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) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,284 | 1,290 | |
/-
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... | add_tsub_cancel_right]
· rintro ⟨l, hl, rfl⟩
refine ⟨?_, hl ▸ Nat.le_add_left j l⟩
rw [add_assoc, add_comm, add_assoc, add_comm j l, hl]
exact Nat.sub_add_cancel h
@[simp] lemma antidiagonal_filter_fst_le_of_le {n k : ℕ} (h : k ≤ n) :
{a ∈ antidiagonal n | a.fst ≤ k} = (antidiagonal k).map
... | Mathlib/Data/Finset/NatAntidiagonal.lean | 108 | 119 |
/-
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.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
/-!
# UnivLE and cardinals
-/
noncomputable section
universe u v
open Cardinal
theorem univLE_iff_cardin... | rw [univLE_iff_cardinal_le]
exact lift_mk_le'
| Mathlib/SetTheory/Cardinal/UnivLE.lean | 30 | 31 |
/-
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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | have hn₂ : 2 < n := three_le_nValue <| hN.trans' <| by norm_num1
have : (2 * dValue N - 1) ^ n ≤ N := le_N (hN.trans' <| by norm_num1)
calc
_ ≤ (N ^ (nValue N : ℝ)⁻¹ / rexp 1 : ℝ) ^ (n - 2) / n := ?_
_ < _ := by gcongr; exacts [(tsub_pos_of_lt hn₂).ne', bound hN]
_ ≤ rothNumberNat ((2 * dValue N - 1) ... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 433 | 446 |
/-
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.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... |
/-- A variant of `inv_mul_eq_iff_eq_mul₀` that moves the nonzero hypothesis to another variable. -/
lemma eq_mul_of_inv_mul_eq₀ (hc : c ≠ 0) (h : b⁻¹ * a = c) : a = b * c := by
| Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 282 | 284 |
/-
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... | theorem isAtom_iff_le_of_ge : IsAtom a ↔ a ≠ ⊥ ∧ ∀ b ≠ ⊥, b ≤ a → a ≤ b :=
and_congr Iff.rfl <|
forall_congr' fun b => by
simp only [Ne, @not_imp_comm (b = ⊥), Classical.not_imp, lt_iff_le_not_le]
| Mathlib/Order/Atoms.lean | 77 | 80 |
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
/-!
# Stirling's formula
Thi... | · ext k
dsimp only [f]
rw [← pow_mul, pow_add]
push_cast
field_simp
ring
· have h : ∀ x ≠ (0 : ℝ), 1 + x⁻¹ = (x + 1) / x := fun x hx ↦ by field_simp [hx]
simp (disch := positivity) only [log_stirlingSeq_formula, log_div, log_mul, log_exp,
factorial_succ, cast_mul, cast_succ, cast_zero,... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 77 | 93 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... |
@[simp]
| Mathlib/Data/Option/NAry.lean | 91 | 92 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] (hg : G ≠ ⊥) :
∑ x : G, (x.val : K) = 0 := by
rw [Subgroup.ne_bot_iff_exists_ne_one] at hg
rcases hg with ⟨a, ha⟩
-- The action of a on G as an embedding
let a_mul_emb : G ↪ G := mulLeftEmbedding a
-- ... and ... | Mathlib/FieldTheory/Finite/Basic.lean | 141 | 163 |
/-
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... | (H : ∀ a, p a ↔ q a) : ComputablePred q :=
(funext fun a => propext (H a) : p = q) ▸ hp
| Mathlib/Computability/Halting.lean | 157 | 159 |
/-
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.Category.MonCat.ForgetCorepresentable
import Mathlib.Algebra.Category.Grp.ForgetCorepresentable
import Mathlib.Algebra.Category.Grp.Preadditive
im... | @[to_additive]
instance hasLimits : HasLimits Grp.{u} :=
Grp.hasLimitsOfSize.{u, u}
/-- The forgetful functor from groups to monoids preserves all limits.
This means the underlying monoid of a limit can be computed as a limit in the category of monoids.
