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) 2019 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.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s)
(bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s :=
hs.lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x hx => (hf x hx).hasFDerivWithinAt) bound
/-- The mean value theorem on a c... | Mathlib/Analysis/Calculus/MeanValue.lean | 491 | 502 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | simp only [Gammaℝ, zero_div, div_zero, Complex.Gamma_zero, mul_zero, cpow_zero, sub_zero]
-- Remains to show completed zeta is `o (s ^ (-1))` near 0.
refine (isBigO_const_of_tendsto claim2 <| one_ne_zero' ℂ).trans_isLittleO ?_
rw [isLittleO_iff_tendsto']
· exact Tendsto.congr (fun x ↦ by rw [← one_d... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 571 | 597 |
/-
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... | | zero => simp
| succ n => rw [take_succ', List.dropLast_concat, Nat.add_one_sub_one]
| Mathlib/Data/Stream/Init.lean | 549 | 550 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem IsLindelof.union (hs : IsLindelof s) (ht : IsLindelof t) : IsLindelof (s ∪ t) := by
rw [union_eq_iUnion]; exact isLindelof_iUnion fun b => by cases b <;> assumption
| Mathlib/Topology/Compactness/Lindelof.lean | 374 | 375 |
/-
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.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
/-!
# `init` and `tail`
Given a Witt vecto... |
end
| Mathlib/RingTheory/WittVector/InitTail.lean | 225 | 226 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | · rw [PosNum.cast_bit1, ← two_mul, ← congr_fun Nat.bit_true, Nat.testBit_bit_succ, IH]
· rw [PosNum.cast_bit0, ← two_mul, ← congr_fun Nat.bit_false, Nat.testBit_bit_succ, IH]
| Mathlib/Data/Num/Lemmas.lean | 864 | 865 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | exact I.mul_mem_left _ H
variable {I : Ideal R[X]}
theorem mem_ofPolynomial (x) : x ∈ I.ofPolynomial ↔ x ∈ I :=
Iff.rfl
| Mathlib/RingTheory/Polynomial/Basic.lean | 477 | 483 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | · rw [← Function.comp_id (((↑) : ℝ → Real.Angle) ∘ arg),
(funext_iff.2 fun _ => (neg_neg _).symm : (id : ℂ → ℂ) = Neg.neg ∘ Neg.neg), ←
Function.comp_assoc]
refine ContinuousAt.comp ?_ continuous_neg.continuousAt
suffices ContinuousAt (Function.update (((↑) ∘ arg) ∘ Neg.neg : ℂ → Real.Angle) 0 π) ... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 625 | 639 |
/-
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... | @[deprecated (since := "2025-02-22")]
alias image_prod_mk_subset_prod := image_prodMk_subset_prod
| Mathlib/Data/Set/Prod.lean | 283 | 285 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 925 | 934 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | theorem length_empty : f.length ∅ = 0 :=
nonpos_iff_eq_zero.1 <| iInf_le_of_le 0 <| iInf_le_of_le 0 <| by simp
@[simp]
theorem length_Ioc (a b : ℝ) : f.length (Ioc a b) = ofReal (f b - f a) := by
| Mathlib/MeasureTheory/Measure/Stieltjes.lean | 178 | 182 |
/-
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... | ext
rfl
@[simp]
theorem comapDomain_single (f : α → β) (a : α) (m : M)
(hif : Set.InjOn f (f ⁻¹' (single (f a) m).support)) :
comapDomain f (Finsupp.single (f a) m) hif = Finsupp.single a m := by
rcases eq_or_ne m 0 with (rfl | hm)
· simp only [single_zero, comapDomain_zero]
· rw [eq_single_iff, coma... | Mathlib/Data/Finsupp/Basic.lean | 657 | 667 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | def skewAdjointMatricesSubmodule : Submodule R (Matrix n n R) :=
pairSelfAdjointMatricesSubmodule (-J) J
@[simp]
theorem mem_skewAdjointMatricesSubmodule :
A₁ ∈ skewAdjointMatricesSubmodule J ↔ J.IsSkewAdjoint A₁ := by
erw [mem_pairSelfAdjointMatricesSubmodule]
simp [Matrix.IsSkewAdjoint, Matrix.IsAdjointPai... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 596 | 608 |
/-
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... | { AdjoinRoot.liftHom g pb.gen h₂ with
toFun := AdjoinRoot.liftHom g pb.gen h₂
invFun := pb.lift (root g) h₁
| Mathlib/RingTheory/AdjoinRoot.lean | 600 | 602 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add (hi _ ij).1 h₁)
have := lt_of_le_of_lt (abv_add abv _ _) (add_lt_add this (h₂ _ ij))
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 454 | 455 |
/-
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.Pi
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite.Basic
/-!
