt, ledx, -⟩ :
∃ d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < R ∧ R / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ R... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 83 | 105 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Combinatorics.Young.YoungDiagram
/-!
# Semistandard Young tableaux
A semistandard Young tableau is a filling of a Young diagram by natural numbers, suc... | rw [if_pos hcell, if_pos (μ.up_left_mem (by rfl) (le_of_lt hj) hcell)]
col_strict' hi hcell := by
rwa [if_pos hcell, if_pos (μ.up_left_mem (le_of_lt hi) (by rfl) hcell)]
zeros' not_cell := if_neg not_cell
| Mathlib/Combinatorics/Young/SemistandardTableau.lean | 129 | 133 |
/-
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.Analysis.SpecialFunctions.Complex.Log
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
/-!
# Complex roots of unity
In this file we show that th... | · convert Complex.arg_cos_add_sin_mul_I _
· push_cast; rfl
· push_cast; rfl
field_simp [hn]
refine ⟨(neg_lt_neg Real.pi_pos).trans_le ?_, ?_⟩
· rw [neg_zero]
exact mul_nonneg (mul_nonneg i.cast_nonneg <| by simp [Real.pi_pos.le])
(by rw [inv_nonneg]; simp only [Nat.cast_nonneg])
... | Mathlib/RingTheory/RootsOfUnity/Complex.lean | 134 | 188 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
/-!
# The W construction as a multivariate polynomial functor.
W types are well-founded tree-like structu... | rw [h, comp_wPathCasesOn]
-- TODO: this technical theorem is used in one place in constructing the initial algebra.
-- Can it be avoided?
/-- Constructor of a value of `P.obj (α ::: β)` from components.
Useful to avoid complicated type annotation -/
abbrev objAppend1 {α : TypeVec n} {β : Type u} (a : P.A) (f' : P.dr... | Mathlib/Data/PFunctor/Multivariate/W.lean | 217 | 228 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | · rw [dual_indep_iff_exists]
exact ⟨B, hB, disjoint_of_subset_left inter_subset_left disjoint_sdiff_left⟩
simp only [diff_inter_self_eq_diff, mem_diff, not_and, not_not, imp_iff_right he.1.1] at he
simp_rw [dual_dep_iff_forall, insert_subset_iff, and_iff_right he.1.1,
and_iff_left (inter_subset_left.trans... | Mathlib/Data/Matroid/Dual.lean | 182 | 201 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | end Decidable
| Mathlib/Data/List/Rotate.lean | 618 | 619 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | theorem sdiff_sdiff_eq_sdiff_sup (h : z ≤ x) : x \ (y \ z) = x \ y ⊔ z := by
rw [sdiff_sdiff_right', inf_eq_right.2 h]
@[simp]
theorem sdiff_sdiff_right_self : x \ (x \ y) = x ⊓ y := by
rw [sdiff_sdiff_right, inf_idem, sdiff_self, bot_sup_eq]
theorem sdiff_sdiff_eq_self (h : y ≤ x) : x \ (x \ y) = y := by
rw [s... | Mathlib/Order/BooleanAlgebra.lean | 347 | 368 |
/-
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.Analysis.MellinTransform
/-!
# Abstract functional equations for Mellin transforms
This file formalises a general version of an argument used to pro... | P.toStrongFEPair.functional_equation s
/-- Functional equation formulated for `Λ`. -/
theorem functional_equation (s : ℂ) :
P.Λ (P.k - s) = P.ε • P.symm.Λ s := by
linear_combination (norm := module) P.functional_equation₀ s - P.Λ₀_eq (P.k - s)
+ congr(P.ε • $(P.symm_Λ₀_eq s)) + congr(($(mul_inv_cancel₀ P.h... | Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean | 435 | 449 |
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | /-! ### Differentiability of linear isometry equivs, and invariance of differentiability -/
| Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 237 | 239 |
/-
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_zero_var (m : ℝ) : gaussianPDFReal m 0 = 0 := by
ext1 x
simp [gaussianPDFReal]
| Mathlib/Probability/Distributions/Gaussian.lean | 51 | 54 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,185 | 1,188 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | simpa using hF (Multiplicative.ofAdd 1)
/-- If two lifts of circle homeomorphisms have the same translation number, then they are
semiconjugate by a `CircleDeg1Lift`. This version uses assumptions `IsUnit f₁` and `IsUnit f₂`
| Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 874 | 877 |
/-
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.Algebra.Category.ModuleCat.Presheaf.ChangeOfRings
/-!
