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/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Floor.Div
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Roots of natural numbers, rounded up and down
This file defines the flooring and... | @[simp] lemma floorRoot_one_right (hn : n ≠ 0) : floorRoot n 1 = 1 := by simp [floorRoot, hn]
| Mathlib/Data/Nat/Factorization/Root.lean | 62 | 62 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Algebra.MvPolynomial.Degrees
/-!
# Variables of polynomials
This file establishes man... | (p + q).vars ⊆ p.vars ∪ q.vars := by
intro x hx
| Mathlib/Algebra/MvPolynomial/Variables.lean | 98 | 99 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | theorem smul_def {i} (m : M i) (w : Word M) :
m • w = rcons { equivPair i w with head := m * (equivPair i w).head } :=
rfl
theorem of_smul_def (i) (w : Word M) (m : M i) :
of m • w = rcons { equivPair i w with head := m * (equivPair i w).head } :=
| Mathlib/GroupTheory/CoprodI.lean | 493 | 498 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Analysis.PSeries
import Mathlib.NumberTheory.SmoothNumbers
/-!
# The sum of the reciprocals of the primes d... | /-- The sum over the reciprocals of the primes diverges. -/
theorem not_summable_one_div_on_primes :
¬ Summable (indicator {p | p.Prime} (fun n : ℕ ↦ (1 : ℝ) / n)) := by
intro h
obtain ⟨k, hk⟩ := h.nat_tsum_vanishing (Iio_mem_nhds one_half_pos : Iio (1 / 2 : ℝ) ∈ 𝓝 0)
specialize hk ({p | Nat.Prime p} ∩ {p | ... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 64 | 79 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Module.Hom
import Mathlib.GroupTheory.GroupAction.Ri... | NonUnitalRingHom.coe_toMulHom, centerToCentroid_apply, ← map_mul_right, one_mul] }
end NonAssocSemiring
section NonUnitalNonAssocRing
variable [NonUnitalNonAssocRing α]
/-- Negation of `CentroidHom`s as a `CentroidHom`. -/
instance : Neg (CentroidHom α) :=
⟨fun f ↦
{ (-f : α →+ α) with
map_mul_lef... | Mathlib/Algebra/Ring/CentroidHom.lean | 530 | 543 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | by_contra hc
let a := g' 0 0
let d := g' 1 1
have had : T ^ (-a) * g' = S * T ^ d := by
rw [g_eq_of_c_eq_one hc]
dsimp [a, d]
group
let w := T ^ (-a) • g' • z
have h₁ : w = S • T ^ d • z := by simp only [w, ← mul_smul, had]
replace h₁ : normSq w < 1 := h₁.symm ▸ normSq_S_sm... | Mathlib/NumberTheory/Modular.lean | 478 | 509 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | rw [add_comm, toIcoMod_add_zsmul', add_comm]
@[simp]
theorem toIocMod_zsmul_add (a b : α) (m : ℤ) : toIocMod hp a (m • p + b) = toIocMod hp a b := by
rw [add_comm, toIocMod_add_zsmul]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 365 | 369 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,156 | 1,159 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... |
@[simp]
theorem filter_single_of_pos {a : α} {b : M} (h : p a) : (single a b).filter p = single a b :=
(filter_eq_self_iff _ _).2 fun _ hx => (single_apply_ne_zero.1 hx).1.symm ▸ h
| Mathlib/Data/Finsupp/Basic.lean | 830 | 834 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | limsup f v = if ∃ (j : Subtype p), s j = ∅ then sInf univ else
if ∀ (j : Subtype p), ¬BddAbove (range (fun (i : s j) ↦ f i)) then sInf ∅
else ⨅ (j : Subtype p), ⨆ (i : s (limsup_reparam f s p j)), f i :=
HasBasis.liminf_eq_ite (α := αᵒᵈ) hv f
end Classical
| Mathlib/Order/LiminfLimsup.lean | 1,061 | 1,067 |
/-
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.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