-/
@[to_additive AddGrp.forget₂AddMonPreservesLimitsOfSize
"... | Mathlib/Algebra/Category/Grp/Limits.lean | 154 | 163 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGrou... | (c • f) ∣[k]A = c • f ∣[k]A := by
simp_rw [← smul_one_smul ℂ c f, ← smul_one_smul ℂ c (f ∣[k]A)]
ext1
simp_rw [slash]
simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]
ring
private theorem zero_slash (k : ℤ) (A : GL(2, ℝ)⁺) : (0 : ℍ → ℂ) ∣[k]A = 0 :=
fune... | Mathlib/NumberTheory/ModularForms/SlashActions.lean | 107 | 122 |
/-
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... | J ^ n ⊔ I = J ^ n := by
classical
by_cases hI : I = ⊥
· simp_all
rw [irreducible_pow_sup hI hJ, min_eq_right]
rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
exact_mod_cast hn
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 908 | 914 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... |
@[to_additive]
theorem prod_Ioi_zero {v : Fin n.succ → M} :
∏ i ∈ Ioi 0, v i = ∏ j : Fin n, v j.succ := by
| Mathlib/Algebra/BigOperators/Fin.lean | 159 | 162 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | have : m a₁ ≤ m b - k :=
Nat.lt_succ.mp (by rw [← Nat.succ_sub this]; simpa using ih a₁ ha₁)
exact lt_of_lt_of_le (Ord a₁ a₂ ha₂) this
List.eq_nil_iff_forall_not_mem.mpr
(by intro b' ha'; by_contra; simpa using bindList_m_lt (m b + 1) b' ha')
have mapGraph_graph {... | Mathlib/Computability/Primrec.lean | 1,047 | 1,057 |
/-
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.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | · intro h X Y f ⟨X', Y', f', hf', ⟨e⟩⟩
simp only [Arrow.iso_w' e, F.map_comp]
have := h _ hf'
infer_instance
| Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 127 | 130 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | theorem Valid'.balance'_lemma {α l l' r r'} (H1 : BalancedSz l' r')
(H2 : Nat.dist (@size α l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l') :
| Mathlib/Data/Ordmap/Ordset.lean | 312 | 313 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,306 | 1,309 | |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... |
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {c : 𝕜 → F →L[𝕜] G} {c' : F →L[𝕜] G}
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 431 | 432 |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a proof that if `G` is a `p`-group ac... | section P2comm
variable [Fact p.Prime] {n : ℕ}
open Subgroup
/-- The cardinality of the `center` of a `p`-group is `p ^ k` where `k` is positive. -/
| Mathlib/GroupTheory/PGroup.lean | 338 | 344 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | Mathlib/Algebra/Order/ToIntervalMod.lean | 1,019 | 1,020 | |
/-
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... | obtain ⟨q, hq, hq'⟩ := isBigO_atTop_F_nat_one (sub_nonneg.mpr ha.2.le)
refine ⟨_, lt_min hp hq, ?_⟩
have : F_int 1 a =ᶠ[atTop] fun t ↦ F_nat 1 a t + F_nat 1 (1 - a) t := by
filter_upwards [eventually_gt_atTop 0] with t ht
exact F_int_eq_of_mem_Icc 1 (Ico_subset_Icc_self ha) ht
refine this.isBigO.trans (... | Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean | 268 | 279 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | /-- A function is `C^(n + 1)` on an open domain if and only if it is
differentiable there, and its derivative (formulated with `deriv`) is `C^n`. -/
theorem contDiffOn_succ_iff_deriv_of_isOpen (hs : IsOpen s₂) :
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,280 | 1,282 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.GDelta.UniformSpace
import Mathlib.Topology.Instances.EReal.Lemmas
import Mathl... | (hs : MeasurableSet s) (hKs : K ⊆ s) : closure K ⊆ s := by
rw [hK.closure_eq_biUnion_inseparable, iUnion₂_subset_iff]
exact fun x hx y hy ↦ (hy.mem_measurableSet_iff hs).1 (hKs hx)
| Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 275 | 277 |
/-
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.Ordering.Basic
import Mathlib.Order.Synonym
/-!
# Comparison
This file provides basic results about orderings and comparison in linear orders.
