Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Order.OrderClosed
/-!
# Topological lattices
In this file we define mixin classes `ContinuousI... |
variable [Max L] [ContinuousSup L] {f g : X → L} {s : Set X} {x : X}
lemma ContinuousAt.sup' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
| Mathlib/Topology/Order/Lattice.lean | 168 | 171 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.W.Basic
/-!
# Polynomial functors
This file defines polynomial functors and the W-type construction as a
polynomial functor. (For the M-type cons... | ⟨x.1.1, fun a₂ => ⟨x.1.2 a₂, fun a₁ => x.2 ⟨a₂, a₁⟩⟩⟩
end PFunctor
/-
| Mathlib/Data/PFunctor/Univariate/Basic.lean | 158 | 162 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Module.Projective
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# ... | theorem rank_comp_le (g : V →ₗ[K] V') (f : V' →ₗ[K] V'₁) :
rank (f.comp g) ≤ min (rank f) (rank g) := by
| Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 72 | 73 |
/-
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, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | rw [trace_mul_comm]
simp
| Mathlib/LinearAlgebra/Trace.lean | 106 | 108 |
/-
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.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | simp [inl, fst]
@[reassoc (attr := simp)]
lemma inl_v_snd_v (p q : ℤ) (hpq : p + (-1) = q) :
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 77 | 80 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... | theorem abs_det_adjustToOrientation [Nonempty ι] (e : Basis ι R M)
(x : Orientation R M ι) (v : ι → M) : |(e.adjustToOrientation x).det v| = |e.det v| := by
rcases e.det_adjustToOrientation x with h | h <;> simp [h]
end Basis
end LinearOrderedCommRing
| Mathlib/LinearAlgebra/Orientation.lean | 311 | 317 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
/-!
# Tagged partitions
A tagged (pre)partition is a (pre)partition `π` enriched with a tagged point for each ... |
theorem disjUnion_tag_of_mem_right (h : Disjoint π₁.iUnion π₂.iUnion) (hJ : J ∈ π₂) :
| Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | 318 | 319 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 394 | 396 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... |
/-- `f.IsSwap` indicates that the permutation `f` is a transposition of two elements. -/
| Mathlib/GroupTheory/Perm/Support.lean | 191 | 192 |
/-
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... |
namespace ENNReal
/-- The real power function `x^y` on extended nonnegative reals, defined for `x : ℝ≥0∞` and
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 431 | 434 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Algebra.Module.LinearMap.Rat
import Mathlib.Tactic.Module
/... | + $(add_left_aux7 𝕜 y z) - $(add_left_aux8 𝕜 y z))
have H := congr(𝓚 $H_re + I * 𝓚 $H_im)
simp only [inner_, map_add, map_sub, map_neg, map_mul, map_ofNat] at H ⊢
linear_combination H / 8
| Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 182 | 185 |
/-
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 | 1,372 | 1,375 | |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | theorem map_eq_of_subset {f : α ↪ α} (h : f '' s ⊆ s) (hs : s.Finite := by toFinite_tac) :
f '' s = s :=
eq_of_subset_of_ncard_le h (ncard_map _).ge hs
| Mathlib/Data/Set/Card.lean | 731 | 734 |
/-
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.Shift.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
/-! Sequences of functors from a category equipped with a shift
Let `F : C... |
/--
When `f : X ⟶ Y⟦m⟧`, `m + n = mn`, `n + a = a'` and `ha'' : m + a' = a''`, this lemma
relates the two morphisms `F.shiftMap f a' a'' ha''` and `(F.shift a).map (f⟦n⟧')`. Indeed,
| Mathlib/CategoryTheory/Shift/ShiftSequence.lean | 207 | 210 |
/-
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.Module.Torsion
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.... | rw [← sub_eq_zero]
exact I.mem_toCotangent_ker
| Mathlib/RingTheory/Ideal/Cotangent.lean | 69 | 71 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Nat.Factors
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
impo... | Mathlib/NumberTheory/FLT/Four.lean | 319 | 326 | |
/-
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 ... | rfl
| Mathlib/Algebra/Lie/Submodule.lean | 938 | 938 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johan Commelin
-/
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
/-!
