Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
/-!
# Midpoint of a segment for characteristic zero
We collect lemmas th... | [Invertible (2 : k)] [∀ i, AddCommGroup (V i)] [∀ i, Module k (V i)]
[∀ i, AddTorsor (V i) (P i)] (f g : ∀ i, P i) (i : ι) :
midpoint k f g i = midpoint k (f i) (g i) :=
| Mathlib/LinearAlgebra/AffineSpace/MidpointZero.lean | 45 | 47 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | exact blim.const_mul _
end Metric
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 175 | 178 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Fintype.Order
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.Quotients
import... | rw [← funMap_quotient_mk'_sigma_mk']
rfl
map_rel' := by
intro n R x
change RelMap R (fun a => (⟦.mk f i (x a)⟧ : DirectLimit G f)) ↔ _
simp only [relMap_quotient_mk'_sigma_mk']
| Mathlib/ModelTheory/DirectLimit.lean | 272 | 278 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | alias not_eq_key := ne_key
/-! ### `NodupKeys` -/
| Mathlib/Data/List/Sigma.lean | 73 | 76 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.LinearAlgebra.Prod
/-!
# Trivial Square-Ze... | (r : R) :
lift f g hg hfg hgf (inl r) = f r :=
show f r + g 0 = f r by rw [map_zero, add_zero]
@[simp]
theorem lift_apply_inr (f : R →ₐ[S] A) (g : M →ₗ[S] A)
(hg : ∀ x y, g x * g y = 0)
| Mathlib/Algebra/TrivSqZeroExt.lean | 965 | 971 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.ShortComplex.Retract
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
#... | lemma quasiIsoAt_iff_exactAt' (f : K ⟶ L) (i : ι) [K.HasHomology i] [L.HasHomology i]
(hL : L.ExactAt i) :
QuasiIsoAt f i ↔ K.ExactAt i := by
simp only [quasiIsoAt_iff, ShortComplex.quasiIso_iff, exactAt_iff,
ShortComplex.exact_iff_isZero_homology] at hL ⊢
constructor
| Mathlib/Algebra/Homology/QuasiIso.lean | 82 | 87 |
/-
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... | · rfl
· simp only [addX]
ring1
· field_simp [hy]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 508 | 511 |
/-
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... |
end Haar
| Mathlib/MeasureTheory/Group/Measure.lean | 874 | 876 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... |
@[to_additive]
theorem div_ae_eq_one {β} [Group β] (f g : α → β) : f / g =ᵐ[μ] 1 ↔ f =ᵐ[μ] g := by
refine ⟨fun h ↦ h.mono fun x hx ↦ ?_, fun h ↦ h.mono fun x hx ↦ ?_⟩
| Mathlib/MeasureTheory/Measure/Restrict.lean | 630 | 633 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | (ht : IsClosed t) :
∃ δ, 0 < δ ∧ Disjoint (thickening δ s) (thickening δ t) := by
obtain ⟨r, hr, h⟩ := exists_pos_forall_lt_edist hs ht hst
refine ⟨r / 2, half_pos (NNReal.coe_pos.2 hr), ?_⟩
rw [disjoint_iff_inf_le]
rintro z ⟨hzs, hzt⟩
rw [mem_thickening_iff_exists_edist_lt] at hzs hzt
rw [← NNReal.... | Mathlib/Topology/MetricSpace/Thickening.lean | 384 | 396 |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on ℕ
This file defines a simple dista... | Or.elim (lt_or_ge i j)
(by intro h; rw [max_eq_right_of_lt h, min_eq_left_of_lt h, dist_eq_sub_of_le (Nat.le_of_lt h)])
(by intro h; rw [max_eq_left h, min_eq_right h, dist_eq_sub_of_le_right h])
theorem dist_succ_succ {i j : Nat} : dist (succ i) (succ j) = dist i j := by
| Mathlib/Data/Nat/Dist.lean | 85 | 89 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Extreme sets
This file defines extreme sets and extreme points for sets in a module.
An extreme s... |
end SMul
| Mathlib/Analysis/Convex/Extreme.lean | 166 | 168 |
/-
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.AlgebraicTopology.SimplicialSet.Monoidal
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
/-!