# Pushforward of presheaves of modules
If `F : C ⥤ D`, the precomposition `F.op ⋙ _` induces a functor fr... | lemma pushforward_obj_map_apply (M : PresheafOfModules.{v} R) {X Y : Cᵒᵖ} (f : X ⟶ Y)
(m : (ModuleCat.restrictScalars (φ.app X).hom).obj (M.obj (Opposite.op (F.obj X.unop)))) :
(((pushforward φ).obj M).map f).hom m = M.map (F.map f.unop).op m := rfl
/-- `@[simp]`-normal form of `pushforward_obj_map_apply`. -... | Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean | 78 | 82 |
/-
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.MeasureTheory.Constructions.Cylinders
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Real
import Mathlib.MeasureTheory.Measurable... | refine measurableSet_generateFrom ⟨{1}, measurableSet_singleton 1, ?_⟩
ext x
simp [Set.indicator_const_preimage_eq_union]
· rintro t ⟨n, ht⟩
suffices MeasurableSpace.generateFrom {t | n ≤ i ∧
MeasurableSet[MeasurableSpace.comap ((s n).indicator (fun _ => 1 : Ω → β)) mβ] t} ≤
MeasurableSp... | Mathlib/Probability/Process/Filtration.lean | 281 | 305 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | theorem continuous_sqrt : Continuous sqrt := sqrt.continuous
| Mathlib/Data/Real/Sqrt.lean | 91 | 91 |
/-
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.... | end Monoid
| Mathlib/Algebra/Group/Basic.lean | 196 | 197 |
/-
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.Data.NNReal.Basic
import Mathlib.Order.Fin.Tuple
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.... | ⟨I.lower ∘ Fin.succAbove i, I.upper ∘ Fin.succAbove i, fun _ ↦ I.lower_lt_upper _⟩
@[simp]
theorem face_mk {n} (l u : Fin (n + 1) → ℝ) (h : ∀ i, l i < u i) (i : Fin (n + 1)) :
face ⟨l, u, h⟩ i = ⟨l ∘ Fin.succAbove i, u ∘ Fin.succAbove i, fun _ ↦ h _⟩ := rfl
@[gcongr, mono]
theorem face_mono {n} {I J : Box (Fin ... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 354 | 363 |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.DenselyOrdered
import Mathlib.Data.Real.Archimedean
import Mathlib.Order.LiminfLimsu... | have h := IsCoboundedUnder.of_frequently_ge (a := 0)
<| (h₁.and_eventually h₃).mono fun x ⟨ux_0, vx_0⟩ ↦ mul_nonneg ux_0 vx_0
have h' := isBoundedUnder_le_mul_of_nonneg h₁ h₂ h₃ h₄
have u0 : 0 ≤ limsup u f := le_limsup_of_frequently_le h₁ h₂
| Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 476 | 479 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | (s : Finset ι) (h : s.prod f ≠ 0) : (s.prod f).rootSet S = ⋃ i ∈ s, (f i).rootSet S := by
classical
simp only [rootSet, aroots, ← Finset.mem_coe]
rw [Polynomial.map_prod, roots_prod, Finset.bind_toFinset, s.val_toFinset, Finset.coe_biUnion]
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 487 | 490 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... |
theorem log_neg_one : log (-1) = π * I := by simp [log]
theorem log_I : log I = π / 2 * I := by simp [log]
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 97 | 101 |
/-
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.Sites.OneHypercover
/-!
# Characterization of sheaves using 1-hypercovers
In this file, given a Grothendieck topology `J` on a category `C`,
we ... | end GrothendieckTopology
namespace Presheaf
lemma isSheaf_iff_of_isGeneratedByOneHypercovers
| Mathlib/CategoryTheory/Sites/IsSheafOneHypercover.lean | 156 | 160 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,561 | 1,562 | |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | and_congr_left_iff]
intro h
| Mathlib/Probability/Process/Stopping.lean | 792 | 793 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
/-!
