Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2022 Sebastian Monnet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Monnet
-/
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
/... |
end KrullT2
section TotallySeparated
instance {K L : Type*} [Field K] [Field L] [Algebra K L] [Algebra.IsIntegral K L] :
TotallySeparatedSpace (L ≃ₐ[K] L) := by
rw [totallySeparatedSpace_iff_exists_isClopen]
intro σ τ h_diff
have hστ : σ⁻¹ * τ ≠ 1 := by rwa [Ne, inv_mul_eq_one]
rcases DFunLike.exists_ne ... | Mathlib/FieldTheory/KrullTopology.lean | 262 | 276 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... | lemma smoothNumbers_compl (N : ℕ) : (N.smoothNumbers)ᶜ \ {0} ⊆ {n | N ≤ n} := by
simpa only [smoothNumbers_eq_factoredNumbers]
using factoredNumbers_compl <| Finset.filter_subset _ (Finset.range N)
/-- If `p` is positive and `n` is `p`-smooth, then every product `p^e * n` is `(p+1)`-smooth. -/
lemma pow_mul_mem_... | Mathlib/NumberTheory/SmoothNumbers.lean | 365 | 375 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
impo... | end IsDomain
end CommRing
end IsPrimitiveRoot
| Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 220 | 231 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | Mathlib/GroupTheory/GroupAction/Basic.lean | 733 | 737 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | congr_arg NNReal.toReal (opNNNorm_pi f)
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 550 | 551 |
/-
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.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | variable {β : Type v} (f : α → β) {x y : FreeGroup α}
/-- Any function from `α` to `β` extends uniquely to a group homomorphism from the free group over
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 672 | 674 |
/-
Copyright (c) 2021 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson, Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Finset.Order
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Ma... |
lemma Set.Finite.exists_minimal_le {s : Set α} (hs : s.Finite) (h : a ∈ s) :
| Mathlib/Data/Fintype/Order.lean | 184 | 185 |
/-
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.Exact
import Mathlib.CategoryTheory.ComposableArrows
/-!
# Exact sequences
A sequence of `n` composable arrows `S : ComposableArr... | for a use of this lemma. -/
lemma exact_of_δ₀ {S : ComposableArrows C (n + 2)}
(h : (mk₂ (S.map' 0 1) (S.map' 1 2)).Exact) (h₀ : S.δ₀.Exact) : S.Exact := by
rw [exact_iff_δ₀]
| Mathlib/Algebra/Homology/ExactSequence.lean | 272 | 275 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | sup_sdiff_mem_of_mem_compression ht ((erase_subset _ _).trans hvt)
(disjoint_erase_comm.2 huvs.1)
rwa [sup_eq_union, sdiff_erase (mem_union_left _ <| hvt hbv), union_erase_of_mem hat, ←
erase_union_of_mem hau] at hsb
· refine mem_shadow_iff.2
⟨(t ⊔ u) \ v,
sup_sdiff... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 319 | 420 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 394 | 394 | |
/-
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... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 749 | 755 | |
/-
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... | rw [← measure_biUnion_eq_iSup hsc]
· congr
simp only [← bex_def] at hst
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,323 | 1,325 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | @[simp]
theorem coe_toReal (θ : Angle) : (θ.toReal : Angle) = θ := by
induction θ using Real.Angle.induction_on
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 454 | 456 |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | {a b : M} (h : a ⤳ b) (n : ℕ) : (a ^ n) ⤳ (b ^ n) :=
Nat.recOn n (by simp only [pow_zero, specializes_rfl]) fun _ ihn ↦ by
simpa only [pow_succ] using ihn.mul h
| Mathlib/Topology/Algebra/Monoid.lean | 150 | 152 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | ⟨fun x y ↦ (x : α) < y⟩
| Mathlib/Order/Basic.lean | 1,001 | 1,001 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Order.Filter.Germ.OrderedMonoid
import Mathlib.Algebra.Order.Ring.Defs
/-!
