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) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 417 | 418 | |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | rw [← χ.pow_apply' two_ne_zero, h, MulChar.one_apply hx]
· exact .inl <| map_nonunit χ hx
end quadratic_and_comp
end Properties
/-!
### Multiplicative characters with finite domain
-/
| Mathlib/NumberTheory/MulChar/Basic.lean | 512 | 521 |
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
import Mathlib.Com... | lemma IsMatching.iSup {ι : Sort _} {f : ι → Subgraph G} (hM : (i : ι) → (f i).IsMatching)
(hd : Pairwise fun i j ↦ Disjoint (f i).support (f j).support) :
(⨆ i, f i).IsMatching := by
intro v hv
| Mathlib/Combinatorics/SimpleGraph/Matching.lean | 127 | 130 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, 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 log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x := by
convert log_opow_mul hb x zero_ne_one.symm using 1
· rw [mul_one]
· rw [log_one_right, add_zero]
theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o ≠ 0) : 0 < o / (b ^ log b o) := by
| Mathlib/SetTheory/Ordinal/Exponential.lean | 465 | 471 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Data.Fin.VecNotati... | rw [goldenRatio, goldenConj]
ring
| Mathlib/Data/Real/GoldenRatio.lean | 64 | 66 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
/-!
# A `norm_num` extension for Jacobi and Legendre symbols
We extend the `norm_num` tactic so that it can be... | Nat.mul_le_mul_left _ (Nat.succ_le.mpr (Nat.pos_of_ne_zero (Nat.ne_of_beq_eq_false hb)))
_ ≤ b := Nat.mul_div_le b 2
theorem jacobiSymNat.one_left (b : ℕ) : jacobiSymNat 1 b = 1 := by
rw [jacobiSymNat, Nat.cast_one, jacobiSym.one_left]
/-- Turn a Legendre symbol into a Jacobi symbol. -/
| Mathlib/Tactic/NormNum/LegendreSymbol.lean | 76 | 82 |
/-
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... | _ = det B := by rw [hC, one_mul]
theorem det_updateRow_add_self (A : Matrix n n R) {i j : n} (hij : i ≠ j) :
det (updateRow A i (A i + A j)) = det A := by
simp [det_updateRow_add,
det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)]
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 464 | 469 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | have : V.ι ⁻¹ᵁ V.ι ''ᵁ W = W := W.map_functor_eq (U := V)
have e₂ := (X.homOfLE e).naturality (eqToIso this).hom.op
have e₃ := e₂.symm.trans e₁
| Mathlib/AlgebraicGeometry/Restrict.lean | 255 | 257 |
/-
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.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | have hmul : g ∈ connectedComponent (g * h) := by
apply Continuous.image_connectedComponent_subset (continuous_mul_left g)
| Mathlib/Topology/Algebra/Group/Basic.lean | 646 | 647 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.... | Mathlib/Data/PNat/Basic.lean | 395 | 399 | |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
/-!
# Stabilisation of gcf Computations Under Termination
## Summary
We show that the continuants and co... | simp only [contsAux, Nat.add_eq, Nat.add_zero, terminatedAt_n]
theorem contsAux_stable_of_terminated (n_lt_m : n < m) (terminatedAt_n : g.TerminatedAt n) :
g.contsAux m = g.contsAux (n + 1) := by
| Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 31 | 34 |
/-
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.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... |
theorem isCaratheodory_iff_le {s : Set α} :
MeasurableSet[OuterMeasure.caratheodory m] s ↔ ∀ t, m (t ∩ s) + m (t \ s) ≤ m t :=
isCaratheodory_iff_le' m
protected theorem iUnion_eq_of_caratheodory {s : ℕ → Set α}
(h : ∀ i, MeasurableSet[OuterMeasure.caratheodory m] (s i)) (hd : Pairwise (Disjoint on s)) :
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 177 | 192 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.Algebra.Homology.ComplexShapeSigns
import Mathlib.Algebra.Homology.HomologicalBicomplex
import Mathlib.Algebra.Module.... | · rw [K.D₁_shape c₁₂ _ _ h₂, K.D₂_shape c₁₂ _ _ h₂, comp_zero, comp_zero, neg_zero]