... | Mathlib/Order/Compare.lean | 246 | 248 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
map `g : X → Π i, π i`. This lemma shows that infimum of the topologies on `X` induced by
| Mathlib/Topology/Constructions.lean | 977 | 979 |
/-
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.MeasureTheory.Measure.Tilted
/-!
# Log-likelihood Ratio
The likelihood ratio between two measures `μ` and `ν` is their Radon-Nikodym derivative
`μ.rnDeri... | · rw [llr, exp_log, if_neg h_zero]
exact ENNReal.toReal_pos h_zero hx.ne
lemma exp_llr_of_ac (μ ν : Measure α) [SigmaFinite μ] [Measure.HaveLebesgueDecomposition μ ν]
(hμν : μ ≪ ν) :
| Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | 46 | 50 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... |
-- Note: we need to make sure `fromEdgeSet_adj` and this lemma are confluent.
-- In particular, both yield `s(u, v) ∈ (fromEdgeSet s).edgeSet` ==> `s(v, w) ∈ s ∧ v ≠ w`.
@[simp]
theorem edgeSet_fromEdgeSet : (fromEdgeSet s).edgeSet = s \ { e | e.IsDiag } := by
ext e
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 567 | 572 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | property that its composition with the canonical `MultilinearMap R s E` is
the given multilinear map `φ`. -/
def lift : MultilinearMap R s E ≃ₗ[R] (⨂[R] i, s i) →ₗ[R] E where
toFun φ := { liftAux φ with map_smul' := liftAux.smul }
invFun φ' := φ'.compMultilinearMap (tprod R)
left_inv φ := by
ext
simp [lif... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 431 | 444 |
/-
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.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... | end CongrLeft
section Sigma
| Mathlib/Algebra/DirectSum/Module.lean | 320 | 323 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic
/-!
# Homomorphisms of semirings and... | end Semiring
| Mathlib/Algebra/Ring/Hom/Defs.lean | 595 | 596 |
/-
Copyright (c) 2020 Bhavik Mehta, Edward Ayers, Thomas Read. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers, Thomas Read
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Cate... | -- TODO: Generalise the below to its commuted variants.
| Mathlib/CategoryTheory/Closed/Cartesian.lean | 308 | 308 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | /-- A generalisation of **Sylow's first theorem**. If `p ^ n` divides
the cardinality of `G`, then there is a subgroup of cardinality `p ^ n` -/
theorem exists_subgroup_card_pow_prime [Finite G] (p : ℕ) {n : ℕ} [Fact p.Prime]
(hdvd : p ^ n ∣ Nat.card G) : ∃ K : Subgroup G, Nat.card K = p ^ n :=
let ⟨K, hK⟩ := e... | Mathlib/GroupTheory/Sylow.lean | 650 | 655 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | lemma polynomial_eq : W'.polynomial = Cubic.toPoly
⟨0, 1, Cubic.toPoly ⟨0, 0, W'.a₁, W'.a₃⟩, Cubic.toPoly ⟨-1, -W'.a₂, -W'.a₄, -W'.a₆⟩⟩ := by
simp only [polynomial, Cubic.toPoly]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 139 | 141 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes, Kevin Buzzard
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units.Basic
/-!