# Exact sequences with free modules
This file proves resu... | ι → ι ⊕ ι' ← ι'
```
where the top row is an exact sequence of modules and the maps on the bottom are `Sum.inl` and
`Sum.inr`. If `u` is injective and `v` and `w` are linearly independent, then `u` is linearly
independent. -/
theorem linearIndependent_leftExact : LinearIndependent R u := by
rw [linearIndepe... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 62 | 68 |
/-
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... | refine (convexJoin_assoc_aux _ _ _).antisymm ?_
simp_rw [convexJoin_comm s, convexJoin_comm _ u]
exact convexJoin_assoc_aux _ _ _
theorem convexJoin_left_comm (s t u : Set E) :
convexJoin 𝕜 s (convexJoin 𝕜 t u) = convexJoin 𝕜 t (convexJoin 𝕜 s u) := by
simp_rw [← convexJoin_assoc, convexJoin_comm]
the... | Mathlib/Analysis/Convex/Join.lean | 132 | 150 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
i... |
variable [HasZeroObject C]
| Mathlib/CategoryTheory/Simple.lean | 110 | 111 |
/-
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.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | lemma normEDS_zero : normEDS b c d 0 = 0 := by
rw [← Nat.cast_zero, normEDS_ofNat, preNormEDS'_zero, zero_mul]
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 305 | 306 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Data.Set.Function
/-!
# Comparing sums and integrals
## Summary
It is often the case ... | ∫ x in a..b, g x ≤ ∑ i ∈ Finset.Ico a b, f i := by
convert neg_le_neg (sum_Ico_le_integral_of_le (f := -f) (g := -g) hab
(fun i hi x hx ↦ neg_le_neg (h i hi x hx)) hg.neg) <;> simp
theorem AntitoneOn.integral_le_sum (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, ... | Mathlib/Analysis/SumIntegralComparisons.lean | 73 | 95 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | · exact fun I hI ↦ ⟨hI.1, hI.2.trans inter_subset_left⟩
exact fun I hI ↦ ⟨I, rfl.le, hI.1, subset_inter hI.2 hI.1.subset_ground⟩
theorem isBasis'_iff_isBasis (hX : X ⊆ M.E := by aesop_mat) : M.IsBasis' I X ↔ M.IsBasis I X := by
rw [isBasis'_iff_isBasis_inter_ground, inter_eq_self_of_subset_left hX]
| Mathlib/Data/Matroid/Basic.lean | 874 | 878 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | nth_rewrite 1 [← H]
rw [Quotient.mk_add]
congr 1
· simp [Algebra.algebraMap_eq_smul_one ((pb.basis.repr b) _)]
· rw [Quotient.mk_zero, Quotient.mk_eq_zero, coe_basis]
| Mathlib/RingTheory/PowerBasis.lean | 139 | 143 |
/-
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.liftCycles k hk ≫ S.leftHomologyπ
@[reassoc]
lemma liftCycles_leftHomologyπ_eq_zero_of_boundary (x : A ⟶ S.X₁) (hx : k = x ≫ S.f) :
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 963 | 966 |
/-
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.Data.List.Basic
/-!
# Double universal quantification on a list
This file provides an API for `List.Forall₂` (definition in `Data.Lis... | theorem rel_filter {p : α → Bool} {q : β → Bool}
(hpq : (R ⇒ (· ↔ ·)) (fun x => p x) (fun x => q x)) :
(Forall₂ R ⇒ Forall₂ R) (filter p) (filter q)
| _, _, Forall₂.nil => Forall₂.nil
| a :: as, b :: bs, Forall₂.cons h₁ h₂ => by
| Mathlib/Data/List/Forall2.lean | 245 | 249 |
/-
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.Group.Action.Pointwise.Finset
import Mathlib.Algebra.Ring.Nat
/-!
