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) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | lemma IsFundamentalDomain.projection_respects_measure {ν : Measure α}
(μ : Measure (Quotient α_mod_G)) [i : QuotientMeasureEqMeasurePreimage ν μ] {t : Set α}
(fund_dom_t : IsFundamentalDomain G t ν) : μ = (ν.restrict t).map π :=
i.projection_respects_measure' t fund_dom_t
@[to_additive addProjection_respects... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 759 | 764 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... |
end List
| Mathlib/Data/List/Range.lean | 87 | 90 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... | apply ofFunction_caratheodory; intro t
rcases t.eq_empty_or_nonempty with h | h
· simp [h, Set.not_nonempty_empty]
· convert le_trans _ (hs t)
· simp [h]
| Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 210 | 214 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... | variable {cs}
variable {t : W} (ht : cs.IsReflection t)
include ht
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 68 | 70 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | IsAscendingCentralSeries (upperCentralSeries G) :=
⟨rfl, fun _x _n h => h⟩
theorem upperCentralSeries_mono : Monotone (upperCentralSeries G) := by
refine monotone_nat_of_le_succ ?_
intro n x hx y
rw [mul_assoc, mul_assoc, ← mul_assoc y x⁻¹ y⁻¹]
exact mul_mem hx (Normal.conj_mem (upperCentralSeries_normal... | Mathlib/GroupTheory/Nilpotent.lean | 219 | 227 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Math... | have hφx := Path.obj_eq_of_cons_eq_cons h'
have hφp := Path.heq_of_cons_eq_cons h'
have hφe := HEq.trans (Hom.cast_heq rfl hφy _).symm (Path.hom_heq_of_cons_eq_cons h')
have h_path_star : φ.pathStar u ⟨x₁, p₁⟩ = φ.pathStar u ⟨x₂, p₂⟩ := by
simp only [Prefunctor.pathStar_apply, Sigma.mk.i... | Mathlib/Combinatorics/Quiver/Covering.lean | 211 | 243 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | · have h : ∀ᵐ x ∂μ, x ∈ s → ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal := by
filter_upwards with a ha using by simp [ha]
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 785 | 786 |
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | · exact fun z hz => ⟨y, yt, dist_le_diam_of_mem (bs.union bt) (subset_union_left hz)
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 707 | 707 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | rw [polyCharpoly, polyCharpolyAux, Polynomial.coeff_map] at H ⊢
set B := (Module.Free.chooseBasis R M).end
set g := toMvPolynomial b' b LinearMap.id
apply_fun (MvPolynomial.bind₁ g) at H
have : toMvPolynomial b' B φ = fun i ↦ (MvPolynomial.bind₁ g) (toMvPolynomial b B φ i) :=
funext <| toMvPolynomial_comp... | Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 410 | 419 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | to_encard_tac; rw [hs.cast_ncard_eq, hs.diff.cast_ncard_eq,
encard_diff_singleton_add_one h]
@[simp] theorem ncard_diff_singleton_of_mem {a : α} (h : a ∈ s) (hs : s.Finite := by toFinite_tac) :
(s \ {a}).ncard = s.ncard - 1 :=
eq_tsub_of_add_eq (ncard_diff_singleton_add_one h hs)
theorem ncard_diff_single... | Mathlib/Data/Set/Card.lean | 628 | 635 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | split_ifs with h
· apply lt_trans (ih _) hp
· calc
_ < p ^ n + p ^ n * (p - 1) := ?_
_ = p ^ (n + 1) := ?_
· apply add_lt_add_of_lt_of_le (ih _)
apply Nat.mul_le_mul_left
apply le_pred_of_lt
apply ZMod.val_lt
· rw [mul_tsub, mul_one, ← _root_.pow_succ]
... | Mathlib/NumberTheory/Padics/RingHoms.lean | 310 | 326 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | rw [terminatedAt_iff_s_none, partNum_none_iff_s_none]
| Mathlib/Algebra/ContinuedFractions/Translations.lean | 41 | 42 |
/-
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.Convex.StrictConvexBetween
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.Normed.Affine.Convex
import Mathlib.Geometry.E... | /-- A single point is cospherical. -/
theorem cospherical_singleton (p : P) : Cospherical ({p} : Set P) := by
use p
simp
| Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 175 | 178 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
/-!