# Monoid homomorphisms and units
This file allows to... | ⟨IsLocalHom.map_nonunit a, IsUnit.map f⟩
| Mathlib/Algebra/Group/Units/Hom.lean | 240 | 241 |
/-
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, Andrew Yang, Yuyang Zhao
-/
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# The... |
theorem integralNormalization_coeff {i : ℕ} :
(integralNormalization p).coeff i =
if p.degree = i then 1 else coeff p i * p.leadingCoeff ^ (p.natDegree - 1 - i) := by
have : p.coeff i = 0 → p.degree ≠ i := fun hc hd => coeff_ne_zero_of_eq_degree hd hc
simp +contextual [sum_def, integralNormalization, coe... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 48 | 53 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 743 | 744 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... |
open NormNum
namespace Tactic
namespace NormNum
/-- A predicate representing partial progress in a proof of `Squarefree`. -/
def SquarefreeHelper (n k : ℕ) : Prop :=
0 < k → (∀ m, Nat.Prime m → m ∣ bit1 n → bit1 k ≤ m) → Squarefree (bit1 n)
theorem squarefree_bit10 (n : ℕ) (h : SquarefreeHelper n 1) : Squarefree... | Mathlib/Data/Nat/Squarefree.lean | 406 | 417 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 918 | 918 | |
/-
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 ΨSq_even_ofNat (m : ℕ) : W.ΨSq (2 * (m + 3)) =
(W.preΨ' (m + 2) ^ 2 * W.preΨ' (m + 3) * W.preΨ' (m + 5) -
W.preΨ' (m + 1) * W.preΨ' (m + 3) * W.preΨ' (m + 4) ^ 2) ^ 2 * W.Ψ₂Sq := by
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 280 | 283 |
/-
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.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | rw [adjoin_eq_span, Submonoid.closure_eq_one_union, span_union, ← NonUnitalAlgebra.adjoin_eq_span,
NonUnitalAlgebra.adjoin_eq]
end CommSemiring
| Mathlib/RingTheory/Adjoin/Basic.lean | 160 | 163 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | (⨂ₜ[R] i, a (Sum.inl i)) ⊗ₜ[R] ⨂ₜ[R] i, a (Sum.inr i) := by
simp [tmulEquiv]
end tmulEquiv
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 910 | 913 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | theorem length_sublists' : ∀ l : List α, length (sublists' l) = 2 ^ length l
| [] => rfl
| a :: l => by
simp +arith only [sublists'_cons, length_append, length_sublists' l,
length_map, length, Nat.pow_succ']
@[simp]
theorem sublists_nil : sublists (@nil α) = [[]] :=
rfl
@[simp]
theorem sublists_single... | Mathlib/Data/List/Sublists.lean | 82 | 93 |
/-
Copyright (c) 2021 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Central bin... | Mathlib/Data/Nat/Choose/Central.lean | 131 | 138 | |
/-
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.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... |
/-- A nonzero Hermitian matrix has an eigenvector with nonzero eigenvalue. -/
lemma exists_eigenvector_of_ne_zero (hA : IsHermitian A) (h_ne : A ≠ 0) :
∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ A *ᵥ v = t • v := by
| Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 146 | 149 |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... |
section
| Mathlib/Order/CompactlyGenerated/Basic.lean | 354 | 356 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | suffices H : Tendsto (fun n ↦ ∫ x in t n, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) by
have A : ∀ n, ∫ x in t n, f x ∂μ = 0 :=
fun n ↦ h'f _ ((isCompact_compactCovering β n).closure.inter_right hs)
simp_rw [A, tendsto_const_nhds_iff] at H
exact H.symm
have B : s = ⋃ n, t n := by
rw [← Set.iUnion_i... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 405 | 424 |
/-
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.Order.Group.Nat
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Data.Nat.Cast.WithTop
/-!
# `WithBot ℕ`
Lemmas about the type... | · simp [WithBot.add_bot]
simp [← WithBot.coe_add, Nat.add_eq_two_iff]
theorem add_eq_three_iff {n m : WithBot ℕ} :
n + m = 3 ↔ n = 0 ∧ m = 3 ∨ n = 1 ∧ m = 2 ∨ n = 2 ∧ m = 1 ∨ n = 3 ∧ m = 0 := by
cases n
· simp [WithBot.bot_add]
| Mathlib/Data/Nat/WithBot.lean | 44 | 50 |
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... | simp [not_nonempty_iff_eq_empty.mp contra] at hx
lemma pow_nilpotencyClass (hx : IsNilpotent x) : x ^ (nilpotencyClass x) = 0 :=
Nat.sInf_mem hx
end ZeroPow
| Mathlib/RingTheory/Nilpotent/Defs.lean | 120 | 126 |
/-
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.Algebra.Group.Pointwise.Set.Card
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Topology.Algebr... | @[to_additive]
lemma tendsto_measure_smul_diff_isCompact_isClosed [LocallyCompactSpace G]
[IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ] {k : Set G}
(hk : IsCompact k) (h'k : IsClosed k) :
| Mathlib/MeasureTheory/Group/Measure.lean | 505 | 508 |
/-
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.Analysis.SpecialFunctions.Gamma.Beta
/-!