# Simplicial categories
A simplicial category is a category ... | Mathlib/AlgebraicTopology/SimplicialCategory/Basic.lean | 79 | 84 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 669 | 670 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | rw [tendsto_iff_forall_toWeakDualBCNN_tendsto]
simp_rw [toWeakDualBCNN_apply _ _, ← testAgainstNN_coe_eq, ENNReal.tendsto_coe,
ENNReal.toNNReal_coe]
end weak_convergence -- section
section Hausdorff
variable [HasOuterApproxClosed Ω] [BorelSpace Ω]
open Function
/-- The mapping `toWeakDualBCNN` from finite ... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 522 | 535 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | lemma map_preΨ' (n : ℕ) : (W.map f).preΨ' n = (W.preΨ' n).map f := by
simp only [preΨ', map_Ψ₂Sq, map_Ψ₃, map_preΨ₄, ← coe_mapRingHom, map_preNormEDS']
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 563 | 564 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Final
/-!
# Conditions for `parallelPair` to be initial
In this file we give sufficient conditions on a category... | lemma parallelPair_initial_mk' {X Y : C} (f g : X ⟶ Y)
(h₁ : ∀ Z, Nonempty (X ⟶ Z))
(h₂ : ∀ ⦃Z : C⦄ (i j : X ⟶ Z),
Zigzag (J := CostructuredArrow (parallelPair f g) Z)
(mk (Y := zero) i) (mk (Y := zero) j)) :
(parallelPair f g).Initial where
out Z := by
have : Nonempty (CostructuredArrow... | Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean | 25 | 40 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | @[to_additive (attr := simp)]
theorem div_div_cancel (a b : G) : a / (a / b) = b :=
| Mathlib/Algebra/Group/Basic.lean | 950 | 951 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 883 | 886 | |
/-
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.Algebra.Group.Embedding
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Order.Interval.Finset.Basic
/-!
# Finite... |
theorem Icc_eq_pair : Finset.Icc a (a + 1) = {a, a + 1} := by
| Mathlib/Data/Int/Interval.lean | 133 | 134 |
/-
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.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generali... | · rw [h.t_id] at this
· convert Subtype.eq this
rw [h.V_id]
trivial
instance (h : MkCore.{u}) (i j : h.J) : IsIso (h.t i j) := by
use h.t j i; constructor <;> ext1; exacts [h.t_inv _ _ _, h.t_inv _ _ _]
/-- (Implementation) the restricted transition map to be fed into `TopCat.GlueData`. -/
def MkCore.t' (... | Mathlib/Topology/Gluing.lean | 304 | 317 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,083 | 1,086 | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... |
theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 194 | 195 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... | _root_.ne_of_lt
(lt_of_le_of_lt (Nat.mul_le_mul_left _ a1)
(Nat.lt_add_of_pos_right <| mul_pos (d_pos a1) (strictMono_y a1 np)))
h
cases this; simp at h; exact ⟨h.symm, rfl⟩
theorem eq_of_xn_modEq_lem3 {i n} (npos : 0 < n) :
∀ {j}, i < j → j ≤ 2 * n → j ≠ n → ¬(a = 2 ∧ n = 1 ∧ i =... | Mathlib/NumberTheory/PellMatiyasevic.lean | 571 | 583 |
/-
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... | section ConstantCoeff
/-- `constantCoeff p` returns the constant term of the polynomial `p`, defined as `coeff 0 p`.
| Mathlib/Algebra/MvPolynomial/Basic.lean | 831 | 833 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... | end MeasureTheory
| Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 427 | 444 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | OuterRegular.smul μ coe_ne_top
open scoped Function in -- required for scoped `on` notation
/-- If the restrictions of a measure to countably many open sets covering the space are
outer regular, then the measure itself is outer regular. -/
lemma of_restrict [OpensMeasurableSpace α] {μ : Measure α} {s : ℕ → Set α}
... | Mathlib/MeasureTheory/Measure/Regular.lean | 388 | 395 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 932 | 934 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | theorem counitInv_app_functor (e : C ≌ D) (X : C) :
e.counitInv.app (e.functor.obj X) = e.functor.map (e.unit.app X) := by
symm
simp only [id_obj, comp_obj, counitInv]
rw [← Iso.app_inv, ← Iso.comp_hom_eq_id (e.counitIso.app _), Iso.app_hom, functor_unit_comp]
| Mathlib/CategoryTheory/Equivalence.lean | 166 | 170 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | μ.real s • ⨍ x in s, f x ∂μ = ∫ x in s, f x ∂μ := by
haveI := Fact.mk h.lt_top
| Mathlib/MeasureTheory/Integral/Average.lean | 365 | 366 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 565 | 570 | |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | This should not be confused with `(measurePreserving_add_right μ g).quasiMeasurePreserving`. "]
theorem quasiMeasurePreserving_mul_right [IsMulLeftInvariant μ] (g : G) :
QuasiMeasurePreserving (fun h : G => h * g) μ μ := by
refine ⟨measurable_mul_const g, AbsolutelyContinuous.mk fun s hs => ?_⟩
rw [map_apply (m... | Mathlib/MeasureTheory/Group/Prod.lean | 440 | 445 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Algebra.Group.Nat.Range
import Mathlib.Data.Set.Finite.Basic
/-!