# Minimal polynomials
This file defines the minimal polynomial of an element `x` of an `A... |
theorem map_ne_one [Nontrivial B] {R : Type*} [Semiring R] [Nontrivial R] (f : A →+* R) :
(minpoly A x).map f ≠ 1 := by
by_cases hx : IsIntegral A x
| Mathlib/FieldTheory/Minpoly/Basic.lean | 100 | 103 |
/-
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 | 952 | 953 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Star
/-!
# Star operations on derivatives
This file contains the usual formulas (and ... | simpa using h.star.hasDerivAtFilter
protected nonrec theorem HasDerivWithinAt.star (h : HasDerivWithinAt f f' s x) :
| Mathlib/Analysis/Calculus/Deriv/Star.lean | 31 | 33 |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | theorem degree_pos_of_irreducible (hp : Irreducible p) : 0 < p.degree :=
lt_of_not_ge fun hp0 =>
have := eq_C_of_degree_le_zero hp0
not_irreducible_C (p.coeff 0) <| this ▸ hp
theorem X_sub_C_mul_divByMonic_eq_sub_modByMonic {K : Type*} [Ring K] (f : K[X]) (a : K) :
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 586 | 591 |
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
import Mathlib.RingTheory.LocalRing.RingHom.Basic
import Mathlib.R... | · rw [ha0, addVal_zero, top_le_iff, addVal_eq_top_iff] at h
rw [h]
apply dvd_zero
| Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 441 | 443 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Hull
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Topology.Bornology.A... | Absorbs 𝕜 s t ↔ ∀ᶠ c : 𝕜 in 𝓝 0, MapsTo (c • ·) t s := by
rw [← nhdsNE_sup_pure, Filter.eventually_sup, Filter.eventually_pure,
← absorbs_iff_eventually_nhdsNE_zero, and_iff_left]
intro x _
simpa only [zero_smul]
| Mathlib/Analysis/LocallyConvex/Basic.lean | 178 | 183 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Seminorm
import Mathlib.GroupTheory.GroupAction.Pointwise
/-!
# The Minkowski functional, normed field version
In this file we define `(eg... | one_smul 𝕜 s ▸ mem_smul_of_egauge_lt hs (by simpa)
lemma egauge_eq_zero_iff : egauge 𝕜 s x = 0 ↔ ∃ᶠ c : 𝕜 in 𝓝 0, x ∈ c • s := by
refine (iInf₂_eq_bot _).trans ?_
rw [(nhds_basis_uniformity uniformity_basis_edist).frequently_iff]
simp [and_comm]
@[simp]
lemma egauge_univ [(𝓝[≠] (0 : 𝕜)).NeBot] : egauge ... | Mathlib/Analysis/Convex/EGauge.lean | 127 | 135 |
/-
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.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
/-!
## `ℤ` forms a conditionally complete linear order
The integer... | have : s.Nonempty ∧ BddBelow s := ⟨Hinh, b, Hb⟩
simp only [sInf, dif_pos this]
| Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 69 | 70 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Bilinear form
This file defines a bilinear form over a module. ... |
@[simp]
theorem neg_apply (x y : M₁) : (-B₁) x y = -B₁ x y :=
rfl
| Mathlib/LinearAlgebra/BilinearForm/Basic.lean | 109 | 112 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | obtain ⟨n, rfl⟩ := hn
rw [pow_add, pow_one, hχ.pow_even (even_two_mul _), one_mul]
/-- A multiplicative character `χ` into an integral domain is quadratic
if and only if `χ^2 = 1`. -/
lemma isQuadratic_iff_sq_eq_one {M R : Type*} [CommMonoid M] [CommRing R] [NoZeroDivisors R]
| Mathlib/NumberTheory/MulChar/Basic.lean | 496 | 501 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | det_eq_elem_of_subsingleton _ _
theorem det_mul_aux {M N : Matrix n n R} {p : n → n} (H : ¬Bijective p) :
(∑ σ : Perm n, ε σ * ∏ x, M (σ x) (p x) * N (p x) x) = 0 := by
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 116 | 119 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Trace.Basic
import Mathlib.RingTheory.Norm.Basic
/-!