# Subsemigroups: membership criteria
In this file we prove various facts about membership in a subsemigroup.
The i... | theorem coe_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) :
(↑(sSup S) : Set M) = ⋃ s ∈ S, ↑s :=
Set.ext fun x => by simp [mem_sSup_of_directed_on hS]
| Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 67 | 70 |
/-
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.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 148 | 150 | |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite.Range
import Mathlib.Order.Partition.Finpartition
/-!
# Equivalenc... | rintro y ⟨⟨_, rfl⟩, ha⟩
ext x
| Mathlib/Data/Setoid/Partition.lean | 97 | 98 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | AEMeasurable (fun x => derivWithin f (Ioi x) x) μ :=
(measurable_derivWithin_Ioi f).aemeasurable
theorem aestronglyMeasurable_derivWithin_Ioi (μ : Measure ℝ) :
AEStronglyMeasurable (fun x => derivWithin f (Ioi x) x) μ :=
(stronglyMeasurable_derivWithin_Ioi f).aestronglyMeasurable
end RightDeriv
section W... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 765 | 783 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... | @[fun_prop]
theorem DifferentiableWithinAt.sum (h : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x) :
DifferentiableWithinAt 𝕜 (fun y => ∑ i ∈ u, A i y) s x :=
HasFDerivWithinAt.differentiableWithinAt <|
HasFDerivWithinAt.sum fun i hi => (h i hi).hasFDerivWithinAt
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 346 | 350 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | @[norm_cast]
theorem Int.natCast_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b :=
| Mathlib/RingTheory/Multiplicity.lean | 219 | 220 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | theorem le_rpow_add {x : ℝ} (hx : 0 ≤ x) (y z : ℝ) : x ^ y * x ^ z ≤ x ^ (y + z) := by
rcases le_iff_eq_or_lt.1 hx with (H | pos)
· by_cases h : y + z = 0
· simp only [H.symm, h, rpow_zero]
calc
(0 : ℝ) ^ y * 0 ^ z ≤ 1 * 1 :=
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 205 | 210 |
/-
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... | -- Split into `NonUnitalNonAssocSemiring`, `One` and `natCast` instances.
cases inst₁; cases inst₂
congr
theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by
intro _ _ _
ext <;> congr
end NonAssocSemiring
/-! ### NonUnitalNonAssocRing -/
namespace No... | Mathlib/Algebra/Ring/Ext.lean | 149 | 169 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | theorem div_mul_le_div_mul_of_div_le_div (h : a / b ≤ c / d) (he : 0 ≤ e) :
a / (b * e) ≤ c / (d * e) := by
| Mathlib/Algebra/Order/Field/Basic.lean | 191 | 192 |
/-
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... | apply extendFrom_eq
· rw [closure_Ioo hab.ne]
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc]
· simpa [hab]
theorem eq_lim_at_right_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b)
| Mathlib/Topology/Order/ExtendFrom.lean | 32 | 38 |
/-
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.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
/-!
