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) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | _ = (factors I).prod := by rw [Multiset.map_id']
_ = I := associated_iff_eq.mp (factors_prod hI)
)
@[simp]
theorem IsDedekindDomain.quotientEquivPiFactors_mk {I : Ideal R} (hI : I ≠ ⊥) (x : R) :
IsDedekindDomain.quotientEquivPiFactors hI (Ideal.Quotient.mk I x) = fun _P =>
Ideal.Quotient.mk... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,271 | 1,280 |
/-
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.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... |
theorem derivative_succ_aux (n ν : ℕ) :
Polynomial.derivative (bernsteinPolynomial R (n + 1) (ν + 1)) =
(n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1)) := by
rw [bernsteinPolynomial]
suffices ((n + 1).choose (ν + 1) : R[X]) * ((↑(ν + 1 : ℕ) : R[X]) * X ^ ν) * (1 - X) ^ (n - ν) -
... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 93 | 99 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limit... |
section
| Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean | 145 | 146 |
/-
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.BigOperators.Group.List.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Data.Fintype.EquivFin
import... | instance Equiv.instFintype [Fintype α] [Fintype β] : Fintype (α ≃ β) :=
if h : Fintype.card β = Fintype.card α then
Trunc.recOnSubsingleton (Fintype.truncEquivFin α) fun eα =>
| Mathlib/Data/Fintype/Perm.lean | 140 | 142 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.Equiv
/-!
# If a semiring is a field, any isomorphic semiring is also a field.
This is in a separate... | protected theorem isField {A : Type*} (B : Type*) [Semiring A] [Semiring B] (hB : IsField B)
(e : A ≃* B) : IsField A where
exists_pair_ne := have ⟨x, y, h⟩ := hB.exists_pair_ne; ⟨e.symm x, e.symm y, e.symm.injective.ne h⟩
mul_comm := fun x y => e.injective <| by rw [map_mul, map_mul, hB.mul_comm]
mul_inv_can... | Mathlib/Algebra/Field/Equiv.lean | 17 | 23 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | rw [mem_Ioi] at hx
rw [iter_deriv_zpow]
refine mul_pos ?_ (zpow_pos hx _)
norm_cast
refine int_prod_range_pos (by decide) fun hm => ?_
rw [← Finset.coe_Ico] at hm
norm_cast at hm
fin_cases hm <;> simp_all
section SqrtMulLog
theorem hasDerivAt_sqrt_mul_log {x : ℝ} (hx : x ≠ 0) :
HasDerivAt (fun x =... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 98 | 110 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.GroupTheory.GroupAction.R... |
lemma pow_dvd_sub_pow_of_pow_eq_zero_right (hp : n + m ≤ p + 1) (h_comm : Commute x y)
(hy : y ^ n = 0) : x ^ m ∣ (x - y) ^ p :=
| Mathlib/Algebra/Ring/Divisibility/Lemmas.lean | 73 | 75 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... |
lemma exact_of_iso (e : S₁ ≅ S₂) (h : S₁.Exact) : S₂.Exact := by
obtain ⟨⟨h, z⟩⟩ := h
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 109 | 111 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | ext
simp [h]
rw [← this]
exact LinearPMap.mk_apply _ _ _
theorem domRestrict_le {f : E →ₗ.[R] F} {S : Submodule R E} : f.domRestrict S ≤ f :=
| Mathlib/LinearAlgebra/LinearPMap.lean | 659 | 664 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | [MulLeftMono H] [MulRightMono H] {f : G → H}
(h : ∀ x, 1 ≤ f x) (a : G) : f |a| ≤ f a * f (-a) :=
| Mathlib/Algebra/Order/Group/Abs.lean | 251 | 252 |
/-
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.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | apply Quotient.sound
simpa using hf'
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 952 | 954 |
/-
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.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | intro x hx
simp only [g', hx, Pi.mul_apply]
· have M : MeasurableSet { x : α | f' x ≠ 0 }ᶜ :=
(hf.measurable_mk (measurableSet_singleton 0).compl).compl
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 384 | 387 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 1,193 | 1,196 | |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 328 | 344 | |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... | (f a ha).bindOnSupport fun b hb =>
g b ((mem_support_bindOnSupport_iff f b).mpr ⟨a, ha, hb⟩) := by
refine PMF.ext fun a => ?_
dsimp only [bindOnSupport_apply]
simp only [← tsum_dite_right, ENNReal.tsum_mul_left.symm, ENNReal.tsum_mul_right.symm]
| Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 255 | 259 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
/-!
