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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | rw [← SetLike.mem_coe, range_coe, Set.mem_range] at h₁
obtain ⟨x, hx⟩ := h₁
have hx' : x ∈ p := h₂ hx
have hxz : z + x ∈ p := by
apply h₂
simp [hx, hz]
suffices z + x - x ∈ p by simpa only [this, add_sub_cancel_right]
exact p.sub_mem hxz hx'
end Ring
section Semifield
variable [Se... | Mathlib/Algebra/Module/Submodule/Range.lean | 235 | 253 |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 759 | 762 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... |
-- A quick check that insertionSort is stable:
| Mathlib/Data/List/Sort.lean | 391 | 392 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | haveI : Nonempty { U // U ∈ c } := ⟨⟨U₀, hU₀⟩⟩
obtain ⟨U₀compl, -, -⟩ := hc hU₀
| Mathlib/Topology/Compactness/Compact.lean | 1,043 | 1,044 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | Mathlib/Algebra/Polynomial/EraseLead.lean | 52 | 52 | |
/-
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.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functor... | lemma tprod_setProd_singleton_right (s : Set β) (c : γ) (f : β × γ → α) :
(∏' x : s ×ˢ {c}, f x) = ∏' b : s, f (b, c) := by
rw [tprod_congr_set_coe _ Set.prod_singleton, tprod_image _ (Prod.mk_left_injective c).injOn]
| Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 50 | 53 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
/-!
# Convex hull
This file defines the convex hull of a set `s` in a module. `con... | exact hx.2 (mem_singleton _)
rintro hx
suffices h : s \ {x} = convexHull 𝕜 (s \ {x}) by
rw [h]
exact convex_convexHull 𝕜 _
| Mathlib/Analysis/Convex/Hull.lean | 127 | 131 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... |
theorem isPiSystem_Ixx {Ixx : α → α → Set α} {p : α → α → Prop}
(Hne : ∀ {a b}, (Ixx a b).Nonempty → p a b)
(Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (f : ι → α)
(g : ι' → α) : @IsPiSystem α { S | ∃ i j, p (f i) (g j) ∧ Ixx (f i) (g j) = S } := by
simpa only [exists_rang... | Mathlib/MeasureTheory/PiSystem.lean | 164 | 170 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... |
theorem not_imp : ¬(a → b) ↔ a ∧ ¬b := open scoped Classical in Decidable.not_imp_iff_and_not
| Mathlib/Logic/Basic.lean | 302 | 303 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,753 | 1,755 | |
/-
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.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Pointwise
import Mathlib.Topology.Order.Basic
/-!
# Strictly convex sets
This file defines st... |
theorem StrictConvex.sub (hs : StrictConvex 𝕜 s) (ht : StrictConvex 𝕜 t) : StrictConvex 𝕜 (s - t) :=
(sub_eq_add_neg s t).symm ▸ hs.add ht.neg
end AddCommGroup
end OrderedRing
| Mathlib/Analysis/Convex/Strict.lean | 344 | 350 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 837 | 839 | |
/-
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.Order.Disjoint
import Mathlib.Order.RelIso.Basic
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Order homomorphisms
This file defines order homomorphi... | /-- Note that this goal could also be stated `(Codisjoint on f) a b` -/
theorem Codisjoint.map_orderIso [SemilatticeSup α] [OrderTop α] [SemilatticeSup β] [OrderTop β]
{a b : α} (f : α ≃o β) (ha : Codisjoint a b) : Codisjoint (f a) (f b) := by
rw [codisjoint_iff_le_sup, ← f.map_sup, ← f.map_top]
| Mathlib/Order/Hom/Basic.lean | 1,125 | 1,128 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... |
This is implemented as a mixin for `Semiring α`.