# Counting on ℕ
Thi... | alias ⟨_, count_succ_eq_count⟩ := count_succ_eq_count_iff
| Mathlib/Data/Nat/Count.lean | 106 | 107 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | MvPowerSeries.coeff_C_mul _ φ a
| Mathlib/RingTheory/PowerSeries/Basic.lean | 310 | 311 |
/-
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.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 953 | 956 | |
/-
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... |
end Equation
section Nonsingular
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 209 | 213 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... |
open Filter
| Mathlib/Order/LiminfLimsup.lean | 1,113 | 1,115 |
/-
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.LinearAlgebra.QuadraticForm.IsometryEquiv
/-! # Quadratic form on product and pi types
## Main definitions
* `QuadraticForm.prod Q₁ Q₂`: the quadratic for... | section Ring
variable [CommRing R]
variable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup P]
| Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 202 | 205 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... | theorem modOf_add_divOf [IsCancelAdd G] (x : k[G]) (g : G) :
x %ᵒᶠ g + of' k G g * (x /ᵒᶠ g) = x := by
| Mathlib/Algebra/MonoidAlgebra/Division.lean | 171 | 172 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
/-!
## `ℤ` forms a conditionally complete linear order
The integer... | @[simp]
theorem csInf_empty : sInf (∅ : Set ℤ) = 0 :=
| Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 73 | 74 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Reid Barton, Simon Hudon, Kenny Lau
-/
import Mathlib.Logic.Equiv.Defs
import Mathlib.Logic.Small.Defs
/-!
# Opposites
In this file we define a structure `Opposite α` con... | rfl
| Mathlib/Data/Opposite.lean | 58 | 58 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.... | @[simp] def _root_.Int.rawCast [Ring α] (n : ℤ) : α := n
| Mathlib/Tactic/NormNum/Result.lean | 165 | 166 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,106 | 1,107 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... |
end BasisDivisor
| Mathlib/LinearAlgebra/Lagrange.lean | 182 | 183 |
/-
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.Integral.DominatedConvergence
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# Measurability of the integral against a kernel
Th... | filter_upwards with y
simp_rw [s', SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
simp only [f', hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
mem_setOf_eq]
refine
tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 102 | 110 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | theorem _root_.Finset.coe_infsep_of_offDiag_empty
[DecidableEq α] {s : Finset α} (hs : s.offDiag = ∅) : (s : Set α).infsep = 0 := by
rw [← Finset.not_nonempty_iff_eq_empty] at hs
rw [Finset.coe_infsep, dif_neg hs]
| Mathlib/Topology/MetricSpace/Infsep.lean | 428 | 432 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.Algebra.Module.LocalizedModule.IsLocalization
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.RingTheory.Ideal.Quotient.Operations
impo... |
@[simp]
lemma mvPolynomialQuotientEquiv_apply (p : MvPolynomial Unit R) :
mvPolynomialQuotientEquiv S r (Ideal.Quotient.mk _ p) = aeval (S₁ := S) (fun _ ↦ invSelf r) p :=
| Mathlib/RingTheory/MvPolynomial/Localization.lean | 120 | 123 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
import Mathlib.CategoryTheory.Shift.Induced
import Mathlib.CategoryTheory.Quotient
/-!