# Lemmas about filters and ordered rings.
-/
namespace Filte... | [MulRightMono β] {l : Filter α} {f₁ f₂ g₁ g₂ : α → β}
(hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) : f₁ * g₁ ≤ᶠ[l] f₂ * g₂ := by
filter_upwards [hf, hg] with x hfx hgx using _root_.mul_le_mul' hfx hgx
| Mathlib/Order/Filter/Ring.lean | 27 | 30 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | (hg : (mulSupport g).Finite) : ∏ᶠ i, f i / g i = (∏ᶠ i, f i) / ∏ᶠ i, g i := by
simp only [div_eq_mul_inv, finprod_mul_distrib hf ((mulSupport_inv g).symm.rec hg),
finprod_inv_distrib]
| Mathlib/Algebra/BigOperators/Finprod.lean | 547 | 549 |
/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... |
variable (T)
/-- A theory models a (bounded) formula when any of its nonempty models realizes that formula on all
inputs. -/
def ModelsBoundedFormula (φ : L.BoundedFormula α n) : Prop :=
| Mathlib/ModelTheory/Satisfiability.lean | 275 | 280 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
... | intro x (hx : x < 0) y (hy : y < 0) hxy a b ha hb hab
have hx' : 0 < -x := by linarith
have hy' : 0 < -y := by linarith
have hxy' : -x ≠ -y := by contrapose! hxy; linarith
| Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 206 | 209 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | theorem tendsto_swap4_prod {h : Filter γ} {k : Filter δ} :
Tendsto (fun p : (α × β) × γ × δ => ((p.1.1, p.2.1), (p.1.2, p.2.2))) ((f ×ˢ g) ×ˢ (h ×ˢ k))
((f ×ˢ h) ×ˢ (g ×ˢ k)) :=
map_swap4_prod.le
theorem prod_map_map_eq.{u, v, w, x} {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
| Mathlib/Order/Filter/Prod.lean | 308 | 313 |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Basic Translation Lemmas Between Functions Defined for Continued... | @[simp]
theorem zeroth_den_eq_one : g.dens 0 = 1 :=
| Mathlib/Algebra/ContinuedFractions/Translations.lean | 116 | 117 |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | Mathlib/GroupTheory/SemidirectProduct.lean | 228 | 228 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | convert ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => 1
simp
end
section
variable {n : ℕ} {A : Type*} [SeminormedRing A] [NormedAlgebra 𝕜 A]
theorem norm_mkPiAlgebraFin_succ_le : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n.succ A‖ ≤ 1 := by
refine opNorm_le_bound zero_le_one fu... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 724 | 751 |
/-
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.Group.Pi.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... |
theorem image_update_Ioo_right (f : ∀ i, α i) (i : ι) (b : α i) :
| Mathlib/Order/Interval/Set/Pi.lean | 160 | 161 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 701 | 702 | |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | (z : Sym2 α) : motive z :=
Quot.rec f h z
/-- Dependent recursion principal for `Sym2` when the target is a `Subsingleton` type.
See `Quot.recOnSubsingleton`. -/
| Mathlib/Data/Sym/Sym2.lean | 138 | 142 |
/-
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... | lemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ :=
Subgroup.coe_eq_univ.1 <| eq_univ_of_forall fun _a ↦ smul_set_empty
@[to_additive (attr := simp)]
| Mathlib/Algebra/Pointwise/Stabilizer.lean | 29 | 32 |
/-
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.Image
import Mathlib.Order.Hom.BoundedLattice
/-!
# Complete lattice homomorphisms
This file defines frame homomorphisms and complete la... | map_bot := fun f => by
rw [← sSup_empty, map_sSup, Set.image_empty, sSup_empty] }
| Mathlib/Order/Hom/CompleteLattice.lean | 138 | 139 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | rw [mem_range] at hu
omega
| Mathlib/Data/List/Range.lean | 66 | 67 |
/-
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.Order.Monotone.Odd
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.Spe... | /-! #### `Real.cos` -/
theorem HasStrictFDerivAt.cos (hf : HasStrictFDerivAt f f' x) :
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 770 | 773 |
/-
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.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | Mathlib/Topology/Order.lean | 906 | 907 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | refl.tail hab
theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by
induction hbc with
| Mathlib/Logic/Relation.lean | 306 | 309 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | theorem card_le_three : #{a, b, c} ≤ 3 :=
| Mathlib/Data/Finset/Card.lean | 112 | 112 |
/-
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... | Mathlib/Topology/Compactness/Compact.lean | 1,180 | 1,186 | |
/-
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... | /-- The infimum edistance is invariant under isometries -/
theorem infEdist_image (hΦ : Isometry Φ) : infEdist (Φ x) (Φ '' t) = infEdist x t := by
simp only [infEdist, iInf_image, hΦ.edist_eq]
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 191 | 193 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Multiset coercion to type