# e-transforms
e-transforms are a family of transformations of pairs of finite ... | Mathlib/Combinatorics/Additive/ETransform.lean | 189 | 190 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 949 | 950 | |
/-
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.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... |
theorem subtype_val {p : α → Prop} [DecidablePred p] {hp : PrimrecPred p} :
haveI := Primcodable.subtype hp
Primrec (@Subtype.val α p) := by
letI := Primcodable.subtype hp
| Mathlib/Computability/Primrec.lean | 1,137 | 1,141 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
# `C^n` monoid
A `C^n` monoid is a monoid that is also a `C^n` m... | variable (I) (g h : G)
variable [ContMDiffMul I ∞ G]
/-- Left multiplication by `g`. It is meant to mimic the usual notation in Lie groups.
| Mathlib/Geometry/Manifold/Algebra/Monoid.lean | 201 | 204 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | rw [← sub_eq_neg_add, ← neg_sub, hxy, ← sub_eq_neg_add, hzx, smul_neg, smul_comm, neg_add_cancel]
open AffineMap
/-- If `z = lineMap x y c` is a point on the line passing through `x` and `y`, then the open
segment `openSegment 𝕜 x y` is included in the union of the open segments `openSegment 𝕜 x z`,
`openSegment ... | Mathlib/Analysis/Convex/Segment.lean | 372 | 381 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... |
variable (K V) in
theorem cardinal_lt_aleph0_of_finiteDimensional [Finite K] [Module.Free K V] [Module.Finite K V] :
#V < ℵ₀ := by
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 241 | 244 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | @[simps]
def cokernelEpiComp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Epi f] [HasCokernel g] :
cokernel (f ≫ g) ≅ cokernel g where
hom := cokernel.desc _ (cokernel.π g) (by simp)
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 916 | 919 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,622 | 1,627 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | rw [← Equiv.symm_trans_swap_trans, mul_def, Equiv.symm_trans_swap_trans, mul_def]
repeat (rw [trans_apply])
simp [swap, swapCore]
split_ifs <;> rfl
have hefx : e x ≠ e (f x) := mt e.injective.eq_iff.1 hfx
rw [if_neg hfx, ← signAux_eq_signAux2 _ _ e hy, this, signAux_mul, sign... | Mathlib/GroupTheory/Perm/Sign.lean | 319 | 344 |
/-
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.RingTheory.WittVector.Truncated
import Mathlib.RingTheory.WittVector.Identities
import Mathlib.NumberTheory.Padics.RingHoms
/-!
# Co... | /-- The ring of Witt vectors over `ZMod p` is isomorphic to the ring of `p`-adic integers. This
equivalence is witnessed by `WittVector.toPadicInt` with inverse `WittVector.fromPadicInt`.
-/
| Mathlib/RingTheory/WittVector/Compare.lean | 192 | 194 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | /-- A polynomial `p` that has as many roots as its degree
can be written `p = p.leadingCoeff * ∏(X - a)`, for `a` in `p.roots`. -/
theorem C_leadingCoeff_mul_prod_multiset_X_sub_C (hroots : Multiset.card p.roots = p.natDegree) :
C p.leadingCoeff * (p.roots.map fun a => X - C a).prod = p :=
(eq_leadingCoeff_mul_of... | Mathlib/Algebra/Polynomial/Roots.lean | 683 | 698 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | (f₁ : A ≃ₐ[S] C) (f₂ : C ≃ₐ[S] E) (g₁ : B ≃ₐ[R] D) (g₂ : D ≃ₐ[R] F) :
congr (f₁.trans f₂) (g₁.trans g₂) = (congr f₁ g₁).trans (congr f₂ g₂) :=
AlgEquiv.coe_algHom_injective <| map_comp f₂.toAlgHom f₁.toAlgHom g₂.toAlgHom g₁.toAlgHom
| Mathlib/RingTheory/TensorProduct/Basic.lean | 1,082 | 1,085 |
/-
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.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | Mathlib/Algebra/GCDMonoid/Finset.lean | 286 | 292 | |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | apply h2.mul_left_cancel
rw [two_mul, two_mul, h x x, QuadraticMap.polar_self, two_smul, map_add]
/-- The symmetric product of vectors is a scalar -/
theorem ι_mul_ι_add_swap (a b : M) :
ι Q a * ι Q b + ι Q b * ι Q a = algebraMap R _ (QuadraticMap.polar Q a b) :=
mul_add_swap_eq_polar_of_forall_mul_self_eq _... | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 233 | 242 |
/-
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.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 232 | 232 | |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Galois.Basic
/-!