@[simp]
| Mathlib/Order/Heyting/Basic.lean | 321 | 322 |
/-
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.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | theorem algEquivOfAlgEquiv_symm : (algEquivOfAlgEquiv S Q h H).symm =
algEquivOfAlgEquiv Q S h.symm (show Submonoid.map h.symm T = M by
rw [← H, ← Submonoid.map_coe_toMulEquiv, AlgEquiv.symm_toMulEquiv,
← Submonoid.comap_equiv_eq_map_symm, ← Submonoid.map_coe_toMulEquiv,
Submonoid.comap_map_eq... | Mathlib/RingTheory/Localization/Basic.lean | 230 | 237 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | exact hx
theorem convexBodyLT_convex : Convex ℝ (convexBodyLT K f) :=
Convex.prod (convex_pi (fun _ _ => convex_ball _ _)) (convex_pi (fun _ _ => convex_ball _ _))
open Fintype MeasureTheory MeasureTheory.Measure ENNReal
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 70 | 75 |
/-
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.Homology
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# Homolo... | lemma leftHomologyMap'_congr (h : Homotopy φ₁ φ₂) (h₁ : S₁.LeftHomologyData)
(h₂ : S₂.LeftHomologyData) : leftHomologyMap' φ₁ h₁ h₂ = leftHomologyMap' φ₂ h₁ h₂ := by
rw [h.eq_add_nullHomotopic, leftHomologyMap'_add, leftHomologyMap'_nullHomotopic, add_zero]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 647 | 649 |
/-
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.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... |
lemma Icc_insert_succ_left (h : a ≤ b) : insert a (Icc (a + 1) b) = Icc a b := by
ext x
| Mathlib/Order/Interval/Finset/Nat.lean | 148 | 150 |
/-
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
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | simp [image_eq_zero_of_nmem_tsupport hx]
rw [← this, indicator_smul_left]
apply Integrable.smul_of_top_left
· rw [integrable_indicator_iff hK.measurableSet]
exact hf.integrableOn_isCompact hK
· exact hg.memLp_top_of_hasCompactSupport h'g μ
open Filter
theorem integrable_iff_integrableAtFilter_cocompac... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 347 | 362 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... |
section CharP
variable [Ring R]
lemma CharP.orderOf_eq_two_iff [Nontrivial R] [NoZeroDivisors R] (p : ℕ)
| Mathlib/Algebra/CharP/Two.lean | 134 | 139 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.Separation.Basic
/-!
# Topology on the set of filters on a type... | simp_rw [nhds_eq, Function.comp_def, sInter_lift'_sets monotone_principal.Iic, Iic,
le_principal_iff, ← setOf_forall, ← Filter.le_def]
| Mathlib/Topology/Filter.lean | 134 | 135 |
/-
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 | 1,776 | 1,788 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... | Filter.inf_isMeasurablyGenerated _ _
instance (priority := 100) OpensMeasurableSpace.separatesPoints [T0Space α] :
SeparatesPoints α := by
| Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 327 | 330 |
/-
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... | Here we formulate it as a `UniformEquiv`. -/
def uniformEquivUniformFun (h : univ ∈ 𝔖) : (α →ᵤ[𝔖] β) ≃ᵤ (α →ᵤ β) where
toFun f := UniformFun.ofFun <| toFun _ f
invFun f := ofFun _ <| UniformFun.toFun f
left_inv _ := rfl
right_inv _ := rfl
uniformContinuous_toFun := by
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 783 | 789 |
/-
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.Module.Submodule.Map
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.RingTheory.Ideal.Quotient.Defs
/-!