# Asymptotic bounds for Jacobi theta functions
The goal of this file is to establish some tech... | lemma f_int_negSucc (k : ℕ) {a : ℝ} (ha : a ≤ 1) (t : ℝ) (n : ℕ) :
f_int k a t (Int.negSucc n) = f_nat k (1 - a) t n := by
have : (Int.negSucc n) + a = -(n + (1 - a)) := by { push_cast; ring }
rw [f_int, f_nat, this, abs_neg, neg_sq, abs_of_nonneg (by linarith)]
| Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 213 | 216 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | StructureSheaf.toOpen (A⁰_ f) U.unop ≫ (toSpec 𝒜 f).c.app U =
awayToΓ 𝒜 f ≫ (Proj| pbo f).presheaf.map (homOfLE le_top).op :=
Eq.trans (by congr) <| ΓSpec.toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app _ U
@[reassoc]
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 694 | 698 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,661 | 1,662 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | simp [imp_iff_not_or]
theorem frequently_imp_distrib_left {f : Filter α} [NeBot f] {p : Prop} {q : α → Prop} :
(∃ᶠ x in f, p → q x) ↔ p → ∃ᶠ x in f, q x := by simp [frequently_imp_distrib]
| Mathlib/Order/Filter/Basic.lean | 787 | 790 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,015 | 2,016 | |
/-
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.PartrecCode
import Mathlib.Data.Set.Subsingleton
/-!
# Computability theory and the halting problem
A universal partial recursive funct... | ⟨by infer_instance,
(cond hf (const false) (const true)).of_eq fun n => by
simp only [Bool.not_eq_true]
cases f n <;> rfl⟩
| Mathlib/Computability/Halting.lean | 176 | 179 |
/-
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 | 1,696 | 1,700 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
/-!
# Infini... | simpa only [div_eq_mul_inv, one_mul] using this
theorem Summable.const_div (h : Summable (fun x ↦ 1 / f x)) (b : α) :
Summable fun i ↦ b / f i :=
(h.hasSum.const_div b).summable
| Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 113 | 117 |
/-
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.Data.NNReal.Basic
import Mathlib.Order.Fin.Tuple
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.... |
theorem injective_coe : Injective ((↑) : Box ι → Set (ι → ℝ)) := by
rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h
simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h
congr
exacts [le_antisymm h.2.1 h.1.1, le_antisymm h.1.2 h.2.2]
@[simp, norm_cast]
theorem coe_inj : (I : Set (ι → ℝ)) = J ↔ I = J :=
inj... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 158 | 169 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | (mulSupport fun v : HeightOneSpectrum R => (v.asIdeal : FractionalIdeal R⁰ K) ^
(-((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ))).Finite := by
rw [mulSupport]
simp_rw [zpow_neg, Ne, inv_eq_one]
exact finite_mulSupport_coe hI
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 122 | 127 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
/-!
