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 |
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/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | /--
If `α` is a type, then `FreeGroup α` is the free group generated by `α`.
| Mathlib/GroupTheory/FreeGroup/Basic.lean | 377 | 378 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Order.Archimedean.IndicatorCard
import Mathlib.Probability.Martingale.Centering
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Prob... | ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) → ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) := by
have ht :
∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, Tendsto (fun n => stoppedValue f (leastGE f i n) ω) atTop (𝓝 c) := by
rw [ae_all_iff]
exact fun i => Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)
... | Mathlib/Probability/Martingale/BorelCantelli.lean | 145 | 150 |
/-
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... |
/-- `count p n` can be expressed as the cardinality of `{k // k < n ∧ p k}`. -/
theorem count_eq_card_fintype (n : ℕ) : count p n = Fintype.card { k : ℕ // k < n ∧ p k } := by
| Mathlib/Data/Nat/Count.lean | 54 | 56 |
/-
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.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | (b.map f).det v = b.det (f.symm ∘ v) := by
rw [Basis.det_apply, Basis.toMatrix_map, Basis.det_apply]
| Mathlib/LinearAlgebra/Determinant.lean | 609 | 611 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | theorem inv_mul_eq_one {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) : I⁻¹ * I = 1 :=
(mul_comm _ _).trans (h.mul_inv_eq_one hI)
protected theorem isUnit {I : FractionalIdeal A⁰ K} (hI : I ≠ 0) : IsUnit I :=
isUnit_of_mul_eq_one _ _ (h.mul_inv_eq_one hI)
theorem isNoetherianRing : IsNoetherianRing A := by
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 274 | 280 |
/-
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.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... | (hp₁p₃ : p₁ ≠ p₃) : (2 : ℤ) • ∡ p₃ p₁ s.center + (2 : ℤ) • ∡ p₁ p₂ p₃ = π := by
rw [← oangle_center_eq_two_zsmul_oangle hp₁ hp₂ hp₃ hp₂p₁ hp₂p₃,
oangle_eq_pi_sub_two_zsmul_oangle_center_right hp₁ hp₃ hp₁p₃, add_sub_cancel]
| Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 128 | 131 |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | · intro h_eq; rw [Subtype.coe_inj] at h_eq; exact h_nab h_eq
· exact h_fab
/-- If the sequence of ordinal-indexed approximations takes a value twice,
then it actually stabilised at that value. -/
lemma lfpApprox_mem_fixedPoints_of_eq {a b c : Ordinal}
(h_init : x ≤ f x) (h_ab : a < b) (h_ac : a ≤ c) (h_fab : l... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 177 | 194 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.Finset.Defs
import Mathlib.Data.Multiset.ZeroCons
import Mathlib.Order.Directed
/-!
# Finsets of ordered types
-/
universe u v w
va... | theorem Directed.finset_le {r : α → α → Prop} [IsTrans α r] {ι} [hι : Nonempty ι] {f : ι → α}
(D : Directed r f) (s : Finset ι) : ∃ z, ∀ i ∈ s, r (f i) (f z) :=
show ∃ z, ∀ i ∈ s.1, r (f i) (f z) from
Multiset.induction_on s.1 (let ⟨z⟩ := hι; ⟨z, fun _ ↦ by simp⟩)
fun i _ ⟨j, H⟩ ↦
let ⟨k, h₁, h₂⟩ ... | Mathlib/Data/Finset/Order.lean | 19 | 26 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... |
@[to_additive]
protected theorem Multipliable.map [CommMonoid γ] [TopologicalSpace γ] (hf : Multipliable f) {G}
[FunLike G α γ] [MonoidHomClass G α γ] (g : G) (hg : Continuous g) : Multipliable (g ∘ f) :=
(hf.hasProd.map g hg).multipliable
@[to_additive]
protected theorem Multipliable.map_iff_of_leftInverse [Co... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 201 | 208 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# First-... | Mathlib/ModelTheory/Satisfiability.lean | 527 | 528 | |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | Mathlib/Data/Finset/NoncommProd.lean | 463 | 484 | |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
/-!