# Fintype instances for pi types
-/
assert_not_exists Order... | (ht : ∀ b, a ≠ b → (t b).Nonempty) : ((piFinset t).image fun f ↦ f a) = t a := by
refine (eval_image_piFinset_subset _ _).antisymm fun x h ↦ mem_image.2 ?_
choose f hf using ht
exact ⟨fun b ↦ if h : a = b then h ▸ x else f _ h, by aesop, by simp⟩
| Mathlib/Data/Fintype/Pi.lean | 94 | 98 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... |
@[simp]
theorem domDomCongr_smul {S : Type*} [Monoid S] [DistribMulAction S N] [SMulCommClass R S N]
(σ : ι ≃ ι') (c : S) (f : M [⋀^ι]→ₗ[R] N) :
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 672 | 675 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | theorem coeff_prod (f : ι → PowerSeries R) (d : ℕ) (s : Finset ι) :
coeff R d (∏ j ∈ s, f j) = ∑ l ∈ finsuppAntidiag s d, ∏ i ∈ s, coeff R (l i) (f i) := by
simp only [coeff]
| Mathlib/RingTheory/PowerSeries/Basic.lean | 637 | 639 |
/-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Gr... | apply fund_dom_s.quotientMeasureEqMeasurePreimage
ext U _
have meas_π : Measurable (QuotientGroup.mk : G → G ⧸ Γ) := continuous_quotient_mk'.measurable
let μ' : Measure (G ⧸ Γ) := (ν.restrict s).map π
haveI has_fund : HasFundamentalDomain Γ.op G ν := ⟨⟨s, fund_dom_s⟩⟩
have i : QuotientMeasureEqMeasurePreima... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 160 | 184 |
/-
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 Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | Mathlib/Data/Rat/Defs.lean | 507 | 508 | |
/-
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.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | theorem mul_noncommProd_erase [DecidableEq α] (s : Multiset α) {a : α} (h : a ∈ s) (comm)
(comm' := fun _ hx _ hy hxy ↦ comm (s.mem_of_mem_erase hx) (s.mem_of_mem_erase hy) hxy) :
a * (s.erase a).noncommProd comm' = s.noncommProd comm := by
induction' s using Quotient.inductionOn with l
| Mathlib/Data/Finset/NoncommProd.lean | 198 | 201 |
/-
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.DirectSum.LinearMap
import Mathlib.Algebra.Lie.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | the trace form. -/
lemma traceForm_apply_lie_apply' (x y z : L) :
traceForm R L M ⁅x, y⁆ z = - traceForm R L M y ⁅x, z⁆ :=
calc traceForm R L M ⁅x, y⁆ z
= - traceForm R L M ⁅y, x⁆ z := by rw [← lie_skew x y, map_neg, LinearMap.neg_apply]
_ = - traceForm R L M y ⁅x, z⁆ := by rw [traceForm_apply_lie_apply... | Mathlib/Algebra/Lie/TraceForm.lean | 80 | 86 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | @[simp]
theorem ofFinsupp_nsmul (a : ℕ) (b) :
| Mathlib/Algebra/Polynomial/Basic.lean | 173 | 174 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | ⟨fun ⟨_, ⟨a, ha, hab⟩, hb⟩ ↦ ⟨a, ha, hab.symm ▸ hb⟩, fun ⟨a, hp, hq⟩ ↦ ⟨f a, ⟨a, hp, rfl⟩, hq⟩⟩
| Mathlib/Logic/Basic.lean | 581 | 582 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Analysis.Specia... | ext fun n => by
rw [RingHom.comp_apply, RingHom.id_apply, RingHom.map_frobenius, coeff_pthRoot,
| Mathlib/RingTheory/Perfection.lean | 137 | 138 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... | conv_rhs => rw [← (prodEquivOfIsCompl p q h).apply_symm_apply x]
rw [coe_prodEquivOfIsCompl', Submodule.add_mem_iff_left _ (Submodule.coe_mem _),
mem_right_iff_eq_zero_of_disjoint h.disjoint]
| Mathlib/LinearAlgebra/Projection.lean | 119 | 121 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Yury Kudryashov
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
/-!