| Mathlib/Algebra/Ring/Defs.lean | 390 | 391 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | instance : SemilatticeSup (FractionalIdeal S P) :=
{ FractionalIdeal.lattice with }
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 447 | 449 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | rw [le_inf_iff, le_himp_comm, sup_le_iff]
| Mathlib/Order/Heyting/Basic.lean | 306 | 306 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | (fun i => by fin_cases i <;> rfl)
lemma mk₁_surjective (X : ComposableArrows C 1) : ∃ (X₀ X₁ : C) (f : X₀ ⟶ X₁), X = mk₁ f :=
⟨_, _, X.map' 0 1, ext₁ rfl rfl (by simp)⟩
variable (F)
| Mathlib/CategoryTheory/ComposableArrows.lean | 277 | 282 |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | intro m₁ m₂ n₁ n₂ h₁
obtain ⟨⟨n₃, y₃, hn₃⟩, h₂⟩ := Localization.r_iff_exists.mp h₁
simp only [Subtype.coe_mk, mul_eq_mul_left_iff] at h₂
cases h₂
· obtain ⟨a₁, ha₁⟩ := n₁.prop
| Mathlib/SetTheory/Surreal/Dyadic.lean | 205 | 209 |
/-
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.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | lemma tilted_zero (μ : Measure α) [IsProbabilityMeasure μ] : μ.tilted 0 = μ := by simp
| Mathlib/MeasureTheory/Measure/Tilted.lean | 78 | 78 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Riccardo Brasca, Eric Rodriguez
-/
import Mathlib.Data.PNat.Prime
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib... | noncomputable def subOnePowerBasis : PowerBasis K L :=
(hζ.powerBasis K).ofGenMemAdjoin
(((integral {n} K L).isIntegral ζ).sub isIntegral_one)
(hζ.powerBasis_gen_mem_adjoin_zeta_sub_one _)
| Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 128 | 131 |
/-
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
-/
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Ring.Hom.Defs
i... | simp_rw [pow_succ _ (n + 1)]
exact WithTop.mul_lt_mul (WithTop.pow_right_strictMono n.succ_ne_zero h) h
| Mathlib/Algebra/Order/Ring/WithTop.lean | 260 | 262 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Topology.Algebra.Monoid
/-!
# Topological group with zero
I... |
/-- The function `f x / g x` is discontinuous when `g x = 0`. However, under appropriate
| Mathlib/Topology/Algebra/GroupWithZero.lean | 210 | 211 |
/-
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.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | hι := hι }
@[simp]
| Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 207 | 209 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | let f : Discrete ι' ≌ Discrete ι := Discrete.equivalence e.symm
have : colimit.ι F i = (colimit.ι F i ≫ (HasColimit.isoOfEquivalence f (Iso.refl _)).inv) ≫
(HasColimit.isoOfEquivalence f (Iso.refl _)).hom := by
simp
rw [this, HasColimit.isoOfEquivalence_inv_π]
infer_instance
end Prod
end SheafedSpac... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 922 | 938 |
/-
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.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | induction' n with _ hn
· rw [Submodule.pow_zero, Submodule.pow_zero]
· rw [Submodule.pow_succ, Submodule.pow_succ]
| Mathlib/RingTheory/Ideal/Operations.lean | 324 | 326 |
/-
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... | apply LinearMap.ext
intro x
apply h'
/-- A dependent version of `ext`. -/
theorem dExt {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g :=
| Mathlib/LinearAlgebra/LinearPMap.lean | 65 | 71 |
/-
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... | 0 < r i n := by exact hrpos_i
_ ≤ u := by exact hu.1
exact rpow_le_rpow_of_exponent_nonpos (by positivity)
(by exact_mod_cast (le_of_lt hu.2)) (le_of_lt hp)
_ ≤ n ^ p a b * #(Ico (r i n) n) • (c₂ * g n / n ^ (p a b + 1)) := by
... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 680 | 768 |
/-
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.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsympt... | (fun z => z ^ m * exp (b₁ * z)) =o[l] fun z => z ^ n * exp (b₂ * z) := by
simpa only [cpow_intCast] using hl.isLittleO_cpow_mul_exp hb m n
| Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 200 | 201 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform
import Mathlib.Data.Real.Basic