# Deligne's archimedean Gamma-factors
In the theory of L-series one frequently encounters the following fun... | apply mul_ne_zero
· simp [pi_ne_zero]
· apply Gamma_ne_zero_of_re_pos
rw [div_ofNat_re]
exact div_pos hs two_pos
| Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean | 66 | 71 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | exact ⟨fun ⟨φ, hφ⟩ => ⟨φ.toRatAlgHom, hφ⟩, fun ⟨φ, hφ⟩ => ⟨φ.toRingHom, hφ⟩⟩
end Roots
section Bounded
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 73 | 77 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Group.Graph
import Mathlib.LinearAlgebra.Span.... | Mathlib/LinearAlgebra/Prod.lean | 981 | 987 | |
/-
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.Set.Countable
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.FunProp.Attr
import Mathlib.Tactic.Mea... |
end MeasurableFunctions
| Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 510 | 512 |
/-
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.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... |
end Classical
@[to_additive]
theorem IsCyclic.exists_monoid_generator [Finite α] [IsCyclic α] :
∃ x : α, ∀ y : α, y ∈ Submonoid.powers x := by
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 346 | 351 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 766 | 767 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finite.Basic
/-!
# Lagrange's four square theorem
The main result in this file is `sum_four_squares`,
a proof that every natural number is th... | (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) * (x ^ 2 + y ^ 2 + z ^ 2 + w ^ 2) := by
rw [← Int.natCast_inj]
push_cast
simp only [sq_abs, _root_.euler_four_squares]
namespace Int
theorem sq_add_sq_of_two_mul_sq_add_sq {m x y : ℤ} (h : 2 * m = x ^ 2 + y ^ 2) :
m = ((x - y) / 2) ^ 2 + ((x + y) / 2) ^ 2 :=
| Mathlib/NumberTheory/SumFourSquares.lean | 34 | 42 |
/-
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 | 630 | 631 | |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | simp only [degree_zero, zero_add]
constructor
· rw [mul_comm]
rw [bot_lt_iff_ne_bot]
intro h
exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)
/-- If `m` is a proper divisor of `n`, then `X ^ m - 1` divides
`∏ i ∈ Nat.properDivisors n, cyclotomic i R`. -/
theorem X_pow_sub_one_dvd_prod_cyclotomic (R : Typ... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 443 | 453 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Logic.Basic
import Mathlib.Data.Set.Basic
/-!
# Cent... | end Mul
section Semigroup
variable [Semigroup M] {a b : M}
| Mathlib/Algebra/Group/Center.lean | 170 | 173 |
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topolog... | tends to a positive constant `C` then `f * g` tends to `Filter.atTop`. -/
theorem Filter.Tendsto.atTop_mul_pos {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))
filter_upwards [... | Mathlib/Topology/Algebra/Order/Field.lean | 63 | 67 |
/-
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.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.RingTheory.AlgebraTower
import Mathli... | finite_of_finrank_pos <| by rw [hn]; exact n.succ_pos
theorem finite_iff_of_rank_eq_nsmul {W} [AddCommMonoid W] [Module R W] [Module.Free R W] {n : ℕ}
(hn : n ≠ 0) (hVW : Module.rank R M = n • Module.rank R W) :
Module.Finite R M ↔ Module.Finite R W := by
| Mathlib/LinearAlgebra/Dimension/Free.lean | 187 | 191 |
/-
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.Logic.Relation
import Mathlib.Logic.Unique
import Mathlib.Util.Notation3
/-!