# Discriminant of a famil... | Mathlib/RingTheory/Discriminant.lean | 321 | 329 | |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
/-!
# Diagonal matrices
This file co... | ext i j
by_cases g : i = j; · rw [g, transpose_apply]
simp [h g, h (Ne.symm g)]
/-- The block matrix `A.fromBlocks 0 0 D` is diagonal if `A` and `D` are diagonal. -/
| Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 131 | 135 |
/-
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.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... |
/-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`. -/
theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) :
cylinder x n = { y | res y n = res x n } := by
ext y
dsimp [cylinder]
rw [res_eq_res]
end Res
/-!
### A distance function on `Π n, E n`
We define a distance fun... | Mathlib/Topology/MetricSpace/PiNat.lean | 223 | 236 |
/-
Copyright (c) 2014 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.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | Mathlib/Data/Bool/Basic.lean | 246 | 246 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... | end CommRing
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 252 | 253 |
/-
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.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... |
@[simp] theorem isClique_map_image_iff {f : α ↪ β} :
(G.map f).IsClique (f '' s) ↔ G.IsClique s := by
rw [isClique_map_iff, f.injective.subsingleton_image_iff]
obtain (hs | hs) := s.subsingleton_or_nontrivial
· simp [hs, IsClique.of_subsingleton]
simp [or_iff_right hs.not_subsingleton, Set.image_eq_image f... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 119 | 130 |
/-
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-... | (eq_of_le_of_not_lt (hle (hs.min (nonempty_of_mem ha))
(hs.min_mem (nonempty_of_mem ha))) (hs.not_lt_min (nonempty_of_mem ha) ha)).symm
| Mathlib/Order/WellFoundedSet.lean | 624 | 626 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | lemma c_relation_of_char_three : W.c₄ ^ 3 = W.c₆ ^ 2 := by
linear_combination -W.c_relation + 576 * W.Δ * CharP.cast_eq_zero R 3
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 213 | 214 |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,581 | 1,582 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | rw [Function.onFun, Set.inter_comm t2, Set.inter_comm t2]
exact Disjoint.inter_left _ (Disjoint.inter_right _ (hf_disj hij))
· exact fun i ↦ (h2 _ ht2).inter (h1 _ (hf_meas i))
theorem IndepSets.indep' {_mΩ : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ]
| Mathlib/Probability/Independence/Kernel.lean | 551 | 556 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.NormNum.Inv
/-!
# Real sign function
This file introduces and contains some results about `Real.sign` wh... | Mathlib/Data/Real/Sign.lean | 40 | 40 | |
/-
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.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
/-!
# Semisimple Lie algebras
The famous Cartan-Dynkin-Killing classification of semisim... | lie_mem := by
rintro x _ ⟨y, hy, rfl⟩
-- We need to show that `⁅x, y⁆ ∈ J` for any `x ∈ L` and `y ∈ J`.
-- Since `L` is semisimple, `x` is contained
-- in the supremum of `I` and the atoms not equal to `I`.