· rw [K.D₁_shape c₁₂ _ _ h₁, K.D₂_shape c₁₂ _ _ h₁, zero_comp, zero_comp, neg_zero]
| Mathlib/Algebra/Homology/TotalComplex.lean | 241 | 243 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... | rw [dif_pos (Nat.pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0]
· dsimp
rw [dif_pos hn, ← Units.eq_iff, ← Units.coeHom_apply, map_prod, Units.val_mk0,
prod_congr rfl _]
intro x hx
rw [dif_pos (Nat.pos_of_mem_divisors (Nat.snd_mem_divisors_of_mem_antidiagonal hx)),
Units.coeHom... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,268 | 1,274 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... |
@[to_additive]
theorem finprod_of_isEmpty [IsEmpty α] (f : α → M) : ∏ᶠ i, f i = 1 := by
rw [← finprod_one]
| Mathlib/Algebra/BigOperators/Finprod.lean | 190 | 193 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... | theorem sub {x y : R} (hx : IsSelfAdjoint x) (hy : IsSelfAdjoint y) : IsSelfAdjoint (x - y) := by
simp only [isSelfAdjoint_iff, star_sub, hx.star_eq, hy.star_eq]
| Mathlib/Algebra/Star/SelfAdjoint.lean | 127 | 128 |
/-
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.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A schem... | rfl
@[reassoc]
| Mathlib/AlgebraicGeometry/Scheme.lean | 132 | 134 |
/-
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... |
lemma δ_neg_one_cochain (z : Cochain F G (-1)) :
δ (-1) 0 z = Cochain.ofHom (Homotopy.nullHomotopicMap'
(fun i j hij => z.v i j (by dsimp at hij; rw [← hij, add_neg_cancel_right]))) := by
ext p
rw [δ_v (-1) 0 (neg_add_cancel 1) _ p p (add_zero p) (p-1) (p+1) rfl rfl]
simp only [neg_add_cancel, one_smul... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 547 | 555 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... |
/-- `Fin.mk_zero` in `Lean` only applies in `Fin (n + 1)`.
This one instead uses a `NeZero n` typeclass hypothesis.
-/
| Mathlib/Data/Fin/Basic.lean | 196 | 199 |
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.IsPrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
import Mathlib.Data.Z... | (mk_le_mk_of_dvd (dvd_of_mem_normalizedFactors hp))
rwa [associated_iff_eq.mp hq']
| Mathlib/RingTheory/ChainOfDivisors.lean | 427 | 452 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Lie algebras with non-degenerate invariant bilinear forms are s... |
open LieSubmodule in
/--
A finite-dimensional Lie algebra over a field is semisimple
if it does not have non-trivial abelian ideals and it admits a
non-degenerate reflexive invariant bilinear form.
Here a form is *invariant* if it is compatible with the Lie bracket: `Φ ⁅x, y⁆ z = Φ x ⁅y, z⁆`.
-/
theorem isSemisimple_o... | Mathlib/Algebra/Lie/InvariantForm.lean | 186 | 202 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | AlgHom.congr_fun involute_comp_involute
| Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 55 | 56 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | apply lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (h i)
end BoundedPi
section FinitePi
| Mathlib/GroupTheory/Nilpotent.lean | 729 | 734 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | Mathlib/Logic/Function/Basic.lean | 1,055 | 1,059 | |
/-
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... | @[simp] theorem compl_lt_self [Nontrivial α] : xᶜ < x ↔ x = ⊤ := by
simpa using lt_compl_self (a := xᶜ)
| Mathlib/Order/BooleanAlgebra.lean | 650 | 651 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... | theorem zero_tensor {W X Y Z : C} (f : Y ⟶ Z) : (0 : W ⟶ X) ⊗ f = 0 := by
simp [tensorHom_def]
| Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 52 | 53 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | ⌈logb b r⌉ = Int.clog b r := by
obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
by_cases hb : 1 < b
· have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 374 | 378 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | /-- Simplification of the integral of `sin x ^ m * cos x ^ n`, case `m` and `n` are both even. -/
theorem integral_sin_pow_even_mul_cos_pow_even (m n : ℕ) :
(∫ x in a..b, sin x ^ (2 * m) * cos x ^ (2 * n)) =
