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/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | use pullback.snd f g y
exact CategoryTheory.congr_fun pullback.condition y
· rintro ⟨y, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_fst_apply]
theorem pullback_snd_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) :
Set.range (pullback.snd f g) = { y :... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 211 | 247 |
/-
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 | 1,067 | 1,072 | |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | lemma shiftLeft'_false : ∀ n, shiftLeft' false m n = m <<< n
| 0 => rfl
| n + 1 => by
have : 2 * (m * 2^n) = 2^(n+1)*m := by
| Mathlib/Data/Nat/Bits.lean | 131 | 134 |
/-
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 zsmul_add] lemma mul_zpow : ∀ n : ℤ, (a * b) ^ n = a ^ n * b ^ n
| (n : ℕ) => by simp_rw [zpow_natCast, mul_pow]
| .negSucc n => by simp_rw [zpow_negSucc, ← inv_pow, mul_inv, mul_pow]
| Mathlib/Algebra/Group/Basic.lean | 610 | 613 |
/-
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
... | variable {𝕜 ι : Type*} {E : ι → Type*} [Finite ι] [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
[Π i, AddCommGroup (E i)] [Π i, Module 𝕜 (E i)] {s : Set ι} {t : Π i, Set (E i)} {x : Π i, E i}
open Finset Fintype
lemma mem_convexHull_pi (h : ∀ i ∈ s, x i ∈ convexHull 𝕜 (t i)) : x ∈ convexHull 𝕜 (s.pi t) :... | Mathlib/Analysis/Convex/Combination.lean | 562 | 585 |
/-
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... | toMatrix_distrib_mul_action_toLinearMap b x
/-- `leftMulMatrix b x` is the matrix corresponding to the linear map `fun y ↦ x * y`.
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 797 | 800 |
/-
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... | imageSubobjectMap sq ≫ (imageSubobjectIso _).hom := by
erw [← Iso.comp_inv_eq, Category.assoc, ← (imageSubobjectIso f).eq_inv_comp,
image_map_comp_imageSubobjectIso_inv sq]
| Mathlib/CategoryTheory/Subobject/Limits.lean | 419 | 422 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 332 | 333 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Nat.Basic
/-!
# Cast of integers (additional theorems)
This ... | | -[m+1], -[n+1] =>
| Mathlib/Data/Int/Cast/Basic.lean | 105 | 105 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | ← List.flatMap_map Nat.succ fun n => sublistsLen n tl,
← flatMap_cons (f := fun n => sublistsLen n tl), ← range_succ_eq_map]
exact l_ih
end List
| Mathlib/Data/List/Sublists.lean | 405 | 411 |
/-
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.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | Mathlib/Data/Sign.lean | 354 | 354 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | lemma divisors_filter_dvd_of_dvd {n m : ℕ} (hn : n ≠ 0) (hm : m ∣ n) :
{d ∈ n.divisors | d ∣ m} = m.divisors := by
| Mathlib/NumberTheory/Divisors.lean | 220 | 221 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.PUnit
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheo... | the multiplication on the tensor product `M.X ⊗ N.X` is defined in the obvious way:
it is the tensor product of the multiplications on `M` and `N`,
except that the tensor factors in the source come in the wrong order,
which we fix by pre-composing with a permutation isomorphism constructed from the braiding.
(There is... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 440 | 447 |
/-
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.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | theorem PredOrder.hasBasis_nhds_Ioc [OrderTopology α] [PredOrder α] [NoMaxOrder α] {a : α} :
(𝓝 a).HasBasis (a < ·) (Set.Ico a ·) :=
PredOrder.hasBasis_nhds_Ioc_of_exists_gt (exists_gt a)
| Mathlib/Topology/Order/Basic.lean | 500 | 502 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Polynomial.Taylor
import Mathlib.FieldTheory.RatFunc.AsPolynomial
/-!
