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 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | | add x y hx hy => simp only [map_add, Subalgebra.coe_add, hx, hy]
/-! ### Constructions needed for `CliffordAlgebra.evenEquivEvenNeg` -/
/-- One direction of `CliffordAlgebra.evenEquivEvenNeg` -/
def evenToNeg (Q' : QuadraticForm R M) (h : Q' = -Q) :
CliffordAlgebra.even Q →ₐ[R] CliffordAlgebra.even Q' :=
ev... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 218 | 227 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 727 | 731 | |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file deals with the following version... | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by
classical
rw [cutExpand_iff]
rintro ⟨_, _, _, ⟨⟩, _⟩
| Mathlib/Logic/Hydra.lean | 104 | 107 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,785 | 1,788 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.Linear
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.... | Mathlib/Algebra/Homology/HomotopyCategory.lean | 302 | 305 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | theorem product_sdiff_offDiag : s ×ˢ s \ s.offDiag = s.diag := by
rw [← diag_union_offDiag, union_sdiff_self, sdiff_eq_self_of_disjoint (disjoint_diag_offDiag _)]
theorem diag_inter : (s ∩ t).diag = s.diag ∩ t.diag :=
| Mathlib/Data/Finset/Prod.lean | 342 | 345 |
/-
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 | 494 | 495 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
@[simp]
theorem Icc_bot [OrderBot α] : Icc (⊥ : α) a = Iic a := rfl
| Mathlib/Order/Interval/Finset/Basic.lean | 514 | 516 |
/-
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.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inr z) := by
simp only [glueDist]
rw [add_right_comm, add_le_add_iff_right]
refine le_ciInf_add fun p => ciInf_le_of_le ⟨0, ?_⟩ p ?_
· exact forall_mem_range.2 fun _ => add_nonneg dist_nonneg dist_nonneg
· linarith [dist_triangle_left z (Ψ p) ... | Mathlib/Topology/MetricSpace/Gluing.lean | 112 | 119 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | rfl
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 427 | 427 |
/-
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... | theorem dist_div_sin_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.sin (∡ p₃ p₁ p₂) = dist p₁ p₃ := by
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, ... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 700 | 704 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... |
theorem natCast_eq_zero_or_one_of_two_eq_zero (n : ℕ) (h : (2 : R) = 0) :
| Mathlib/Algebra/Ring/Parity.lean | 382 | 383 |
/-
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... | · rintro ⟨sU, hU⟩ x hx
rcases h.mem_nhds_iff.mp (IsOpen.mem_nhds hU hx) with ⟨sV, ⟨⟨V, H₁, H₂⟩, hsV⟩⟩
| Mathlib/Topology/Sets/Opens.lean | 276 | 277 |
/-
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... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,116 | 2,119 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheory... | refine ⟨N, fun s hs ↦ ?_⟩
have := hN _ <| factoredNumbers_compl hs
rwa [← hsum.tsum_subtype_add_tsum_subtype_compl (factoredNumbers s),
add_sub_cancel_left, tsum_eq_tsum_diff_singleton (factoredNumbers s)ᶜ hf₀]
-- Versions of the three lemmas above for `smoothNumbers N`
| Mathlib/NumberTheory/EulerProduct/Basic.lean | 128 | 134 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | IsStronglyAtomic αᵒᵈ ↔ IsStronglyCoatomic α := by
simpa [isStronglyAtomic_iff, OrderDual.exists, OrderDual.forall,
OrderDual.toDual_le_toDual, and_comm, isStronglyCoatomic_iff] using forall_comm
@[simp] theorem isStronglyCoatomic_dual_iff_is_stronglyAtomic :
| Mathlib/Order/Atoms.lean | 382 | 386 |
/-
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.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... | subst hab
exact (h hb.isMax).elim
theorem IsSuccLimit.succ_lt (hb : IsSuccLimit b) (ha : a < b) : succ a < b :=
| Mathlib/Order/SuccPred/Limit.lean | 216 | 219 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 720 | 723 | |
/-
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.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | /-! ### Deducing differentiability from smoothness -/
variable {n : WithTop ℕ∞}
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 460 | 462 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | Mathlib/CategoryTheory/Limits/IsLimit.lean | 1,036 | 1,043 | |