# Galois Groups of Polynomials
In this file, we introduce the Galois group of a polynomial... | variable (p E)
/-- `Polynomial.Gal.galAction` as a permutation representation -/
def galActionHom [Fact (p.Splits (algebraMap F E))] : p.Gal →* Equiv.Perm (rootSet p E) :=
MulAction.toPermHom _ _
theorem galActionHom_restrict [Fact (p.Splits (algebraMap F E))] (ϕ : E ≃ₐ[F] E) (x : rootSet p E) :
| Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 197 | 203 |
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Kernel.Composition.Comp
/-!
# Invariance of measures along a kernel
We say that a measure `μ` is invariant with respect to a kernel `κ` if its ... | Mathlib/Probability/Kernel/Invariance.lean | 87 | 92 | |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
/-!
# Euclidean algorithm for ℕ
This file sets up a version of the Euclidean algorithm that only works w... | fun h => by
have : SizeOf.sizeOf u.step < SizeOf.sizeOf u := u.step_wf h
rw [reduce_b h, flip_isReduced]
apply reduce_isReduced
| Mathlib/Data/PNat/Xgcd.lean | 315 | 318 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | @[simp]
theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by
unfold stdBasisMatrix
ext
| Mathlib/Data/Matrix/Basis.lean | 51 | 54 |
/-
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 | 219 | 219 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Associator
import Mathlib.CategoryTheory.Monoi... | apply TensorProduct.ext_threefold
intro x y z
-- Porting note (https://github.com/leanprover-community/mathlib4/pull/10934): used to be dsimp [tensorHom, associator]
change x ⊗ₜ[R] ((leftUnitor N).hom) (y ⊗ₜ[R] z) = ((rightUnitor M).hom) (x ⊗ₜ[R] y) ⊗ₜ[R] z
erw [TensorProduct.lid_tmul, TensorProduct.rid_tmul]... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 150 | 154 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathli... | theorem isInteger_of_isUnit_den {x : K} (h : IsUnit (den A x : A)) : IsInteger A x := by
obtain ⟨d, hd⟩ := h
have d_ne_zero : algebraMap A K (den A x) ≠ 0 :=
IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A x).2
use ↑d⁻¹ * num A x
refine _root_.trans ?_ (mk'_num_den A x)
rw [map_mul, map_units_... | Mathlib/RingTheory/Localization/NumDen.lean | 97 | 105 |
/-
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... | "A version of `Pi.continuousNeg` for non-dependent functions. It is needed
because sometimes Lean fails to use `Pi.continuousNeg` for non-dependent functions."]