# modular equivalence for submodule
-... | (Submodule.Quotient.eq U₂).2 <| HU <| (Submodule.Quotient.eq U₁).1 hxy
| Mathlib/LinearAlgebra/SModEq.lean | 51 | 52 |
/-
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... |
variable {Γ : Type*} [AddCommMonoid Γ] [LinearOrder Γ] [IsOrderedCancelAddMonoid Γ]
[NonUnitalNonAssocSemiring R]
theorem coeff_mul_order_add_order (x y : HahnSeries Γ R) :
| Mathlib/RingTheory/HahnSeries/Multiplication.lean | 497 | 501 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Semisimple
/-!
# Eigenspaces of semisimple linear endomorphisms
This file contains basic result... | simpa [LinearMap.restrict_sub h₄ h₃] using (LinearMap.restrict_commute hfg h₄ h₃).isNilpotent_sub
(f.isNilpotent_restrict_sub_algebraMap μ l) (Module.End.isNilpotent.restrict h₃ hnil)
lemma IsFinitelySemisimple.genEigenspace_eq_eigenspace
(hf : f.IsFinitelySemisimple) (μ : R) {k : ℕ∞} (hk : 0 < k) :
| Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean | 49 | 53 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | subst_vars; simp [_root_.pow_add]
| Mathlib/Tactic/Ring/Basic.lean | 873 | 874 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... |
/-- An equipartition of a finset with `n` elements into `k` parts has
`n - n % k` parts of size `n / k`. -/
theorem IsEquipartition.card_small_parts_eq_mod (hP : P.IsEquipartition) :
#{p ∈ P.parts | #p = #s / #P.parts} = #P.parts - #s % #P.parts := by
conv_rhs =>
arg 1
rw [← filter_card_add_filter_neg_ca... | Mathlib/Order/Partition/Equipartition.lean | 89 | 100 |
/-
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.LocallySurjective
import Mathlib.CategoryTheory.Sites.Localization
/-!
# Locally bijective morphisms of presheaves
Let `C` a be category e... | J.W f ↔ Presheaf.IsLocallyInjective J f ∧ Presheaf.IsLocallySurjective J f := by
apply WEqualsLocallyBijective.iff
lemma W_of_isLocallyBijective [Presheaf.IsLocallyInjective J f]
| Mathlib/CategoryTheory/Sites/LocallyBijective.lean | 112 | 115 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... | · obtain ⟨x, y, hxy, rfl⟩ := h
exact card_support_swap hxy
theorem Disjoint.card_support_mul (h : Disjoint f g) :
#(f * g).support = #f.support + #g.support := by
rw [← Finset.card_union_of_disjoint]
· congr
| Mathlib/GroupTheory/Perm/Support.lean | 631 | 637 |
/-
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.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | have h : {(1, x)} ⊆ divisorsAntidiagonal x := by simp [x0]
| Mathlib/NumberTheory/ArithmeticFunction.lean | 296 | 296 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,321 | 1,333 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,904 | 1,907 | |
/-
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 | 2,062 | 2,069 | |
/-
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... | have hnl : n ≤ l.length := le_of_succ_le_succ h
have hnl' : n ≤ (l ++ [a]).length := by
rw [length_append, length_cons, List.length]; exact le_of_succ_le h
rw [rotate'_cons_succ, rotate'_eq_drop_append_take hnl', drop, take,
| Mathlib/Data/List/Rotate.lean | 61 | 64 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Preserves.Shapes... | have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)
simp_rw [F.map_id] at this
apply PullbackCone.mono_of_isLimitMkIdId _ this
instance (priority := 100) preservesMonomorphisms_of_preservesLimitsOfShape
| Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 32 | 36 |
/-
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.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
@[simp]
| Mathlib/Order/Heyting/Basic.lean | 727 | 728 |
/-
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.QuadraticDiscriminant
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
/-!