# Diffeomorphisms
This file implem... | h.toHomeomorph.toPartialHomeomorph.MDifferentiable I J :=
⟨h.mdifferentiableOn _ hn, h.symm.mdifferentiableOn _ hn⟩
theorem uniqueMDiffOn_image_aux (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M}
(hs : UniqueMDiffOn I s) : UniqueMDiffOn J (h '' s) := by
| Mathlib/Geometry/Manifold/Diffeomorph.lean | 335 | 339 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint... | (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : E) ‖x₀‖) x₀) :
HasEigenvector (T : E →ₗ[𝕜] E) (↑(T.rayleighQuotient x₀)) x₀ := by
refine ⟨?_, hx₀⟩
rw [Module.End.mem_eigenspace_iff]
exact hT.eq_smul_self_of_isLocalExtrOn hextr
| Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 170 | 175 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
/-!
# Natural isomorphisms
For the most ... | -- Porting note: the next lemma used to be in `ext`, but that is no longer allowed.
-- We've added an aesop apply rule;
-- it would be nice to have a hook to run those without aesop warning it didn't close the goal.
apply IsIso.eq_inv_of_hom_inv_id
rw [← NatTrans.comp_app]
simp
| Mathlib/CategoryTheory/NatIso.lean | 196 | 202 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | SameRay.sameRay_comm.trans sameRay_neg_smul_right_iff
theorem sameRay_neg_smul_left_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
| Mathlib/LinearAlgebra/Ray.lean | 489 | 491 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | simp [Ne.symm (succ_ne_zero i)]
· let j' := pred j h
have : j'.succ = j := succ_pred j h
| Mathlib/Data/Fin/Tuple/Basic.lean | 130 | 132 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib... | then the norm of `ζ - 1` is `p`. -/
theorem norm_sub_one_of_prime_ne_two' [hpri : Fact (p : ℕ).Prime]
[hcyc : IsCyclotomicExtension {p} K L] (hζ : IsPrimitiveRoot ζ p)
(hirr : Irreducible (cyclotomic p K)) (h : p ≠ 2) : norm K (ζ - 1) = p := by
replace hirr : Irreducible (cyclotomic (p ^ (0 + 1) : ℕ) K) := by... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 468 | 478 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,157 | 1,160 | |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | intro n
simp_rw [upperCrossingTime_succ_eq]
exact isStoppingTime_hitting_isStoppingTime ih₂ (fun _ => lowerCrossingTime_le)
measurableSet_Ici hf _
refine ⟨this, ?_⟩
intro n
exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)
measurableSet_Iic hf _... | Mathlib/Probability/Martingale/Upcrossing.lean | 314 | 325 |
/-
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.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | @[simp]
theorem succ_eq_succ_iff : succ a = succ b ↔ a = b :=
succ_eq_succ_iff_of_not_isMax (not_isMax a) (not_isMax b)
| Mathlib/Order/SuccPred/Basic.lean | 484 | 486 |
/-
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.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 _).f j := by
have : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr :=
biprod.hom_ext _ _ (by simp) (by simp)
obtain ⟨i, hij⟩ := hc j
dsimp [nullHomotopicMap]
by_cases hj : ∃ (k : ι), c.Rel j k
· obtain ⟨k, hjk⟩ := hj
simp only... | Mathlib/Algebra/Homology/HomotopyCofiber.lean | 480 | 505 |
/-
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.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | (μ.restrict A : Measure Ω) = (μ : Measure Ω).restrict A := rfl
theorem restrict_apply_measure (μ : FiniteMeasure Ω) (A : Set Ω) {s : Set Ω}
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 262 | 264 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | @[simp]
theorem KaehlerDifferential.map_D (x : A) :
KaehlerDifferential.map R S A B (KaehlerDifferential.D R A x) =
KaehlerDifferential.D S B (algebraMap A B x) :=
Derivation.congr_fun (KaehlerDifferential.map_compDer R S A B) x
theorem KaehlerDifferential.ker_map :
LinearMap.ker (KaehlerDifferential.m... | Mathlib/RingTheory/Kaehler/Basic.lean | 701 | 716 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | toNat_eq_ofNat.trans mk_eq_two_iff
theorem card_eq_two_iff' (x : α) : Nat.card α = 2 ↔ ∃! y, y ≠ x :=
toNat_eq_ofNat.trans (mk_eq_two_iff' x)
| Mathlib/SetTheory/Cardinal/Finite.lean | 197 | 200 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,358 | 2,360 | |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.ModelTheory... | Mathlib/ModelTheory/Syntax.lean | 901 | 911 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,152 | 2,154 | |
/-
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... | exact (-o).tan_oangle_add_right_of_oangle_eq_pi_div_two h