This module defines a `CoeSort` instance for multi... | simp [hv]
· convert x.2.2 using 1
exact (count_cons_of_ne hv _).symm
⟩⟩
| Mathlib/Data/Multiset/Fintype.lean | 255 | 258 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,101 | 1,101 | |
/-
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.Group.Pointwise.Set.Scalar
import Mathlib.Algebra.Ring.Subsemiring.Basic
import Mathlib.R... | exist_integer_multiples M s id
/-- A choice of a common multiple of the denominators of a `Finset`-indexed family of fractions. -/
noncomputable def commonDenom {ι : Type*} (s : Finset ι) (f : ι → S) : M :=
(exist_integer_multiples M s f).choose
| Mathlib/RingTheory/Localization/Integer.lean | 107 | 111 |
/-
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.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | Mathlib/Algebra/GeomSum.lean | 65 | 65 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | ∏ x ∈ s, f x = ∏' x, Set.mulIndicator (↑s) f x := by
rw [tprod_eq_prod' (Set.mulSupport_mulIndicator_subset),
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 401 | 402 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,219 | 2,227 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | L.mapArrow.EssSurj where
mem_essImage f := by
have := Localization.essSurj_mapArrow L.op W.op
obtain ⟨g, ⟨e⟩⟩ : ∃ (g : _), Nonempty (L.op.mapArrow.obj g ≅ Arrow.mk f.hom.op) :=
⟨_, ⟨Functor.objObjPreimageIso _ _⟩⟩
exact ⟨Arrow.mk g.hom.unop, ⟨Arrow.isoMk (Arrow.rightFunc.mapIso e.symm).unop
... | Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 975 | 982 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio, Yongle Hu
-/
import Mathlib.RingTheory.DiscreteValuationRing.TFAE
import Mathlib.RingTheory.LocalProperties.IntegrallyClosed
/-!
# D... | IsScalarTower.of_algebraMap_eq fun x => (IsLocalization.lift_eq h x).symm
haveI : IsFractionRing Aₘ (FractionRing A) :=
IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M _ _
refine (isDedekindDomain_iff _ (FractionRing A)).mpr ⟨?_, ?_, ?_, ?_⟩
· infer_instance
· exact IsLocalization.isNoethe... | Mathlib/RingTheory/DedekindDomain/Dvr.lean | 88 | 108 |
/-
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 | 1,399 | 1,401 | |
/-
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.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Norm... | · have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
· exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
replace h := h.isBigO_sub_partialSum_pow k.succ
simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
exact h.continuou... | Mathlib/Analysis/Analytic/Uniqueness.lean | 96 | 101 |
/-
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... |
theorem toType_noMax_of_succ_lt {o : Ordinal} (ho : ∀ a < o, succ a < o) : NoMaxOrder o.toType :=
⟨has_succ_of_type_succ_lt (type_toType _ ▸ ho)⟩
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 334 | 336 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | protected theorem hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear :
HasFiniteFPowerSeriesOnBall (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2)
f.toFormalMultilinearSeriesOfMultilinear 0 (Fintype.card (Option ι) + 1) ⊤ := by
apply HasFiniteFPowerSeriesOnBall.mk' ?_ ENNReal.zero_lt_top ?_
· intro m hm
appl... | Mathlib/Analysis/Analytic/CPolynomial.lean | 155 | 160 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,064 | 1,066 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | refine ⟨wbtw_pointReflection _ _ _, h, ?_⟩
nth_rw 1 [← pointReflection_self R x]
exact (pointReflection_involutive R x).injective.ne h
theorem wbtw_midpoint (x y : P) : Wbtw R x (midpoint R x y) y := by
convert wbtw_pointReflection R (midpoint R x y) x
rw [pointReflection_midpoint_left]
theorem sbtw_midpoin... | Mathlib/Analysis/Convex/Between.lean | 851 | 873 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | rcases Decidable.eq_or_ne n 0 with (rfl | hn)
· simp
rcases le_or_lt b 1 with hb | hb
· interval_cases b <;> simp +arith [digits_zero_succ', hn]
simpa [digits_len, hb, hn] using log_mono_right (le_succ _)
theorem le_digits_len_le (b n m : ℕ) (h : n ≤ m) : (digits b n).length ≤ (digits b m).length :=
monoto... | Mathlib/Data/Nat/Digits.lean | 463 | 474 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Pointwise
/-! # Open mapping theorem for morphisms of topological groups
We prove t... | (f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
IsOpenMap f := by
let A : MulAction G H := MulAction.compHom _ f
have : ContinuousSMul G H := continuousSMul_compHom h'f
have : IsPretransitive G H := isPretransitive_compHom hf
have : f = (fun (g : G) ↦ g • (1 : H)) := by simp [A, MulActi... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 112 | 121 |
/-
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.HasLimits
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
/-!