# Factoring through subobjects
The predicate `h : P.Fact... | Classical.choose_spec ((factors_iff _ _).mp h)
@[simp]
theorem factorThru_self {X : C} (P : Subobject X) (h) : P.factorThru P.arrow h = 𝟙 (P : C) := by
| Mathlib/CategoryTheory/Subobject/FactorThru.lean | 107 | 110 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | elements. -/
@[to_additive
"An element is in the closure of a two-element set if it is a linear combination of
| Mathlib/Algebra/Group/Submonoid/Membership.lean | 490 | 492 |
/-
Copyright (c) 2022 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.Finite.Sum
import Mathlib.Data.Fintype.Order
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.Quotients
import... | theorem equiv_lift_of {i : ι} (x : G i) :
equiv_lift L ι G f G' f' g H_commuting (of L ι G f i x) = of L ι G' f' i (g i x) := rfl
| Mathlib/ModelTheory/DirectLimit.lean | 399 | 401 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | Mathlib/MeasureTheory/Group/Prod.lean | 501 | 511 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | theorem rename_isHomogeneous_iff {f : σ → τ} (hf : f.Injective) :
(rename f φ).IsHomogeneous n ↔ φ.IsHomogeneous n := by
refine ⟨fun h d hd ↦ ?_, rename_isHomogeneous⟩
convert ← @h (d.mapDomain f) _
· simp only [weight_apply, Pi.one_apply, smul_eq_mul, mul_one]
exact Finsupp.sum_mapDomain_index_inj (h := ... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 271 | 278 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Topology.Neighborhoods
/-!
# Neighborhoods of a set
In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all ne... | subset_compl_iff_disjoint_left]
| Mathlib/Topology/NhdsSet.lean | 52 | 53 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | foldrm f x xs = List.foldrM f x (toList xs) := by
change _ = foldrM.ofFreeMonoid f (FreeMonoid.ofList <| toList xs) x
simp only [foldrm, toList_spec, foldMap_hom_free (foldrM.ofFreeMonoid f),
foldrm.ofFreeMonoid_comp_of, foldrM.get, FreeMonoid.ofList_toList]
| Mathlib/Control/Fold.lean | 363 | 367 |
/-
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.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
/-!
# Study of face maps for the Dold-Kan correspondence
In this file, we obtain the technical... | · simp only [Fin.val_zero, Fin.val_succ, Fin.coe_castSucc, zero_add, pow_zero, one_smul,
pow_one, neg_smul, comp_neg, ← Fin.castSucc_zero (n := n + 1), δ_comp_σ_self, δ_comp_σ_succ,
add_neg_cancel]
· intro j
rw [comp_zsmul, comp_zsmul, δ_comp_σ_of_gt', v.comp_δ_eq_zero_assoc, zero_comp, zs... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 143 | 168 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara, Rida Hamadani
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.ENat.Lattice
/-!
# Graph metric
This module defines the `SimpleGraph.edi... | | inl hvw => simp [hvw]
| inr hvw =>
obtain ⟨p, hp⟩ := exists_walk_of_edist_ne_top huv
obtain ⟨q, hq⟩ := exists_walk_of_edist_ne_top hvw
| Mathlib/Combinatorics/SimpleGraph/Metric.lean | 99 | 102 |
/-
Copyright (c) 2020 Anne Baanen. 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.Group.Hom.Instances
import Mathlib.Algebra.Modul... |
instance (priority := 100) instAddMonoidHomClass [FunLike F M M₃] [SemilinearMapClass F σ M M₃] :
AddMonoidHomClass F M M₃ :=
{ SemilinearMapClass.toAddHomClass with
map_zero := fun f ↦
| Mathlib/Algebra/Module/LinearMap/Defs.lean | 135 | 139 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x := by
have := (HasStrictFDerivAt.mul' hc hd).hasStrictDerivAt
rwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply,
ContinuousLinearMap.smulRight_apply, C... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 206 | 212 |
/-
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.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... |
theorem derivative_C_mul_X_sq (a : R) : derivative (C a * X ^ 2) = C (a * 2) * X := by
rw [derivative_C_mul_X_pow, Nat.cast_two, pow_one]
| Mathlib/Algebra/Polynomial/Derivative.lean | 97 | 99 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,660 | 2,661 | |
/-
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 | 420 | 422 | |
/-
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... | add_comm, sin_angle_add_of_inner_eq_zero h h0]
/-- The tangent of an angle in a right-angled triangle as a ratio of sides. -/
theorem tan_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
Real.tan (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₃ p₂ := by
rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 398 | 404 |
/-
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.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | { Equiv.piCongrRight fun _a => e.toEquiv with
uniformContinuous_toFun := UniformOnFun.postcomp_uniformContinuous e.uniformContinuous
uniformContinuous_invFun := UniformOnFun.postcomp_uniformContinuous e.symm.uniformContinuous }
/-- Let `f : γ → α`, `𝔖 : Set (Set α)`, `𝔗 : Set (Set γ)`, and assume that `∀ T... | Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 888 | 895 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | _ ≤ ∑' n, μ (Td n) := measure_iUnion_le _
_ = ⨆ I : Finset ℕ, ∑ n ∈ I, μ (Td n) := ENNReal.tsum_eq_iSup_sum
_ ≤ ⨆ n, μ (t n) := iSup_le fun I => by
rcases hd.finset_le I with ⟨N, hN⟩
calc
(∑ n ∈ I, μ (Td n)) = μ (⋃ n ∈ I, Td n) :=
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 460 | 465 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Yoneda
/-!