have hx : x ∈ I ⊔ sSup ({I' : LieIdeal R L | IsAtom I'} \ {I}) := b... | Mathlib/Algebra/Lie/Semisimple/Basic.lean | 119 | 182 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | theorem sum_choose_succ_mul (f : ℕ → ℕ → R) (n : ℕ) :
(∑ i ∈ range (n + 2), ((n + 1).choose i : R) * f i (n + 1 - i)) =
(∑ i ∈ range (n + 1), (n.choose i : R) * f i (n + 1 - i)) +
∑ i ∈ range (n + 1), (n.choose i : R) * f (i + 1) (n - i) := by
simpa only [nsmul_eq_mul] using sum_choose_succ_nsmul f ... | Mathlib/Data/Nat/Choose/Sum.lean | 218 | 222 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | rintro (rfl | ⟨r, rfl⟩) <;>
simp [inner_smul_right, norm_smul, inner_self_eq_norm_sq_to_K, inner_self_eq_norm_mul_norm,
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 682 | 683 |
/-
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... |
attribute [mk_iff] TransGen
attribute [refl] ReflGen.refl
| Mathlib/Logic/Relation.lean | 278 | 280 |
/-
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.Group.Units.Basic
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
import Mathl... | ⟨fun h => by simpa using congr_arg (constantCoeff σ k) h, fun h =>
ext fun n => by
| Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 217 | 218 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
/-!... | theorem PInfty_add_QInfty : (PInfty : K[X] ⟶ _) + QInfty = 𝟙 _ := by
dsimp only [QInfty]
simp only [add_sub_cancel]
| Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 123 | 125 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... | apply vars_pow
· intro j
| Mathlib/Algebra/MvPolynomial/Monad.lean | 314 | 315 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Notation
/-!
# Positive & negative parts
Mathematical structures posse... | lemma div_mabs_eq_inv_leOnePart_sq (a : α) : a / |a|ₘ = (a⁻ᵐ ^ 2)⁻¹ := by
rw [← mabs_div_eq_leOnePart_sq, inv_div]
| Mathlib/Algebra/Order/Group/PosPart.lean | 206 | 207 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 989 | 1,013 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | /-! Bundled morphisms can be down-cast to weaker bundlings -/
attribute [coe] MonoidHom.toOneHom
attribute [coe] AddMonoidHom.toZeroHom
| Mathlib/Algebra/Group/Hom/Defs.lean | 492 | 495 |
/-
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, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,747 | 1,768 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... |
lemma SupClosed.preimage [FunLike F β α] [SupHomClass F β α] (hs : SupClosed s) (f : F) :
SupClosed (f ⁻¹' s) :=
| Mathlib/Order/SupClosed.lean | 62 | 64 |
/-
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.SetLike.Basic
import Mathlib.ModelTheory.Semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure t... | theorem Definable.compl {s : Set (α → M)} (hf : A.Definable L s) : A.Definable L sᶜ := by
rcases hf with ⟨φ, hφ⟩
refine ⟨φ.not, ?_⟩
ext v
| Mathlib/ModelTheory/Definability.lean | 141 | 144 |
/-
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.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | @[gcongr]
theorem image₂_subset_right (hs : s ⊆ s') : image₂ f s t ⊆ image₂ f s' t :=
image₂_subset hs Subset.rfl
| Mathlib/Data/Finset/NAry.lean | 77 | 79 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | @[simp]
theorem image2_eq_empty_iff : image2 f s t = ∅ ↔ s = ∅ ∨ t = ∅ := by
| Mathlib/Data/Set/NAry.lean | 127 | 128 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.Group.Prod
/-!