# Basic definitions about impartial (pre-)games
We will d... | Note that this is a slightly more general definition than the one that's usually in the literature,
as we don't require `G ≡ -G`. Despite this, the Sprague-Grundy theorem still holds: see
`SetTheory.PGame.equiv_nim_grundyValue`.
| Mathlib/SetTheory/Game/Impartial.lean | 35 | 38 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
/-!
# Phragmen-Lindelöf principle
In this file we prove several ve... | for some `A`, `B`, and `c < 2`;
* `f` is equal to zero on the boundary of the third quadrant.
Then `f` is equal to zero on the closed third quadrant. -/
theorem eq_zero_on_quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
(hB : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 529 | 550 |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | found at the end of the path if it is a valid path for that value and
return a default tree -/
def isubtree [DecidableEq F.A] [Inhabited (M F)] : Path F → M F → M F
| [], x => x
| ⟨a, i⟩ :: ps, x =>
| Mathlib/Data/PFunctor/Univariate/M.lean | 396 | 400 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... | · exact (Algebra.smul_def _ _).symm
variable {A K}
@[simp]
| Mathlib/RingTheory/DedekindDomain/Different.lean | 283 | 287 |
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle, Jake Levinson
-/
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathli... | (-1 : ℝ) ^ n * aeval x (hermite n) * Real.exp (-(x ^ 2 / 2)) := by
rw [mul_assoc]
induction' n with n ih generalizing x
· rw [Function.iterate_zero_apply, pow_zero, one_mul, hermite_zero, C_1, map_one, one_mul]
· replace ih : deriv^[n] _ = _ := _root_.funext ih
have deriv_gaussian :
deriv (fun y =... | Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 40 | 55 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# Matrix results for barycentric co-ordinates
Results about the ... | variable {ι' : Type*}
theorem toMatrix_row_sum_one [Fintype ι] (q : ι' → P) (i : ι') : ∑ j, b.toMatrix q i j = 1 := by
| Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 48 | 50 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | hrm
⟨n + k, by
| Mathlib/RingTheory/Ideal/Operations.lean | 849 | 850 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | ∀ z : ℂ, z ∈ Metric.ball s v → ‖F' z t‖ ≤ bound t := by
refine (ae_restrict_mem measurableSet_Ioi).mono fun t ht z hz => ?_
simp_rw [F', bound, norm_smul, norm_mul, norm_real, mul_assoc]
gcongr
rw [norm_cpow_eq_rpow_re_of_pos ht]
rcases le_or_lt 1 t with h | h
· refine le_add_of_le_of_nonn... | Mathlib/Analysis/MellinTransform.lean | 339 | 414 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | obtain ⟨T', hT', hrun⟩ := tr_respects_aux₂ (Λ := Λ) hT o
have := hgo.tail' rfl
rw [tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_snd,
stk_nth_val _ (hT k), List.getElem?_eq_none (le_of_eq List.length_reverse),
Option.isNone, cond, hrun, TM1.stepAux] at this
| Mathlib/Computability/TuringMachine.lean | 663 | 667 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, David Kurniadi Angdinata, Devon Tuma, Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathlib.GroupTheory.GroupAction.... | rw [eval₂_sum]
refine Finset.sum_eq_zero fun n _ => ?_
dsimp
rw [eval₂_monomial (C.comp (Quotient.mk I)) X]
refine mul_eq_zero_of_left (Polynomial.ext fun m => ?_) (X ^ n)
rw [RingHom.comp_apply, coeff_C]
by_cases h : m = 0
· simpa [h] using Quotient.eq_zero_iff_mem.2 ((mem_map_C_iff.1 ha) n)
· simp [... | Mathlib/RingTheory/Polynomial/Quotient.lean | 94 | 107 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Patrick Massot
-/
import Mathlib.Topology.Neighborhoods
/-!
# Neighborhoods of a set
In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all ne... | Mathlib/Topology/NhdsSet.lean | 216 | 218 | |
/-
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... | simp [h₂]
· subst h₂
simp [h]
· simp [h₁, h₂, ih]
| Mathlib/Data/List/Sigma.lean | 483 | 487 |
/-
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... |
section LocallyFiniteOrderBot
| Mathlib/Order/Interval/Finset/Basic.lean | 915 | 916 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | Mathlib/Analysis/Convex/Between.lean | 911 | 913 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
/-!