# Ind- and ... | of `F` evaluated at `D`. This variant is for contravariant diagrams, see
`colimitCoyonedaHomIsoLimit'` for a covariant version. -/
noncomputable def colimitCoyonedaHomIsoLimit :
(colimit (D.rightOp ⋙ coyoneda) ⟶ F) ≅ limit (D ⋙ F ⋙ uliftFunctor.{u₁}) :=
colimitHomIsoLimitYoneda _ F ≪≫
HasLimit.isoOfNatIso (is... | Mathlib/CategoryTheory/Limits/IndYoneda.lean | 140 | 146 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... | theorem coeff_smul [Zero R] (x : HahnSeries Γ R) (y : HahnModule Γ' R V) (a : Γ') :
((of R).symm <| x • y).coeff a =
∑ ij ∈ VAddAntidiagonal x.isPWO_support ((of R).symm y).isPWO_support a,
x.coeff ij.fst • ((of R).symm y).coeff ij.snd :=
rfl
@[deprecated (since := "2025-01-31")] alias smul_coeff :... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 164 | 174 |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | cons_injective2.right _
/-- A sequence has terminated at position `n` if the value at position `n` equals `none`. -/
| Mathlib/Data/Seq/Seq.lean | 101 | 103 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | Mathlib/Analysis/Convex/Intrinsic.lean | 340 | 351 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | @[simp]
theorem mk_eq_iff {φ ψ : K →+* ℂ} : mk φ = mk ψ ↔ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ := by
constructor
· -- We prove that the map ψ ∘ φ⁻¹ between φ(K) and ℂ is uniform continuous, thus it is either the
-- inclusion or the complex conjugation using `Complex.uniformContinuous_ringHom_eq_id_or_conj`
... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 325 | 356 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 1,568 | 1,569 | |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
import Mathlib.Order.Directed
/-!
# Monotone functions on intervals
This file sho... | lemma AntitoneOn.mapsTo_Ici (h : AntitoneOn f (Ici a)) : MapsTo f (Ici a) (Iic (f a)) :=
fun _ _ ↦ by aesop
| Mathlib/Order/Interval/Set/Image.lean | 34 | 35 |
/-
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... |
end AdaptedStoppedProcess
section Nat
/-! ### Filtrations indexed by `ℕ` -/
| Mathlib/Probability/Process/Stopping.lean | 944 | 951 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Constructions
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
/-! ## Equivalence of smoothne... | hg.contMDiffAt.comp_contMDiffWithinAt x hf
theorem ContDiffAt.comp_contMDiffAt {g : F → F'} {f : M → F} {x : M} (hg : ContDiffAt 𝕜 n g (f x))
| Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 72 | 74 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | -/
section IntegrationByPartsBilinear
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,174 | 1,176 |
/-
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... | theorem tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass {γ : Type*}
{F : Filter γ} {μs : γ → FiniteMeasure Ω}
(μs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize))
(mass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)) (f : Ω →ᵇ ℝ≥0) :
Tendsto (fun i ↦ (μs i).testAg... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 451 | 467 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.I... | rw [setIntegral_comp_smul μ f s hR.ne', abs_of_nonneg (inv_nonneg.2 (pow_nonneg hR.le _))]
theorem integral_comp_mul_left (g : ℝ → F) (a : ℝ) :
| Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 145 | 147 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | rw [RCLike.le_iff_re_im] at hab
rw [RCLike.le_iff_re_im, smul_re, smul_re, smul_im, smul_im]
| Mathlib/Analysis/RCLike/Basic.lean | 862 | 863 |
/-
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... | StrictConcaveOn ℝ D f :=
((strictAntiOn_of_deriv_neg hD.interior fun z hz =>
(differentiableAt_of_deriv_ne_zero
(hf'' z hz).ne).differentiableWithinAt.continuousWithinAt) <|
by rwa [interior_interior]).strictConcaveOn_of_deriv
hD hf
/-- If a function `f` is twice differentia... | Mathlib/Analysis/Convex/Deriv.lean | 279 | 286 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | _ ≤ ‖f‖ * ‖x‖ := le_opNorm _ _