# Laurent expansions of rational functions
## Main declarations
* `R... | def laurent : RatFunc R →ₐ[R] RatFunc R :=
RatFunc.mapAlgHom (.ofLinearMap (taylor r) (taylor_one _) (taylor_mul _))
| Mathlib/FieldTheory/Laurent.lean | 74 | 75 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Typeclasses... | · simpa only [x_in_A, indicator_of_not_mem, Eq.comm (a := (0 : ℝ≥0∞)), indicator_apply_eq_zero,
false_iff, not_false_eq_true, Pi.one_apply, one_ne_zero, imp_false] using obs
· intro h
ext A A_mble
by_cases x_in_A : x ∈ A
· simp only [Measure.dirac_apply' _ A_mble, x_in_A, indicator_o... | Mathlib/MeasureTheory/Measure/Dirac.lean | 194 | 203 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | simpa using coeff_list_prod_of_natDegree_le _ _ hl
theorem coeff_prod_of_natDegree_le (f : ι → R[X]) (n : ℕ) (h : ∀ p ∈ s, natDegree (f p) ≤ n) :
coeff (∏ i ∈ s, f i) (#s * n) = ∏ i ∈ s, coeff (f i) n := by
| Mathlib/Algebra/Polynomial/BigOperators.lean | 219 | 222 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
theorem exists_finset_piecewise_mem_of_mem_nhds [DecidableEq ι] {s : Set (∀ a, π a)} {x : ∀ a, π a}
(hs : s ∈ 𝓝 x) (y : ∀ a, π a) : ∃ I : Finset ι, I.piecewise x y ∈ s := by
simp only [nhds_pi, Filter.mem_pi'] at hs
| Mathlib/Topology/Constructions.lean | 924 | 927 |
/-
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.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... |
theorem edgeDensity_sub_edgeDensity_le_one_sub_mul (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁) (hs₂ : s₂.Nonempty)
(ht₂ : t₂.Nonempty) :
edgeDensity r s₂ t₂ - edgeDensity r s₁ t₁ ≤ 1 - #s₂ / #s₁ * (#t₂ / #t₁) := by
refine (sub_le_sub_left (mul_edgeDensity_le_edgeDensity r hs ht hs₂ ht₂) _).trans ?_
refine le_trans ?_ (m... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 171 | 176 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 535 | 548 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,021 | 1,024 | |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma P3.trans (h₁ : P3 x₁ x₂ y₁ y₂) (h₂ : P3 x₂ x₃ y₁ y₂) : P3 x₁ x₃ y₁ y₂ := by
rw [P3] at h₁ h₂
rw [P3, ← add_lt_add_iff_left (⟦x₂ * y₁⟧ + ⟦x₂ * y₂⟧)]
convert add_lt_add h₁ h₂ using 1 <;> abel
| Mathlib/SetTheory/Surreal/Multiplication.lean | 100 | 103 |
/-
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 | 627 | 630 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.Dilation
/-!
# Dilation equivalence
In this file we define `DilationEquiv X Y`, a type of bundled equivalences between `X` and... | instance : Group (X ≃ᵈ X) where
mul e e' := e'.trans e
| Mathlib/Topology/MetricSpace/DilationEquiv.lean | 135 | 136 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Order.Group.Multiset
import Mathlib.Data.Setoid.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Tactic.Ap... |
/-- Inverse of `Sym_option_succ_equiv.decode`. -/
def decode : Sym (Option α) n ⊕ Sym α n.succ → Sym (Option α) n.succ
| Sum.inl s => none ::ₛ s
| Sum.inr s => s.map Embedding.some
@[simp]
theorem decode_inl (s : Sym (Option α) n) : decode (Sum.inl s) = none ::ₛ s :=
rfl
| Mathlib/Data/Sym/Basic.lean | 594 | 603 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | variable [PseudoMetricSpace X]
@[simp] theorem nndist_ofMul (a b : X) : nndist (ofMul a) (ofMul b) = nndist a b := rfl
@[simp] theorem nndist_ofAdd (a b : X) : nndist (ofAdd a) (ofAdd b) = nndist a b := rfl
@[simp] theorem nndist_toMul (a b : Additive X) : nndist a.toMul b.toMul = nndist a b := rfl
@[simp]
theorem ... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,168 | 1,178 |
/-
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... | ← AlgHom.card_of_splits F E A hA, Fintype.card, toFinset_range, Finset.card_image_iff,
Finset.coe_univ, ← injective_iff_injOn_univ]
theorem primitive_element_iff_algHom_eq_of_eval (α : E)
| Mathlib/FieldTheory/PrimitiveElement.lean | 386 | 389 |
/-
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, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... |
@[simp]
theorem Matrix.toLinearMapₛₗ₂'_single (M : Matrix n m N₂) (i : n) (j : m) :
Matrix.toLinearMapₛₗ₂' R σ₁ σ₂ M (Pi.single i 1) (Pi.single j 1) = M i j :=
Matrix.toLinearMap₂'Aux_single σ₁ σ₂ M i j
@[simp]
| Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 194 | 200 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
/-!