imp... | private lemma good_vertices_triangle_card [DecidableEq α] (dst : 2 * ε ≤ G.edgeDensity s t)
(dsu : 2 * ε ≤ G.edgeDensity s u) (dtu : 2 * ε ≤ G.edgeDensity t u) (utu : G.IsUniform ε t u)
(x : α) (hx : x ∈ s \ (badVertices G ε s t ∪ badVertices G ε s u)) :
ε ^ 3 * #t * #u ≤ #((({y ∈ t | G.Adj x y} ×ˢ {y ∈ u |... | Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean | 72 | 95 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... |
/-- Bound for `|log x * x ^ t|` in the interval `(0, 1]`, for positive real `t`. -/
theorem abs_log_mul_self_rpow_lt (x t : ℝ) (h1 : 0 < x) (h2 : x ≤ 1) (ht : 0 < t) :
|log x * x ^ t| < 1 / t := by
rw [lt_div_iff₀ ht]
have := abs_log_mul_self_lt (x ^ t) (rpow_pos_of_pos h1 t) (rpow_le_one h1.le h2 ht.le)
rwa... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 869 | 875 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | simpa only [Set.prod_subset_iff] using @mem_prod_iff α β p f g
theorem tendsto_fst : Tendsto Prod.fst (f ×ˢ g) f :=
| Mathlib/Order/Filter/Prod.lean | 117 | 119 |
/-
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 Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | end Disjoint
section Sigma
| Mathlib/Data/Set/Image.lean | 1,373 | 1,375 |
/-
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.Algebra
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.RingTheory.Algebra... | map (AdjoinRoot.of f.factor) f /ₘ (X - C (AdjoinRoot.root f.factor))
theorem X_sub_C_mul_removeFactor (f : K[X]) (hf : f.natDegree ≠ 0) :
(X - C (AdjoinRoot.root f.factor)) * f.removeFactor = map (AdjoinRoot.of f.factor) f := by
let ⟨g, hg⟩ := factor_dvd_of_natDegree_ne_zero hf
apply (mul_divByMonic_eq_iff_i... | Mathlib/FieldTheory/SplittingField/Construction.lean | 88 | 93 |
/-
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.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | Mathlib/Data/Nat/ModEq.lean | 527 | 531 | |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
/-!
# The coevaluation map on finite dimensional vector spaces
Given a finite dimensional vector space `V` over a field... | apply TensorProduct.ext
apply LinearMap.ext_ring; apply (Basis.ofVectorSpace K V).ext; intro j
rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLinearMap]
rw [lid_tmul, one_smul, rid_symm_apply]
simp only [Linea... | Mathlib/LinearAlgebra/Coevaluation.lean | 81 | 95 |
/-
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... | induction' m with m hm
· exact Nat.modEq_one
· rw [Nat.ModEq.comm, add_comm, Nat.modEq_iff_dvd' (Nat.one_le_pow' _ _)] at hm
obtain ⟨d, hd⟩ := hm
rw [tsub_eq_iff_eq_add_of_le (Nat.one_le_pow' _ _), add_comm] at hd
rw [pow_succ, pow_mul, hd, add_pow, Finset.sum_range_succ', pow_zero, one_mul, one_pow,
... | Mathlib/FieldTheory/Finite/Basic.lean | 623 | 644 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
/-!
# Integrals against peak functions
A sequence of peak functions is a sequence of functions with a... |
/-- If a continuous function `c` realizes its maximum at a unique point `x₀` in a compact set `s`,
then the sequence of functions `(c x) ^ n / ∫ (c x) ^ n` is a sequence of peak functions
concentrating around `x₀`. Therefore, `∫ (c x) ^ n * g / ∫ (c x) ^ n` converges to `g x₀` if `g` is
integrable on `s` and continuou... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 235 | 247 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
/-!
# Comparison with the normalized Moore complex functor
In this file, we show that when the category `A` is abelian,
the... | /-- `inclusionOfMooreComplexMap X` is a split mono. -/
def splitMonoInclusionOfMooreComplexMap (X : SimplicialObject A) :
SplitMono (inclusionOfMooreComplexMap X) where
retraction := PInftyToNormalizedMooreComplex X
id := by
simp only [← cancel_mono (inclusionOfMooreComplexMap X), assoc, id_comp,
| Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 97 | 102 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | cases this (two_le_of_mem_cycleType (mem_of_mem_erase hm)) (two_le_of_mem_cycleType hn) h
theorem isCycle (h : IsThreeCycle σ) : IsCycle σ := by
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 612 | 614 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.ModelTheory.Basic
/-!