# Intersecting families
This file defines intersecting families and proves their basic proper... | classical
rintro a b hab ha
rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]
· exact mem_insert_self _ _
rw [coe_insert]
exact
hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab
| Mathlib/Combinatorics/SetFamily/Intersecting.lean | 111 | 118 |
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | exact h.smul _
| Mathlib/Algebra/Lie/Free.lean | 99 | 100 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
/-!
# Basic definitions about impartial (pre-)games
We will d... |
theorem forall_rightMoves_fuzzy_iff_equiv_zero : (∀ j, G.moveRight j ‖ 0) ↔ G ≈ 0 := by
refine ⟨fun hb => ?_, fun hp i => ?_⟩
· rw [equiv_zero_iff_ge G, zero_le_lf]
| Mathlib/SetTheory/Game/Impartial.lean | 173 | 176 |
/-
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.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | @[simp]
protected theorem add_iff_right :
| Mathlib/Algebra/ModEq.lean | 161 | 162 |
/-
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... | abel
lemma δ_zero_cochain_comp {n₂ : ℤ} (z₁ : Cochain F G 0) (z₂ : Cochain G K n₂)
(m₂ : ℤ) (h₂ : n₂ + 1 = m₂) :
δ n₂ m₂ (z₁.comp z₂ (zero_add n₂)) =
z₁.comp (δ n₂ m₂ z₂) (zero_add m₂) +
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 505 | 510 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,271 | 1,273 | |
/-
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 | 746 | 747 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy
rcases hxy with ⟨rfl, hxy⟩; rw [heq_iff_eq] at hxy; subst y
exact ⟨x, hys, rfl⟩
/-- Indexed sum of sets. `s.sigma t` is the set of dependent pairs `⟨i, a⟩` such that `i ∈ s` and
`a ∈ t i`. -/
protected def sigma (s : Set ι) (t : ∀ i, Set (α i)) : Set (Σ i, α... | Mathlib/Data/Set/Sigma.lean | 43 | 50 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 542 | 562 | |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | theorem hom_right_act_hom' :
((regular R).tensorBimod P).actRight ≫ hom P = (hom P ▷ S.X) ≫ P.actRight := by
dsimp; dsimp [hom, TensorBimod.actRight, regular]
refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
dsimp
slice_lhs 1 4 => rw [π_tensor_id_preserves_coequalizer_inv_colimMap_desc]
... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 631 | 643 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | simp only [subset_def, ← forall_and]
refine forall_congr' fun x => ?_
| Mathlib/Data/Set/Basic.lean | 1,385 | 1,386 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono... | intro s m hm
apply small_k.isLimit.hom_ext
apply PullbackCone.equalizer_ext small_k.cone _ _
· exact (hm WalkingCospan.left).trans (by simp)
· exact (hm WalkingCospan.right).trans (by simp)⟩ }
/--
If `(a,b)` is the kernel pair of `f`, and `f` is a coequalizer morphism for some parallel pa... | Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean | 139 | 150 |
/-
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.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... |
theorem IsIntegral.exists_multiple_integral_of_isLocalization [Algebra Rₘ S] [IsScalarTower R Rₘ S]
(x : S) (hx : IsIntegral Rₘ x) : ∃ m : M, IsIntegral R (m • x) := by
rcases subsingleton_or_nontrivial Rₘ with _ | nontriv
· haveI := (_root_.algebraMap Rₘ S).codomain_trivial
exact ⟨1, Polynomial.X, Polynom... | Mathlib/RingTheory/Localization/Integral.lean | 259 | 279 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Construction of the inverse functor of the Dold-Kan equivalence
... | simpa only [left_eq_add, Nat.one_ne_zero] using hi.1
@[simp]
| Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 105 | 107 |
/-
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... | (add : ∀ x y, motive x → motive y → motive (x + y)) :
motive z := by
simp_rw [← tprodCoeff_eq_smul_tprod] at smul_tprod
exact PiTensorProduct.induction_on' z smul_tprod add
@[ext]
theorem ext {φ₁ φ₂ : (⨂[R] i, s i) →ₗ[R] E}
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 358 | 364 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 787 | 789 | |
/-
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.Order.SuccPred.LinearLocallyFinite
import Mathlib.Probability.Martingale.Basic
/-!