# Linear equivalences of tensor products as isometrie... | section tensorRId
theorem comp_tensorRId_eq (Q₁ : QuadraticForm R M₁) :
Q₁.comp (TensorProduct.rid R M₁) = Q₁.tmul (sq (R := R)) := by
refine (QuadraticMap.associated_rightInverse R).injective ?_
ext m₁ m₁'
dsimp [-associated_apply]
| Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 153 | 159 |
/-
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 | 676 | 677 | |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sigma
import Mathlib.... | refine i.lastCases ?_ fun _ ↦ by dsimp only [smash]; apply Fin.addCases_left
show p.smash q h (Fin.natAdd p.length (0 : Fin (q.length + 1))) = _
simpa only [smash, Fin.addCases_right] using h.symm
lemma smash_castAdd {p q : RelSeries r} (h : p.last = q.head) (i : Fin p.length) :
p.smash q h (i.castAdd q.leng... | Mathlib/Order/RelSeries.lean | 543 | 551 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... | ∏ c ∈ t, f c ∈ unitInterval := _root_.prod_mem (S := unitInterval.submonoid) h
instance : LinearOrderedCommMonoidWithZero I where
zero_mul i := zero_mul i
mul_zero i := mul_zero i
zero_le_one := nonneg'
| Mathlib/Topology/UnitInterval.lean | 232 | 237 |
/-
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,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | theorem inter_reverse {xs ys : List α} : xs.inter ys.reverse = xs.inter ys := by
| Mathlib/Data/List/Lattice.lean | 147 | 147 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.Instances.AddCircle
/-!
# The additive circle as a norm... | obtain rfl | ht := eq_or_ne t 0
· simp
simp only [norm_eq_infDist, Real.norm_eq_abs, ← Real.norm_eq_abs, ← infDist_smul₀ ht, smul_zero]
congr with m
simp only [zmultiples, eq_iff_sub_mem, zsmul_eq_mul, mem_mk, mem_setOf_eq,
mem_smul_set_iff_inv_smul_mem₀ ht, smul_eq_mul]
simp_rw [mul_left_comm, ← smul_e... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 44 | 68 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison
-/
import Mathlib.CategoryTheory.Equivalence
/-!
# Opposite categories
We provide a category instance on `Cᵒᵖ`.
The morphisms `X ⟶ Y` are defined to be the... | functor_unitIso_comp X := by
apply Quiver.Hom.unop_inj
dsimp
| Mathlib/CategoryTheory/Opposites.lean | 484 | 486 |
/-
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... | theorem sdiff_le_comm : a \ b ≤ c ↔ a \ c ≤ b := by rw [sdiff_le_iff, sdiff_le_iff']
| Mathlib/Order/Heyting/Basic.lean | 388 | 389 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | -/
variable [FunLike F M N]
| Mathlib/Algebra/Group/Hom/Defs.lean | 213 | 215 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
/-!