∫ x in a..b, ((1 - cos (2 * x)) / 2) ^ m * ((1 + cos (2 * x)) / 2) ^ n := by
field_simp [pow_mul, si... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 796 | 809 |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | def tail (s : Seq α) : Seq α :=
⟨s.1.tail, fun n' => by
| Mathlib/Data/Seq/Seq.lean | 129 | 130 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 1,008 | 1,011 | |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | def fourierSubalgebra : StarSubalgebra ℂ C(AddCircle T, ℂ) where
toSubalgebra := Algebra.adjoin ℂ (range fourier)
star_mem' := by
show Algebra.adjoin ℂ (range (fourier (T := T))) ≤
star (Algebra.adjoin ℂ (range (fourier (T := T))))
refine adjoin_le ?_
rintro - ⟨n, rfl⟩
exact subset_adjoin ⟨-n,... | Mathlib/Analysis/Fourier/AddCircle.lean | 184 | 193 |
/-
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.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... |
@[deprecated (since := "2024-12-22")]
| Mathlib/Topology/Order/DenselyOrdered.lean | 187 | 188 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | s.affineCombination k p₁ w -ᵥ s.affineCombination k p₂ w := by
simp_rw [affineCombination_apply, vadd_vsub_vadd_cancel_right]
exact s.sum_smul_vsub_eq_weightedVSubOfPoint_sub _ _ _ _
/-- A weighted sum of pairwise subtractions, where the point on the right is constant and the
sum of the weights is 1. -/
theo... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 472 | 482 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... |
set_option linter.deprecated false in
| Mathlib/Order/Interval/Finset/Fin.lean | 124 | 125 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | {μ ν : FiniteMeasure Ω} (h : ∀ (f : Ω →ᵇ ℝ), ∫ x, f x ∂μ = ∫ x, f x ∂ν) :
μ = ν := by
apply ext_of_forall_lintegral_eq
intro f
apply (ENNReal.toReal_eq_toReal_iff' (lintegral_lt_top_of_nnreal μ f).ne
(lintegral_lt_top_of_nnreal ν f).ne).mp
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 325 | 330 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 579 | 582 | |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | simp_rw [← two_mul, pow_mul, neg_sq]
lemma Even.neg_one_pow (h : Even n) : (-1 : α) ^ n = 1 := by rw [h.neg_pow, one_pow]
| Mathlib/Algebra/Ring/Parity.lean | 41 | 43 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | variable {R : Type*} [NonAssocSemiring R] {p : Nat} (f : ∀ k : ℕ, R →+* ZMod (p ^ k))
/-- Given a family of ring homs `f : Π n : ℕ, R →+* ZMod (p ^ n)`,
`nthHom f r` is an integer-valued sequence
whose `n`th value is the unique integer `k` such that `0 ≤ k < p ^ n`
and `f n r = (k : ZMod (p ^ n))`.
| Mathlib/NumberTheory/Padics/RingHoms.lean | 465 | 471 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... |
variable (l l' : List α)
| Mathlib/Data/List/Rotate.lean | 352 | 353 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
i... | · suffices IsFiltered (StructuredArrow (op X) F.op)ᵒᵖ from isCofiltered_of_isFiltered_op _
suffices IsFiltered (CostructuredArrow F X)ᵒᵖᵒᵖ from
| Mathlib/CategoryTheory/Functor/Flat.lean | 131 | 132 |
/-
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-... |
theorem ModelsBoundedFormula.realize_boundedFormula
{φ : L.BoundedFormula α n} (h : T ⊨ᵇ φ) (M : Type*)
[L.Structure M] [M ⊨ T] [Nonempty M] {v : α → M} {xs : Fin n → M} : φ.Realize v xs := by
have h' : φ.toFormula.Realize (Sum.elim v xs) := (models_toFormula_iff.2 h).realize_formula M
simp only [BoundedFo... | Mathlib/ModelTheory/Satisfiability.lean | 355 | 362 |
/-
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... | theorem reflection_orthogonalComplement_singleton_eq_neg (v : E) : reflection (𝕜 ∙ v)ᗮ v = -v :=
reflection_mem_subspace_orthogonal_precomplement_eq_neg (Submodule.mem_span_singleton_self v)
theorem reflection_sub {v w : F} (h : ‖v‖ = ‖w‖) : reflection (ℝ ∙ (v - w))ᗮ v = w := by
set R : F ≃ₗᵢ[ℝ] F := reflection (... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 967 | 972 |
/-
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.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Tactic.AdaptationNote
/-!