... | simp only [Int.cast_add, Int.cast_two, Int.cast_mul, Int.cast_one, hx, hy] at hx hy hxy
rw [hx, hy, add_zero, zero_div, mul_div_assoc, mul_div_assoc, ←
add_mul (2 * (k : ℂ) + 1) (2 * l + 1) (π / 2), ← mul_div_assoc, hxy]
theorem tan_add' {x y : ℂ}
(h : (∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 126 | 141 |
/-
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.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... |
instance : OrderedSub (Π₀ i, α i) :=
| Mathlib/Data/DFinsupp/Order.lean | 242 | 243 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... |
/-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_oangle_add_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.sin (o.oangle (x + y) y) * ‖x + y‖ = ‖x‖ := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 155 | 159 |
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
/-!
# Stirling's formula
Thi... | exact absurd h (by positivity)
· have : sqrt π ≠ 0 := by positivity
nth_rewrite 2 [← div_self this]
convert tendsto_stirlingSeq_sqrt_pi.div tendsto_const_nhds this using 1
ext n
field_simp [stirlingSeq, mul_right_comm]
end Stirling
| Mathlib/Analysis/SpecialFunctions/Stirling.lean | 238 | 247 |
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
import Mathlib.Com... | (hM <| M.edge_vert huv).unique huv huw
lemma IsMatching.eq_of_adj_right (hM : M.IsMatching) (huw : M.Adj u w) (hvw : M.Adj v w) : u = v :=
| Mathlib/Combinatorics/SimpleGraph/Matching.lean | 96 | 98 |
/-
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... | @[simp]
| Mathlib/Algebra/Polynomial/Roots.lean | 192 | 192 |
/-
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.RingTheory.Noetherian.Basic
/-!
# Finiteness of `IsScalarTower`
We prove that given `IsScalarTower F K A`, if `A` is finite as a module over `F` then
`A` is fi... | Mathlib/FieldTheory/Tower.lean | 71 | 82 | |
/-
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.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... | rw [eq_comm, Hom.cast_eq_iff_heq]
exact ⟨HEq.symm, HEq.symm⟩
/-!
| Mathlib/Combinatorics/Quiver/Cast.lean | 63 | 66 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | @[simps]
def equivHom : (F ⟶ G) ≃+ Cocycle F G 0 where
toFun := ofHom
invFun := homOf
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 686 | 689 |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | def wordProd (ω : List B) : W := prod (map cs.simple ω)
| Mathlib/GroupTheory/Coxeter/Basic.lean | 358 | 358 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | simp only [mul_def]
exact coeToSubmodule_injective (coeSubmodule_mul _ _ _)
theorem mul_left_mono (I : FractionalIdeal S P) : Monotone (I * ·) := by
intro J J' h
simp only [mul_def]
exact mul_le.mpr fun x hx y hy => mul_mem_mul hx (h hy)
theorem mul_right_mono (I : FractionalIdeal S P) : Monotone fun J => J... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 537 | 549 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... | refine ⟨f ⁻¹' t, t.finite_toSet.preimage f.injective.injOn, ?_⟩
have hf : Function.Injective f.toHom := f.injective
refine map_injective_of_injective hf ?_
rw [← h, map_closure, Embedding.coe_toHom, image_preimage_eq_of_subset]
intro x hx
have h' := subset_closure (L := L) hx
rw [h] at h'
exact Hom.map_... | Mathlib/ModelTheory/FinitelyGenerated.lean | 87 | 98 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Int.Sqrt
/-!... | theorem exists_mul_self (x : ℚ) : (∃ q, q * q = x) ↔ Rat.sqrt x * Rat.sqrt x = x :=
⟨fun ⟨n, hn⟩ => by rw [← hn, sqrt_eq, abs_mul_abs_self], fun h => ⟨Rat.sqrt x, h⟩⟩
| Mathlib/Data/Rat/Sqrt.lean | 30 | 31 |
/-
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.Category.ModuleCat.Adjunctions
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Al... | simpa using LinearMap.ext_iff.1 (hom_comm_apply f g y) (A.ρ g x) }
left_inv _ := Action.Hom.ext (ModuleCat.hom_ext <| TensorProduct.ext' fun _ _ => rfl)
right_inv _ := by ext; rfl
variable {A B C}