# Basic theory of bornology
We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining
bor... | theorem isCobounded_biInter_finset (s : Finset ι) {f : ι → Set α} :
IsCobounded (⋂ i ∈ s, f i) ↔ ∀ i ∈ s, IsCobounded (f i) :=
biInter_finset_mem s
@[simp]
theorem isCobounded_iInter [Finite ι] {f : ι → Set α} :
IsCobounded (⋂ i, f i) ↔ ∀ i, IsCobounded (f i) :=
iInter_mem
| Mathlib/Topology/Bornology/Basic.lean | 226 | 233 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 723 | 725 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,912 | 1,915 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra... | exact zero_mem (A 0)
| succ _ n_ih =>
rw [Nat.cast_succ]
exact add_mem n_ih (SetLike.one_mem_graded _)
| Mathlib/Algebra/DirectSum/Internal.lean | 66 | 69 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... |
lemma take_drop : (a.drop m).take n = (a.take (m + n)).drop m := by
| Mathlib/Data/Stream/Init.lean | 586 | 587 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
/-!
# Contractible spaces
In this file, we define `ContractibleSpace`, a space ... | exact Homotopic.hcomp hy (Homotopic.refl g)
theorem Nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) :
(f.comp g).Nullhomotopic := by
obtain ⟨y, hy⟩ := hf
| Mathlib/Topology/Homotopy/Contractible.lean | 32 | 36 |
/-
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... | Mathlib/Computability/TMToPartrec.lean | 1,364 | 1,367 | |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | (fun x ↦ a x • u x) ~[l] fun x ↦ b x • v x := by
rcases hab.exists_eq_mul with ⟨φ, hφ, habφ⟩
have : ((fun x ↦ a x • u x) - (fun x ↦ b x • v x)) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x) := by
convert (habφ.comp₂ (· • ·) <| EventuallyEq.refl _ u).sub
| Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 218 | 221 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | theorem hasDerivAt_rpow_const {x p : ℝ} (h : x ≠ 0 ∨ 1 ≤ p) :
HasDerivAt (fun x => x ^ p) (p * x ^ (p - 1)) x := by
rcases ne_or_eq x 0 with (hx | rfl)
· exact (hasStrictDerivAt_rpow_const_of_ne hx _).hasDerivAt
replace h : 1 ≤ p := h.neg_resolve_left rfl
apply hasDerivAt_of_hasDerivAt_of_ne fun x hx =>
... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 396 | 403 |
/-
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 zsmul_zero_sub]
| Mathlib/Algebra/Group/Basic.lean | 474 | 474 |
/-
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.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal + b.toReal := by
lift a to ℝ≥0 using ha
lift b to ℝ≥0 using hb
rfl
| Mathlib/Data/ENNReal/Real.lean | 37 | 40 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | theorem mem_sublists' {s t : List α} : s ∈ sublists' t ↔ s <+ t := by
induction' t with a t IH generalizing s
· simp only [sublists'_nil, mem_singleton]
exact ⟨fun h => by rw [h], eq_nil_of_sublist_nil⟩
simp only [sublists'_cons, mem_append, IH, mem_map]
| Mathlib/Data/List/Sublists.lean | 68 | 72 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 755 | 762 | |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | obtain ⟨γ, hγ⟩ : ∃ γ : Path (p' 0) (p' n), (∀ i ≤ n, p' i ∈ range γ) ∧ range γ ⊆ s := by
have hp' : ∀ i ≤ n, p' i ∈ s := by
intro i hi
simp [p', Nat.lt_succ_of_le hi, hp]
clear_value p'
clear hp p
induction n with
| zero =>
use Path.refl (p' 0)
constructor
· rintro i ... | Mathlib/Topology/Connected/PathConnected.lean | 365 | 399 |
/-
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.Fold
import Mathlib.Data.Multiset.Bind
import Mathlib.Order.SetNotation
/-!
# Unions of finite sets
This file defines the union of a fami... | lemma biUnion_biUnion [DecidableEq γ] (s : Finset α) (f : α → Finset β) (g : β → Finset γ) :
(s.biUnion f).biUnion g = s.biUnion fun a ↦ (f a).biUnion g := by
ext
simp only [Finset.mem_biUnion, exists_prop]
simp_rw [← exists_and_right, ← exists_and_left, and_assoc]
rw [exists_comm]
| Mathlib/Data/Finset/Union.lean | 181 | 186 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.List.InsertIdx
/-! # Some lemmas about lists involving sets
Split out from `Data.List.Basic` to... |
theorem mapAccumr₂_eq_foldr {σ φ : Type*} (f : α → β → σ → σ × φ) :
∀ (as : List α) (bs : List β) (s : σ),
mapAccumr₂ f as bs s = foldr (fun ab s =>
let r := f ab.1 ab.2 s.1
(r.1, r.2 :: s.2)
) (s, []) (as.zip bs)
| [], [... | Mathlib/Data/List/Lemmas.lean | 90 | 97 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | @[to_additive (attr := simp)]
theorem inv_le_self_iff : a⁻¹ ≤ a ↔ 1 ≤ a := by simp [inv_le_iff_one_le_mul']
| Mathlib/Algebra/Order/Group/Defs.lean | 138 | 139 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Action.Pointwise.Finset
import Mathlib.Algebra.Ring.Nat
/-!