/-- The cosine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
adjacent side. -/
theorem cos_oangle_add_right_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle x (x + y)) * ‖... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 126 | 131 |
/-
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... | refine ⟨max (Group.nilpotencyClass G₁) (Group.nilpotencyClass G₂), ?_⟩
rw [lowerCentralSeries_prod,
lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (le_max_left _ _),
lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (le_max_right _ _), bot_prod_bot]
| Mathlib/GroupTheory/Nilpotent.lean | 687 | 690 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... | classical
suffices H : (fun i : ι => f i • b i) = fun b_1 : ι => b.repr.symm.toContinuousLinearEquiv <|
(fun i : ι => lp.single 2 i (f i) (E := (fun _ : ι => 𝕜))) b_1 by
rw [H]
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 424 | 427 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
(f : X → Z) (hf : Continuous f) (g : Y → Z) (hg : IsEmbedding g) :
{ p : X × Y // f p.1 = g p.2 } ≃ₜ f ⁻¹' Set.range g where
toFun := fun x ↦ ⟨x.1.1, _, x.2.symm⟩
invFun := fun x ↦ ⟨⟨x.1, Exists.choose x.2⟩, (Exists.choose_sp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 146 | 154 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | theorem GrothendieckTopology.sheafify_isSheaf (P : Cᵒᵖ ⥤ D) : Presheaf.IsSheaf J (J.sheafify P) :=
GrothendieckTopology.Plus.isSheaf_plus_plus _ _
variable (D)
/-- The sheafification functor, as a functor taking values in `Sheaf`. -/
@[simps]
| Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 555 | 561 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.RelIso.Set
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order... |
theorem mem_range_iff_rel (f : r ≺i s) : ∀ {b : β}, b ∈ Set.range f ↔ s b f.top :=
f.mem_range_iff_rel' _
| Mathlib/Order/InitialSeg.lean | 287 | 290 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
/-! # Euler's infinite product for the sine function
This file... | have a : HasDerivAt (fun y : ℂ => y * (2 * z)) _ x := hasDerivAt_mul_const _
have b : HasDerivAt (Complex.sin ∘ fun y : ℂ => (y * (2 * z))) _ x :=
HasDerivAt.comp (x : ℂ) (Complex.hasDerivAt_sin (x * (2 * z))) a
have c := b.comp_ofReal.div_const (2 * z)
field_simp at c; simp only [fun y => mul_comm y (2 * z... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 39 | 46 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,732 | 1,735 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | IsCoprime b.re b.im := by
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 319 | 319 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,285 | 1,289 | |
/-
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.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Additive Haar measure constructed from a basis
Given a ba... |
theorem parallelepiped_basis_eq (b : Basis ι ℝ E) :
parallelepiped b = {x | ∀ i, b.repr x i ∈ Set.Icc 0 1} := by
| Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 52 | 54 |
/-
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... | conv_lhs => rw [jacobiTheta₂, ← Equiv.tsum_eq (Equiv.neg ℤ)]
refine tsum_congr (fun n ↦ ?_)
simp_rw [jacobiTheta₂_term, Equiv.neg_apply, Int.cast_neg, neg_sq, mul_assoc, neg_mul_neg]
lemma jacobiTheta₂_conj (z τ : ℂ) :
conj (jacobiTheta₂ z τ) = jacobiTheta₂ (conj z) (-conj τ) := by
| Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 407 | 412 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... |
theorem nextCoeff_of_natDegree_pos (hp : 0 < p.natDegree) :
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 316 | 317 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | simp [mul_comm (j + 1)]
/-- Odd Bernoulli numbers (greater than 1) are zero. -/
theorem bernoulli'_odd_eq_zero {n : ℕ} (h_odd : Odd n) (hlt : 1 < n) : bernoulli' n = 0 := by
let B := mk fun n => bernoulli' n / (n ! : ℚ)
suffices (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1) by
rcases mul_eq_mul_right_if... | Mathlib/NumberTheory/Bernoulli.lean | 158 | 177 |
/-
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... | · simp
theorem measure_diff_null' (h : μ (s₁ ∩ s₂) = 0) : μ (s₁ \ s₂) = μ s₁ :=
measure_congr <| diff_ae_eq_self.2 h
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 220 | 223 |
/-
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... | constructor
· rintro ⟨h0|h0, h1|h1, h2⟩ <;> simp [h0, h1, h2]
| Mathlib/Data/Set/Prod.lean | 599 | 600 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
/-!