# Martingales
A family of functions `f : ι → Ω → E` is a martingale wit... | (hnonneg : ∀ n ω, 0 ≤ ξ n ω) :
Submartingale (fun n => ∑ k ∈ Finset.range n, ξ k * (f (k + 1) - f k)) 𝒢 μ := by
have hξbdd : ∀ i, ∃ C, ∀ ω, |ξ i ω| ≤ C := fun i =>
⟨R, fun ω => (abs_of_nonneg (hnonneg i ω)).trans_le (hbdd i ω)⟩
have hint : ∀ m, Integrable (∑ k ∈ Finset.range m, ξ k * (f (k + 1) - f k))... | Mathlib/Probability/Martingale/Basic.lean | 477 | 482 |
/-
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.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
/-!
# Remov... | suffices DifferentiableOn ℂ F (s \ {c}) ∧ ContinuousAt F c by
rw [differentiableOn_compl_singleton_and_continuousAt_iff hc, ← differentiableOn_dslope hc,
dslope_sub_smul] at this
have hc : Tendsto f (𝓝[≠] c) (𝓝 (deriv F c)) :=
continuousAt_update_same.mp (this.continuousOn.continuousAt hc)
r... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 71 | 87 |
/-
Copyright (c) 2022 Michael Blyth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Blyth
-/
import Mathlib.LinearAlgebra.Projectivization.Basic
/-!
# Independence in Projective Space
In this file we define independence and dependence of families of elements ... | rw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2]
dsimp only [Function.comp_def, Matrix.cons_val]
| Mathlib/LinearAlgebra/Projectivization/Independence.lean | 103 | 104 |
/-
Copyright (c) 2022 Alex J. Best, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Ring.Equiv
/-!
# Ordered ring homomorphisms
Homomorphisms between ordered (se... |
@[simp]
| Mathlib/Algebra/Order/Hom/Ring.lean | 416 | 417 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | rw [zero_add, pow_one]
infer_instance
rw [absdiscr_prime_pow_succ p 0 K]
simp only [Int.reduceNeg, pow_zero, one_mul, zero_add, mul_one, mul_eq_mul_left_iff, gt_iff_lt,
Nat.cast_pos, PNat.pos, pow_eq_zero_iff', neg_eq_zero, one_ne_zero, ne_eq, false_and, or_false]
| Mathlib/NumberTheory/Cyclotomic/Rat.lean | 603 | 607 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Batteries.Data.List.Basic
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two... | @[simp]
theorem levenshtein_nil_cons (y) (ys) :
levenshtein C [] (y :: ys) = C.insert y + levenshtein C [] ys := by
dsimp +unfoldPartialApp [levenshtein, suffixLevenshtein, impl]
congr
rw [List.getLast_eq_getElem]
congr
rw [show (List.length _) = 1 from _]
| Mathlib/Data/List/EditDistance/Defs.lean | 278 | 285 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Un... |
/-- If monoid with zero homs `f` and `g` from `ℚ≥0` agree on the positive naturals then they are
| Mathlib/Data/Rat/Cast/Defs.lean | 240 | 241 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... |
/-- A point is in an `Ioo` iff it can be expressed as a strict convex combination of the endpoints.
-/
theorem Convex.mem_Ioo (h : x < y) :
z ∈ Ioo x y ↔ ∃ a b, 0 < a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z := by
simp only [← openSegment_eq_Ioo h, openSegment, smul_eq_mul, exists_and_left, mem_setOf_eq]
/-- A p... | Mathlib/Analysis/Convex/Segment.lean | 517 | 527 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Order.Antidiag.Prod
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.NatAntidiagonal
/-!