# e-transforms
e-transforms are a family of transformations of pairs of finite ... | x.1.card + x.2.card + (x.1.card + x.2.card) := by
rw [add_add_add_comm, mulETransformLeft.card, mulETransformRight.card, ← mul_add, two_mul]
end Group
| Mathlib/Combinatorics/Additive/ETransform.lean | 150 | 153 |
/-
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, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... |
theorem cos_add_int_mul_two_pi (x : ℂ) (n : ℤ) : cos (x + n * (2 * π)) = cos x :=
cos_periodic.int_mul n x
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,093 | 1,095 |
/-
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.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Comb... | {B ∈ (atomise U <| P.nonuniformWitnesses G ε U).parts |
B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty}.biUnion
fun B => B \ {A ∈ (chunk hP G ε hU).parts | A ⊆ B}.biUnion id := by
intro x hx
rw [← biUnion_filter_atomise hX (G.nonuniformWitness_subset h₂), star, mem_sdiff,
mem_biUnion] at... | Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 92 | 124 |
/-
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.Topology.PartialHomeomorph
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Charted spaces
A smooth manifold is a topological space `M`... |
variable {ι : Type*} {Hi : ι → Type*}
instance modelPiInhabited [∀ i, Inhabited (Hi i)] : Inhabited (ModelPi Hi) :=
Pi.instInhabited
| Mathlib/Geometry/Manifold/ChartedSpace.lean | 835 | 840 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,691 | 1,692 | |
/-
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... | | 1, b => by rw [one_add b, succ_to_nat, add_comm, cast_one]
| a, 1 => by rw [add_one a, succ_to_nat, cast_one]
| Mathlib/Data/Num/Lemmas.lean | 69 | 70 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbs
import Mathlib.NumberTheory.ClassNumber.Finite
import Mathlib.NumberTheory.NumberField.Discriminant.Basic
/-!
# Class... | rw [← Real.sqrt_lt (by positivity) (by positivity), mul_assoc, ← inv_mul_lt_iff₀' (by positivity),
mul_inv, ← inv_pow, inv_div, inv_div, mul_assoc, Int.cast_abs] at h
rw [← classNumber_eq_one_iff, classNumber, Fintype.card_eq_one_iff]
refine ⟨1, fun C ↦ ?_⟩
obtain ⟨I, rfl, hI⟩ := exists_ideal_in_class_of_no... | Mathlib/NumberTheory/NumberField/ClassNumber.lean | 77 | 92 |
/-
Copyright (c) 2021 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.SetFamily.Shadow
/-!
# UV-compressions
This file defines UV-compression. It is an operation on a set family that reduces its ... | · rfl
lemma _root_.Set.Sized.uvCompression (huv : #u = #v) (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
(𝓒 u v 𝒜 : Set (Finset α)).Sized r := by
simp_rw [Set.Sized, mem_coe, mem_compression]
rintro s (hs | ⟨huvt, t, ht, rfl⟩)
· exact h𝒜 hs.1
· rw [card_compress huv, h𝒜 ht]
private theorem aux (huv : ∀ x ... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 276 | 285 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | simp only [comp_zero_cochain_v, ofHom_v, HomologicalComplex.id_f, comp_id]
@[simp]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 379 | 381 |
/-
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.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.Analysis.NormedSpace.Multiline... | variable [IsTopologicalAddGroup V]
/-- In a topological vector space with separating dual, the group of continuous linear equivalences
acts transitively on the set of nonzero vectors: given two nonzero vectors `x` and `y`, there
exists `A : V ≃L[R] V` mapping `x` to `y`. -/
theorem exists_continuousLinearEquiv_apply_e... | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 117 | 144 |
/-
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.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | refine (tendsto_congr' ?_).mpr (tendsto_log_atTop.atTop_add (tendsto_const_nhds (x := 1)))
rw [EventuallyEq, eventually_atTop]
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 91 | 92 |
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Algebra.Pi
import Mathlib.RingTheory.AdicCompletion.Basic
/-!