_ = _ := by rw [hn, zero_mul]))
(by
rintro rfl
exact opNorm_zero)
/-- If a normed space is non-trivial, then the norm of the identity equals `1`. -/
@[simp]
| Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 99 | 107 |
/-
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... | | _, .cons _ h => h
| Mathlib/Data/List/Basic.lean | 517 | 518 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | mk_eq_zero.not.2 <| hf.not_dvd_of_natDegree_lt h0 hd
| Mathlib/RingTheory/AdjoinRoot.lean | 193 | 194 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 143 | 144 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | exact degree_interpolate_lt _ (Set.InjOn.mono (coe_subset.mpr (erase_subset _ _)) hvs)
| Mathlib/LinearAlgebra/Lagrange.lean | 337 | 338 |
/-
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.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | RCLike.hasSum_conj' _
theorem summable_conj {f : α → ℂ} : (Summable fun x => conj (f x)) ↔ Summable f :=
RCLike.summable_conj _
theorem conj_tsum (f : α → ℂ) : conj (∑' a, f a) = ∑' a, conj (f a) :=
| Mathlib/Analysis/Complex/Basic.lean | 509 | 514 |
/-
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.Max
import Mathlib.Data.Fintype.Basic
/-!
# Lemmas relating fintypes and order/lattice structure.
-/
open Function
open Nat
universe u... | Mathlib/Data/Fintype/Lattice.lean | 62 | 65 | |
/-
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... | [T2Space X] : Compactum where
A := X
a := Ultrafilter.lim
unit := by
ext x
exact lim_eq (pure_le_nhds _)
assoc := by
ext FF
| Mathlib/Topology/Category/Compactum.lean | 368 | 375 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | Mathlib/Data/Finmap.lean | 626 | 629 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | (hd : DifferentiableWithinAt 𝕜 d s x) :
derivWithin (fun y => (c y).comp (d y)) s x =
(derivWithin c s x).comp (d x) + (c x).comp (derivWithin d s x) := by
by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (hc.hasDerivWithinAt.clm_comp hd.hasDerivWithinAt).derivWithin hsx
· simp [derivWithin_zero_of... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 454 | 459 |
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... |
theorem hasGradientAtFilter_const : HasGradientAtFilter (fun _ => c) 0 x L := by
| Mathlib/Analysis/Calculus/Gradient/Basic.lean | 304 | 305 |
/-
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.FieldTheory.Finiteness
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.... | rw [f.apply_ind x hx] at h₂
rw [← h₂, f.apply_ind x hx, one_smul, one_smul] at h₁
have := f.mem_extChartAt_source_of_eq_one h₂.symm
exact (extChartAt I (f.c _)).injOn (f.mem_extChartAt_ind_source x hx) this h₁
theorem embeddingPiTangent_injective (f : SmoothBumpCovering ι I M) :
Injective f.embeddingPiTang... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 68 | 75 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 353 | 358 | |
/-
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... | rw [iff_iff_eq]; congr with x; simp [and_comm]
theorem ae_restrict_iff {p : α → Prop} (hp : MeasurableSet { x | p x }) :
(∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ, x ∈ s → p x :=
| Mathlib/MeasureTheory/Measure/Restrict.lean | 557 | 560 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... |
lemma summable_jacobiTheta₂'_term_iff (z τ : ℂ) :
Summable (jacobiTheta₂'_term · z τ) ↔ 0 < im τ := by
constructor
· rw [← summable_jacobiTheta₂_term_iff (z := z)]
refine fun h ↦ (h.norm.mul_left (2 * π)⁻¹).of_norm_bounded_eventually _ ?_
have : ∀ᶠ (n : ℤ) in cofinite, n ≠ 0 :=
Int.cofinite_eq ▸... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 217 | 239 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: the quadratic for... | end QuadraticMap
| Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 370 | 372 |
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
/-!