# Wide equalizers and wide coequalizers
This file defines wide... | pt := t.pt
ι :=
{ app := fun X => eqToHom (by cases X <;> aesop_cat) ≫ t.ι.app X
| Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean | 436 | 438 |
/-
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.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-20")
| Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 146 | 149 | |
/-
Copyright (c) 2023 Junyan Xu, Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.G... | namespace DomMulAct
lemma mem_stabilizer_iff {g : (Perm α)ᵈᵐᵃ} :
| Mathlib/GroupTheory/Perm/DomMulAct.lean | 41 | 43 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
## Deprecation note
Many lemmas in this file have been replaced by the... | (∃ x ∈ l.zipIdx, p x) ↔ ∃ (i : ℕ) (_ : i < length l), p (l[i], i) :=
exists_mem_zipIdx.trans <| by simp
| Mathlib/Data/List/Enum.lean | 39 | 40 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,727 | 1,730 | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma
import Mathlib.Combinatorics.Simp... |
lemma triangleRemovalBound_pos (hε : 0 < ε) (hε₁ : ε ≤ 1) : 0 < triangleRemovalBound ε := by
have : 0 < 1 - ε / 4 := by linarith
unfold triangleRemovalBound
| Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean | 35 | 38 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
/-!
# Stalks
For a presheaf `F` on a topol... | lemma Γgerm_res_apply (F : X.Presheaf C)
{U : Opens X} {i : U ⟶ ⊤} (x : X) (hx : x ∈ U) [ConcreteCategory C FC] (s) :
F.germ U x hx (F.map i.op s) = F.Γgerm x s := F.germ_res_apply i x hx s
| Mathlib/Topology/Sheaves/Stalks.lean | 122 | 125 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
import Mathlib.GroupTheory.FreeAbelianGroup
/-!
# Free rings
The theory of the free ring over a type.
## Main definitions
* `FreeRing α` : the free (not com... | inferInstanceAs <| Nontrivial (FreeAbelianGroup _)
namespace FreeRing
variable {α : Type u}
/-- The canonical map from α to `FreeRing α`. -/
def of (x : α) : FreeRing α :=
FreeAbelianGroup.of (FreeMonoid.of x)
theorem of_injective : Function.Injective (of : α → FreeRing α) :=
FreeAbelianGroup.of_injective.com... | Mathlib/RingTheory/FreeRing.lean | 60 | 71 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 2,250 | 2,254 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.S... | theorem isOpen_iff {s : Set Γ₀} : IsOpen s ↔ (0 : Γ₀) ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s := by
rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]
| Mathlib/Topology/Algebra/WithZeroTopology.lean | 120 | 121 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Module.Basic
/-!
# Group and ring filter bases
A `GroupFilterBasis` is a `FilterBasis` on a group with some properties relating
the ... | HasBasis (@nhds G B.topology 1) (fun V : Set G ↦ V ∈ B) id := by
rw [B.nhds_one_eq]
exact B.toFilterBasis.hasBasis
@[to_additive]
theorem mem_nhds_one (B : GroupFilterBasis G) {U : Set G} (hU : U ∈ B) :
U ∈ @nhds G B.topology 1 := by
rw [B.nhds_one_hasBasis.mem_iff]
exact ⟨U, hU, rfl.subset⟩
-- See no... | Mathlib/Topology/Algebra/FilterBasis.lean | 171 | 183 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.BorelCantelli
import Mathlib.Probability.ConditionalExpectation
import Mathlib.Probability.Independence.Basic
/-!