# Algebr... | let f : AdicCompletion I R → ∀ n, R ⧸ (I ^ n • ⊤ : Ideal R) := Subtype.val
Subtype.val_injective.commRing f rfl rfl
(fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
| Mathlib/RingTheory/AdicCompletion/Algebra.lean | 87 | 89 |
/-
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.Order.Hom.CompleteLattice
import Mathlib.Topology.Compactness.Bases
import Mathlib.Topology.ContinuousMap.Basic
impor... | · obtain ⟨t, ht⟩ :=
H (fun i => U i) (fun i => (U i).isOpen) (by simpa using show (s : Set α) ⊆ ↑(iSup U) from hU)
refine ⟨t, Set.Subset.trans ht ?_⟩
simp only [Set.iUnion_subset_iff]
show ∀ i ∈ t, U i ≤ t.sup U
exact fun i => Finset.le_sup
/-- The preimage of an open set, as an open set. -/
| Mathlib/Topology/Sets/Opens.lean | 332 | 339 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Finsupp
import Mathlib.Order.Filter.AtTopBot.Defs
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Gra... | change Localization.mk _ _ = Localization.mk _ _
simp only [num_add, den_add, ← Localization.add_mk]
| Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 367 | 368 |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Tactic.Abel
/-!
# Circulant matrices
This file contains the defini... |
theorem Fin.circulant_injective : ∀ n, Injective fun v : Fin n → α => circulant v
| 0 => by simp [Injective]
| _ + 1 => Matrix.circulant_injective
| Mathlib/LinearAlgebra/Matrix/Circulant.lean | 61 | 64 |
/-
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... | [Module S N] [IsScalarTower S R N] (a : S)
(x : M) : Q (a • x) = (a * a) • Q x := by
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 236 | 237 |
/-
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.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | (C) For any closed set F, the limsup of the measures of F under μs is at most
the measure of F under μ, i.e., limsupᵢ μsᵢ(F) ≤ μ(F).
Combining with a earlier proven implications, we get that (B) implies also
(O) For any open set G, the liminf of the measures of G under μs is at least
the measure of G ... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 360 | 367 |
/-
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.AlgebraicGeometry.EllipticCurve.Affine
import Mathlib.LinearAlgebra.FreeModule.Norm
import Mathlib.RingTheory.ClassGroup
import Mat... | mk_XYIdeal'_neg_mul
lemma mk_XYIdeal'_mul_mk_XYIdeal' {x₁ x₂ y₁ y₂ : F} (h₁ : W.Nonsingular x₁ y₁)
(h₂ : W.Nonsingular x₂ y₂) (hxy : ¬(x₁ = x₂ ∧ y₁ = W.negY x₂ y₂)) :
ClassGroup.mk (XYIdeal' h₁) * ClassGroup.mk (XYIdeal' h₂) =
ClassGroup.mk (XYIdeal' <| nonsingular_add h₁ h₂ hxy) := by
rw [← map_mul]... | Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean | 395 | 435 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 944 | 945 | |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
/-!