# Optional sampling theorem
If `τ` is a bounded stopping time and `σ` ... | alias condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const :=
condExp_stopping_time_ae_eq_restrict_eq_const_of_le_const
theorem stoppedValue_ae_eq_restrict_eq (h : Martingale f ℱ μ) (hτ : IsStoppingTime ℱ τ)
(hτ_le : ∀ x, τ x ≤ n) [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] (i : ι) :
stop... | Mathlib/Probability/Martingale/OptionalSampling.lean | 77 | 85 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | rcases Int.emod_two_eq_zero_or_one y with hy | hy
-- x even, y even
· exfalso
apply Nat.not_coprime_of_dvd_of_dvd (by decide : 1 < 2) _ _ hc
· apply Int.natCast_dvd.1
apply Int.dvd_of_emod_eq_zero hx
· apply Int.natCast_dvd.1
apply Int.dvd_of_emod_eq_zero hy
-- x even, y odd
· left
| Mathlib/NumberTheory/PythagoreanTriples.lean | 120 | 129 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
/-!
# Euclidean algorithm for ℕ
This file sets up a version of the Euclidean algorithm that only works w... | ring
/-- The reduction step does not change the product vector. -/
| Mathlib/Data/PNat/Xgcd.lean | 275 | 277 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | exact h₁ w h_ne
theorem convexBodyLT'_neg_mem (x : mixedSpace K) (hx : x ∈ convexBodyLT' K f w₀) :
-x ∈ convexBodyLT' K f w₀ := by
simp only [Set.mem_prod, Set.mem_pi, Set.mem_univ, mem_ball, dist_zero_right, Real.norm_eq_abs,
true_implies, Subtype.forall, Prod.fst_neg, Pi.neg_apply, norm_neg, Prod.snd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 169 | 186 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
/-!
# Angles between vectors
This fil... |
/-- The cosine of the angle between two vectors. -/
theorem cos_angle (x y : V) : Real.cos (angle x y) = ⟪x, y⟫ / (‖x‖ * ‖y‖) :=
Real.cos_arccos (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1
| Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 64 | 67 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | Mathlib/Data/Nat/PartENat.lean | 772 | 773 | |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.List.Sym
/-! # Unordered tuples of elements of a multiset
Defines `Multiset.sym` and the specialized `Multiset.sym2` for computing multisets of all
un... | theorem sym2_eq_zero_iff {m : Multiset α} : m.sym2 = 0 ↔ m = 0 :=
m.inductionOn fun xs => by simp
| Mathlib/Data/Multiset/Sym.lean | 47 | 48 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | end IsTopologicalAddGroup
end SeminormedRing
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 151 | 154 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.GroupWithZero.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
im... | theorem map_neg_one : f (-1) = -1 :=
f.map_one ▸ f.map_neg 1
| Mathlib/Algebra/Ring/Equiv.lean | 640 | 642 |
/-
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.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | /-- An element `a` is `AddLECancellable` if `a + b ≤ a + c` implies `b ≤ c` for all `b` and `c`.