# The second Borel... | (⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) μ by rwa [iSup_singleton] at this
exact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa)
theorem iIndepFun.condExp_natural_ae_eq_of_lt [SecondCountableTopology β] [CompleteSpace β]
[NormedSpace ℝ β] (hf : ∀ i, StronglyMeasurable... | Mathlib/Probability/BorelCantelli.lean | 43 | 48 |
/-
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.Order.Hom.CompleteLattice
import Mathlib.Topology.Compactness.Bases
import Mathlib.Topology.ContinuousMap.Basic
impor... | rw [Set.range_inclusion i]
exact U.isOpen.preimage continuous_subtype_val
| Mathlib/Topology/Sets/Opens.lean | 251 | 252 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | @[simp]
theorem get?_tail (s : Seq α) (n) : get? (tail s) n = get? s (n + 1) :=
rfl
/-- Recursion principle for sequences, compare with `List.recOn`. -/
| Mathlib/Data/Seq/Seq.lean | 237 | 241 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,554 | 1,558 | |
/-
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.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... |
@[simp]
theorem Ico_sub_Ico_right (a b c : α) : Ico a b - Ico c b = Ico a (min b c) := by
rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_diff_Ico_right]
| Mathlib/Order/Interval/Multiset.lean | 293 | 296 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | def image (f : α → β) (s : Finset α) : Finset β :=
(s.1.map f).toFinset
| Mathlib/Data/Finset/Image.lean | 268 | 269 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | add_le_add (le_restrict_apply _ _) (le_restrict_apply _ _)
_ = 0 := by rw [ae_iff.1 ht, ae_iff.1 htc, zero_add]
theorem mem_map_restrict_ae_iff {β} {s : Set α} {t : Set β} {f : α → β} (hs : MeasurableSet s) :
| Mathlib/MeasureTheory/Measure/Restrict.lean | 608 | 611 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction.Ring
/-!
# Relating unital and non-unital subs... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 390 | 394 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: the quadratic for... | (Q₁.prod Q₂).IsOrtho (0, m₂) (0, m₂') ↔ Q₂.IsOrtho m₂ m₂' := by
simp [isOrtho_def]
end Semiring
| Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 197 | 200 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# The pow... | ext
simp [mem_powersetCard]
| Mathlib/Data/Finset/Powerset.lean | 229 | 231 |
/-
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... | /-- An angle in a right-angled triangle expressed using `arcsin`. -/
theorem oangle_left_eq_arcsin_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arcsin (dist p₃ p₂ / dist p₁ p₃) := by
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 550 | 557 |
/-
Copyright (c) 2024 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.Analysis.Calculus.Deriv.Pi
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Analysis.InnerProductSpace.NormP... | rw [finrank_fintype_fun_eq_card, Fintype.card_fin (finrank ℝ E)]
let e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this
let c := addHaarScalarFactor μ ((volume : Measure (ι → ℝ)).map e.symm)
exact (c * ‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ ^ p) * (c ^ p)⁻¹
/-- The **Gagliardo-Nirenberg-Sobolev inequality**.... | Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean | 362 | 425 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | Iff.trans ⟨fun h h' => by simp [logb, h'] at h, div_self⟩ log_ne_zero
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 64 | 64 |
/-
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.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.WithBot
/-!
# Intervals in `WithTop α` and `WithBot α`
In this file we ... | theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 75 | 76 |
/-
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... |
theorem comp_tail {α : Sort*} {β : Sort*} (g : α → β) (q : Fin n.succ → α) :
g ∘ tail q = tail (g ∘ q) := by
ext j
simp [tail]
| Mathlib/Data/Fin/Tuple/Basic.lean | 277 | 282 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | variable {ι : Type*}
| Mathlib/Data/Real/Sqrt.lean | 413 | 413 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | @[simp]
theorem interior_Ici' {a : α} (ha : (Iio a).Nonempty) : interior (Ici a) = Ioi a := by
rw [← compl_Iio, interior_compl, closure_Iio' ha, compl_Iic]
theorem interior_Ici [NoMinOrder α] {a : α} : interior (Ici a) = Ioi a :=
| Mathlib/Topology/Order/DenselyOrdered.lean | 75 | 79 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.NormNum
/-!
# Convex and concave functions
This... | theorem ConvexOn.add (hf : ConvexOn 𝕜 s f) (hg : ConvexOn 𝕜 s g) : ConvexOn 𝕜 s (f + g) :=
⟨hf.1, fun x hx y hy a b ha hb hab =>
calc
f (a • x + b • y) + g (a • x + b • y) ≤ a • f x + b • f y + (a • g x + b • g y) :=
add_le_add (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab)
_ = a • (f x + g ... | Mathlib/Analysis/Convex/Function.lean | 189 | 195 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | LeftHomologyData.liftK_π_eq_zero_of_boundary _ k x hx
@[reassoc (attr := simp)]
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 968 | 970 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.LinearMap
import Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions
import Mathlib.MeasureTheory.Function.StronglyMeasur... |
@[deprecated (since := "2025-02-21")]
alias Memℒp.inner_const := MemLp.inner_const
| Mathlib/MeasureTheory/Function/L2Space.lean | 81 | 83 |
/-
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.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | section PartialOrder
variable [PartialOrder α] {s : Set α} {a : α}
| Mathlib/Order/WellFoundedSet.lean | 618 | 620 |
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Discriminant
/-!