# The `arctan` function.
Inequalit... | univ_subset_iff.1 surjOn_tan
/-- `Real.tan` as an `OrderIso` between `(-(π / 2), π / 2)` and `ℝ`. -/
def tanOrderIso : Ioo (-(π / 2)) (π / 2) ≃o ℝ :=
(strictMonoOn_tan.orderIso _ _).trans <|
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 89 | 93 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | classical
ext1 s
dsimp [extend]
by_cases h : P s
· simp [h]
· simp [h, ENNReal.smul_top, hc]
theorem le_extend {s : α} (h : P s) : m s h ≤ extend m s := by
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 55 | 62 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | /-- `1` is the multiplicative homomorphism sending all elements to `1`. -/
@[to_additive "`0` is the additive homomorphism sending all elements to `0`"]
instance [Mul M] [MulOneClass N] : One (M →ₙ* N) :=
⟨⟨fun _ => 1, fun _ _ => (one_mul 1).symm⟩⟩
| Mathlib/Algebra/Group/Hom/Defs.lean | 958 | 961 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | Mathlib/Probability/Kernel/Basic.lean | 746 | 750 | |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
i... | simp only [Path.trans_apply, not_le, coe_reparam, Function.comp_apply, one_div, Path.refl_apply]
split_ifs
| Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 143 | 144 |
/-
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
import Mathlib.LinearAlgebra.AffineSpace.Independent
/-!
# Simplicial complexes
In this file, we define ... | theorem convexHull_subset_space (hs : s ∈ K.faces) : convexHull 𝕜 ↑s ⊆ K.space := by
convert subset_biUnion_of_mem hs
| Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 86 | 87 |
/-
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
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... |
/-- **Minimum principle** for concave functions on an interval. If a function `f` is concave on the
interval `[x, y]`, then the eventual minimum of `f` on `[x, y]` is at `x` or `y`. -/
lemma ConcaveOn.min_le_of_mem_Icc {s : Set 𝕜} {f : 𝕜 → β} {x y z : 𝕜} (hf : ConcaveOn 𝕜 s f)
(hx : x ∈ s) (hy : y ∈ s) (hz : z... | Mathlib/Analysis/Convex/Jensen.lean | 313 | 317 |
/-
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.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
import Mathlib.CategoryTheory.Limits.Shape... | /-- A mono factorisation coming from a strong epi-mono factorisation always has the universal
| Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 797 | 797 |
/-
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.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.WithBot
/-!
# Intervals in `WithTop α` and `WithBot α`
In this file we ... |
theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by
| Mathlib/Order/Interval/Set/WithBotTop.lean | 80 | 81 |
/-
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.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | def IsEllDivSequence : Prop :=
IsEllSequence W ∧ IsDivSequence W
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 83 | 84 |
/-
Copyright (c) 2022 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
/-!
# Subtraction of measures
In this file we define `μ - ν` to be the least measure `τ` such that `μ ≤ ... | end Measure
end MeasureTheory
| Mathlib/MeasureTheory/Measure/Sub.lean | 137 | 139 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | lemma surjOn_of_injOn_of_card_le (f : α → β) (hf : Set.MapsTo f s t) (hinj : Set.InjOn f s)
(hst : #t ≤ #s) : Set.SurjOn f s t := by
classical
suffices s.image f = t by simp [← this, Set.SurjOn]
have : s.image f ⊆ t := by aesop (add simp Finset.subset_iff)
| Mathlib/Data/Finset/Card.lean | 433 | 437 |
/-
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... | theorem funMap_equiv_unify {n : ℕ} (F : L.Functions n) (x : Fin n → Σˣ f) (i : ι)
(hi : i ∈ upperBounds (range (Sigma.fst ∘ x))) :
funMap F x ≈ .mk f _ (funMap F (unify f x i hi)) :=
funMap_unify_equiv G f F x (Classical.choose (Finite.bddAbove_range fun a => (x a).1)) i _ hi
theorem relMap_equiv_unify {n : ... | Mathlib/ModelTheory/DirectLimit.lean | 199 | 204 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.GroupTheory.Subgroup.Center
import Mathlib.GroupTheory.Submonoid.Centralizer
/-!