# The shift on a quotient category
Let `C... | @[simp]
lemma iso_hom_app (a : A) (X : C) :
(iso F r hF a).hom.app ((functor r).obj X) =
(lift r F hF).map (((functor r).commShiftIso a).inv.app X) ≫
(F.commShiftIso a).hom.app X := by
dsimp only [iso, natIsoLift]
rw [natTransLift_app]
dsimp
erw [comp_id, id_comp, id_comp, id_comp, Functor.map_i... | Mathlib/CategoryTheory/Shift/Quotient.lean | 77 | 85 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... |
@[to_additive] lemma IsComplement.pairwiseDisjoint_smul (hst : IsComplement S T) :
S.PairwiseDisjoint (· • T) := fun a ha b hb hab ↦ disjoint_iff_forall_ne.2 <| by
rintro _ ⟨c, hc, rfl⟩ _ ⟨d, hd, rfl⟩
exact hst.1.ne (a₁ := (⟨a, ha⟩, ⟨c, hc⟩)) (a₂:= (⟨b, hb⟩, ⟨d, hd⟩)) (by simp [hab])
@[to_additive AddSubgroup... | Mathlib/GroupTheory/Complement.lean | 158 | 167 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | ⟨@Injective.subsingleton _ _ _ C_injective, by
intro
| Mathlib/Algebra/Polynomial/Basic.lean | 699 | 700 |
/-
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.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | /-- Taking kernels is an order-reversing map from the quotient objects of `X` to the subobjects of
`X`. -/
@[simps]
| Mathlib/CategoryTheory/Subobject/Limits.lean | 232 | 234 |
/-
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... |
@[deprecated (since := "2025-04-10")] alias range_prod_map := range_prodMap
| Mathlib/Data/Set/Prod.lean | 243 | 245 |
/-
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.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# Infinitely smooth "bump" func... | {x : X} (hc : ContDiffAt ℝ n c x) (hr : ContDiffAt ℝ n (fun x => (f x).rIn) x)
(hR : ContDiffAt ℝ n (fun x => (f x).rOut) x) (hg : ContDiffAt ℝ n g x) :
| Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 187 | 188 |
/-
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... | map_add' x y := piTensorHomMapFun₂_add x y
map_smul' x y := piTensorHomMapFun₂_smul x y
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 671 | 673 |
/-
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... | mul_one]
· intro i' _ i'_ne
rw [Finsupp.single_eq_of_ne i'_ne.symm, mul_zero]
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 621 | 623 |
/-
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.Idempotents.Karoubi
/-!
# Biproducts in the idempotent completion of a preadditive category
In this file, we define an instance expressing that... | apply hasBiproduct_of_total (Biproducts.bicone F)
simp only [hom_ext_iff]
refine biproduct.hom_ext' _ _ (fun j => ?_)
simp only [Biproducts.bicone_pt_X, sum_hom, comp_f, Biproducts.bicone_π_f,
biproduct.bicone_π, biproduct.map_π, Biproducts.bicone_ι_f, biproduct.ι_map... | Mathlib/CategoryTheory/Idempotents/Biproducts.lean | 74 | 88 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... |
/-- `U n` is the `n`-th Chebyshev polynomial of the second kind. -/
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 159 | 160 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.LinearAlgebra.SesquilinearForm
/-!
# Orthogonal complements of su... | theorem orthogonalFamily_self :
OrthogonalFamily 𝕜 (fun b => ↥(cond b K Kᗮ)) fun b => (cond b K Kᗮ).subtypeₗᵢ
| true, true => absurd rfl
| true, false => fun _ x y => inner_right_of_mem_orthogonal x.prop y.prop
| false, true => fun _ x y => inner_left_of_mem_orthogonal y.prop x.prop
| false, false => absur... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 185 | 191 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | lemma zpow_one_sub_natCast (a : G) (n : ℕ) : a ^ (1 - n : ℤ) = a / a ^ n := by
| Mathlib/Algebra/Group/Basic.lean | 863 | 863 |
/-
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.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | rw [whisker_exchange]; monoidal
_ = η_ (ᘁX) X ≫ (ᘁX) ◁ f ≫ (ᘁX) ◁ g := by
rw [coevaluation_evaluation'']; monoidal
/-- Given an exact pairing on `Y Y'`,
we get a bijection on hom-sets `(Y' ⊗ X ⟶ Z) ≃ (X ⟶ Y ⊗ Z)`
by "pulling the string on the left" up or down.