/-
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)... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 217 | 220 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Co... | theorem hasDerivWithinAt_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : Set ℝ}
(ht : UniqueDiffWithinAt ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y) :
HasDerivWithinAt
(fun z => (((k + 1 : ℝ) * k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWith... | Mathlib/Analysis/Calculus/Taylor.lean | 141 | 146 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | namespace Submodule
open LinearMap
variable [Semiring R]
variable [AddCommMonoid M] [AddCommMonoid M₂]
variable [Module R M] [Module R M₂]
| Mathlib/LinearAlgebra/Prod.lean | 466 | 473 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | dsimp at h
| Mathlib/NumberTheory/LucasLehmer.lean | 390 | 390 |
/-
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.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
impo... |
/-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series is differentiable and its derivative is the sum of the derivatives. -/
theorem hasFDerivAt_tsum (hu : Summable u) (hf : ∀ n ... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 115 | 119 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Junyan Xu, Sophie Morel
-/
import Mathlib.CategoryTheory.Limits.Creates
import Mathlib.CategoryTheory.Limits.Types.Limits
import Mathlib.CategoryTheory.Limits.Types... | Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean | 126 | 133 | |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... | have := Classical.choose_spec h
set b' := Classical.choose h
simp_rw [const, mk_eq_mk, EventuallyEq, ← const_def, eventually_const] at this
rw [Function.const, this]
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 581 | 585 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | (hM : Commute (stdBasisMatrix i j 1) M) : M i i = M j j := by
have := ext_iff.mpr hM i j
aesop
/-- `M` is a scalar matrix if it commutes with every non-diagonal `stdBasisMatrix`. -/
theorem mem_range_scalar_of_commute_stdBasisMatrix {M : Matrix n n α}
| Mathlib/Data/Matrix/Basis.lean | 272 | 277 |
/-
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... | theorem exists_nhds_ne_inf_principal_neBot (hs : IsCompact s) (hs' : s.Infinite) :
∃ z ∈ s, (𝓝[≠] z ⊓ 𝓟 s).NeBot :=
hs'.exists_accPt_of_subset_isCompact hs Subset.rfl
| Mathlib/Topology/Compactness/Compact.lean | 901 | 904 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | rw [Submodule.mem_sup]
let v : K₁ := orthogonalProjection K₁ x
have hvm : x - v ∈ K₁ᗮ := sub_orthogonalProjection_mem_orthogonal x
constructor
· rintro ⟨y, hy, z, hz, rfl⟩
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 743 | 747 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Initial and terminal objects in a category.
##... | Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | 702 | 706 | |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | | n+1 => ((pell n).1 * a + d a1 * (pell n).2, (pell n).1 + (pell n).2 * a)
| Mathlib/NumberTheory/PellMatiyasevic.lean | 97 | 98 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
/-!
# Locally integrable functions
A function is called *locally integrable* (`MeasureTheory.LocallyIntegrabl... | theorem IntegrableOn.mul_continuousOn_of_subset (hg : IntegrableOn g A μ) (hg' : ContinuousOn g' K)
(hA : MeasurableSet A) (hK : IsCompact K) (hAK : A ⊆ K) :
IntegrableOn (fun x => g x * g' x) A μ := by
rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
rw [IntegrableOn, ← memLp_one_iff_integ... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 597 | 609 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | /-- The exponential characteristic is unique. -/
lemma ExpChar.eq {p q : ℕ} (hp : ExpChar R p) (hq : ExpChar R q) : p = q := by
rcases hp with ⟨hp⟩ | ⟨hp'⟩
| Mathlib/Algebra/CharP/Defs.lean | 324 | 326 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
theorem Ico_subset_Ico_iff {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁) :
Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := by
| Mathlib/Order/Interval/Finset/Basic.lean | 816 | 818 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 142 | 142 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Category.MonCat.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
/-!