instance Pi.has_continuous_inv' : ContinuousInv (ι → G) :=
Pi.continuousInv
| Mathlib/Topology/Algebra/Group/Basic.lean | 208 | 211 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... |
set_option linter.deprecated false in
| Mathlib/Order/Interval/Finset/Fin.lean | 104 | 105 |
/-
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 | 1,631 | 1,632 | |
/-
Copyright (c) 2022 Matej Penciak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matej Penciak, Moritz Doll, Fabien Clery
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# The Symplectic Group
This file defines the symplectic group and proves element... |
variable [Fintype l]
theorem J_squared : J l R * J l R = -1 := by
| Mathlib/LinearAlgebra/SymplecticGroup.lean | 43 | 46 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... |
end Unions
variable (m)
/-- Given an arbitrary function on a subset of sets, we can define the outer measure corresponding
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 150 | 155 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertIdx
/-! # Some lemmas about lists involving sets
Split out from `Data.List.Basic` to... |
@[deprecated (since := "2024-12-10")] alias tail_reverse_eq_reverse_dropLast := tail_reverse
theorem injOn_insertIdx_index_of_not_mem (l : List α) (x : α) (hx : x ∉ l) :
Set.InjOn (fun k => l.insertIdx k x) { n | n ≤ l.length } := by
induction' l with hd tl IH
· intro n hn m hm _
simp_all [Set.mem_singlet... | Mathlib/Data/List/Lemmas.lean | 23 | 41 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | ‖b i‖ = 1 := b.orthonormal.norm_eq_one i
@[simp]
lemma nnnorm_eq_one (b : OrthonormalBasis ι 𝕜 E) (i : ι) :
‖b i‖₊ = 1 := b.orthonormal.nnnorm_eq_one i
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 386 | 390 |
/-
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... | exact degree_mul_le_of_le hn hp
@[simp]
theorem leadingCoeff_monomial (a : R) (n : ℕ) : leadingCoeff (monomial n a) = a := by
classical
by_cases ha : a = 0
· simp only [ha, (monomial n).map_zero, leadingCoeff_zero]
· rw [leadingCoeff, natDegree_monomial, if_neg ha, coeff_monomial]
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 418 | 425 |
/-
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 | 942 | 946 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... |
theorem EuclideanSpace.inner_single_right (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι) :
⟪v, EuclideanSpace.single i (a : 𝕜)⟫ = a * conj (v i) := by simp [apply_ite conj]
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 271 | 273 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Multiset.Bind
import Mathlib.Order.SetNotation
/-!
# Unions of finite sets
This file defines the union of a fami... | @[simp]
lemma disjiUnion_eq_biUnion (s : Finset α) (f : α → Finset β) (hf) :
s.disjiUnion f hf = s.biUnion f := eq_of_veq (s.disjiUnion f hf).nodup.dedup.symm
| Mathlib/Data/Finset/Union.lean | 158 | 160 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... |
end
section NeedsGroup
namespace Submodule
variable [CommRing R] [AddCommGroup M] [Module R M]
variable {ι : Type*} [DecidableEq ι] {S : Finset ι}
/-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
| Mathlib/Algebra/Module/Torsion.lean | 452 | 462 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... |
theorem KaehlerDifferential.tensorProductTo_surjective :
Function.Surjective (KaehlerDifferential.D R S).tensorProductTo := by
intro x; obtain ⟨x, rfl⟩ := (KaehlerDifferential.ideal R S).toCotangent_surjective x
exact ⟨x, KaehlerDifferential.D_tensorProductTo x⟩
/-- The `S`-linear maps from `Ω[S⁄R]` to `M` ar... | Mathlib/RingTheory/Kaehler/Basic.lean | 316 | 327 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... |
theorem eq_of_uniformity {α : Type*} [UniformSpace α] [T0Space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y :=
| Mathlib/Topology/UniformSpace/Separation.lean | 150 | 152 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... |
theorem dvd_mod_iff {a b c : R} (h : c ∣ b) : c ∣ a % b ↔ c ∣ a := by
| Mathlib/Algebra/EuclideanDomain/Basic.lean | 63 | 64 |
/-
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.Shapes.BinaryBiproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Preservation of biproducts
We define the... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 445 | 447 | |
/-
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.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | @[reassoc (attr := simp)]
lemma homologyπ_naturality :
K.homologyπ i ≫ homologyMap φ i = cyclesMap φ i ≫ L.homologyπ i :=
ShortComplex.homologyπ_naturality _
@[reassoc (attr := simp)]
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 382 | 387 |
/-
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.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... | theorem DirectedOn.strictConvex_sUnion {S : Set (Set E)} (hdir : DirectedOn (· ⊆ ·) S)
(hS : ∀ s ∈ S, StrictConvex 𝕜 s) : StrictConvex 𝕜 (⋃₀ S) := by
rw [sUnion_eq_iUnion]
exact (directedOn_iff_directed.1 hdir).strictConvex_iUnion fun s => hS _ s.2
end SMul
section Module
| Mathlib/Analysis/Convex/Strict.lean | 85 | 92 |
/-
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.Data.Complex.ExponentialBounds
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.SpecialF... | lemma eulerMascheroniSeq_zero : eulerMascheroniSeq 0 = 0 := by
simp [eulerMascheroniSeq, harmonic_zero]
| Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean | 47 | 48 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | theorem le_essInf_of_ae_le {f : α → β} (c : β) (hf : (fun _ => c) ≤ᵐ[μ] f) : c ≤ essInf f μ :=
@essSup_le_of_ae_le α βᵒᵈ _ _ _ _ c hf
theorem essSup_const_bot : essSup (fun _ : α => (⊥ : β)) μ = (⊥ : β) :=
| Mathlib/MeasureTheory/Function/EssSup.lean | 178 | 181 |
/-
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.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
/-!