This is true in `ℝ≥0∞` for all elements except `∞`. -/
@[simp]
theorem addLECancellable_iff_ne {a : ℝ≥0∞} : AddLECancellable a ↔ a ≠ ∞ := by
| Mathlib/Data/ENNReal/Operations.lean | 252 | 255 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | (hG.liftPropWithinAt_congr_iff h₁ (h₁ _ hx)).mpr h
theorem liftPropAt_congr_iff_of_eventuallyEq (h₁ : g' =ᶠ[𝓝 x] g) :
LiftPropAt P g' x ↔ LiftPropAt P g x :=
hG.liftPropWithinAt_congr_iff_of_eventuallyEq (by simp_rw [nhdsWithin_univ, h₁]) h₁.eq_of_nhds
theorem liftPropAt_congr_of_eventuallyEq (h : LiftPropAt... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 403 | 411 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 300 | 301 | |
/-
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... | theorem average_add_measure [IsFiniteMeasure μ] {ν : Measure α} [IsFiniteMeasure ν] {f : α → E}
(hμ : Integrable f μ) (hν : Integrable f ν) :
| Mathlib/MeasureTheory/Integral/Average.lean | 350 | 351 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | section IsOfFinOrder
| Mathlib/GroupTheory/OrderOfElement.lean | 48 | 49 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | one_gcd
@[simp]
theorem coprime_one {n : ℕ+} : n.Coprime 1 :=
| Mathlib/Data/PNat/Prime.lean | 222 | 225 |
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Equal and mixed characteristic
In commutative ... | ⟨to_not_mixedCharZero R, of_not_mixedCharZero R⟩
/-- A ring is a `ℚ`-algebra iff it has equal characteristic zero. -/
theorem nonempty_algebraRat_iff :
Nonempty (Algebra ℚ R) ↔ ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) := by
constructor
· intro h_alg
| Mathlib/Algebra/CharP/MixedCharZero.lean | 261 | 267 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Pointwise.Set.Sc... | /-- `preimage_smul_setₛₗ` when both scalars act by unit -/
@[to_additive]
theorem preimage_smul_setₛₗ_of_isUnit_isUnit (f : F)
(hc : IsUnit c) (hc' : IsUnit (σ c)) (t : Set β) : f ⁻¹' (σ c • t) = c • f ⁻¹' t :=
preimage_smul_setₛₗ' hc.smul_bijective.surjective hc'.smul_bijective.injective
| Mathlib/GroupTheory/GroupAction/Pointwise.lean | 87 | 91 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | /-- A topological module over a ring has continuous negation.
This cannot be an instance, because it would cause search for `[Module ?R M]` with unknown `R`. -/
theorem ContinuousNeg.of_continuousConstSMul [ContinuousConstSMul R M] : ContinuousNeg M where
continuous_neg := by simpa using continuous_const_smul (T := ... | Mathlib/Topology/Algebra/Module/Basic.lean | 44 | 50 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | classical
rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
← Embeddings.card K ℂ, Nat.add_sub_of_le]
exact Fintype.card_subtype_le _
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 651 | 655 |
/-
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.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | (∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))) ≤ ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k := by
induction n with
| zero => simp
| succ n ihn =>
suffices (u (n + 1) - u n) • f (u (n + 1)) ≤ ∑ k ∈ Ico (u n + 1) (u (n + 1) + 1), f k by
rw [sum_range_succ, ← sum_Ico_consecutive]
exacts [add_le_add i... | Mathlib/Analysis/PSeries.lean | 84 | 98 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
@[simp]
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 137 | 138 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | rw [← Complex.rotation, ← hf, o.rotation_map, LinearIsometryEquiv.symm_apply_apply]
/-- Negating the orientation negates the angle in `rotation`. -/
theorem rotation_neg_orientation_eq_neg (θ : Real.Angle) : (-o).rotation θ = o.rotation (-θ) :=
LinearIsometryEquiv.ext <| by simp [rotation_apply]
/-- The inner pro... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 365 | 373 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... |
theorem le_log_iff_exp_le (hy : 0 < y) : x ≤ log y ↔ exp x ≤ y := by rw [← exp_le_exp, exp_log hy]
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 155 | 156 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... |
@[deprecated (since := "2024-11-03")] alias det_updateRow_smul' := det_updateRow_smul_left
theorem det_updateCol_smul_left (M : Matrix n n R) (j : n) (s : R) (u : n → R) :
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 405 | 408 |
/-
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.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | Mathlib/Order/Interval/Finset/Nat.lean | 109 | 109 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,181 | 2,186 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... |
theorem iff_comap_eq :
MeasurableEmbedding f ↔
Injective f ∧ MeasurableSpace.comap f ‹_› = ‹_› ∧ MeasurableSet (range f) :=
⟨fun hf ↦ ⟨hf.injective, hf.comap_eq, hf.measurableSet_range⟩, fun hf ↦
{ injective := hf.1
measurable := by rw [← hf.2.1]; exact comap_measurable f
measurableSet_imag... | Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 654 | 663 |
/-
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... | protected theorem trichotomy₂ {p q : ℝ≥0∞} (hpq : p ≤ q) :
p = 0 ∧ q = 0 ∨
p = 0 ∧ q = ∞ ∨
p = 0 ∧ 0 < q.toReal ∨
| Mathlib/Data/ENNReal/Real.lean | 344 | 347 |
/-
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, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 3,406 | 3,408 | |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Antoine Labelle
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlge... | Recall that `X →₀ R` is Lean's way of talking about the free `R`-module
on a type `X`. The universal property `Finsupp.linearCombination` says that to a map
`X → N` from a type to an `R`-module, we get an associated R-module map
`(X →₀ R) →ₗ N`. Apply this to a (noncomputable) map `P → M` coming from th... | Mathlib/Algebra/Module/Projective.lean | 98 | 116 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
/-!