# The category... | | Prequotient.of j x => (s.ι.app j) x
| one => 1
| mul x y => descFunLift _ x * descFunLift _ y
/-- The function from the colimit monoid to the cone point of any other cocone. -/
| Mathlib/Algebra/Category/MonCat/Colimits.lean | 179 | 183 |
/-
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... |
/-- The real logarithm base b is continuous as a function from nonzero reals. -/
@[fun_prop]
theorem continuous_logb : Continuous fun x : { x : ℝ // x ≠ 0 } => logb b x :=
continuous_log.div_const _
/-- The real logarithm base b is continuous as a function from positive reals. -/
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 448 | 454 |
/-
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... | /-- The type of finite measures is a measurable space when equipped with the Giry monad. -/
instance : MeasurableSpace (FiniteMeasure Ω) := Subtype.instMeasurableSpace
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 282 | 284 |
/-
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.GroupTheory.MonoidLocalization.Away
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Ma... | rw [← Ideal.unit_mul_mem_iff_mem _ (IsLocalization.map_units S s), IsLocalization.mk'_spec',
IsLocalization.mem_map_algebraMap_iff M]
simp_rw [← map_mul, IsLocalization.eq_iff_exists M, mul_comm x, ← mul_assoc, ← Submonoid.coe_mul]
exact ⟨fun ⟨⟨y, t⟩, c, h⟩ ↦ ⟨_, (c * t).2, h ▸ I.mul_mem_left c.1 y.2⟩, fun ⟨s... | Mathlib/RingTheory/Localization/Ideal.lean | 78 | 89 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | theorem card_edgeFinset_induce_support :
#(G.induce G.support).edgeFinset = #G.edgeFinset :=
card_edgeFinset_induce_of_support_subset subset_rfl
theorem map_neighborFinset_induce (v : s) :
((G.induce s).neighborFinset v).map (.subtype s)
= G.neighborFinset v ∩ s.toFinset := by
ext; simp [Set.mem_def]... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 467 | 481 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,896 | 1,899 | |
/-
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.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
/-!
# Schwarz lemma
In this file we prove several versions of the Schw... | simpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv, ← div_eq_inv_mul, ←
dist_eq_norm, div_le_iff₀ (dist_pos.2 hne)] using norm_dslope_le_div_of_mapsTo_ball hd h_maps hz
end Space
variable {f : ℂ → ℂ} {c z : ℂ} {R R₁ R₂ : ℝ}
| Mathlib/Analysis/Complex/Schwarz.lean | 154 | 160 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | -- (using `LinearMap.det_toMatrix`)
theorem det_eq_det_toMatrix_of_finset [DecidableEq M] {s : Finset M} (b : Basis s A M)
(f : M →ₗ[A] M) : LinearMap.det f = Matrix.det (LinearMap.toMatrix b b f) := by
have : ∃ s : Finset M, Nonempty (Basis s A M) := ⟨s, ⟨b⟩⟩
rw [LinearMap.coe_det, dif_pos, detAux_def'' _ b] <... | Mathlib/LinearAlgebra/Determinant.lean | 181 | 190 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | ‖x‖ ≤ √(Fintype.card ι) * ⨆ i, ‖⟪b i, x⟫‖ := by
calc ‖x‖
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 448 | 449 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... |
section Prod
| Mathlib/GroupTheory/OrderOfElement.lean | 1,083 | 1,084 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... | ∃ t ⊆ s, #t = mulRothNumber s ∧ ThreeGPFree (t : Set α) :=
Nat.findGreatest_spec (P := fun m ↦ ∃ t ⊆ s, #t = m ∧ ThreeGPFree (t : Set α))
(Nat.zero_le _) ⟨∅, empty_subset _, card_empty, by norm_cast; exact threeGPFree_empty⟩
variable {s t} {n : ℕ}
@[to_additive]
theorem ThreeGPFree.le_mulRothNumber (hs : Th... | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 270 | 277 |
/-
Copyright (c) 2018 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Data.Nat.Find
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Basic properties of holors
Holors... | simp [hbi']
· intro (hid' : Subtype.mk i _ ∉ Finset.attach (Finset.range d))
exfalso
exact absurd (Finset.mem_attach _ _) hid'
-- CP rank
/-- `CPRankMax1 x` means `x` has CP rank at most 1, that is,
| Mathlib/Data/Holor.lean | 231 | 237 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Mario Carneiro, Sean Leather
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finite sets in `Option α`
In this file we define
* `Option.t... | @[simp]
theorem insertNone_eraseNone [DecidableEq (Option α)] (s : Finset (Option α)) :
insertNone (eraseNone s) = insert none s := by ext (_ | x) <;> simp
| Mathlib/Data/Finset/Option.lean | 142 | 144 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,215 | 1,217 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | exact (o.two_zsmul_oangle_smul_left_of_ne_zero _ _ (Units.ne_zero _)).symm