# Language Maps
Maps between first-order languages in... | Mathlib/ModelTheory/LanguageMap.lean | 411 | 411 | |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | (same_add_div h).symm
| Mathlib/Algebra/Field/Basic.lean | 37 | 37 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | rw [Sbtw, Sbtw, wbtw_comm, ← and_assoc, ← and_assoc, and_right_comm]
alias ⟨Sbtw.symm, _⟩ := sbtw_comm
| Mathlib/Analysis/Convex/Between.lean | 150 | 152 |
/-
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.... | · intro h
| Mathlib/LinearAlgebra/Prod.lean | 597 | 597 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
/-!
# Met... |
theorem sinh_half_dist_add_dist (a b c : ℍ) : sinh ((dist a b + dist b c) / 2) =
(dist (a : ℂ) b * dist (c : ℂ) (conj ↑b) + dist (b : ℂ) c * dist (a : ℂ) (conj ↑b)) /
| Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 66 | 68 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Action.Prod
import Mathlib.Algebra.Group.Subg... | Mathlib/GroupTheory/GroupAction/Basic.lean | 739 | 743 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.DFinsupp.BigOperators
import Mathlib.Data.DFinsupp.Order
import Mathlib.Order.Interval.Finset.Basic
import Ma... |
lemma card_Iic : #(Iic f) = ∏ i ∈ f.support, #(Iic (f i)) := by
| Mathlib/Data/DFinsupp/Interval.lean | 189 | 190 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... |
theorem Disjoint.inv_right (h : Disjoint f g) : Disjoint f g⁻¹ :=
h.symm.inv_left.symm
| Mathlib/GroupTheory/Perm/Support.lean | 87 | 90 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | simpa only [and_assoc, and_left_comm] using (nhdsWithin_basis_open a s).mem_iff
theorem mem_nhdsWithin_iff_exists_mem_nhds_inter {t : Set α} {a : α} {s : Set α} :
| Mathlib/Topology/ContinuousOn.lean | 89 | 91 |
/-
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.Data.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | rw [dual_isBase_iff, diff_diff_cancel_left hB]
theorem ground_not_isBase (M : Matroid α) [h : RankPos M✶] : ¬M.IsBase M.E := by
| Mathlib/Data/Matroid/Dual.lean | 209 | 211 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Wieser
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
impor... | Mathlib/Data/Real/Pointwise.lean | 136 | 137 | |
/-
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... | rcases a with ⟨⟨a₁, a₂⟩, a₃⟩
rcases b with ⟨⟨b₁, b₂⟩, b₃⟩
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 598 | 599 |
/-
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 ncard_insert_of_mem {a : α} (h : a ∈ s) : ncard (insert a s) = s.ncard := by
| Mathlib/Data/Set/Card.lean | 601 | 602 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 812 | 818 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | @[to_additive]
theorem norm_le_norm_add_norm_div (u v : E) : ‖v‖ ≤ ‖u‖ + ‖u / v‖ := by
| Mathlib/Analysis/Normed/Group/Basic.lean | 527 | 528 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.... | theorem T_pow_mul_apply_one (n : ℤ) (g : SL(2, ℤ)) : (T ^ n * g) 1 = g 1 := by
ext j
simp [coe_T_zpow, Matrix.vecMul, dotProduct, Fin.sum_univ_succ, vecTail]
@[simp]
| Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 470 | 474 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | @[simp, mfld_simps]
theorem refl_trans : (PartialEquiv.refl α).trans e = e :=
PartialEquiv.ext (fun _ => rfl) (fun _ => rfl) (by simp [trans_source, preimage_id])
| Mathlib/Logic/Equiv/PartialEquiv.lean | 639 | 641 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | theorem mk_wcovBy_mk_iff_right : (a, b₁) ⩿ (a, b₂) ↔ b₁ ⩿ b₂ :=
swap_wcovBy_swap.trans mk_wcovBy_mk_iff_left
| Mathlib/Order/Cover.lean | 530 | 531 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | @[simp]
theorem toMatrix_lmul_eq (x : S) :
LinearMap.toMatrix b b (LinearMap.mulLeft R x) = leftMulMatrix b x :=
rfl
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 833 | 836 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotent
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.... | /-- The inclusion map `I ⧸ I ^ 2` to `R ⧸ I ^ 2`. -/
def cotangentToQuotientSquare : I.Cotangent →ₗ[R] R ⧸ I ^ 2 :=
Submodule.mapQ (I • ⊤) (I ^ 2) I.subtype
(by
rw [← Submodule.map_le_iff_le_comap, Submodule.map_smul'', Submodule.map_top,
Submodule.range_subtype, smul_eq_mul, pow_two] )
theorem to_... | Mathlib/RingTheory/Ideal/Cotangent.lean | 88 | 96 |
/-
Copyright (c) 2016 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.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... |
theorem injective_surjInv (h : Surjective f) : Injective (surjInv h) :=
| Mathlib/Logic/Function/Basic.lean | 441 | 442 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Projective.Basic
import Mathlib.Algebra.Category.Grp.EpiM... | rw [projective_iff_preservesEpimorphisms_coyoneda_obj]
refine ⟨fun h : (preadditiveCoyonedaObj P ⋙ forget _).PreservesEpimorphisms => ?_, ?_⟩
· exact Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveCoyonedaObj P)
(forget _)
· intro
exact (inferInstance : (preadditiveCoyonedaObj P ... | Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean | 42 | 50 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... |
@[simp]
theorem div_zero (a : R) : a / 0 = 0 :=
EuclideanDomain.quotient_zero a
| Mathlib/Algebra/EuclideanDomain/Defs.lean | 141 | 144 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | Mathlib/LinearAlgebra/Ray.lean | 731 | 735 | |
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.Da... |
/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
`f` and `g`, which monovary together on `s`, is strictly decreased by a permutation if and only if
| Mathlib/Algebra/Order/Rearrangement.lean | 285 | 287 |
/-
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... | W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂) = W.addX x₁ x₂ (W.slope x₁ x₂ y₁ <| W.negY x₂ y₂) -
(y₁ - W.negY x₁ y₁) * (y₂ - W.negY x₂ y₂) / (x₂ - x₁) ^ 2 := by
simp_rw [slope_of_X_ne hx, addX, negY, ← neg_sub x₁, neg_sq]
field_simp [sub_ne_zero.mpr hx]
ring1
/-- The formula `y(P₁)(x(P₂) - x(P₃)) + y(P₂)(x(P₃) -... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 582 | 588 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | Mathlib/Order/Bounds/Basic.lean | 1,714 | 1,720 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | IsPreconnected (connectedComponentIn F x) := by
rw [connectedComponentIn]; split_ifs
· exact IsInducing.subtypeVal.isPreconnected_image.mpr isPreconnected_connectedComponent
· exact isPreconnected_empty
theorem isConnected_connectedComponent {x : α} : IsConnected (connectedComponent x) :=
| Mathlib/Topology/Connected/Basic.lean | 503 | 508 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,906 | 1,908 | |
/-
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... | size (.node' l x r) = size (eraseMin (.node' l x r)) + 1 := by
have := H.dual.eraseMax_aux
| Mathlib/Data/Ordmap/Ordset.lean | 390 | 391 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
/-!
# Equivalences and sets
In this file we pr... |
theorem Equiv.swap_bijOn_exchange (ha : a ∈ s) (hb : b ∉ s) :
BijOn (Equiv.swap a b) s (insert b (s \ {a})) := by
refine ⟨fun x hx ↦ ?_, (Equiv.injective _).injOn, fun x hx ↦ ?_⟩
· obtain (rfl | hxa) := eq_or_ne x a
· simp [swap_apply_left]
rw [swap_apply_of_ne_of_ne hxa (by rintro rfl; contradiction)]... | Mathlib/Logic/Equiv/Set.lean | 642 | 648 |
/-
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... | (f : E →L[𝕜] F).contDiff
theorem LinearIsometry.contDiff (f : E →ₗᵢ[𝕜] F) : ContDiff 𝕜 n f :=
f.toContinuousLinearMap.contDiff
theorem LinearIsometryEquiv.contDiff (f : E ≃ₗᵢ[𝕜] F) : ContDiff 𝕜 n f :=
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 140 | 145 |
/-
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.... | rw [coe_bot, univ_pi_eq_empty]
exact Ioc_eq_empty hi
| Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 300 | 302 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | | bit1
deriving DecidableEq, Inhabited, Fintype
| Mathlib/Computability/TMToPartrec.lean | 158 | 160 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Encodable
impor... | Mathlib/FieldTheory/Cardinality.lean | 80 | 85 | |
/-
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.MeasurableSpace.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Sequence of measurable functions associated to a sequence o... | (hp : ∀ᵐ x ∂μ, p x fun n => f n x) (n : ι) : aeSeq hf p n =ᵐ[μ] f n :=
ae_all_iff.mp (aeSeq_eq_fun_ae hf hp) n
theorem iSup [SupSet β] [Countable ι] (hf : ∀ i, AEMeasurable (f i) μ)
(hp : ∀ᵐ x ∂μ, p x fun n => f n x) : ⨆ n, aeSeq hf p n =ᵐ[μ] ⨆ n, f n := by
| Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | 108 | 112 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | (hg : IsMultiplicative g) : IsMultiplicative (pdiv f g) :=
⟨by simp [hf, hg], fun {m n} cop => by
simp only [pdiv_apply, map_mul_of_coprime hf cop, map_mul_of_coprime hg cop,
div_eq_mul_inv, mul_inv]
| Mathlib/NumberTheory/ArithmeticFunction.lean | 663 | 666 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... |
@[to_additive]
theorem exponent_ne_zero_of_finite : exponent G ≠ 0 :=
ExponentExists.of_finite.exponent_ne_zero
| Mathlib/GroupTheory/Exponent.lean | 393 | 396 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... | Mathlib/ModelTheory/Basic.lean | 989 | 991 | |
/-
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.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... | simp
lemma coeff_natTrailingDegree_ne_zero : coeff p p.natTrailingDegree ≠ 0 ↔ p ≠ 0 :=
coeff_natTrailingDegree_eq_zero.not
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 128 | 131 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... | simp [rankPos_iff, hE.ne_empty.symm]
| Mathlib/Data/Matroid/Constructions.lean | 195 | 196 |
/-
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 | 1,932 | 1,939 | |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... |
@[simp]
theorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by
cases xs
| Mathlib/Data/Vector/Snoc.lean | 42 | 45 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | obtain ⟨k, hk⟩ : ∃ k : ℕ, (p : ℚ) ^ (-((k : ℕ) : ℤ)) < ε := exists_pow_neg_lt_rat p hε
use k
intro j hj
| Mathlib/NumberTheory/Padics/RingHoms.lean | 498 | 500 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,711 | 1,714 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | HasFPowerSeriesOnBall (fun w => (2 * π * I : ℂ)⁻¹ • ∮ z in C(c, R), (z - w)⁻¹ • f z)
(cauchyPowerSeries f c R) c R :=
{ r_le := le_radius_cauchyPowerSeries _ _ _
r_pos := ENNReal.coe_pos.2 hR
hasSum := fun hy ↦ hasSum_cauchyPowerSeries_integral hf <| by simpa using hy }
namespace circleIntegral
/-... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 564 | 578 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | rw [← not_nonempty_iff_eq_empty, pi_nonempty_iff]
push_neg
refine exists_congr fun i => ?_
cases isEmpty_or_nonempty (α i) <;> simp [*, forall_and, eq_empty_iff_forall_not_mem]
| Mathlib/Data/Set/Prod.lean | 665 | 668 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Yongle Hu
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.RingTheory.Ideal.Pointwise
import Mathlib.RingTheory.Ideal.Quot... | exact (injective_iff_map_eq_zero (algebraMap (R ⧸ p) (S ⧸ P))).mp h _ hx
· intro hx
rw [hx, RingHom.map_zero]
variable [Algebra R S]
/-- `R / p` has a canonical map to `S / pS`. -/
instance Quotient.algebraQuotientMapQuotient : Algebra (R ⧸ p) (S ⧸ map (algebraMap R S) p) :=
Ideal.Quotient.algebraQuotient... | Mathlib/RingTheory/Ideal/Over.lean | 56 | 70 |
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | end rangeRestrict
end Semiring
| Mathlib/Algebra/Module/Submodule/Range.lean | 422 | 425 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.Convex.TotallyBounded
/-!