# Minimal... | over a field. -/
def _root_.Algebra.adjoin.powerBasis' (hx : IsIntegral R x) :
PowerBasis R (Algebra.adjoin R ({x} : Set S)) :=
PowerBasis.map (AdjoinRoot.powerBasis' (minpoly.monic hx)) (minpoly.equivAdjoin hx)
| Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 154 | 158 |
/-
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.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A schem... | Scheme.forgetToLocallyRingedSpace.op ⋙ LocallyRingedSpace.Γ
theorem Γ_def : Γ = Scheme.forgetToLocallyRingedSpace.op ⋙ LocallyRingedSpace.Γ :=
rfl
@[simp]
theorem Γ_obj (X : Schemeᵒᵖ) : Γ.obj X = Γ(unop X, ⊤) :=
| Mathlib/AlgebraicGeometry/Scheme.lean | 452 | 458 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | lemma zpow_add' {m n : ℤ} (h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0) :
a ^ (m + n) = a ^ m * a ^ n := by
by_cases hm : m = 0
| Mathlib/Algebra/GroupWithZero/Basic.lean | 427 | 429 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,327 | 1,329 | |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | Mathlib/Data/List/Indexes.lean | 322 | 328 | |
/-
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.Group.PUnit
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheo... | def laxMonoidalToMon : LaxMonoidalFunctor (Discrete PUnit.{u + 1}) C ⥤ Mon_ C where
obj F := (F.mapMon : Mon_ _ ⥤ Mon_ C).obj (trivial (Discrete PUnit))
map α := ((Functor.mapMonFunctor (Discrete PUnit) C).map α).app _
| Mathlib/CategoryTheory/Monoidal/Mon_.lean | 369 | 371 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... |
/-- **Line test** for equivalences.
| Mathlib/Data/Set/Prod.lean | 1,005 | 1,007 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... | theorem toOuterMeasure_apply_le_toMeasure_apply : p.toOuterMeasure s ≤ p.toMeasure s :=
le_toMeasure_apply p.toOuterMeasure _ s
| Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 215 | 217 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Order.Filter.ZeroAndBoundedAtFilt... | def IsBoundedAtImInfty {α : Type*} [Norm α] (f : ℍ → α) : Prop :=
BoundedAtFilter atImInfty f
| Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean | 37 | 38 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Algebra.Module.LinearMapPiProd
import Mathlib.LinearAlgebra.Multilinear.Basic
/-!
# Continuous multilinear maps
We define continuous m... | theorem mkPiRing_eq_zero_iff (z : M) : ContinuousMultilinearMap.mkPiRing R ι z = 0 ↔ z = 0 := by
rw [← mkPiRing_zero, mkPiRing_eq_iff]
end CommRing
| Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean | 676 | 680 |
/-
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... | rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩
obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ a ∈ Ioo x w, deriv f a < (f w - f x) / (w - x) := by
apply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _
· exact hf.mono (Icc_subset_Icc le_rfl hwy.le)
· exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)
· intro z hz
... | Mathlib/Analysis/Convex/Deriv.lean | 124 | 154 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,281 | 2,284 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | · exact add_nonneg two_pos.le (sqrtTwoAddSeries_zero_nonneg _)
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 677 | 678 |
/-
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.Countable.Defs
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Equiv
import Mathlib.... | /-- An encodable type has decidable equality. Not set as an instance because this is usually not the
best way to infer decidability. -/
def decidableEqOfEncodable (α) [Encodable α] : DecidableEq α
| Mathlib/Logic/Encodable/Basic.lean | 77 | 79 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.Order.CompleteBooleanAlgebra
/-!