/-- If the spans of two vectors are equal, twice angles with those vectors on the right are
equal. -/
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 329 | 332 |
/-
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... | Mathlib/Data/Real/Sqrt.lean | 516 | 519 | |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... |
theorem sum_divisors_eq_sum_properDivisors_add_self :
∑ i ∈ divisors n, i = (∑ i ∈ properDivisors n, i) + n := by
| Mathlib/NumberTheory/Divisors.lean | 372 | 374 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | · norm_cast at hfm; simp [hfm] }
@[simp] theorem coe_map : (N.map f : Set M') = f '' N := rfl
@[simp]
theorem toSubmodule_map : (N.map f : Submodule R M') = (N : Submodule R M).map (f : M →ₗ[R] M') :=
rfl
@[deprecated (since := "2024-12-30")] alias coeSubmodule_map := toSubmodule_map
/-- A morphism of Lie m... | Mathlib/Algebra/Lie/Submodule.lean | 793 | 805 |
/-
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.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equaliz... | simp only [map_mul, map_pow, ← op_comp, ← F.map_comp, homOfLE_comp, ← CommRingCat.comp_apply]
at e₁ e₂ ⊢
rw [e₁, e₂, mul_left_comm]
| Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 211 | 214 |
/-
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.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 119 | 123 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | antisymm ((pairwise_append.1 hs₂).2.2 _ m a mem_cons_self) (h₁ _ (by simp [m]))
rw [(@eq_replicate_iff _ a (length u₂ + 1) (a :: u₂)).2,
(@eq_replicate_iff _ a (length u₂ + 1) (u₂ ++ [a])).2] <;>
constructor <;>
| Mathlib/Data/List/Sort.lean | 123 | 126 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | | succ _, _, _, Fin2.fz => Prod.mk
| Mathlib/Data/TypeVec.lean | 441 | 442 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Data.Complex.Trigonometric
/-!
# Trigonometric functions as sums of infinite series
In t... |
/-- The power series expansion of `Real.cosh`. -/
lemma hasSum_cosh (r : ℝ) : HasSum (fun n ↦ r ^ (2 * n) / ↑(2 * n)!) (cosh r) :=
mod_cast Complex.hasSum_cosh r
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 134 | 137 |
/-
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.Probability.IdentDistrib
import Mathlib.Probability.Independence.Integrable
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mat... | aesop (add simp Nat.lt_succ_iff)
_ ≤ ∑ i ∈ range N, (i + 1) * ∫ _ in i..(i + 1 : ℕ), (1 : ℝ) ∂ρ := by
apply sum_le_sum fun i _ => ?_
simp only [Nat.cast_add, Nat.cast_one, sum_const, card_range, nsmul_eq_mul, Nat.cast_min]
refine mul_le_mul_of_nonneg_right (min_le_left _ _) ?_
... | Mathlib/Probability/StrongLaw.lean | 230 | 301 |
/-
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.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | /-- Two partial equivs that have the same `source`, same `toFun` and same `invFun`, coincide. -/
@[ext]
protected theorem ext {e e' : PartialEquiv α β} (h : ∀ x, e x = e' x)
(hsymm : ∀ x, e.symm x = e'.symm x) (hs : e.source = e'.source) : e = e' := by
have A : (e : α → β) = e' := by
ext x
exact h x
| Mathlib/Logic/Equiv/PartialEquiv.lean | 458 | 464 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Algebra.Modul... | @[simp]
theorem zero_comp (f : M →ₛₗ[σ₁₂] M₂) : ((0 : M₂ →ₛₗ[σ₂₃] M₃).comp f : M →ₛₗ[σ₁₃] M₃) = 0 :=
rfl
instance : Inhabited (M →ₛₗ[σ₁₂] M₂) :=
| Mathlib/Algebra/Module/LinearMap/Defs.lean | 785 | 789 |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... | refine' x.induction_on _ (fun x y => _) fun y z hy hz => _
· simp only [smul_zero]
· simp only [TensorProduct.smul_tmul']
show (r * x) ⊗ₜ y = _
rw [← ofMulAction_self_smul_eq_mul, smul_tprod_one_asModule]
· rw [smul_add, hz, hy, smul_add] -/
show _ = Representation.asAlgebr... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 246 | 264 |
/-
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.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... | Mathlib/Data/Int/ModEq.lean | 274 | 276 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | Measure.FiniteAtFilter.integrableAtFilter_of_tendsto
lemma Measure.integrableOn_of_bounded (s_finite : μ s ≠ ∞) (f_mble : AEStronglyMeasurable f μ)
{M : ℝ} (f_bdd : ∀ᵐ a ∂(μ.restrict s), ‖f a‖ ≤ M) :
IntegrableOn f s μ :=
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 495 | 499 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.StructurePolynomial
/-!