# Quotient types
This module extends the core library's treatment of quo... | Note that in later files the preferred spelling is `⊤ : Setoid α`. -/
def trueSetoid : Setoid α :=
⟨_, true_equivalence⟩
/-- `Trunc α` is the quotient of `α` by the always-true relation. This
| Mathlib/Data/Quot.lean | 441 | 445 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | lemma neg'_eq (f : L.obj X ⟶ L.obj Y) (φ : W.LeftFraction X Y)
(hφ : f = φ.map L (inverts L W)) :
neg' W f = φ.neg.map L (inverts L W) := by
obtain ⟨φ₀, rfl, hφ₀⟩ : ∃ (φ₀ : W.LeftFraction X Y)
(_ : f = φ₀.map L (inverts L W)),
neg' W f = φ₀.neg.map L (inverts L W) :=
⟨_, (exists_leftFraction L W... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 102 | 116 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Andrew Yang, Joël Riou
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Monoid... |
lemma shiftFunctorAdd'_zero_add (a : A) :
shiftFunctorAdd' C 0 a a (zero_add a) = (Functor.leftUnitor _).symm ≪≫
isoWhiskerRight (shiftFunctorZero C A).symm (shiftFunctor C a) := by
ext X
dsimp [shiftFunctorAdd', shiftFunctorZero, shiftFunctor]
simp only [eqToHom_app, obj_ε_app, Discrete.addMonoidal_left... | Mathlib/CategoryTheory/Shift/Basic.lean | 220 | 227 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... |
theorem isIso_arrow_iff_eq_top {Y : C} (P : Subobject Y) : IsIso P.arrow ↔ P = ⊤ := by
rw [isIso_iff_mk_eq_top, mk_arrow]
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 233 | 235 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
/-!
# The ideal class group
This file defines the ideal class group `ClassGroup R` of fractional ideals of `R`
inside its f... | · rintro ⟨x, y, hx, hy, h⟩
have : IsUnit (mk' (FractionRing R) x ⟨y, mem_nonZeroDivisors_of_ne_zero hy⟩) := by
simpa only [isUnit_iff_ne_zero, ne_eq, mk'_eq_zero_iff_eq_zero] using hx
refine ⟨this.unit, ?_⟩
rw [mul_comm, ← Units.eq_iff, Units.val_mul, coe_toPrincipalIdeal]
convert
(mk'_mul... | Mathlib/RingTheory/ClassGroup.lean | 126 | 144 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | @[continuity]
lemma continuous_sumCoeffsExp : Continuous (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) := by
refine continuous_finset_sum Finset.univ fun i _ => Continuous.mul (by fun_prop) ?_
exact Continuous.rpow continuous_const continuous_id (fun x => Or.inl (ne_of_gt (R.b_pos i)))
lemma strictAnti_sumCoeffsExp : Stri... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 513 | 520 |
/-
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 ... | Mathlib/Algebra/CubicDiscriminant.lean | 567 | 575 | |
/-
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.Algebra.Group.Pointwise.Set.Card
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Topology.Algebr... | "]
instance _root_.ContinuousMulEquiv.isHaarMeasure_map [BorelSpace G] [IsTopologicalGroup G]
{H : Type*} [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H]
[IsTopologicalGroup H] (e : G ≃ₜ* H) : (μ.map e).IsHaarMeasure :=
e.toMulEquiv.isHaarMeasure_map μ e.continuous e.symm.continuous
/-- A c... | Mathlib/MeasureTheory/Group/Measure.lean | 807 | 814 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | (f : ι → ι) (hh : ∀ x : ι, R (f x) x) : LiftR' (RelLast' _ R) ((id ::: f) <$$> x) x := by
have := liftR_map_last R x f id hh
rwa [appendFun_id_id, MvFunctor.id_map] at this
end LiftRMap
variable {F : TypeVec (n + 1) → Type u} [q : MvQPF F]
theorem Cofix.abs_repr {α} (x : Cofix F α) : Quot.mk _ (Cofix.repr x)... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 390 | 419 |
/-
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... | t.ite s₁ s₂ = s₁ ∪ (s₂ \ t) := by
ext x
| Mathlib/Data/Set/Basic.lean | 1,390 | 1,391 |
/-
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.Polynomial.AlgebraMap
import Mathlib.RingTheory.IsTensorProduct
/-!
# Base change of polynomial algebras
Given `[CommSemiring R] [Semiring A] [Al... |
The bare function `A[X] → A ⊗[R] R[X]`.