# Properties of morphisms
We provide the basic framework for talking about prope... | Mathlib/CategoryTheory/MorphismProperty/Basic.lean | 313 | 313 | |
/-
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.Data.Finset.Lattice.Fold
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
/-!
# Symmetric powers of a finset
This file defines the sym... | theorem sym_mono (h : s ⊆ t) (n : ℕ) : s.sym n ⊆ t.sym n := fun _m hm ↦
mem_sym_iff.2 fun _a ha ↦ h <| mem_sym_iff.1 hm _ ha
| Mathlib/Data/Finset/Sym.lean | 244 | 245 |
/-
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... | theorem mapRange_comp (f : N → P) (hf : f 0 = 0) (f₂ : M → N) (hf₂ : f₂ 0 = 0) (h : (f ∘ f₂) 0 = 0)
(g : α →₀ M) : mapRange (f ∘ f₂) h g = mapRange f hf (mapRange f₂ hf₂ g) :=
| Mathlib/Data/Finsupp/Defs.lean | 309 | 310 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.MeasurableSpace.Prod
import Mathlib.MeasureTheory.Measure.Ty... | theorem borel_eq_generateFrom_Ioi_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ioi (a : ℝ)}) := by
rw [borel_eq_generateFrom_Ioi]
refine le_antisymm
(generateFrom_le ?_)
(generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
rintro _ ⟨a, rfl⟩
have : IsGLB (range ((↑) : ℚ →... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 56 | 66 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | variable {K}
theorem mem_inv_iff (hI : I ≠ 0) {x : K} : x ∈ I⁻¹ ↔ ∀ y ∈ I, x * y ∈ (1 : FractionalIdeal R₁⁰ K) :=
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 76 | 78 |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | @[simp]
theorem coprod_map_prod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (S : Submodule R M)
| Mathlib/LinearAlgebra/Prod.lean | 234 | 235 |
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | section
variable [SecondCountableTopology X]
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 212 | 215 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Powerset
import Mathlib.Algebra.NoZeroSMulDivisors.Pi
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Finty... | ext j
by_cases h : j = i
· rw [h]
simp [m'', hit]
| Mathlib/LinearAlgebra/Multilinear/Basic.lean | 423 | 426 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Module.LinearMap.Prod
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Analysis... | end OrderedSemiring
section OrderedCommSemiring
| Mathlib/Analysis/Convex/Star.lean | 259 | 262 |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... |
lemma Even.not_odd [NeZero (2 : S)] (hψ : Even ψ) : ¬Odd ψ :=
not_and.mp ψ.not_even_and_odd hψ
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 306 | 308 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | h.mono (fun _ ↦ And.left) ⟨h.prop, hQ⟩
theorem Maximal.and_left (h : Maximal P x) (hQ : Q x) : Maximal (fun x ↦ (Q x ∧ P x)) x :=
h.mono (fun _ ↦ And.right) ⟨hQ, h.prop⟩
| Mathlib/Order/Minimal.lean | 119 | 122 |
/-
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, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,099 | 1,109 | |
/-
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, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... |
theorem IsUniformInducing.comp {g : β → γ} (hg : IsUniformInducing g) {f : α → β}
(hf : IsUniformInducing f) : IsUniformInducing (g ∘ f) :=
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 62 | 64 |
/-
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 | 219 | 222 | |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 647 | 647 | |
/-
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 | 2,625 | 2,625 |
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