# Witt vectors
In this file we define the type of `p`-typical Witt vectors and ring op... | Mathlib/RingTheory/WittVector/Defs.lean | 389 | 390 | |
/-
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.Data.Nat.Fib.Basic
/-!
# Zeckendorf's Theorem
This file proves Zeckendorf's theorem: Every natural number can be written uniquely as a sum of
distinct no... | @[simp] lemma sum_zeckendorf_fib (n : ℕ) : (n.zeckendorf.map fib).sum = n := by
induction n using zeckendorf.induct <;> simp_all [fib_greatestFib_le]
| Mathlib/Data/Nat/Fib/Zeckendorf.lean | 167 | 168 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Types.Shapes
import Mathlib.Tactic.ApplyFun
/-!
# The equalizer diagram sheaf conditi... | theorem compatible_iff (x : FirstObj P X) : (Arrows.Compatible P π ((Types.productIso _).hom x)) ↔
firstMap P X π x = secondMap P X π x := by
rw [Arrows.pullbackCompatible_iff]
constructor
| Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 320 | 323 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne, Adam Topaz
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.Connected.T... | @[simp]
theorem ofLE_proj (h : A ≤ B) (x : X) : ofLE h (A.proj x) = B.proj x :=
Quotient.sound' (B.refl _)
| Mathlib/Topology/DiscreteQuotient.lean | 211 | 213 |
/-
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... | theorem ppred_to_nat : ∀ n : Num, (↑) <$> ppred n = Nat.ppred n
| 0 => rfl
| Mathlib/Data/Num/Lemmas.lean | 723 | 724 |
/-
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... | rw [neg_def]
rfl
| Mathlib/Algebra/RingQuot.lean | 234 | 235 |
/-
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.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | obtain ⟨k, hk⟩ := k
suffices genWeightSpace M ((k + 1) • α + χ) ≤ genWeightSpaceChain M α χ p q by
apply this
-- was `simpa using [...]` and very slow
-- (https://github.com/leanprover-community/mathlib4/issues/19751)
simpa only [zsmul_eq_mul, Int.cast_add, Pi.intCast_def, Int.cast_one] ... | Mathlib/Algebra/Lie/Weights/Chain.lean | 143 | 150 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 469 | 471 | |
/-
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 | 432 | 441 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.ShortComplex.Retract
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
#... | Mathlib/Algebra/Homology/QuasiIso.lean | 474 | 475 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | where the inverse is `fun x : ℝ≥0 => x ^ (1 / y)`. -/
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 415 | 415 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 992 | 996 | |
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno, Junyan Xu
-/
import Mathlib.CategoryTheory.PathCategory.Basic
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Bicategory.Free
import Mathlib... | map f := ⟨normalizeAux nil f⟩
map₂ η := eqToHom <| Discrete.ext <| normalizeAux_congr nil η
mapId _ := eqToIso <| Discrete.ext rfl
mapComp f g := eqToIso <| Discrete.ext <| normalizeAux_nil_comp f g
/-- Auxiliary definition for `normalizeEquiv`. -/
| Mathlib/CategoryTheory/Bicategory/Coherence.lean | 188 | 193 |
/-
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... | addBottom L' = L₁ ∧
(∀ k, L'.map (proj k) = ListBlank.mk ((S k).map some).reverse) ∧ S k = L₂ := by
obtain ⟨c₁, h₁, rfl⟩ := (Part.mem_map_iff _).1 H₁
obtain ⟨c₂, h₂, rfl⟩ := (Part.mem_map_iff _).1 H₂
obtain ⟨_, ⟨L', hT⟩, h₃⟩ := Turing.tr_eval (tr_respects M) (trCfg_init k L) h₂
| Mathlib/Computability/TuringMachine.lean | 725 | 729 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 764 | 767 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,862 | 1,879 | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 812 | 812 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.LocalProperties.Reduced
/-!