(We don't need to show that it's an algebra map, thankfully --- just that it's an inverse.)
| Mathlib/RingTheory/PolynomialAlgebra.lean | 109 | 111 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... |
-- The additive version is also a `linarith` lemma.
| Mathlib/Algebra/Order/Group/Defs.lean | 165 | 166 |
/-
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.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | theorem algHom_ext {A : Type*} [Semiring A] [Algebra R A] {f g : MvPolynomial σ R →ₐ[R] A}
(hf : ∀ i : σ, f (X i) = g (X i)) : f = g :=
AddMonoidAlgebra.algHom_ext' (mulHom_ext' fun X : σ => MonoidHom.ext_mnat (hf X))
@[simp]
theorem algHom_C {A : Type*} [Semiring A] [Algebra R A] (f : MvPolynomial σ R →ₐ[R] A) ... | Mathlib/Algebra/MvPolynomial/Basic.lean | 460 | 470 |
/-
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.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
simp only [V, mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and]⟩)
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 430 | 431 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.Yoneda
import Mat... | For example, if the source `J` of `F` is a partially ordered set, then to give
`c : Cocone F` is to give a collection of morphisms `ιⱼ : F j ⟶ c.pt` and, for
all `j ≤ k` in `J`, morphisms `ιⱼₖ : F j ⟶ F k` such that `Fⱼₖ ≫ Fₖ = Fⱼ` for all `j ≤ k`.
| Mathlib/CategoryTheory/Limits/Cones.lean | 143 | 146 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... |
theorem inv_add_inv_ennreal : (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1 := by
| Mathlib/Data/Real/ConjExponents.lean | 163 | 164 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes, Antoine Chambert-Loir
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator.Basic
import Mathlib.GroupTheory.Coset.Basic
import Mathlib.GroupThe... | Finite (commutatorSet (closureCommutatorRepresentatives G)) := by
apply Nat.finite_of_card_ne_zero
rw [card_commutatorSet_closureCommutatorRepresentatives]
exact Finite.card_pos.ne'
| Mathlib/GroupTheory/Abelianization.lean | 319 | 323 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes, Kevin Buzzard
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units.Basic
/-!
# Monoid homomorphisms and units
This file allows to... |
/-- If `f` is a homomorphism from a group `G` to a monoid `M`,
then its image lies in the units of `M`,
| Mathlib/Algebra/Group/Units/Hom.lean | 150 | 152 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
/-! # Optional stopping theorem (fair game theorem)
The optional stopping theor... | Submartingale f 𝒢 μ := by
refine submartingale_of_setIntegral_le hadp hint fun i j hij s hs => ?_
classical
specialize hf (s.piecewise (fun _ => i) fun _ => j) _ (isStoppingTime_piecewise_const hij hs)
(isStoppingTime_const 𝒢 j) (fun x => (ite_le_sup _ _ (x ∈ s)).trans (max_eq_right hij).le)
⟨j, fun... | Mathlib/Probability/Martingale/OptionalStopping.lean | 69 | 80 |
/-
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.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 247 | 258 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Module.OrderedSMul
/-!
# Ordered algebras
An ordered algebra is an ordered semiring, which is an algebr... | Mathlib/Algebra/Order/Algebra.lean | 44 | 48 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | ⟨n * m, by simp [zpow_mul, h]⟩
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 162 | 163 |
/-
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.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
/-!
# Triangle inequality for `Lp`-seminorm
In this file w... | calc
(∫⁻ a, (‖(f + g) a‖₊ : ℝ≥0∞) ^ q ∂μ) ^ (1 / q) ≤
(∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0∞)) a ^ q ∂μ) ^ (1 / q) := by
gcongr with a
simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
_ ≤ eLpNorm' f q μ + eLpNorm' g q μ := ENNReal.lintegr... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 26 | 33 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | exact csInf_le' ⟨ι, f, rfl, rfl⟩
theorem cof_lsub_le_lift {ι} (f : ι → Ordinal) :
cof (lsub.{u, v} f) ≤ Cardinal.lift.{v, u} #ι := by
rw [← mk_uLift.{u, v}]
convert cof_lsub_le.{max u v} fun i : ULift.{v, u} ι => f i.down
exact
lsub_eq_of_range_eq.{u, max u v, max u v}
(Set.ext fun x => ⟨fun ⟨i, ... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 228 | 258 |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,506 | 1,512 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... |
section
| Mathlib/Data/Real/ConjExponents.lean | 132 | 133 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Yury Kudryashov
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Nat.ModEq
/-!