# Surreal numbers
The basic theory of surreal numbers, built on top ... | /-- A small set of surreals is bounded below. -/
lemma bddBelow_of_small (s : Set Surreal.{u}) [Small.{u} s] : BddBelow s := by
simpa using bddBelow_range_of_small (Subtype.val : s → Surreal.{u})
| Mathlib/SetTheory/Surreal/Basic.lean | 435 | 437 |
/-
Copyright (c) 2023 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
/-!
# The binomial distributio... | theorem binomial_apply (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) (i : Fin (n + 1)) :
binomial p h n i = p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ) := by
lift p to ℝ≥0 using ne_top_of_lt <| h.trans_lt one_lt_top
| Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 40 | 42 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | rw [neg_lt, ← inv_gold]
exact inv_lt_one_of_one_lt₀ one_lt_gold
| Mathlib/Data/Real/GoldenRatio.lean | 112 | 114 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,047 | 2,048 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Set.Sigma
import Mathlib.Order.CompleteLattice.Finset
/-!
# Finite sets in a ... |
end Finset
| Mathlib/Data/Finset/Sigma.lean | 211 | 216 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.Preserve... | instance hasColimitsEssentiallySmallSite
[HasColimits <| Sheaf ((equivSmallModel C).inverse.inducedTopology J) A] :
HasColimitsOfSize.{max v₃ w, max v₃ w} <| Sheaf J A :=
Adjunction.has_colimits_of_equivalence ((equivSmallModel C).sheafCongr J
((equivSmallModel C).inverse.inducedTopology J) A).functor
en... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 204 | 212 |
/-
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... | /-- The RHS is a noncomputable version of `Nat.floorRoot` with better order theoretical
properties. -/
lemma floorRoot_def :
floorRoot n a = if n = 0 ∨ a = 0 then 0 else (a.factorization ⌊/⌋ n).prod (· ^ ·) := by
unfold floorRoot; split_ifs with h <;> simp [Finsupp.floorDiv_def, prod_mapRange_index pow_zero]
| Mathlib/Data/Nat/Factorization/Root.lean | 52 | 56 |
/-
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.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... |
section Const
/-! #### Constants -/
| Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 167 | 172 |
/-
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 | 786 | 787 | |
/-
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... |
alias ⟨_, IsRadical.radical⟩ := radical_eq_iff
| Mathlib/RingTheory/Ideal/Operations.lean | 718 | 720 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Polynomial.Module.AEval
/-!
# Polynomial module
In this file, we define the polynomial module for an `R`-module `M`, i.e. the `R[X]`-module `M[X]`.... | lemma aeval_equivPolynomial {S : Type*} [CommRing S] [Algebra S R]
(f : PolynomialModule S S) (x : R) :
aeval x (equivPolynomial f) = eval x (map R (Algebra.linearMap S R) f) := by
induction f using induction_linear with
| zero => simp
| add f g e₁ e₂ => simp_rw [map_add, e₁, e₂]
| single i m => rw [equ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 296 | 303 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Lattice
/-!