# Antidiagonals in ℕ × ℕ as finset... | rw [← map_prodComm_antidiagonal]
simp_rw [aux₁, ← map_filter, antidiagonal_filter_le_fst_of_le h, map_map]
ext ⟨i, j⟩
simpa using aux₂ i j
/-- The set `antidiagonal n` is equivalent to `Fin (n+1)`, via the first projection. -/
@[simps]
def antidiagonalEquivFin (n : ℕ) : antidiagonal n ≃ Fin (n + 1) where
toF... | Mathlib/Data/Finset/NatAntidiagonal.lean | 145 | 155 |
/-
Copyright (c) 2022 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.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | theorem prod_X_sub_C_eq [CommRing S] {x y z : S} :
(X - C x) * (X - C y) * (X - C z) =
toPoly ⟨1, -(x + y + z), x * y + x * z + y * z, -(x * y * z)⟩ := by
rw [← one_mul <| X - C x, ← C_1, C_mul_prod_X_sub_C_eq, one_mul, one_mul, one_mul]
| Mathlib/Algebra/CubicDiscriminant.lean | 75 | 78 |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Spectrum mapping theorem
This file develops proves the spectral mappi... | rw [← Multiset.prod_toList, map_list_prod] at h
replace h := mt List.prod_isUnit h
simp only [not_forall, exists_prop, aeval_C, Multiset.mem_toList, List.mem_map, aeval_X,
exists_exists_and_eq_and, Multiset.mem_map, map_sub] at h
rcases h with ⟨r, r_mem, r_nu⟩
exact ⟨r, by rwa [mem_iff, ← IsUnit.sub_iff],... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 55 | 63 |
/-
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... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 1,014 | 1,015 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 3,069 | 3,071 | |
/-
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.CategoryTheory.Abelian.Basic
/-!
# Idempotent complete categories
In this file, we define the notion of idempotent complete categories
(also known as Karoubian... | constructor
· exact isIdempotentComplete_of_isIdempotentComplete_opposite
· intro h
apply isIdempotentComplete_of_isIdempotentComplete_opposite
rw [isIdempotentComplete_iff_of_equivalence (opOpEquivalence C)]
exact h
instance [IsIdempotentComplete C] : IsIdempotentComplete Cᵒᵖ := by
rwa [isIdempote... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 184 | 195 |
/-
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 | 1,071 | 1,072 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | theorem mk_range_eq (f : α → β) (h : Injective f) : #(range f) = #α :=
mk_congr (Equiv.ofInjective f h).symm
| Mathlib/SetTheory/Cardinal/Basic.lean | 659 | 661 |
/-
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.Order.Ring.Defs
import Mathlib.Algebra.Field.Defs
/-!
# Linear ordered (semi)fields
A ... | Mathlib/Algebra/Order/Field/Defs.lean | 87 | 88 | |
/-
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.Algebra.Group.Conj
import Mathlib.Algebra.GroupWithZero.Units.Basic
/-!
# Conjugacy in a group with zero
-/
assert_not_exists Multiset
-- TODO
-- ass... | @[simp] lemma isConj_iff₀ : IsConj a b ↔ ∃ c : α, c ≠ 0 ∧ c * a * c⁻¹ = b := by
rw [IsConj, Units.exists_iff_ne_zero (p := (SemiconjBy · a b))]
congr! 2 with c
exact and_congr_right (mul_inv_eq_iff_eq_mul₀ · |>.symm)
| Mathlib/Algebra/GroupWithZero/Conj.lean | 21 | 24 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Multiset.Sort
import Mathlib.Order.RelIso.Set
/-!
# Construct a sorted list f... | /-- The bijection `orderEmbOfFin s h` sends `0` to the minimum of `s`. -/
theorem orderEmbOfFin_zero {s : Finset α} {k : ℕ} (h : s.card = k) (hz : 0 < k) :
orderEmbOfFin s h ⟨0, hz⟩ = s.min' (card_pos.mp (h.symm ▸ hz)) := by
simp only [orderEmbOfFin_apply, Fin.getElem_fin, sorted_zero_eq_min']
| Mathlib/Data/Finset/Sort.lean | 211 | 214 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Abs
/-!
# Lemmas about units in `ℤ`, which interact with the order structure.
-/
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} :... | Mathlib/Data/Int/Order/Units.lean | 63 | 67 | |
/-
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.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
/-!