This gives the adjunction `tensorLeftAdju... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 258 | 275 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Disjointed
import Mathlib.Algebra.Order.Ring.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!... |
namespace Prod
variable {α β : Type*} [Ring α] [PartialOrder α] [IsOrderedRing α]
[Ring β] [PartialOrder β] [IsOrderedRing β] [LocallyFiniteOrder α] [LocallyFiniteOrder β]
[DecidableEq α] [DecidableEq β] [DecidableLE (α × β)]
| Mathlib/Order/Interval/Finset/Box.lean | 65 | 69 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v := by
ext
simp
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 505 | 508 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | convex_Icc _ _
end LinearOrderedAddCommMonoid
end Module
end AddCommMonoid
section LinearOrderedAddCommMonoid
| Mathlib/Analysis/Convex/Basic.lean | 311 | 320 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | (hf' : UniformCauchySeqOn f' l s) (hf : ∀ n : ι, ∀ y : E, y ∈ s → HasFDerivAt (f n) (f' n y) y)
{x₀ x : E} (hx₀ : x₀ ∈ s) (hx : x ∈ s) (hfg : Cauchy (map (fun n => f n x₀) l)) :
Cauchy (map (fun n => f n x) l) := by
have : NeBot l := (cauchy_map_iff.1 hfg).1
let t := { y | y ∈ s ∧ Cauchy (map (fun n => ... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 226 | 251 |
/-
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
/-... | (f : ℂ → E) (c : ℂ) (R : ℝ) : (∮ z in C(c, R), a • f z) = a • ∮ z in C(c, R), f z := by
simp only [circleIntegral, ← smul_comm a (_ : ℂ) (_ : E), intervalIntegral.integral_smul]
@[simp]
theorem integral_smul_const [CompleteSpace E] (f : ℂ → ℂ) (a : E) (c : ℂ) (R : ℝ) :
(∮ z in C(c, R), f z • a) = (∮ z in C(c... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 390 | 400 |
/-
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.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.Normed.Group.Pointwise
import Mat... | rw [Real.nndist_eq', Real.nnabs_of_nonneg (sub_nonneg_of_le (‹_ ≤ y› i : y₂ i ≤ y i)),
Real.nndist_eq', Real.nnabs_of_nonneg (sub_nonneg_of_le (‹_ ≤ y› i : y₁ i ≤ y i))]
exact Real.toNNReal_mono (sub_le_sub_left (hy _) _)
| Mathlib/Analysis/Normed/Order/UpperLower.lean | 136 | 139 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.OrderIso
/-!
# Lattice ordered groups
Lattice ordered groups were introduced by [Birkhoff][birkhoff1942]. T... | lemma inf_mul_sup [MulLeftMono α] (a b : α) : (a ⊓ b) * (a ⊔ b) = a * b :=
calc
(a ⊓ b) * (a ⊔ b) = (a ⊓ b) * (a * b * (b⁻¹ ⊔ a⁻¹)) := by
rw [mul_sup b⁻¹ a⁻¹ (a * b), mul_inv_cancel_right, mul_inv_cancel_comm]
_ = (a ⊓ b) * (a * b * (a ⊓ b)⁻¹) := by rw [inv_inf, sup_comm]
_ = a * b := by rw [mul_com... | Mathlib/Algebra/Order/Group/Lattice.lean | 109 | 115 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Vincent Beffara, Rida Hamadani
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.ENat.Lattice
/-!
# Graph metric
This module defines the `SimpleGraph.edi... | theorem reachable_of_edist_ne_top (h : G.edist u v ≠ ⊤) :
G.Reachable u v :=
| Mathlib/Combinatorics/SimpleGraph/Metric.lean | 86 | 87 |
/-
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.Range
/-! ### Linear equivalences involvi... |
end Submodule
namespace LinearMap
| Mathlib/Algebra/Module/Submodule/Equiv.lean | 247 | 250 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | have h₂ := U_add_two R n
have h₃ := T_add_two R (n + 1)
linear_combination (norm := ring_nf) -h₃ - (X : R[X]) * h₂ + h₁ + 2 * (X : R[X]) * ih1 - ih2
| neg_add_one n ih1 ih2 =>
have h₁ := U_add_two R (-n - 1)
have h₂ := U_add_two R (-n)
have h₃ := T_add_two R (-n)
linear_combination (norm :... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 266 | 284 |
/-
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.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.TensorProduct.Finite
/-!