# Formal concept analysis
This file defines concept lattices. A concept of a relation `r : α → β → Prop` is a pair of sets
`s : Set ... | bot_le _ := snd_subset_snd_iff.1 <| subset_univ _
instance : SupSet (Concept α β r) :=
⟨fun S =>
{ fst := extentClosure r (⋂ c ∈ S, (c : Concept _ _ _).snd)
snd := ⋂ c ∈ S, (c : Concept _ _ _).snd
| Mathlib/Order/Concept.lean | 235 | 240 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Congruence subgroups
This defines congruence subgroups of `SL(2, ℤ)` such as `Γ(N)`, `Γ₀(N)` and `Γ₁(N)... | (h : IsCongruenceSubgroup Γ) : IsCongruenceSubgroup (g • Γ) := by
obtain ⟨N, HN⟩ := h
refine ⟨N, ?_⟩
rw [← Gamma_cong_eq_self N g, Subgroup.pointwise_smul_le_pointwise_smul_iff]
| Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 203 | 206 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.Int
/-! # ℤ as a normed group -/
open NNReal
variable {... | _ = ‖n‖ := (norm_eq_abs n).symm
theorem abs_le_floor_nnreal_iff (z : ℤ) (c : ℝ≥0) : |z| ≤ ⌊c⌋₊ ↔ ‖z‖₊ ≤ c := by
rw [Int.abs_eq_natAbs, Int.ofNat_le, Nat.le_floor_iff (zero_le c), NNReal.natCast_natAbs z]
end Int
| Mathlib/Analysis/Normed/Group/Int.lean | 36 | 41 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 3,353 | 3,353 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | theorem pow_lt_ascFactorial (n : ℕ) : ∀ {k : ℕ}, 2 ≤ k → (n + 1) ^ k < (n + 1).ascFactorial k
| 0 => by rintro ⟨⟩
| 1 => by intro; contradiction
| k + 2 => fun _ => pow_lt_ascFactorial' n k
theorem ascFactorial_le_pow_add (n : ℕ) : ∀ k : ℕ, (n+1).ascFactorial k ≤ (n + k) ^ k
| Mathlib/Data/Nat/Factorial/Basic.lean | 277 | 282 |
/-
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
-/
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | (mul_lt_mul_of_pos_left (pow_lt_pow_right_of_lt_one₀ (T_pos hnorm hnsol)
(T_lt_one hnorm) (by norm_num)) (deriv_norm_pos hnorm))
_ = ‖F.eval a‖ / ‖F.derivative.eval a‖ := by
rw [T_gen, sq, pow_one, norm_div, ← mul_div_assoc, PadicInt.padic_norm_e_of_padicInt,
PadicInt.coe_mul, padicNormE... | Mathlib/NumberTheory/Padics/Hensel.lean | 357 | 364 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | | n + 1 => by rw [Nat.mul_succ, Nat.cast_add, natCast_mul m n, Nat.cast_succ, mul_add_one]
@[simp, norm_cast]
theorem natCast_sub (m n : ℕ) : ((m - n : ℕ) : Ordinal) = m - n := by
rcases le_total m n with h | h
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,064 | 1,068 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... |
@[simp]
theorem coe_one : ((1 : L₁ →ₗ⁅R⁆ L₁) : L₁ → L₁) = _root_.id :=
rfl
| Mathlib/Algebra/Lie/Basic.lean | 401 | 404 |
/-
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.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 623 | 634 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Algebra.Group.Hom.Basic
import Mathlib.Logic.Equiv.Basic
/-!
# Multiplicative ... | Mathlib/Algebra/Group/Equiv/Basic.lean | 567 | 569 | |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 790 | 791 | |
/-
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.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... | @[reassoc]
theorem yang_baxter (X Y Z : C) :
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫
(α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom := by
rw [← braiding_tensor_right_assoc... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 146 | 153 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | /-! ### Descents -/
| Mathlib/GroupTheory/Coxeter/Length.lean | 269 | 269 |
/-
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.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... |
@[simp]
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 180 | 181 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... |
theorem neg_one_left (a : R) : Commute (-1) a :=
SemiconjBy.neg_one_left a
| Mathlib/Algebra/Ring/Commute.lean | 82 | 85 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,098 | 1,102 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Cast.Field
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In t... | `PowerSeries.invOneSubPow S d` is equal to `(1 - X)⁻¹ ^ d`.
-/
theorem invOneSubPow_eq_inv_one_sub_pow :
invOneSubPow S d =
(Units.mkOfMulEqOne (1 - X) (mk 1 : S⟦X⟧) <|
Eq.trans (mul_comm _ _) (mk_one_mul_one_sub_eq_one S))⁻¹ ^ d := by
induction d with
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 132 | 138 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | exact Finsupp.sum_add_index (fun i _ ↦ hf i) (fun a _ b₁ b₂ ↦ h_add a b₁ b₂)
| Mathlib/Algebra/Polynomial/Basic.lean | 880 | 881 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | theorem gcd_least_linear {a b : ℤ} (ha : a ≠ 0) :
| Mathlib/Data/Int/GCD.lean | 324 | 324 |
/-
Copyright (c) 2023 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.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
/-!