# Additive operations on derivative... |
@[simp]
theorem differentiableWithinAt_sub_const_iff (c : F) :
| Mathlib/Analysis/Calculus/FDeriv/Add.lean | 627 | 629 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... | @[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_eq_zero_mul :=
Multipliable.tprod_eq_zero_mul
/-- For `f : ℕ → G`, the product `∏' k, f (k + i)` tends to one. This does not require a
multipliability assumption on `f`, as otherwise all such products are one. -/
@[to_additive "For `f : ℕ → G`, ... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 254 | 263 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... |
Since `BoundedOrder`, `OrderBot`, and `OrderTop` are mixins that require `LE`
to be present at define-time, the `extends` mechanism does not work with them.
Instead, we extend using the underlying `Bot` and `Top` data typeclasses, and replicate the
order axioms of those classes here. A "forgetful" instance back to `Bo... | Mathlib/Order/BooleanAlgebra.lean | 486 | 490 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... |
theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c := by
rcases lt_or_eq_of_le (one_le_iff_pos.2 h₁) with h₁ | h₁
· exact (opow_le_opow_iff_right h₁).2 h₂
· subst a
-- Porting note: `le_refl` is required.
| Mathlib/SetTheory/Ordinal/Exponential.lean | 139 | 144 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
/-!
# Equivalences and sets
In this file we pr... | sumCongr_apply, Equiv.coe_trans, Subtype.coe_eta, Subtype.coe_mk, Trans.trans,
Set.sumCompl_symm_apply_compl]
/-- The set product of two sets is equivalent to the type product of their coercions to types. -/
protected def prod {α β} (s : Set α) (t : Set β) : ↥(s ×ˢ t) ≃ s × t :=
| Mathlib/Logic/Equiv/Set.lean | 393 | 397 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | swap; · rw [condExp_of_not_sigmaFinite hm hμm]
exact condExp_of_stronglyMeasurable hm stronglyMeasurable_zero (integrable_zero _ _ _)
@[deprecated (since := "2025-01-21")] alias condexp_zero := condExp_zero
theorem stronglyMeasurable_condExp : StronglyMeasurable[m] (μ[f|m]) := by
by_cases hm : m ≤ m₀
swap; · ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 192 | 200 |
/-
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... | PseudoMetricSpace.dist_triangle x y z
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 189 | 190 |
/-
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
... | intro hf
refine le_trans ?_ (extend_iUnion_le_tsum_nat' _ msU _)
refine le_iInf ?_
intro h2f
exact iInf_le_of_le _ (iInf_le_of_le h2f <| iInf_le _ hf)
theorem inducedOuterMeasure_preimage (f : α ≃ α) (Pm : ∀ s : Set α, P (f ⁻¹' s) ↔ P s)
(mm : ∀ (s : Set α) (hs : P s), m (f ⁻¹' s) ((Pm _).mpr h... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 196 | 210 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
/-!