# Volume forms and measures on inner product spa... | F ≃ᵐ EuclideanSpace ℝ ι := b.repr.toHomeomorph.toMeasurableEquiv
/-- The measurable equivalence defined by an orthonormal basis is volume preserving. -/
theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ι ℝ F) :
MeasurePreserving b.measurableEquiv volume volume := by
convert (b... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 102 | 108 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | is `# m * # n`. -/
theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = #m * #n := by simp
open Fintype
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 241 | 246 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | /--
The $S_n^m$ theorem: There is a computable function, namely `Nat.Partrec.Code.curry`, that takes a
program and a ℕ `n`, and returns a new program using `n` as the first argument.
-/
theorem smn :
| Mathlib/Computability/PartrecCode.lean | 514 | 518 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | P (f x - x) :=
f.map_fract_sub_fract_eq x ▸ h _ ⟨fract_nonneg _, le_of_lt (fract_lt_one _)⟩
theorem translationNumber_lt_of_forall_lt_add (hf : Continuous f) {z : ℝ} (hz : ∀ x, f x < x + z) :
τ f < z := by
obtain ⟨x, -, hx⟩ : ∃ x ∈ Icc (0 : ℝ) 1, ∀ y ∈ Icc (0 : ℝ) 1, f y - y ≤ f x - x :=
isCompact_Icc.... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 773 | 783 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | @[to_additive sum_antidiagonal_choose_succ_nsmul]
theorem prod_antidiagonal_pow_choose_succ {M : Type*} [CommMonoid M] (f : ℕ → ℕ → M) (n : ℕ) :
(∏ ij ∈ antidiagonal (n + 1), f ij.1 ij.2 ^ (n + 1).choose ij.1) =
(∏ ij ∈ antidiagonal n, f ij.1 (ij.2 + 1) ^ n.choose ij.1) *
∏ ij ∈ antidiagonal n, f (ij.... | Mathlib/Data/Nat/Choose/Sum.lean | 203 | 213 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 892 | 897 | |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Floris van Doorn
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# `norm_num` plugin for ... |
lemma Finset.range_succ' {n nn n' : ℕ} (pn : NormNum.IsNat n nn) (pn' : nn = Nat.succ n') :
| Mathlib/Tactic/NormNum/BigOperators.lean | 247 | 248 |
/-
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... |
@[simp]
| Mathlib/ModelTheory/Substructures.lean | 874 | 875 |
/-
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.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | simp
@[reassoc]
lemma d_fst_v (i j k : ℤ) (hij : i + 1 = j) (hjk : j + 1 = k) :
(mappingCone φ).d i j ≫ (fst φ).1.v j k hjk =
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 247 | 251 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
/-!
# Grönwall's inequality
The main technical result of this file is the Grönwall-like inequality
`norm_le_gron... | rw [← dist_eq_norm, ← dist_eq_norm]
refine this.trans ((add_le_add (add_le_add (f_bound t ht) (g_bound t ht)) hv).trans ?_)
rw [add_comm]
/-- If `f` and `g` are two approximate solutions of the same ODE, then the distance between them
can't grow faster than exponentially. This is a simple corollary of Grönwall's... | Mathlib/Analysis/ODE/Gronwall.lean | 156 | 176 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | truncatedInf (s ∪ t) a = truncatedInf s a ⊓ truncatedInf t a := by
simpa only [truncatedInf_of_mem, hs, ht, upper_aux.2 (Or.inl hs), filter_union] using
inf'_union _ _ _
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 228 | 230 |
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