# The meta properties of... | open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
| Mathlib/RingTheory/RingHom/Finite.lean | 28 | 31 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | @[to_additive]
theorem MonoidHom.map_finprod {f : α → M} (g : M →* N) (hf : (mulSupport f).Finite) :
g (∏ᶠ i, f i) = ∏ᶠ i, g (f i) :=
g.map_finprod_plift f <| hf.preimage Equiv.plift.injective.injOn
@[to_additive]
| Mathlib/Algebra/BigOperators/Finprod.lean | 591 | 596 |
/-
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.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | def get : α :=
Option.get _ (Nat.find_spec <| (terminates_def _).1 h)
theorem get_mem : get s ∈ s :=
Exists.intro (length s) (Option.eq_some_of_isSome _).symm
| Mathlib/Data/Seq/Computation.lean | 383 | 387 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta, Adam Topaz
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.Functor.Reflect... | slice_lhs 4 5 => rw [← T.map_comp]
simp only [hom_inv_id_app, Functor.map_id, id_comp]
slice_lhs 1 2 => rw [← T.map_comp]
simp only [Functor.map_comp, Category.assoc]
| Mathlib/CategoryTheory/Monad/Basic.lean | 320 | 323 |
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Constructi... | obtain ⟨α, hα⟩ := @IsCyclic.exists_generator Eˣ _ _
use α
rw [eq_top_iff]
rintro x -
by_cases hx : x = 0
· rw [hx]
exact F⟮α.val⟯.zero_mem
· obtain ⟨n, hn⟩ := Set.mem_range.mp (hα (Units.mk0 x hx))
rw [show x = α ^ n by norm_cast; rw [hn, Units.val_mk0]]
exact zpow_mem (mem_adjoin_simple_self ... | Mathlib/FieldTheory/PrimitiveElement.lean | 56 | 66 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 871 | 871 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | variable [CommRing R] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N]
variable {B : BilinMap R M N}
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 771 | 773 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | | | |
v v v
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 792 | 793 |
/-
Copyright (c) 2023 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.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | lemma in₂₁_iff' :
(graph t).Adj (in₂ c) (in₁ b) ↔ ∃ x : α × β × γ, x ∈ t ∧ x.2.2 = c ∧ x.2.1 = b where
mp := by rintro ⟨⟩; exact ⟨_, ‹_›, by simp⟩
mpr := by rintro ⟨⟨a, b, c⟩, h, rfl, rfl⟩; constructor; assumption
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 105 | 108 |
/-
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... | map f ((reindex R s e).symm x) = (reindex R t e).symm (map (fun i ↦ f (e.symm i)) x) :=
DFunLike.congr_fun (map_comp_reindex_symm _ _) _
variable (ι)
attribute [local simp] eq_iff_true_of_subsingleton in
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 787 | 792 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
/-!
# Association Lists
This file defines association lists. An association list is a list where every element consists o... |
@[simp]
theorem empty_union {s : AList β} : (∅ : AList β) ∪ s = s :=
ext rfl
@[simp]
theorem union_empty {s : AList β} : s ∪ (∅ : AList β) = s :=
ext <| by simp
@[simp]
theorem mem_union {a} {s₁ s₂ : AList β} : a ∈ s₁ ∪ s₂ ↔ a ∈ s₁ ∨ a ∈ s₂ :=
mem_keys_kunion
| Mathlib/Data/List/AList.lean | 388 | 399 |
/-
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 ... | Mathlib/Topology/Connected/Basic.lean | 1,148 | 1,222 | |
/-
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.Algebra.Category.ModuleCat.Presheaf.Colimits
import Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits
/-!
# Free sheaves of modules
In this file, we construct ... | @[simp]
lemma sectionMap_freeMap_freeSection (i : I) :
sectionsMap (freeMap (R := R) f) (freeSection i) = freeSection (f i) := by
simp [← freeHomEquiv_comp_apply]
| Mathlib/Algebra/Category/ModuleCat/Sheaf/Free.lean | 73 | 76 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... | h.transpose.map _ fun _ => rfl
@[simp]
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 74 | 76 |
/-
Copyright (c) 2021 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Topology.Separation.Hausdorff
/-!