# Typeclasses for power-associative structures
In this... | theorem npow_mul_assoc (k m n : ℕ) (x : M) :
(x ^ k * x ^ m) * x ^ n = x ^ k * (x ^ m * x ^ n) := by
simp only [← npow_add, add_assoc]
| Mathlib/Algebra/Group/NatPowAssoc.lean | 65 | 67 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Adjoint... | exact hasEigenvalue_of_hasEigenvector (T'.prop.hasEigenvector_of_isMinOn hx₀_ne this)
theorem subsingleton_of_no_eigenvalue_finiteDimensional (hT : T.IsSymmetric)
(hT' : ∀ μ : 𝕜, Module.End.eigenspace (T : E →ₗ[𝕜] E) μ = ⊥) : Subsingleton E :=
(subsingleton_or_nontrivial E).resolve_right fun _h =>
absurd... | Mathlib/Analysis/InnerProductSpace/Rayleigh.lean | 262 | 277 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | /-- If `s` is a convex set, then `a • s + b • interior s ⊆ interior s` for all `0 ≤ a`, `0 < b`,
`a + b = 1`. See also `Convex.combo_closure_interior_subset_interior` for a stronger version. -/
theorem Convex.combo_self_interior_subset_interior {s : Set E} (hs : Convex 𝕜 s) {a b : 𝕜}
(ha : 0 ≤ a) (hb : 0 < b) (ha... | Mathlib/Analysis/Convex/Topology.lean | 157 | 160 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | /-- For any circle integrable function `f`, the power series `cauchyPowerSeries f c R`, `R > 0`,
converges to the Cauchy integral `(2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z` on the open
disc `Metric.ball c R`. -/
theorem hasSum_cauchyPowerSeries_integral {f : ℂ → E} {c : ℂ} {R : ℝ} {w : ℂ}
(hf : CircleInt... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 541 | 561 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | | _, _, ⟨_, rfl⟩, ⟨_, rfl⟩ => trivial
open Int in
protected theorem nonneg_total : ∀ a : ℤ√d, Nonneg a ∨ Nonneg (-a)
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 612 | 615 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.A... | have : h = 0 :=
by_contra fun hne =>
have : F.derivative.eval a + q * h = 0 :=
(eq_zero_or_eq_zero_of_mul_eq_zero this).resolve_right hne
have : F.derivative.eval a = -q * h := by simpa using eq_neg_of_add_eq_zero_left this
lt_irrefl ‖F.derivative.eval a‖
(calc
‖F.deriv... | Mathlib/NumberTheory/Padics/Hensel.lean | 444 | 466 |
/-
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... | @[deprecated (since := "2025-03-13")] alias CompleteLattice.isStronglyCoatomic :=
Lattice.isStronglyCoatomic
end Lattice
section IsModularLattice
variable [Lattice α] [BoundedOrder α] [IsModularLattice α]
namespace IsCompl
variable {a b : α} (hc : IsCompl a b)
| Mathlib/Order/Atoms.lean | 1,042 | 1,053 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | · intro g₁ hg₁ Hg₁ g₂ hg₂ Hg₂
simpa only [eq_comm] using this p hf hp n₂ n₁ (le_of_not_le hn) g₂ hg₂ Hg₂ g₁ hg₁ Hg₁
have hf0 : f ≠ 0 := hf.ne_zero
intros g₁ hg₁ hgf₁ g₂ hg₂ hgf₂
rw [le_iff_exists_add] at hn
rcases hn with ⟨k, rfl⟩
rw [← hgf₁, pow_add, expand_mul, expand_inj (pow_pos hp n₁)] at hgf₂
su... | Mathlib/FieldTheory/Separable.lean | 406 | 415 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.Kronecker
import Mathlib.RingTheory.TensorProduct.Basic
... |
/-- (Implementation detail).
The function underlying `(A ⊗[R] Matrix n n R) →ₐ[R] Matrix n n A`,
| Mathlib/RingTheory/MatrixAlgebra.lean | 93 | 95 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Logic.Small.Basic
import Mathlib.RingTheory.Ideal.Defs
/-!
# Injecti... | Mathlib/Algebra/Module/Injective.lean | 456 | 459 | |
/-
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.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... |
open scoped Classical in
theorem condExp_of_sigmaFinite (hm : m ≤ m₀) [hμm : SigmaFinite (μ.trim hm)] :
| Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 126 | 128 |
/-
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, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | have lintegral_norm_tendsto_zero :
Tendsto (fun n => ENNReal.toReal <| ∫⁻ a, ENNReal.ofReal ‖fs n a - f a‖ ∂μ) atTop (𝓝 0) :=
(tendsto_toReal zero_ne_top).comp
(tendsto_lintegral_norm_of_dominated_convergence fs_measurable
bound_integrable.hasFiniteIntegral h_bound h_lim)
convert lintegral_no... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,045 | 1,051 |
/-
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.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... |
theorem trim_sum_ge {ι} (m : ι → OuterMeasure α) : (sum fun i => (m i).trim) ≤ (sum m).trim :=
fun s => by
simp only [sum_apply, trim_eq_iInf, le_iInf_iff]
exact fun t st ht =>
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 363 | 367 |
/-
Copyright (c) 2022 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
/-!