# Finite intervals in a disjoint union
This file provides the... | Mathlib/Data/Sum/Interval.lean | 421 | 423 | |
/-
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 | 1,172 | 1,185 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | section general
variable [Module (ZMod n) G] {x : G}
lemma zmod_smul_mem (hx : x ∈ K) : ∀ a : ZMod n, a • x ∈ K := by
| Mathlib/Data/ZMod/Basic.lean | 1,166 | 1,169 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | lemma nnnorm_id [Nontrivial E] : ‖id 𝕜 E‖₊ = 1 := NNReal.eq norm_id
instance normOneClass [Nontrivial E] : NormOneClass (E →L[𝕜] E) :=
⟨norm_id⟩
| Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 114 | 117 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... |
variable {S : Type*}
| Mathlib/Algebra/GeomSum.lean | 411 | 413 |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Sh... | refine ⟨⟨f, hf.subset fun i => Subset.trans (hsupp i) (hVU' i), fun i x => (hf01 i x).1,
fun i x => (hf01 i x).2, fun x hx => ?_⟩,
hfp, fun i => Subset.trans (closure_mono (hsupp i)) (hVU i)⟩
rcases mem_iUnion.1 (hsW hx) with ⟨i, hi⟩
exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).m... | Mathlib/Topology/PartitionOfUnity.lean | 387 | 406 |
/-
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 ... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 535 | 537 | |
/-
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... | · exact ⟨e '' s, by rwa [mem_map, he.injective.preimage_image]⟩
theorem _root_.MeasurableEmbedding.integrableAtFilter_iff_comap [MeasurableSpace β] {e : α → β}
(he : MeasurableEmbedding e) {f : β → E} {μ : Measure β} :
IntegrableAtFilter f (l.map e) μ ↔ IntegrableAtFilter (f ∘ e) l (μ.comap e) := by
simp_r... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 373 | 379 |
/-
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 exists_superset_subset_encard_eq {k : ℕ∞}
(hst : s ⊆ t) (hsk : s.encard ≤ k) (hkt : k ≤ t.encard) :
| Mathlib/Data/Set/Card.lean | 384 | 385 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
| Mathlib/RingTheory/Coprime/Lemmas.lean | 69 | 70 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | theorem index_inf_ne_zero (hH : H.index ≠ 0) (hK : K.index ≠ 0) : (H ⊓ K).index ≠ 0 := by
rw [← relindex_top_right] at hH hK ⊢
exact relindex_inf_ne_zero hH hK
| Mathlib/GroupTheory/Index.lean | 326 | 328 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... |
theorem iSup_smul {ι : Sort*} {t : ι → Submodule R A} {N : Submodule R M} :
(⨆ i, t i) • N = ⨆ i, t i • N :=
le_antisymm (smul_le.mpr fun t ht s hs ↦ iSup_induction _ (motive := (· • s ∈ _)) ht
(fun i t ht ↦ mem_iSup_of_mem i <| smul_mem_smul ht hs)
(by simp_rw [zero_smul]; apply zero_mem) fun x y ↦ by s... | Mathlib/Algebra/Algebra/Operations.lean | 184 | 192 |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,
Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser
-/
import Mathlib.NumberTheory.FLT.Basic
import... | H : a ^ 3 + b ^ 3 = u * c ^ 3
attribute [nolint docBlame] Solution'.a
attribute [nolint docBlame] Solution'.b
attribute [nolint docBlame] Solution'.c
attribute [nolint docBlame] Solution'.u
| Mathlib/NumberTheory/FLT/Three.lean | 194 | 199 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Coherent
import Mathlib.Topology.UniformSpace.Equiv
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformAp... | variable {α β}
/-- Let `t i` be a nonempty directed subfamily of `𝔖`
such that every `s ∈ 𝔖` is included in some `t i`.