| Mathlib/RepresentationTheory/Rep.lean | 344 | 349 |
/-
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.LeftHomology
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# Right Homology of short complexes
In this file, we define the ... | lemma isIso_ι (hg : S.g = 0) : IsIso h.ι := by
have ⟨φ, hφ⟩ := KernelFork.IsLimit.lift' h.hι' (𝟙 _)
(by rw [← cancel_epi h.p, id_comp, p_g', comp_zero, hg])
dsimp at hφ
exact ⟨φ, by rw [← cancel_mono h.ι, assoc, hφ, comp_id, id_comp], hφ⟩
| Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 133 | 137 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.C... | (TopologicalSpace.Opens.map f.base).obj (imageBasicOpen f g U s) =
(TopologicalSpace.Opens.map g.base).obj (imageBasicOpen f g U s)
by injection this
delta imageBasicOpen
rw [preimage_basicOpen f, preimage_basicOpen g]
dsimp only [Functor.op, unop_op]
-- Porting note (https://github.... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | 185 | 211 |
/-
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.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... | /-- If `A` and `B` are isomorphic as `R`-algebras, then so are their polynomial rings -/
def mapAlgEquiv (f : A ≃ₐ[R] B) : Polynomial A ≃ₐ[R] Polynomial B :=
AlgEquiv.ofAlgHom (mapAlgHom f.toAlgHom) (mapAlgHom f.symm.toAlgHom) (by simp) (by simp)
@[simp]
| Mathlib/Algebra/Polynomial/AlgebraMap.lean | 185 | 189 |
/-
Copyright (c) 2021 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak, Kyle Miller, Eric Wieser
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Lin... |
@[simp]
| Mathlib/LinearAlgebra/CrossProduct.lean | 75 | 76 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | have f_meas : ∀ c : ℂ, AEStronglyMeasurable (f c) volume := fun c => by
apply Continuous.aestronglyMeasurable
exact Complex.continuous_exp.comp (continuous_const.mul (continuous_ofReal.pow 2))
have f_cts : ∀ x : ℝ, ContinuousAt (fun c => f c x) b := fun x =>
(Complex.continuous_exp.comp (continuous_id'.... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 238 | 249 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | rw [inits_core_eq]; rfl
| Mathlib/Data/Stream/Init.lean | 660 | 661 |
/-
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.Data.Opposite
import Mathlib.Data.Set.Operations
/-!
# The opposite of a set
The opposite of a set `s` is simply the set obtained by taking the opposit... | Mathlib/Data/Set/Opposite.lean | 39 | 39 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | rw [← setOf_inter_eq_sep, ← Measure.restrict_apply₀
(hf.ennreal_toReal.aestronglyMeasurable.nullMeasurableSet_le aestronglyMeasurable_const),
← measure_diff_null (measure_eq_top_of_lintegral_ne_top hf h)] at this
| Mathlib/MeasureTheory/Integral/Average.lean | 616 | 618 |
/-
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, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... |
/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/
theorem shiftComm' (i j : A) :
f⟦i⟧'⟦j⟧' = (shiftComm _ _ _).hom ≫ f⟦j⟧'⟦i⟧' ≫ (shiftComm _ _ _).hom := by
erw [← shiftComm_symm Y i j, ← ((shiftFunctorComm C i j).hom.naturality_assoc f)]
| Mathlib/CategoryTheory/Shift/Basic.lean | 600 | 604 |
/-
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.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... |
lemma Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact
{K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K) :
∃ x ∈ K, AccPt x (cofinite ⊓ 𝓟 s) :=
| Mathlib/Topology/Compactness/Compact.lean | 748 | 751 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | lemma rnDeriv_eq_top_iff' (κ η : Kernel α γ) (a : α) (x : γ) :
rnDeriv κ η a x = ∞ ↔ x ∈ mutuallySingularSetSlice κ η a := by
rw [rnDeriv_eq_top_iff, mutuallySingularSet, mutuallySingularSetSlice, mem_setOf, mem_setOf]
| Mathlib/Probability/Kernel/RadonNikodym.lean | 252 | 255 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | theorem radical_inf : radical (I ⊓ J) = radical I ⊓ radical J :=
le_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right))