# Bounded monotone sequences converge
In this file we prove a few theorems of the form “if the range of a... | Tendsto f atTop (𝓝 (⨆ i, f i)) := by
cases isEmpty_or_nonempty ι
| Mathlib/Topology/Order/MonotoneConvergence.lean | 113 | 114 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | | rel _ _ hab =>
refine (Relation.EqvGen.rel _ _ hab.add_left).trans _ _ _ ?_
| Mathlib/Algebra/RingQuot.lean | 81 | 82 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | `TopologicalSpace.SeparableSpace` from `SecondCountableTopology` but it can't
deduce `SecondCountableTopology` from `TopologicalSpace.SeparableSpace`.
Porting note (https://github.com/leanprover-community/mathlib4/issues/11215): TODO: the previous paragraph describes the state of the art in Lean 3.
| Mathlib/Topology/Bases.lean | 296 | 299 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Construction of the inverse functor of the Dold-Kan equivalence
... | by_contra h
exact h₂ (Fin.succAbove_ne_zero_zero h)
· rintro rfl
exact ⟨rfl, by dsimp; exact Fin.succ_ne_zero (0 : Fin (j + 1))⟩
theorem eq_δ₀ {n : ℕ} {i : ⦋n⦌ ⟶ ⦋n + 1⦌} [Mono i] (hi : Isδ₀ i) :
i = SimplexCategory.δ 0 := by
| Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 55 | 61 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | lemma exists_ideal_comap_le_prime {S} [CommSemiring S] [FunLike F R S] [RingHomClass F R S]
{f : F} (P : Ideal R) [P.IsPrime] (I : Ideal S) (le : I.comap f ≤ P) :
| Mathlib/RingTheory/Ideal/Maps.lean | 84 | 85 |
/-
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.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
/-!
# Factoring through subobjects
The predicate `h : P.Fact... | (w : P.Factors (f + g)) (wf : P.Factors f) :
P.factorThru (f + g) w - P.factorThru f wf =
P.factorThru g (factors_right_of_factors_add f g w wf) := by
ext
simp
| Mathlib/CategoryTheory/Subobject/FactorThru.lean | 188 | 192 |
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
/-!
# Gram-Schmidt Orthogonalization and Orthonor... | rintro u ⟨k, hk, rfl⟩ v (rfl : v = f i)
apply hf
exact (lt_of_le_of_lt hk (Finset.mem_Iio.mp hj)).ne
· simp
variable {𝕜}
theorem gramSchmidt_ne_zero_coe {f : ι → E} (n : ι)
(h₀ : LinearIndependent 𝕜 (f ∘ ((↑) : Set.Iic n → ι))) : gramSchmidt 𝕜 f n ≠ 0 := by
by_contra h
have h₁ : f n ∈ span 𝕜... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 177 | 195 |
/-
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... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,393 | 1,397 | |
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | @[simps!]
def universalEnvelopingEquivFreeAlgebra :
UniversalEnvelopingAlgebra R (FreeLieAlgebra R X) ≃ₐ[R] FreeAlgebra R X :=
AlgEquiv.ofAlgHom (UniversalEnvelopingAlgebra.lift R <| FreeLieAlgebra.lift R <| FreeAlgebra.ι R)
| Mathlib/Algebra/Lie/Free.lean | 262 | 265 |
/-
Copyright (c) 2021 Patrick Stevens. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Stevens, Thomas Browning
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
/-!