Let `V` bounded by `p` be a basis of entourages of `β`.
| Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean | 659 | 664 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | lemma map_Ψ (n : ℤ) : (W.map f).Ψ n = (W.Ψ n).map (mapRingHom f) := by
simp only [Ψ, map_preΨ, map_ψ₂, ← coe_mapRingHom]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 575 | 576 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,047 | 2,050 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Li... | let e₂ := (Pi.basisFun R n).equiv v₂ (Equiv.refl _)
have range_e₂ : LinearMap.range e₂ = ⊤ := by
rw [LinearMap.range_eq_top]
exact e₂.surjective
refine LinearEquiv.finrank_eq (e₁.ofSubmodules _ _ ?_)
rw [← LinearMap.range_comp, ← LinearMap.range_comp_of_range_eq_top (toLin v₂ v₁ A) range_e₂]
congr 1
... | Mathlib/Data/Matrix/Rank.lean | 255 | 264 |
/-
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.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | theorem derivative_of_natDegree_zero {p : R[X]} (hp : p.natDegree = 0) : derivative p = 0 := by
rw [eq_C_of_natDegree_eq_zero hp, derivative_C]
| Mathlib/Algebra/Polynomial/Derivative.lean | 115 | 116 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | calc
‖f x‖ ≤ M * ‖x‖ := h x
_ ≤ max M 1 * ‖x‖ := by gcongr; apply le_max_left
⟩
namespace NormedAddGroupHom
variable {V V₁ V₂ V₃ : Type*} [SeminormedAddCommGroup V] [SeminormedAddCommGroup V₁]
| Mathlib/Analysis/Normed/Group/Hom.lean | 68 | 75 |
/-
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 ... | theorem lintegral_mono_measure {f : α →ₛ ℝ≥0∞} (h : μ ≤ ν) : f.lintegral μ ≤ f.lintegral ν := by
simp only [lintegral]
gcongr
apply h
/-- `SimpleFunc.lintegral` is monotone both in function and in measure. -/
@[mono, gcongr]
theorem lintegral_mono {f g : α →ₛ ℝ≥0∞} (hfg : f ≤ g) (hμν : μ ≤ ν) :
f.lintegral μ... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 991 | 1,002 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... | Mathlib/GroupTheory/Index.lean | 265 | 265 | |
/-
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.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... |
section Orthogonal
/-- The orthogonal complement of a submodule `N` with respect to some bilinear form is the set of
elements `x` which are orthogonal to all elements of `N`; i.e., for all `y` in `N`, `B x y = 0`.
Note that for general (neither symmetric nor antisymmetric) bilinear forms this definition has a
chiral... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 119 | 130 |
/-
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 | 2,318 | 2,320 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | simp [zpow_add, ih])
(fun i ih => by
simp only [zpow_neg, zpow_natCast, map_inv, map_pow, map.of, sum.map_inv, neg_inj] at ih
simp [zpow_add, ih, sub_eq_add_neg])
section Category
variable {β : Type u}
@[to_additive]
instance : Monad FreeGroup.{u} where
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 842 | 852 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the langua... | theorem const_add_term {α} [AddCommMonoid α] (k n x a a') (h : k + a = a') :
k + @term α _ n x a = term n x a' := by
| Mathlib/Tactic/Abel.lean | 162 | 163 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | Mathlib/Data/Nat/PartENat.lean | 799 | 800 | |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | ext1 ω
simp only [Set.mem_inter_iff, Set.mem_setOf_eq, and_congr_right_iff]
intro hxi
rw [hxi]
constructor <;> intro h
· specialize h i
simpa only [Set.inter_assoc, this, le_refl, Set.setOf_true, Set.inter_univ] using h
· intro j
rw [Set.inter_assoc, this]
by_cases hij : i ≤ j
| Mathlib/Probability/Process/Stopping.lean | 373 | 382 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,041 | 1,048 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,861 | 1,868 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.Normed.Group.Completion
/-!