fun r ⟨⟨m, hrm⟩, ⟨n, hrn⟩⟩ =>
⟨m + n, (pow_add r m n).symm ▸ I.mul_mem_right _ hrm,
| Mathlib/RingTheory/Ideal/Operations.lean | 778 | 781 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Principal ordinals
We define principal or indecomposable ordinals, and we prove the standa... | dsimp only
rw [← one_add_one_eq_two, mul_add, mul_one]
exact add_le_add (le_max_left a b) (le_max_right a b)
theorem isLimit_of_principal_mul (ho₂ : 2 < o) (ho : Principal (· * ·) o) : o.IsLimit :=
isLimit_of_principal_add ((lt_succ 1).trans (succ_one ▸ ho₂))
| Mathlib/SetTheory/Ordinal/Principal.lean | 274 | 279 |
/-
Copyright (c) 2023 Mohanad ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.SemiringInverse
/-! # Block Matrices from Rows and Columns
This file provides the basic definiti... | /-- A column partitioned matrix when transposed gives a row partitioned matrix with columns of the
initial matrix transposed to become rows. -/
lemma transpose_fromCols (A₁ : Matrix m n₁ R) (A₂ : Matrix m n₂ R) :
transpose (fromCols A₁ A₂) = fromRows (transpose A₁) (transpose A₂) := by
ext (i | i) j <;> simp
| Mathlib/Data/Matrix/ColumnRowPartitioned.lean | 147 | 151 |
/-
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... | /-- `gcdD a b = gcd a b` -/
def gcdD : ℕ+ :=
(xgcd a b).a
| Mathlib/Data/PNat/Xgcd.lean | 353 | 355 |
/-
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... | constructor
· show ∀ x ∈ s, infEdist x u ≤ hausdorffEdist s t + hausdorffEdist t u
exact fun x xs =>
calc
infEdist x u ≤ infEdist x t + hausdorffEdist t u :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ hausdorffEdist s t + hausdorffEdist t u :=
add_le_add_right (infEdis... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 336 | 352 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 796 | 799 | |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... |
end PerfectRing
variable [DecidableEq R]
| Mathlib/FieldTheory/Perfect.lean | 326 | 329 |
/-
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.... | see `AddSubgroup.closure_induction_right`."]
| Mathlib/Algebra/Group/Basic.lean | 905 | 905 |
/-
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 | 2,084 | 2,094 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... |
theorem isNoetherian_iff {I : FractionalIdeal R₁⁰ K} :
IsNoetherian R₁ I ↔ ∀ J ≤ I, (J : Submodule R₁ K).FG :=
isNoetherian_submodule.trans ⟨fun h _ hJ => h _ hJ, fun h J hJ => h ⟨J, isFractional_of_le hJ⟩ hJ⟩
theorem isNoetherian_coeIdeal [IsNoetherianRing R₁] (I : Ideal R₁) :
IsNoetherian R₁ (I : Fraction... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 828 | 848 |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | approx ennrealRatEmbed
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 762 | 762 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | exact h
· intro h
use #σ.support, σ
simp [h]
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 122 | 126 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,256 | 2,258 | |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | lemma eapprox_mono {m n : ℕ} (hmn : m ≤ n) : eapprox f m ≤ eapprox f n := monotone_eapprox _ hmn
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 782 | 783 |
/-
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 ... |
@[to_additive (attr := simp)]
theorem mulDysonETransform_idem :
mulDysonETransform e (mulDysonETransform e x) = mulDysonETransform e x := by
ext : 1 <;> dsimp
| Mathlib/Combinatorics/Additive/ETransform.lean | 66 | 70 |
/-
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.... | Mathlib/Algebra/Group/Basic.lean | 1,311 | 1,311 | |
/-
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.... | Mathlib/Control/Fold.lean | 409 | 415 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... | variable (S) in
/-- The seminormed group structure on the quotient by a subgroup. -/
@[to_additive "The seminormed group structure on the quotient by an additive subgroup."]