# Subtraction of measures
In this file we define `μ - ν` to be the least measure `τ` such that `μ ≤ ... | -- Now, we demonstrate `μ - ν = measure_sub`, and apply it.
have h_measure_sub_add : ν + measure_sub = μ := by
ext1 t h_t_measurable_set
simp only [Pi.add_apply, coe_add]
rw [MeasureTheory.Measure.ofMeasurable_apply _ h_t_measurable_set, add_comm,
tsub_add_cancel_of_le (h₂ t)]
have h_measure_sub... | Mathlib/MeasureTheory/Measure/Sub.lean | 71 | 97 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | (N.comap f).IsBase B ↔ N.IsBasis (f '' B) (f '' (f ⁻¹' N.E)) ∧ B.InjOn f ∧ B ⊆ f ⁻¹' N.E := by
rw [← isBasis_ground_iff, comap_isBasis_iff]; rfl
| Mathlib/Data/Matroid/Map.lean | 216 | 218 |
/-
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 ... | le_trans (le_add_of_nonneg_right dist_nonneg) <|
add_le_add_right (le_add_of_nonneg_left dist_nonneg) _
| Mathlib/Topology/MetricSpace/Gluing.lean | 228 | 229 |
/-
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 | 922 | 926 | |
/-
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... |
@[simp]
theorem map_apply_ι (f : M →ₗ[R] N) (m : M) : map f (ι R m) = ι R (f m) :=
CliffordAlgebra.map_apply_ι _ m
@[simp]
theorem map_apply_ιMulti {n : ℕ} (f : M →ₗ[R] N) (m : Fin n → M) :
map f (ιMulti R n m) = ιMulti R n (f ∘ m) := by
rw [ιMulti_apply, ιMulti_apply, map_list_prod]
simp only [List.map_ofF... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 361 | 373 |
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... |
theorem IsPrimitive.ne_zero [Nontrivial R] {p : R[X]} (hp : p.IsPrimitive) : p ≠ 0 := by
rintro rfl
| Mathlib/RingTheory/Polynomial/Content.lean | 56 | 58 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | rw [norm_le_pow_iff_le_valuation x hx, mem_span_pow_iff_le_valuation x hx]
theorem norm_le_pow_iff_norm_lt_pow_add_one (x : ℤ_[p]) (n : ℤ) :
‖x‖ ≤ (p : ℝ) ^ n ↔ ‖x‖ < (p : ℝ) ^ (n + 1) := by
rw [norm_def]; exact Padic.norm_le_pow_iff_norm_lt_pow_add_one _ _
theorem norm_lt_pow_iff_norm_le_pow_sub_one (x : ℤ_[... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 402 | 409 |
/-
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... |
/-- **Triangle Removal Lemma**. If there are not too many triangles (on the order of `(card α)^3`)
then they can all be removed by removing a few edges (on the order of `(card α)^2`). -/
lemma triangle_removal (hG : #(G.cliqueFinset 3) < triangleRemovalBound ε * card α ^ 3) :
∃ G' ≤ G, ∃ _ : DecidableRel G'.Adj,
... | Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean | 145 | 153 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Nontrivial.Defs
import Mathlib.Tactic.SplitIfs
import Mathlib.Logic.Basic
/-!
# Typeclasses for groups with an... | theorem mul_eq_zero_of_left {a : M₀} (h : a = 0) (b : M₀) : a * b = 0 := h.symm ▸ zero_mul b
theorem mul_eq_zero_of_right (a : M₀) {b : M₀} (h : b = 0) : a * b = 0 := h.symm ▸ mul_zero a
| Mathlib/Algebra/GroupWithZero/Defs.lean | 252 | 255 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection.Basic
/-!
# Heyting regular elements
This file defines Heyting regular elements, elements of a Heyting algebra that are their own ... | rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
| Mathlib/Order/Heyting/Regular.lean | 63 | 63 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Set.UnorderedInterval
/-!