# Basic properties of schemes
We provide some basic properties of sche... | induction U using reduce_to_affine_global generalizing hX with
| h₁ X U H =>
obtain ⟨V, hx, i, H⟩ := H ⟨x, hx⟩
specialize H (X.presheaf.map i.op s)
rw [Scheme.basicOpen_res, hs] at H
specialize H (inf_bot_eq _) x hx
rw [← X.sheaf.presheaf.germ_res_apply i x hx s]
exact H
| h₂ X Y f =>
... | Mathlib/AlgebraicGeometry/Properties.lean | 154 | 184 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | def coprod.functorLeftComp (X Y : C) :
coprod.functor.obj (X ⨿ Y) ≅ coprod.functor.obj Y ⋙ coprod.functor.obj X :=
NatIso.ofComponents (coprod.associator _ _)
end CoprodFunctor
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,048 | 1,052 |
/-
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... | rw [prod_eq_inf, comap_inf, Filter.comap_comap, Filter.comap_comap]
theorem comap_prodMap_prod (f : α → β) (g : γ → δ) (lb : Filter β) (ld : Filter δ) :
comap (Prod.map f g) (lb ×ˢ ld) = comap f lb ×ˢ comap g ld := by
| Mathlib/Order/Filter/Prod.lean | 95 | 98 |
/-
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.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | dist p₁ s.circumcenter * dist p₁ s.circumcenter = r * r - s.circumradius * s.circumradius :=
s.dist_circumcenter_sq_eq_sq_sub_circumradius h₁ h₁' h
have hd₂ :
dist p₂ s.circumcenter * dist p₂ s.circumcenter = r * r - s.circumradius * s.circumradius :=
s.dist_circumcenter_sq_eq_sq_sub_circumradius h₂ h... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 792 | 797 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Normed.Algebra.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Data.Complex.Exponential
import Ma... | (hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x := by
have hpos : 0 < (expSeries 𝕂 𝔸).radius := (zero_le _).trans_lt hx
rw [hasFDerivAt_iff_isLittleO_nhds_zero]
suffices
(fun h => exp 𝕂 x * (exp 𝕂 (0 + h) - exp 𝕂 0 - ContinuousLin... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 92 | 107 |
/-
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 | 1,812 | 1,816 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,
Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimensi... | have hs : ⊤ ≤ span K (Set.range b') := by
intro x
have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f
have hf : f '' Set.range b' = Set.range b := by
ext x
simp [f, b', Set.mem_image, Set.mem_range]
rw [hf] at h
have hx : (x : V) ∈ span K... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 171 | 193 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# The canonical topology on a category
We define the finest (largest) Grothendieck topology for which a given presheaf `P`... | variable (J : GrothendieckTopology C)
/--
If `J` is subcanonical, we obtain a "Yoneda" functor from the defining site
into the sheaf category.
-/
| Mathlib/CategoryTheory/Sites/Canonical.lean | 235 | 240 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... | map_eq_one_iff (algebraMap _ _) (algebraMap_leftInverse _).injective
@[instance]
| Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 163 | 165 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,500 | 1,504 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... | ext
simp
| Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 114 | 116 |
/-
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.Order.Zorn
/-!
# Extend a partial order to a linear order
This file constructs a linear order which is an extension of the given partial order, using Zor... | haveI := (hc₁ hs).1
refine ⟨sSup c, ?_, fun z hz => le_sSup hz⟩
refine
{ refl := ?_
trans := ?_
antisymm := ?_ } <;>
simp_rw [binary_relation_sSup_iff]
· intro x
exact ⟨s, hs, refl x⟩
· rintro x y z ⟨s₁, h₁s₁, h₂s₁⟩ ⟨s₂, h₁s₂, h₂s₂⟩
haveI : IsPartialOrde... | Mathlib/Order/Extension/Linear.lean | 26 | 73 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Ideal
/-!
# Ideal operations for Lie algebras
Given a Lie module `M` over a Lie algebra `L`, there is a natural action of the Lie ideals of `L`... | variable [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L']
variable (f : L →ₗ⁅R⁆ L') (I : LieIdeal R L) (J : LieIdeal R L')
/-- Note that the inequality can be strict; e.g., the inclusion of an Abelian subalgebra of a
| Mathlib/Algebra/Lie/IdealOperations.lean | 235 | 238 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Complex.ext_iff.2 <| by simp [two_mul, ofReal]
| Mathlib/Data/Complex/Basic.lean | 572 | 573 |
/-
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]`.... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 336 | 339 | |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Hom.Basic
/-!
# Closure operators between preorders
We defin... | Mathlib/Order/Closure.lean | 527 | 529 | |
/-
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, Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
/-!
# Tensor products of direct sums
This file shows that... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 164 | 169 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
/-!
# Limits involving zero objects
Binary p... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 221 | 222 |
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