# Completion of normed group homs
Given two (semi) normed groups `G` an... | SeminormedAddCommGroup.mem_closure_iff.mp (Completion.isDenseInducing_coe.dense hatg) δ δ_pos
obtain ⟨g' : G, hgg' : f g' = f g, hfg : ‖g'‖ ≤ C' * ‖f g‖⟩ := hC' (f g) (mem_range_self _ g)
have mem_ker : g - g' ∈ f.ker := by rw [f.mem_ker, map_sub, sub_eq_zero.mpr hgg'.symm]
| Mathlib/Analysis/Normed/Group/HomCompletion.lean | 160 | 162 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... |
end NonzeroSemiring
| Mathlib/Algebra/Polynomial/Basic.lean | 1,151 | 1,152 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Data.Set.Function
/-!
# Comparing sums and integrals
## Summary
It is often the case ... |
lemma sum_Ico_le_integral_of_le
(hab : a ≤ b) (h : ∀ i ∈ Ico a b, ∀ x ∈ Ico (i : ℝ) (i + 1 : ℕ), f i ≤ g x)
(hg : IntegrableOn g (Set.Ico a b)) :
∑ i ∈ Finset.Ico a b, f i ≤ ∫ x in a..b, g x := by
have A i (hi : i ∈ Finset.Ico a b) : IntervalIntegrable g volume i (i + 1 : ℕ) := by
rw [intervalIntegra... | Mathlib/Analysis/SumIntegralComparisons.lean | 47 | 70 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,126 | 1,144 | |
/-
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.Group.Action.Opposite
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.GroupWithZero.U... | toFun := star
map_mul' := star_mul' }
| Mathlib/Algebra/Star/Basic.lean | 172 | 173 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | Mathlib/Data/Multiset/Powerset.lean | 322 | 328 | |
/-
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.Data.Complex.Trigonometric
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
/-!
# Multiple angle formulas in terms of... | @algebraMap_eval_S ℝ ℂ _ _ _
/-! ### Complex versions -/
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 39 | 41 |
/-
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.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# The opposite of an ab... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 186 | 188 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | theorem birkhoffAverage_zero (f : α → α) (g : α → M) (x : α) :
birkhoffAverage R f g 0 x = 0 := by simp [birkhoffAverage]
| Mathlib/Dynamics/BirkhoffSum/Average.lean | 44 | 45 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | apply Set.mem_range_self
theorem invEmbedding_comp (i : Fin c.length) (j : Fin (c.blocksFun i)) :
(c.invEmbedding (c.embedding i j) : ℕ) = j := by
simp_rw [coe_invEmbedding, index_embedding, coe_embedding, add_tsub_cancel_left]
/-- Equivalence between the disjoint union of the blocks (each of them seen as
`Fi... | Mathlib/Combinatorics/Enumerative/Composition.lean | 405 | 413 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | lemma ofReal_nnratCast (q : ℚ≥0) : ((q : ℝ) : K) = q := map_nnratCast (algebraMap ℝ K) _
| Mathlib/Analysis/RCLike/Basic.lean | 542 | 543 |
/-
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... |
/-- To cancel a common factor `c` from a `ModEq` we must divide the modulus `m` by `gcd m c`. -/
| Mathlib/Data/Int/ModEq.lean | 174 | 175 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | · obtain ⟨a, ha⟩ := gcd_dvd_left k m
refine ⟨gcd k m, a, gcd_dvd_right _ _, ?_, ha⟩
| Mathlib/Algebra/GCDMonoid/Basic.lean | 483 | 484 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
/-!
# Invertible elements
This file defines a typeclass `Invertible a` for elements `a` with a two-sided
multiplicative inverse.