# Field/algebra norm / trace and localizatio... | cases subsingleton_or_nontrivial R
· haveI : Subsingleton Rₘ := Module.subsingleton R Rₘ
simp [eq_iff_true_of_subsingleton]
let b := Module.Free.chooseBasis R S
letI := Classical.decEq (Module.Free.ChooseBasisIndex R S)
rw [Algebra.norm_eq_matrix_det (b.localizationLocalization Rₘ M Sₘ),
Algebra.norm_... | Mathlib/RingTheory/Localization/NormTrace.lean | 61 | 69 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Basic
/-! # Re... | Mathlib/Analysis/Complex/RealDeriv.lean | 135 | 136 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 261 | 262 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,408 | 1,410 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Data.Fintype.Inv
import Mathlib.GroupTheory.SpecificGroups.Cyclic
... |
variable [CommRing R] [IsDomain R] [Group G]
| Mathlib/RingTheory/IntegralDomain.lean | 111 | 113 |
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | @ext_of_Iic αᵒᵈ _ _ _ _ _ ‹_› _ _ _ h
end MeasureTheory.Measure
@[measurability]
theorem measurableSet_uIcc : MeasurableSet (uIcc a b) :=
measurableSet_Icc
@[measurability]
theorem measurableSet_uIoc : MeasurableSet (uIoc a b) :=
measurableSet_Ioc
| Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 436 | 447 |
/-
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.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | -- see Note [lower instance priority]
| Mathlib/Algebra/Ring/Defs.lean | 347 | 347 |
/-
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... | instance algebraPolynomial : Algebra R[X] A⟦X⟧ :=
| Mathlib/RingTheory/PowerSeries/Basic.lean | 940 | 940 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.Order.Nonneg.Ring
import Mathlib.Data.Nat.Cast.Order.Ring
... | rw [NNRat.smul_def]; exact mul_nonneg q.cast_nonneg ha
| Mathlib/Algebra/Order/Nonneg/Field.lean | 37 | 38 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | rw [← comap_bot', ← comap_bot', ← comap_bot', ← prod_map_comap_prod', bot_prod_bot]
| Mathlib/Algebra/Group/Submonoid/Operations.lean | 800 | 801 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... |
theorem coeff_mul_single_add [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
{b : Γ} : (x * single b r).coeff (a + b) = x.coeff a * r := by
by_cases hr : r = 0
· simp [hr, coeff_mul]
simp only [hr, coeff_smul, coeff_mul, support_single_of_ne, Ne, not_false_iff, smul_eq_mul]
by_cases hx :... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 444 | 454 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,118 | 1,120 | |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 817 | 819 | |
/-
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.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | replace this : s = 1 := by simpa [root_apply_coroot hα, key] using congr_arg α hs
rwa [this, one_smul] at hs
set α' := (cartanEquivDual H).symm α with hα'
have h_eq : h = killingForm K L e f • α' := by
simp only [hα', Subtype.ext_iff, ← ht.lie_e_f, hef]
rw [Submodule.coe_smul_of_tower]
use (2 • (α... | Mathlib/Algebra/Lie/Weights/Killing.lean | 522 | 543 |
/-
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... |
theorem mem_roots_sub_C' {p : R[X]} {a x : R} : x ∈ (p - C a).roots ↔ p ≠ C a ∧ p.eval x = a := by
rw [mem_roots', IsRoot.def, sub_ne_zero, eval_sub, sub_eq_zero, eval_C]
theorem mem_roots_sub_C {p : R[X]} {a x : R} (hp0 : 0 < degree p) :
| Mathlib/Algebra/Polynomial/Roots.lean | 154 | 158 |
/-
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.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... | (by simp [e, left_distrib]) fun m m2 d => ?_
rcases Nat.eq_or_lt_of_le (ih m m2 (dvd_trans d nd')) with me | ml
· subst me
contradiction
apply (Nat.eq_or_lt_of_le ml).resolve_left
intro me
rw [← me, e] at d
change 2 * (i + 2) ∣ n' at d
have := ih _ prime_two (dvd_trans (dvd_of_... | Mathlib/Data/Nat/Squarefree.lean | 164 | 200 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Control.Combinators
import Mathlib.Logic.Function.Defs
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
/-!
Extends the theory on functors... | bind_assoc := by
| Mathlib/Control/Basic.lean | 183 | 183 |
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