# Absolutely convex sets
A set `s` in an commutative monoid `E` i... |
variable (𝕜) [SeminormedRing 𝕜] [SMul 𝕜 E] [SMul ℝ E] [AddCommMonoid E]
/-- A set is absolutely convex if it is balanced and convex. Mathlib's definition of `Convex`
requires the scalars to be an `OrderedSemiring` whereas the definition of `Balanced` requires the
scalars to be a `SeminormedRing`. Mathlib doesn't cu... | Mathlib/Analysis/LocallyConvex/AbsConvex.lean | 52 | 60 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... |
theorem snoc_rev {α n} (a : α) (f : Fin n → α) (i : Fin <| n + 1) :
snoc (α := fun _ => α) f a i.rev = cons (α := fun _ => α) a (f ∘ Fin.rev : Fin _ → α) i := by
simpa using insertNth_rev (last n) a f i
theorem snoc_comp_rev {α n} (a : α) (f : Fin n → α) :
| Mathlib/Data/Fin/Tuple/Basic.lean | 915 | 920 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | Mathlib/Topology/UniformSpace/Separation.lean | 306 | 307 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Prod
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.Bases.Basic
/... | theorem lift_map_le {g : Set β → Filter γ} {m : α → β} : (map m f).lift g ≤ f.lift (g ∘ image m) :=
le_lift.2 fun _s hs => lift_le (image_mem_map hs) le_rfl
| Mathlib/Order/Filter/Lift.lean | 106 | 108 |
/-
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.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.AffineSpace.Pointwise
import Mathlib.LinearAlgebra.Basis.SMul
/-!
# Affine bases and bary... | @[simp]
theorem linear_eq_sumCoords (i : ι) : (b.coord i).linear = -(b.basisOf i).sumCoords :=
rfl
@[simp]
| Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 139 | 143 |
/-
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... | (toSpec 𝒜 f).stalkMap x =
(specStalkEquiv 𝒜 f x f_deg hm).hom ≫ (Proj.stalkIso' 𝒜 x.1).toCommRingCatIso.inv ≫
((Proj.toLocallyRingedSpace 𝒜).restrictStalkIso (Opens.isOpenEmbedding _) x).inv :=
CommRingCat.hom_ext <|
IsLocalization.ringHom_ext (R := A⁰_ f) ((toSpec 𝒜 f).base x).asIdeal.primeC... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 798 | 805 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | {x ∈ s | pred x}.weightedVSubOfPoint p b w := by
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply, ← sum_subtype_eq_sum_filter]
/-- A weighted sum over `{x ∈ s | pred x}` equals one over `s` if all the weights at indices in `s`
not satisfying `pred` are zero. -/
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 208 | 212 |
/-
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 one_lt_one_div (h1 : 0 < a) (h2 : a < 1) : 1 < 1 / a := by
rwa [lt_one_div (@zero_lt_one α _ _ _ _ _) h1, one_div_one]
| Mathlib/Algebra/Order/Field/Basic.lean | 131 | 133 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Partition.Equipartition
/-!
# Turán's theorem
In this file we prove Turán's theorem, the first importan... | lemma IsTuranMaximal.le_iff_eq (hG : G.IsTuranMaximal r) (hH : H.CliqueFree (r + 1)) :
G ≤ H ↔ G = H := by
classical exact ⟨fun hGH ↦ edgeFinset_inj.1 <| eq_of_subset_of_card_le
(edgeFinset_subset_edgeFinset.2 hGH) (hG.2 _ hH), le_of_eq⟩
| Mathlib/Combinatorics/SimpleGraph/Turan.lean | 58 | 61 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | Mathlib/Data/Part.lean | 824 | 825 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | space structure on `Σ n, X n`, where the predistance `dist x y` is obtained by pushing `x` and `y`
in a common `X k` using composition by the `f n`, and taking the distance there. This does not
depend on the choice of `k` as the `f n` are isometries. The metric space associated to this
premetric space is the desired in... | Mathlib/Topology/MetricSpace/Gluing.lean | 515 | 522 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | · rintro _ ⟨ι, t, w, z, hw₀, hw₁, hz, rfl⟩
exact t.centerMass_mem_convexHull hw₀ (hw₁.symm ▸ zero_lt_one) hz
/-- Universe polymorphic version of the reverse implication of `mem_convexHull_iff_exists_fintype`.
-/
lemma mem_convexHull_of_exists_fintype {s : Set E} {x : E} [Fintype ι] (w : ι → R) (z : ι → E)
(h... | Mathlib/Analysis/Convex/Combination.lean | 329 | 350 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.