# Derivatives of `x ↦ x⁻¹` and `f x / g x`
In this... | refine (isBigO_refl (fun p : 𝕜 × 𝕜 => p.1 - p.2) _).mul_isLittleO ((isLittleO_one_iff 𝕜).2 ?_)
rw [← sub_self (x * x)⁻¹]
exact tendsto_const_nhds.sub ((continuous_mul.tendsto (x, x)).inv₀ <| mul_ne_zero hx hx)
theorem hasDerivAt_inv (x_ne_zero : x ≠ 0) : HasDerivAt (fun y => y⁻¹) (-(x ^ 2)⁻¹) x :=
(hasStric... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 48 | 61 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | theorem Antitone.alternating_series_le_tendsto
(hfl : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i) atTop (𝓝 l))
(hfa : Antitone f) (k : ℕ) : ∑ i ∈ range (2 * k), (-1) ^ i * f i ≤ l := by
have hm : Monotone (fun n ↦ ∑ i ∈ range (2 * n), (-1) ^ i * f i) := by
refine monotone_nat_of_le_succ (fun n ↦ ?_)
... | Mathlib/Analysis/SpecificLimits/Normed.lean | 779 | 789 |
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.RingTheory.DedekindDom... | `FaithfulSMul.algebraMap_injective`.
-/
lemma coe_ne_zero_iff {x : 𝓞 K} : algebraMap _ K x ≠ 0 ↔ x ≠ 0 :=
map_ne_zero_iff _ coe_injective
| Mathlib/NumberTheory/NumberField/Basic.lean | 225 | 229 |
/-
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 | 789 | 793 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... |
@[simp]
| Mathlib/Logic/Equiv/Basic.lean | 633 | 634 |
/-
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.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | instance isEmpty_zero_toPGame_leftMoves : IsEmpty (toPGame 0).LeftMoves := by
rw [toPGame_leftMoves]; infer_instance
instance isEmpty_toPGame_rightMoves (o : Ordinal) : IsEmpty o.toPGame.RightMoves := by
| Mathlib/SetTheory/Game/Ordinal.lean | 46 | 49 |
/-
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.Lie.InvariantForm
import Mathlib.Algebra.Lie.Semisimple.Basic
import Mathlib.Algebra.Lie.TraceForm
/-!
# Lie algebras with non-degenerate Killing fo... | lemma killingForm_nondegenerate :
(killingForm R L).Nondegenerate := by
simp [LinearMap.BilinForm.nondegenerate_iff_ker_eq_bot]
| Mathlib/Algebra/Lie/Killing.lean | 63 | 65 |
/-
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.GroupTheory.QuotientGroup.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Stabiliz... |
variable {s : Set G}
open scoped RightActions in
| Mathlib/Algebra/Pointwise/Stabilizer.lean | 103 | 106 |
/-
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.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... |
open Finset in
/-- If an affine independent finset is contained in the affine span of another finset, then its
cardinality is at most the cardinality of that finset. -/
lemma AffineIndependent.card_le_card_of_subset_affineSpan {s t : Finset V}
(hs : AffineIndependent k ((↑) : s → V)) (hst : (s : Set V) ⊆ affineSpa... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 224 | 230 |
/-
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.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | · rintro y ⟨a, b, h, su⟩
symm at su
rw [← sub_eq_iff_eq_add] at su
rw [← su, RingHom.map_sub, mkRingHom_rel h, sub_self]
· simp
| Mathlib/Algebra/RingQuot.lean | 493 | 497 |
/-
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.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | exact LocallyConnectedSpace.open_connected_basis (f x) |>.restrict_subset
(h.isOpen_range.mem_nhds <| mem_range_self _) |>.comap _
theorem IsOpen.locallyConnectedSpace [LocallyConnectedSpace α] {U : Set α} (hU : IsOpen U) :
LocallyConnectedSpace U :=
hU.isOpenEmbedding_subtypeVal.locallyConnectedSpace
end... | Mathlib/Topology/Connected/LocallyConnected.lean | 135 | 142 |
/-
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... | a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
def HasFPowerSeriesAt (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (x : E) :=
∃ r, HasFPowerSeriesOnBall f p x r
/-- Analogue of `HasFPowerSeriesAt` where convergence is required only on a set `s`. -/
def HasFPowerSeries... | Mathlib/Analysis/Analytic/Basic.lean | 426 | 433 |
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