# Locally... | variable [LocallyFiniteOrderBot α] (a : Subtype p)
theorem subtype_Iic_eq : Iic a = (Iic (a : α)).subtype p :=
rfl
| Mathlib/Order/Interval/Finset/Defs.lean | 1,170 | 1,174 |
/-
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... | rw [cancel_mono]
@[ext]
lemma cycles_ext {A : C} (f₁ f₂ : A ⟶ S.cycles) (h : f₁ ≫ S.iCycles = f₂ ≫ S.iCycles) :
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 435 | 438 |
/-
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.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | refine ⟨fun hp ↦ ?_, ?_⟩
· obtain ⟨q, hq₁, hq₂, hq₃⟩ := Ideal.exists_le_prime_disjoint _ _
(disjoint_map_primeCompl_iff_comap_le.mpr hp.le)
exact ⟨q, hq₁, le_antisymm (disjoint_map_primeCompl_iff_comap_le.mp hq₃)
| Mathlib/RingTheory/Ideal/Maps.lean | 630 | 633 |
/-
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 repeat_zero (a : Fin n → α) :
Fin.repeat 0 a = Fin.elim0 ∘ Fin.cast (Nat.zero_mul _) :=
funext fun x => (x.cast (Nat.zero_mul _)).elim0
@[simp]
theorem repeat_one (a : Fin n → α) : Fin.repeat 1 a = a ∘ Fin.cast (Nat.one_mul _) := by
generalize_proofs h
| Mathlib/Data/Fin/Tuple/Basic.lean | 429 | 435 |
/-
Copyright (c) 2020 Mathieu Guay-Paquet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mathieu Guay-Paquet
-/
import Mathlib.Order.Ideal
/-!
# Order filters
## Main definitions
Throughout this file, `P` is at least a preorder, but some sections require more struc... | bot := ⟨⊥⟩
| Mathlib/Order/PFilter.lean | 120 | 120 |
/-
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 | 917 | 934 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | theorem dimH_ball_pi_fin {n : ℕ} (x : Fin n → ℝ) {r : ℝ} (hr : 0 < r) :
dimH (Metric.ball x r) = n := by rw [dimH_ball_pi x hr, Fintype.card_fin]
theorem dimH_univ_pi (ι : Type*) [Fintype ι] : dimH (univ : Set (ι → ℝ)) = Fintype.card ι := by
simp only [← Metric.iUnion_ball_nat_succ (0 : ι → ℝ), dimH_iUnion,
... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 432 | 438 |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... | TopologicalSpace.Opens.ext <| by simp
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 350 | 351 |
/-
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.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | alias tangentCone_nonempty_of_properSpace := tangentConeAt_nonempty_of_properSpace
/-- The tangent cone at a non-isolated point in dimension 1 is the whole space. -/
theorem tangentConeAt_eq_univ {s : Set 𝕜} {x : 𝕜} (hx : AccPt x (𝓟 s)) :
tangentConeAt 𝕜 s x = univ := by
apply eq_univ_iff_forall.2 (fun y ↦ ?... | Mathlib/Analysis/Calculus/TangentCone.lean | 367 | 379 |
/-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Algebra.Order.Invertible
import Mathlib.... | rw [NormNum.IsNat.to_eq h rfl]
exact Nat.cast_nonneg n
lemma nz_of_isNegNat {n : ℕ} [Ring A] [PartialOrder A] [IsStrictOrderedRing A]
| Mathlib/Tactic/Positivity/Core.lean | 131 | 134 |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,528 | 1,529 | |
/-
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... | theorem coe_filterMap : (s.filterMap f f_inj : Set β) = {b | ∃ a ∈ s, f a = some b} :=
Set.ext (by simp only [mem_coe, mem_filterMap, Option.mem_def, Set.mem_setOf_eq, implies_true])
| Mathlib/Data/Finset/Image.lean | 547 | 549 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
theorem insert_pos [DecidableEq α] {x : α} {l : List α} (h : x ∈ l) : Insert.insert x l = l :=
| Mathlib/Data/List/Basic.lean | 137 | 138 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | section NontriviallyNormedField
variable [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
/-- Let `p i` be a family of seminorms on `E`. Let `s` be an absorbent set in `𝕜`.