# Pigeonhole principles
Given pigeons ... | exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum (M := Mᵒᵈ) hf ht hb
/-- The pigeonhole principle for finitely many pigeons counted by weight, non-strict inequality
| Mathlib/Combinatorics/Pigeonhole.lean | 164 | 166 |
/-
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.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | It's given by `x ↦ x ^ (p ^ n)`, see `rootsExpandPowToRoots_apply`. -/
noncomputable def rootsExpandPowToRoots :
(expand R (p ^ n) f).roots.toFinset ↪ f.roots.toFinset where
toFun x := ⟨x ^ p ^ n,
| Mathlib/FieldTheory/Perfect.lean | 344 | 347 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
/-!
# Basis on a quaternion-like algebra
## Main definitions
* `QuaternionAlgebra.Basis A c₁ c₂ c₃`: a basis... | @[simp]
theorem j_mul_k : q.j * q.k = (c₂ * c₃) • 1 - c₃ • q.i := by
| Mathlib/Algebra/QuaternionBasis.lean | 94 | 95 |
/-
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... | lemma ContinuousOn.finset_inf' (hne : s.Nonempty) (hs : ∀ i ∈ s, ContinuousOn (f i) t) :
ContinuousOn (s.inf' hne f) t := fun x hx ↦
ContinuousWithinAt.finset_inf' hne fun i hi ↦ hs i hi x hx
| Mathlib/Topology/Order/Lattice.lean | 342 | 344 |
/-
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.Logic.Function.Defs
import Mathlib.Order.Defs.Unbundled
/-!
# Lexicographic order on a sigma type
This defines the lexicographical order of two arbitrary... | ⟨fun ⟨_, _⟩ => Lex.right _ _ <| refl _⟩
instance [IsIrrefl ι r] [∀ i, IsIrrefl (α i) (s i)] : IsIrrefl _ (Lex r s) :=
⟨by
| Mathlib/Data/Sigma/Lex.lean | 80 | 83 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | @[simp]
theorem generate_sieve (S : Sieve X) : generate S = S :=
giGenerate.l_u_eq S
| Mathlib/CategoryTheory/Sites/Sieves.lean | 420 | 423 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | theorem blockDiag'_conjTranspose {α : Type*} [Star α]
(M : Matrix (Σ i, m' i) (Σ i, n' i) α) (k : o) : blockDiag' Mᴴ k = (blockDiag' M k)ᴴ :=
ext fun _ _ => rfl
section Zero
| Mathlib/Data/Matrix/Block.lean | 733 | 737 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... |
variable [NumberField K]
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 387 | 389 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | · rw [h, revAt_invol, coeff_X_pow_self]
· rw [not_mem_support_iff.mp]
| Mathlib/Algebra/Polynomial/Reverse.lean | 128 | 129 |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Damiano Testa, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mat... | theorem natDegree_divX_eq_natDegree_tsub_one : p.divX.natDegree = p.natDegree - 1 := by
apply map_natDegree_eq_sub (φ := divX_hom)
· intro f
simpa [divX_hom, divX_eq_zero_iff] using eq_C_of_natDegree_eq_zero
· intros n c c0
| Mathlib/Algebra/Polynomial/Inductions.lean | 88 | 92 |
/-
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.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | let e : OrthonormalBasis (Fin n.succ) ℝ E := o.finOrthonormalBasis n.succ_pos Fact.out
have h₁ : e.toBasis.orientation = o := o.finOrthonormalBasis_orientation _ _
have h₂ : e.toBasis.orientation ≠ -o := by
symm
rw [e.toBasis.orientation_ne_iff_eq_neg, h₁]
rw [o.volumeForm_robust e h₁, (-o).volumeForm_r... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 223 | 233 |
/-
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.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Prime.Pow
import Mathlib.NumberTheory.Divisors
/-!
# Prime powers a... | refine ⟨p, pp, ?_⟩
refine Nat.eq_of_factorization_eq (Nat.ordCompl_pos p hn0).ne' (by simp) fun q => ?_
rw [Nat.factorization_ordCompl n p, h1]
simp
@[deprecated (since := "2024-10-24")]
alias IsPrimePow.exists_ord_compl_eq_one := IsPrimePow.exists_ordCompl_eq_one
theorem exists_ordCompl_eq_one_iff_isPrimePow... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 63 | 73 |
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