The in... |
theorem invertible_unique {α : Type u} [Monoid α] (a b : α) [Invertible a] [Invertible b]
| Mathlib/Algebra/Group/Invertible/Defs.lean | 144 | 145 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
/-... | [IsLocallyInjective J f₃] [IsLocallySurjective J f₁] :
IsLocallyInjective J f₂ := by
subst fac
exact isLocallyInjective_of_isLocallyInjective_of_isLocallySurjective J f₁ f₂
lemma isLocallySurjective_of_isLocallySurjective_of_isLocallyInjective
| Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 211 | 216 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 756 | 756 | |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... | protected lemma id {p : 𝒳 ⥤ 𝒮} {R : 𝒮} {a : 𝒳} (ha : p.obj a = R) : p.IsHomLift (𝟙 R) (𝟙 a) := by
cases ha; infer_instance
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 69 | 70 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Cast.Field
import Mathlib.NumberTheory.ArithmeticFunction
/-!
# The von Mangoldt Function
In ... | ∑ i ∈ n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
| Mathlib/NumberTheory/VonMangoldt.lean | 131 | 131 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.AlexandrovDiscrete
import Mathlib.Topology.ContinuousMap.Basic
import Mathlib.Topology.Order.LowerUpperTopology
/-!
# Upper and lower... | lemma isClosed_iff_isLower : IsClosed s ↔ IsLowerSet s := by
rw [← isOpen_compl_iff, isOpen_iff_isUpperSet,
isLowerSet_compl.symm, compl_compl]
| Mathlib/Topology/Order/UpperLowerSetTopology.lean | 233 | 235 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
import Mathlib.MeasureTh... | · refine hf i ?_
simpa using hi
· simp_rw [sum_insertNone, Option.elim]
exact hpq
· rintro (_|i) hi
· exact hq
· refine hp i ?_
simpa using hi
theorem lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top {p : ℝ} {f g : α → ℝ≥0∞}
(hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) < ⊤... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 253 | 273 |
/-
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.NumberTheory.EulerProduct.ExpLog
import Mathlib.NumberTheory.LSeries.Dirichlet
/-!
# The Euler Product for the Riemann Zeta Function and Dirichlet L-Ser... | have hsum := summable_riemannZetaSummand hs
rw [zeta_eq_tsum_one_div_nat_add_one_cpow hs, hsum.of_norm.tsum_eq_zero_add, map_zero, zero_add]
simp only [riemannZetaSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
Nat.cast_add, Nat.cast_one, one_div]
/-- When `s.re > 1`, the map `n ↦ χ(n) * n^(-... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 63 | 68 |
/-
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... |
theorem div_neg_of_neg_of_pos (ha : a < 0) (hb : 0 < b) : a / b < 0 :=
| Mathlib/Algebra/Order/Field/Basic.lean | 323 | 324 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
/-!
# Extensionality lemmas for monoid and group structures
In this file we prove extensionality lemmas... | Function.Injective (@CancelMonoid.toLeftCancelMonoid M) := by
rintro ⟨⟩ ⟨⟩ h
congr
| Mathlib/Algebra/Group/Ext.lean | 87 | 90 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | Mathlib/RingTheory/Ideal/Basic.lean | 922 | 932 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp... |
@[simp]
lemma adj_homEquiv (X : Type u) (M : ModuleCat.{u} R) :
(adj R).homEquiv X M = freeHomEquiv := by
simp only [adj, Adjunction.mkOfHomEquiv_homEquiv]
instance : (forget (ModuleCat.{u} R)).IsRightAdjoint :=
(adj R).isRightAdjoint
end
section Free
open MonoidalCategory
variable [CommRing R]
namespac... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 89 | 109 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | by_cases c : i = j
· simp only [c, if_true, eq_self_iff_true, degreeOf_def, degrees_X, Multiset.count_singleton]
simp [c, if_false, degreeOf_def, degrees_X]
theorem degreeOf_add_le (n : σ) (f g : MvPolynomial σ R) :
degreeOf n (f + g) ≤ max (degreeOf n f) (degreeOf n g) := by
simp_rw [degreeOf_eq_sup]; exa... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 245 | 251 |
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