If all seminorms are uniformly bounded at every point of `s`,
| Mathlib/Analysis/Seminorm.lean | 1,258 | 1,263 |
/-
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... | · exact (Nat.mul_div_cancel' (mem_properDivisors.mp h).1).symm
· rintro ⟨k, hk, rfl⟩
rw [mul_ne_zero_iff] at hn
exact mem_properDivisors.mpr ⟨⟨k, rfl⟩, lt_mul_of_one_lt_right (Nat.pos_of_ne_zero hn.1) hk⟩
| Mathlib/NumberTheory/Divisors.lean | 286 | 289 |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
/-!
# Homotopy between paths
In this file,... | Mathlib/Topology/Homotopy/Path.lean | 336 | 340 | |
/-
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.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | [Zero E] [TopologicalSpace E]
[SMul 𝕜 E] [MulAction 𝕜' E] [IsScalarTower 𝕜 𝕜' E] {s : Set E}
(h : IsVonNBounded 𝕜' s) : IsVonNBounded 𝕜 s := by
intro V hV
refine (h hV).restrict_scalars <| AntilipschitzWith.tendsto_cobounded (K := ‖(1 : 𝕜')‖₊⁻¹) ?_
refine AntilipschitzWith.of_le_mul_nndist fun ... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 408 | 413 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Antoine Chambert-Loir
-/
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.LinearAlgebra.Finsupp.SumProd
/-!... | simp [finsuppTensorFinsupp]
@[simp]
theorem finsuppTensorFinsupp_apply (f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ) :
finsuppTensorFinsupp R S M N ι κ (f ⊗ₜ g) (i, k) = f i ⊗ₜ g k := by
induction f using Finsupp.induction_linear with
| zero => simp
| add f₁ f₂ hf₁ hf₂ => simp [add_tmul, hf₁, hf₂]
| single i... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 263 | 277 |
/-
Copyright (c) 2021 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
/-! # The Wallis formula for Pi
This file establishes the Wallis product for `π` (`Real.tendsto_prod_pi_div_two`). ... | norm_num
· unfold W at IH ⊢
rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.div_mul_div_comm]
refine (div_eq_div_iff ?_ ?_).mpr ?_
any_goals exact ne_of_gt (by positivity)
simp_rw [Nat.mul_succ, Nat.factorial_succ, pow_succ]
push_cast
ring_nf
theorem W_eq_integral_sin_pow_div_int... | Mathlib/Data/Real/Pi/Wallis.lean | 62 | 75 |
/-
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... | · simp only [zero_lt_one, rpow_zero]
· rw [← neg_neg p, rpow_neg, inv_pos]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 333 | 334 |
/-
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... | -/
scoped macro (name := transfer_rw) "transfer_rw" : tactic => `(tactic|
(repeat first | rw [← to_nat_inj] | rw [← lt_to_nat] | rw [← le_to_nat]
repeat first | rw [add_to_nat] | rw [mul_to_nat] | rw [cast_one] | rw [cast_zero]))
/--
This tactic tries to prove (in)equalities about `Num`s by transferring them ... | Mathlib/Data/Num/Lemmas.lean | 323 | 329 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | def leftUnitorNatIso (a b : B) : (precomposing _ _ b).obj (𝟙 a) ≅ 𝟭 (a ⟶ b) :=
NatIso.ofComponents (λ_ ·)
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 469 | 470 |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | rcases Ordinal.lt_lift_iff.1 h with ⟨o, h', rfl⟩
rw [lt_ord, ← lift_card, lift_lt_aleph0, ← typein_enum (· < ·) h']
exact lt_aleph0_iff_fintype.2 ⟨Set.fintypeLTNat _⟩
| Mathlib/SetTheory/Ordinal/Basic.lean | 1,121 | 1,123 |
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