# Central bin... | 2 ≤ 2 * n := Nat.le_mul_of_pos_right _ n_pos
_ = (2 * n).choose 1 := (choose_one_right (2 * n)).symm
_ ≤ centralBinom n := choose_le_centralBinom 1 n
| Mathlib/Data/Nat/Choose/Central.lean | 57 | 60 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
simp only [lowerCrossingTime, hitting_le ω]
theorem upperCrossingTime_le_lowerCrossingTime :
| Mathlib/Probability/Martingale/Upcrossing.lean | 186 | 189 |
/-
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, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | theorem rank_span_finset_le (s : Finset M) : Module.rank R (span R (s : Set M)) ≤ s.card := by
simpa using rank_span_le s.toSet
theorem rank_span_of_finset (s : Finset M) : Module.rank R (span R (s : Set M)) < ℵ₀ :=
(rank_span_finset_le s).trans_lt (Cardinal.nat_lt_aleph0 _)
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 440 | 444 |
/-
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.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | rw [RingHom.map_mul, RingHom.map_mul, ← div_mul_div_comm, num_div_denom, num_div_denom]
theorem denom_add_dvd (x y : RatFunc K) : denom (x + y) ∣ denom x * denom y := by
rw [denom_dvd (mul_ne_zero (denom_ne_zero x) (denom_ne_zero y))]
refine ⟨x.num * y.denom + x.denom * y.num, ?_⟩
| Mathlib/FieldTheory/RatFunc/Basic.lean | 962 | 966 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.... | · refine ⟨0, by simp, ?_⟩
symm
simp [hx]
· rcases exists_dual_vector 𝕜 x hx with ⟨g, g_norm, g_eq⟩
exact ⟨g, g_norm.le, g_eq⟩
end DualVector
| Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 182 | 188 |
/-
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.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | @[simp] lemma ofIsLimitKernelFork_f' (hf : S.f = 0) (c : KernelFork S.g) (hc : IsLimit c) :
(ofIsLimitKernelFork S hf c hc).f' = 0 := by
rw [← cancel_mono (ofIsLimitKernelFork S hf c hc).i, f'_i, hf, zero_comp]
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 185 | 187 |
/-
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, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | · apply Cardinal.lift_le.mpr
refine le_ciSup (Cardinal.bddAbove_range _) ⟨rangeSplitting f '' s, ?_⟩
apply LinearIndependent.of_comp f.rangeRestrict
convert li.comp (Equiv.Set.rangeSplittingImageEquiv f s) (Equiv.injective _) using 1
· exact (Cardinal.lift_mk_eq'.mpr ⟨Equiv.Set.rangeSplittingImageEquiv ... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 296 | 300 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalizatio... | (Localization t.asHomogeneousIdeal.toIdeal.primeCompl) ⟨c, ht'⟩).mul_left_cancel
rw [← map_mul, ← map_mul, hc']
lemma homogeneousLocalizationToStalk_stalkToFiberRingHom (x z) :
homogeneousLocalizationToStalk 𝒜 x (stalkToFiberRingHom 𝒜 x z) = z := by
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean | 306 | 310 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | rcases isOpen_pi_iff.1 ho f hfo with ⟨I, u, huo, hI⟩
rsuffices ⟨f, hf⟩ : ∃ f : (i : I) → c i, g ⟨I, f⟩ ∈ Set.pi I u
| Mathlib/Topology/Bases.lean | 472 | 473 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Shift.CommShift
import Mathlib.CategoryTheory.Shift.ShiftSequence
/-! # Induced shift sequences
When `G : C ⥤ A` is a functor from a category eq... |
lemma shiftIso_hom_app_obj (n a a' : M) (ha' : n + a = a') (X : C) :
(shiftIso L G M F' e' n a a' ha').hom.app (L.obj X) =
(F' a).map ((L.commShiftIso n).inv.app X) ≫
(e' a).hom.app (X⟦n⟧) ≫ (G.shiftIso n a a' ha').hom.app X ≫ (e' a').inv.app X :=
| Mathlib/CategoryTheory/Shift/InducedShiftSequence.lean | 64 | 68 |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | rintro _ ⟨⟨w, hw⟩, rfl⟩
exact hx w hw
lemma ker_restrict_eq_of_codisjoint {p q : Submodule R M} (hpq : Codisjoint p q)
{B : LinearMap.BilinForm R M} (hB : ∀ x ∈ p, ∀ y ∈ q, B x y = 0) :
LinearMap.ker (B.restrict p) = (LinearMap.ker B).comap p.subtype := by
ext ⟨z, hz⟩
simp only [LinearMap.mem_ker, ... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 281 | 298 |
/-
Copyright (c) 2022 Alex J. Best, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Ring.Equiv
/-!
# Ordered ring homomorphisms
Homomorphisms between ordered (se... | Mathlib/Algebra/Order/Hom/Ring.lean | 514 | 515 |
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