noncomputable instance instSeminormedCommGroup : SeminormedCommGroup (M ⧸ S) where
toUniformSpace := IsTopologicalGroup.toUniformSpace (M ⧸ S)
... | Mathlib/Analysis/Normed/Group/Quotient.lean | 199 | 205 |
/-
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.Limits.Shapes.Pullback.CommSq
/-!
# Relation between mono/epi and pullback/pushout squares
In this file, monomorphisms and epimorphisms are char... | lemma mono_iff_fst_eq_snd (hc : IsLimit c) : Mono f ↔ c.fst = c.snd := by
constructor
· intro hf
simpa only [← cancel_mono f] using c.condition
· intro hf
constructor
intro Z g g' h
obtain ⟨φ, rfl, rfl⟩ := PullbackCone.IsLimit.lift' hc g g' h
rw [hf]
| Mathlib/CategoryTheory/Limits/EpiMono.lean | 38 | 46 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | /-- `toNNRat` and `(↑) : ℚ≥0 → ℚ` form a Galois insertion. -/
protected def gi : GaloisInsertion toNNRat (↑) :=
| Mathlib/Data/NNRat/Defs.lean | 179 | 180 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
/-!
# Strong epimorphisms
In this file, we define strong epimorphisms. A ... | theorem strongMono_of_mono [StrongMonoCategory C] (f : P ⟶ Q) [Mono f] : StrongMono f :=
StrongMonoCategory.strongMono_of_mono _
| Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 192 | 193 |
/-
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... | obtain ⟨seq : ℕ → ι, h_seq_tendsto⟩ := (atTop : Filter ι).exists_seq_tendsto
rw [(_ : s = ⋃ n, s ∩ {ω | τ ω ≤ seq n})]
· exact MeasurableSet.iUnion fun i => f.le (seq i) _ (hs (seq i))
· ext ω; constructor <;> rw [Set.mem_iUnion]
· intro hx
suffices ∃ i, τ ω ≤ seq i from ⟨this.choose, hx, ... | Mathlib/Probability/Process/Stopping.lean | 334 | 342 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Diagonal matrices
This fi... | rw [LinearMap.rank, range_diagonal, B₁, ← @rank_fun' K]
apply LinearEquiv.rank_eq
apply B₂
end Field
end LinearMap
| Mathlib/LinearAlgebra/Matrix/Diagonal.lean | 86 | 94 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | ∃ C, ∀ᵉ (y ∈ insert x s ∩ EMetric.ball x r') (z ∈ insert x s ∩ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
have := hf.isBigO_image_sub_image_sub_deriv_principal hr
| Mathlib/Analysis/Analytic/Basic.lean | 1,131 | 1,133 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... |
theorem lintegral_le_of_lmarginal_le [Fintype δ] (s : Finset δ) {f g : (∀ i, X i) → ℝ≥0∞}
(hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ) :
∫⁻ x, f x ∂Measure.pi μ ≤ ∫⁻ x, g x ∂Measure.pi μ := by
rcases isEmpty_or_nonempty (∀ i, X i) with h|⟨⟨x⟩⟩
· simp_rw [lintegral_of_isEmpty,... | Mathlib/MeasureTheory/Integral/Marginal.lean | 243 | 248 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... | instance : FunLike (LocallyConstant X Y) X Y where
coe := LocallyConstant.toFun
coe_injective' := by rintro ⟨_, _⟩ ⟨_, _⟩ _; congr
| Mathlib/Topology/LocallyConstant/Basic.lean | 214 | 217 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Normed.Module.Convex
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neum... | rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by
| Mathlib/Analysis/Convex/GaugeRescale.lean | 48 | 52 |
/-
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... | by_cases hF : CompleteSpace F; swap
· simp [integral, hF]
rcases eq_or_ne R 0 with (rfl | hR)
| Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 89 | 91 |
/-
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 only [mem_def, disjiUnion_val, Multiset.mem_bind, exists_prop]
| Mathlib/Data/Finset/Union.lean | 50 | 50 |
/-
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.Logic.OpClass
import Mathlib.Order.Lattice
/-!