# Centralizers of subgroups
-/
ass... | simp only [mem_centralizer_iff, Set.mem_singleton_iff, forall_eq]
exact eq_comm
| Mathlib/GroupTheory/Subgroup/Centralizer.lean | 42 | 44 |
/-
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.... | [AddCommMonoid M₃] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) :
(ker f).prod (ker g) ≤ ker (f.coprod g) := by
rintro ⟨y, z⟩
simp +contextual
theorem ker_coprod_of_disjoint_range {M₂ : Type*} [AddCommGroup M₂] [Module R M₂] {M₃ : Type*}
[AddCommGroup M₃] [Module R M₃] (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R... | Mathlib/LinearAlgebra/Prod.lean | 445 | 455 |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
import Mathlib.MeasureTheory.Function.LpSpace.Complete
import Mathlib.MeasureThe... |
theorem tendsto_approxOn_Lp_eLpNorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
{s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
(hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : eLpNorm (fun x => f x - y₀) p μ < ∞) :
Tendsto (fun n => eLpNorm (⇑(approxOn f hf s y₀... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 85 | 90 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | Mathlib/Analysis/Convex/Side.lean | 881 | 888 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Grou... | hom := mapMap e.hom p
inv := mapMap e.inv p
variable (C)
| Mathlib/CategoryTheory/GradedObject.lean | 421 | 424 |
/-
Copyright (c) 2020 Thomas Browning and Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
/-... | (h : (X ^ n - 1 : F[X]).Splits (RingHom.id F)) : IsSolvable (X ^ n - C a).Gal := by
by_cases ha : a = 0
· rw [ha, C_0, sub_zero]
exact gal_X_pow_isSolvable n
have ha' : algebraMap F (X ^ n - C a).SplittingField a ≠ 0 :=
mt ((injective_iff_map_eq_zero _).mp (RingHom.injective _) a) ha
by_cases hn : n... | Mathlib/FieldTheory/AbelRuffini.lean | 98 | 114 |
/-
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.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | /-- Lift a permutation-respecting function on 2 `AList`s to 2 `Finmap`s. -/
def liftOn₂ {γ} (s₁ s₂ : Finmap β) (f : AList β → AList β → γ)
(H : ∀ a₁ b₁ a₂ b₂ : AList β,
| Mathlib/Data/Finmap.lean | 118 | 120 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... |
@[simp] lemma coe_nonZeroDivisorsRight_eq [NoZeroDivisors M₀] [Nontrivial M₀] :
nonZeroDivisorsRight M₀ = {x : M₀ | x ≠ 0} := by
ext x
simp only [SetLike.mem_coe, mem_nonZeroDivisorsRight_iff, mul_eq_zero, Set.mem_setOf_eq]
refine ⟨fun h ↦ ?_, fun hx y hx' ↦ by aesop⟩
contrapose! h
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 77 | 83 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | | monomial n a =>
rw [smeval_monomial, smul_mul_assoc, smul_mul_assoc, npow_mul_assoc, ← smul_mul_assoc]
theorem smeval_mul_X_pow : ∀ (n : ℕ), (p * X^n).smeval x = p.smeval x * x^n
| 0 => by
simp only [npow_zero, mul_one]
| n + 1 => by
rw [npow_add, ← mul_assoc, npow_one, smeval_mul_X, smeval_mul_X_p... | Mathlib/Algebra/Polynomial/Smeval.lean | 255 | 262 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]
/-- **Weighted Hölder inequality**. -/
lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0∞) :
∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ := by
obtain rfl | hp := hp.eq_or_lt
· simp
... | Mathlib/Analysis/MeanInequalities.lean | 909 | 924 |
/-
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
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | (prod.mapIso (asIso f) (asIso g)).isIso_hom
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 706 | 707 |
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Mul
/-!
# Incidence matrix of a si... |
theorem incMatrix_of_mem_incidenceSet (h : e ∈ G.incidenceSet a) : G.incMatrix R a e = 1 := by
rw [incMatrix_apply, Set.indicator_of_mem h, Pi.one_apply]
variable [Nontrivial R]
| Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 85 | 89 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | /-- The orthogonal projection is self-adjoint. -/
theorem inner_orthogonalProjection_left_eq_right [K.HasOrthogonalProjection] (u v : E) :
⟪↑(K.orthogonalProjection u), v⟫ = ⟪u, K.orthogonalProjection v⟫ := by
rw [← inner_orthogonalProjection_eq_of_mem_left, inner_orthogonalProjection_eq_of_mem_right]
/-- The or... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,058 | 1,066 |
/-
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... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 316 | 319 |
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