# Inversion in an affine sp... | have hy' : inversion c R y ≠ c := by simp [*]
conv in dist _ y => rw [← inversion_inversion c hR y]
rw [dist_inversion_inversion hx hy', dist_inversion_center]
have : dist x c ≠ 0 := dist_ne_zero.2 hx
-- used to be `field_simp`, but was really slow; replaced by `simp only ...` to speed up
-- TODO(https://gi... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 162 | 167 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
/-!
# Extending an alternating map to the exterior algebra
## Main de... | rw [List.ofFn_succ, List.prod_cons, liftAlternating_ι_mul, ih,
AlternatingMap.curryLeft_apply_apply]
congr
exact Matrix.cons_head_tail _
| Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 95 | 98 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le
(hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
rw [← hasFPowerSeriesWithinOnBall_univ] at hf
simpa only [mem_un... | Mathlib/Analysis/Analytic/Basic.lean | 1,142 | 1,148 |
/-
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, Mario Carneiro
-/
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Data.Nat.Prime.Basic
/-!
# `norm_num` extensions on natural numbers
This... | refine minFacHelper_1 e h fun h2 ↦ ?_
rw [← h2] at nk
exact nk <| minFac_prime h.one_lt.ne'
theorem minFacHelper_3 {n k k' : ℕ} (e : k + 2 = k') (nk : (n % k).beq 0 = false)
| Mathlib/Tactic/NormNum/Prime.lean | 84 | 88 |
/-
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.HomotopyCategory.Pretriangulated
import Mathlib.CategoryTheory.Triangulated.Triangulated
import Mathlib.CategoryTheory.ComposableArrows
/-! The... | rw [← Functor.map_comp, ← Functor.map_comp, ← Functor.map_comp,
mappingConeCompHomotopyEquiv_hom_inv_id, comp_id,
mappingConeCompHomotopyEquiv_comm₁ f g,
mappingConeCompTriangle_mor₂]
end CochainComplex
namespace HomotopyCategory
open CochainComplex
variable [HasZeroObject C]
| Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean | 162 | 173 |
/-
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.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,508 | 1,510 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | measurable subsets of `s ∪ t`. -/
theorem restrict_union_congr :
μ.restrict (s ∪ t) = ν.restrict (s ∪ t) ↔
μ.restrict s = ν.restrict s ∧ μ.restrict t = ν.restrict t := by
refine ⟨fun h ↦ ⟨restrict_congr_mono subset_union_left h,
| Mathlib/MeasureTheory/Measure/Restrict.lean | 318 | 322 |
/-
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.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.Measure... | Mathlib/Analysis/Convex/Integral.lean | 359 | 364 | |
/-
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.HomologicalComplex
import Mathlib.Algebra.Homology.ShortComplex.SnakeLemma
import Mathlib.Algebra.Homology.ShortComplex.ShortExact
... | x₃ ≫ hS.δ i j hij = x₁ ≫ S.X₁.homologyπ j :=
(snakeInput hS i j hij).δ_eq x₃ x₂ x₁ h₂ h₁
lemma δ_eq {A : C} (x₃ : A ⟶ S.X₃.X i) (hx₃ : x₃ ≫ S.X₃.d i j = 0)
(x₂ : A ⟶ S.X₂.X i) (hx₂ : x₂ ≫ S.g.f i = x₃)
(x₁ : A ⟶ S.X₁.X j) (hx₁ : x₁ ≫ S.f.f j = x₂ ≫ S.X₂.d i j)
(k : ι) (hk : c.next j = k) :
S.X₃.l... | Mathlib/Algebra/Homology/HomologySequence.lean | 325 | 342 |
/-
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... |
theorem fill_filterNe [DecidableEq α] (a : α) (m : Sym α n) :
(m.filterNe a).2.fill a (m.filterNe a).1 = m :=
Sym.ext
| Mathlib/Data/Sym/Basic.lean | 525 | 528 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | {f f' : ℂ → E} {c : ℂ} {R : ℝ} (hR : 0 ≤ R)
(h : ∀ z ∈ sphere c R, HasDerivWithinAt f (f' z) (sphere c R) z) : (∮ z in C(c, R), f' z) = 0 :=