# `max` and `min`
This file proves basic properties about maxima and minima on a `LinearOrder`.
## Ta... | Mathlib/Order/MinMax.lean | 271 | 272 | |
/-
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... | section LinearOrderedField
| Mathlib/Analysis/Convex/Segment.lean | 362 | 363 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | filter_upwards [eventually_ne_atTop (0 : ℝ)] with x h₁ h₂ using (h₁ h₂).elim
theorem isLittleO_log_id_atTop : log =o[atTop] id :=
isLittleO_pow_log_id_atTop.congr_left fun _ => pow_one _
theorem isLittleO_const_log_atTop {c : ℝ} : (fun _ => c) =o[atTop] log := by
refine Asymptotics.isLittleO_of_tendsto' ?_
... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 397 | 404 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,725 | 1,726 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | simp [← prev_reverse_eq_next]
@[simp]
theorem next_reverse_eq_prev' (s : Cycle α) (hs : Nodup s.reverse) (x : α) (hx : x ∈ s.reverse) :
| Mathlib/Data/List/Cycle.lean | 745 | 748 |
/-
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... | end Lattice
/-! ### Closed intervals in `α × β` -/
section Prod
variable {β : Type*} [Preorder α] [Preorder β]
| Mathlib/Order/Interval/Set/Basic.lean | 941 | 947 |
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Antidiag.Pi
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
im... | -/
def multinomial [DecidableEq α] (m : Multiset α) : ℕ :=
m.toFinsupp.multinomial
theorem multinomial_filter_ne [DecidableEq α] (a : α) (m : Multiset α) :
m.multinomial = m.card.choose (m.count a) * (m.filter (a ≠ ·)).multinomial := by
dsimp only [multinomial]
convert Finsupp.multinomial_update a _
· rw [... | Mathlib/Data/Nat/Choose/Multinomial.lean | 177 | 187 |
/-
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 ... | simp_all only [degree_eq_natDegree hf]
· first | exact WithBot.coe_lt_coe.1 h | exact WithBot.coe_lt_coe.2 h
theorem dim_ne_zero [Nontrivial S] (pb : PowerBasis R S) : pb.dim ≠ 0 := fun h =>
not_nonempty_iff.mpr (h.symm ▸ Fin.isEmpty : IsEmpty (Fin pb.dim)) pb.basis.index_nonempty
theorem dim_pos [Nontr... | Mathlib/RingTheory/PowerBasis.lean | 106 | 117 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | · intro
rw [(HomologyData.ofIsColimitCokernelCofork S hg _
(CokernelCofork.IsColimit.ofEpiOfIsZero (CokernelCofork.ofπ (0 : S.X₂ ⟶ 0) comp_zero)
inferInstance (isZero_zero C))).exact_iff]
exact isZero_zero C
variable {S}
lemma Exact.epi_f' (hS : S.Exact) (h : LeftHomologyData S) : Epi h.f' :=
... | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 292 | 306 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | section mapRange
variable {β β₁ β₂ : ι → Type*}
| Mathlib/LinearAlgebra/DFinsupp.lean | 170 | 172 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 621 | 622 |
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