# Topological space structure on the opposite monoid and on the units group
In this file we define `Topological... |
@[to_additive]
theorem continuous_embedProduct : Continuous (embedProduct M) :=
continuous_induced_dom
@[to_additive]
theorem continuous_val : Continuous ((↑) : Mˣ → M) :=
(@continuous_embedProduct M _ _).fst
@[to_additive]
protected theorem continuous_iff {f : X → Mˣ} :
Continuous f ↔ Continuous (val ∘ f) ∧... | Mathlib/Topology/Algebra/Constructions.lean | 136 | 148 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | calc f ω
_ ≤ g ω + nndist (f ω) (g ω) := NNReal.le_add_nndist (f ω) (g ω)
_ ≤ g ω + nndist f g := (add_le_add_iff_left (g ω)).mpr (le_dist ω)
have le : (f ω : ℝ≥0∞) ≤ (g ω : ℝ≥0∞) + nndist f g := by
simpa only [← ENNReal.coe_add] using (by exact_mod_cast le')
rwa [coe_nnreal_ennrea... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 407 | 422 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... |
theorem AEMeasurable.restrict (hfm : AEMeasurable f μ) {s} : AEMeasurable f (μ.restrict s) :=
⟨AEMeasurable.mk f hfm, hfm.measurable_mk, ae_restrict_of_ae hfm.ae_eq_mk⟩
| Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 278 | 280 |
/-
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.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | Mathlib/Data/Nat/GCD/Basic.lean | 259 | 259 | |
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Eric Wieser, Bhavik Mehta,
Yaël Dillies
-/
import Mathlib.Algebra.Order.Antidiag.Pi
i... | Mathlib/Algebra/Order/Antidiag/Finsupp.lean | 196 | 210 | |
/-
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.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... | ∀ x₁ x₂, f x₁ = f x₂ → x₁ ≠ x₂ → Disjoint (𝓝 x₁) (𝓝 x₂) :=
forall₃_congr fun x x' _ ↦ by simp only [(nhds_basis_opens x).disjoint_iff (nhds_basis_opens x'),
exists_prop, ← exists_and_left, and_assoc, and_comm, and_left_comm]
| Mathlib/Topology/SeparatedMap.lean | 74 | 76 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
import Mathlib.Algebra.MvPolynomial.Eval
/-!
# Formal (multivariate) ... | theorem coeff_truncFun (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
(truncFun n φ).coeff m = if m < n then coeff R m φ else 0 := by
classical
| Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 71 | 73 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | Ico_subset_Ico le_rfl h
| Mathlib/Order/Interval/Finset/Basic.lean | 176 | 177 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Order.Hom.Basic
/-!
# Preimages of intervals under order embeddings
In this file we prove that ... | @[simp] theorem preimage_Icc : e ⁻¹' Icc (e x) (e y) = Icc x y := by ext; simp
| Mathlib/Order/Interval/Set/OrderEmbedding.lean | 33 | 33 |
/-
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... |
/-- A common factor that's coprime with the modulus can be cancelled from a `ModEq` -/
lemma cancel_right_of_coprime (hmc : gcd m c = 1) (h : a * c ≡ b * c [MOD m]) : a ≡ b [MOD m] :=
cancel_left_of_coprime hmc <| by simpa [mul_comm] using h
| Mathlib/Data/Nat/ModEq.lean | 270 | 273 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | simpa
| Mathlib/Data/Matroid/Map.lean | 614 | 615 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... |
open Classical in
lemma count_ne_zero {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) :
count K v I = ((Associates.mk v.asIdeal).count (Associates.mk
(choose (choose_spec (exists_eq_spanSingleton_mul I)))).factors -
(Associates.mk v.asIdeal).count
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 285 | 290 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... |
end LieSubmoduleMapAndComap
namespace LieModuleHom
| Mathlib/Algebra/Lie/Submodule.lean | 921 | 924 |
/-
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
/-!
... |
@[simp]
lemma map_f' : (h.map F).f' = F.map h.f' := by
| Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 126 | 128 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | theorem associator_inv_naturality_middle (X : C) {Y Y' : C} (f : Y ⟶ Y') (Z : C) :
X ◁ f ▷ Z ≫ (α_ X Y' Z).inv = (α_ X Y Z).inv ≫ (X ◁ f) ▷ Z := by simp
@[reassoc]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 452 | 455 |
/-
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... | congr
refine (affineCombination_of_eq_one_of_eq_zero _ _ _ (mem_singleton_self _) ?_ ?_).symm
· simp [hwi]
· simp
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 504 | 508 |
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