integral_eq_zero_of_hasDerivWithinAt' <| (abs_of_nonneg hR).symm ▸ h
/-- If `n < 0` and `|w - c| = |R|`, then `(z - w) ^ n` is not circle integrable on the circle with
... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 423 | 440 |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,269 | 1,271 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... |
@[simp]
theorem fromSpec_preimage_self :
| Mathlib/AlgebraicGeometry/AffineScheme.lean | 493 | 495 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | (∀ x, P x) ↔ P finZeroElim :=
⟨fun h ↦ h _, fun h x ↦ Subsingleton.elim finZeroElim x ▸ h⟩
@[simp]
theorem exists_fin_zero_pi {α : Fin 0 → Sort*} {P : (∀ i, α i) → Prop} :
(∃ x, P x) ↔ P finZeroElim :=
⟨fun ⟨x, h⟩ ↦ Subsingleton.elim x finZeroElim ▸ h, fun h ↦ ⟨_, h⟩⟩
| Mathlib/Data/Fin/Tuple/Basic.lean | 239 | 245 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Tactic.Basic
import Mathlib.Order.Filter.Basic
/-!
# The `peel` tactic
`peel h with h' idents*` tries to apply `forall_imp` (or `Exists.imp`, or `Filte... | private theorem frequently_congr {α : Type*} {p q : α → Prop} {f : Filter α}
(hq : ∀ (x : α), p x ↔ q x) : (∃ᶠ (x : α) in f, p x) ↔ ∃ᶠ (x : α) in f, q x := by
congr! 2; exact hq _
| Mathlib/Tactic/Peel.lean | 101 | 103 |
/-
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
... | MellinConvergent f s := by
rw [MellinConvergent,
mellin_convergent_iff_norm Subset.rfl measurableSet_Ioi hfc.aestronglyMeasurable]
exact mellin_convergent_of_isBigO_scalar hfc.norm hf_top.norm_left hs_top hf_bot.norm_left hs_bot
end MellinConvergent
section MellinDiff
/-- If `f` is `O(x ^ (-a))` as `x → ... | Mathlib/Analysis/MellinTransform.lean | 270 | 285 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | rfl
end NormedGroup
| Mathlib/Analysis/Normed/Group/Basic.lean | 1,301 | 1,303 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Option.Defs
import Mathlib.Control.Functor
import Batteries.Data.List.Basic
import Mathlib.Control.Basic
/-!
# Traversable type class
Type classes for... | ⟨idTransformation⟩
universe s t
| Mathlib/Control/Traversable/Basic.lean | 152 | 154 |
/-
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... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 967 | 1,013 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 1,202 | 1,211 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... | intro f₁
erw [inf_def, inf_def, inf_eq_map_pullback', inf_eq_map_pullback', ← pullback_comp, ←
map_pullback pullback.condition (pullbackIsPullback f₁.arrow g), ← pullback_comp,
pullback.condition]
rfl
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 435 | 440 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Al... | /-- The canonical ring homomorphism interpreting `s ∈ R_f` as a section of the structure sheaf
on the basic open defined by `f ∈ R`. -/
def toBasicOpen (f : R) :
Localization.Away f →+* (structureSheaf R).1.obj (op <| PrimeSpectrum.basicOpen f) :=
| Mathlib/AlgebraicGeometry/StructureSheaf.lean | 560 | 563 |
/-
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 | 2,395 | 2,397 | |
/-
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... | Mathlib/Data/Nat/Bits.lean | 304 | 321 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic... | -- helper lemma for the next three results
theorem x_mul_y_le_y_mul_x {a₁ : Solution₁ d} (h : IsFundamental a₁) {a : Solution₁ d}
(hax : 1 < a.x) (hay : 0 < a.y) : a.x * a₁.y ≤ a.y * a₁.x := by
rw [← abs_of_pos <| zero_lt_one.trans hax, ← abs_of_pos hay, ← abs_of_pos h.x_pos, ←
abs_of_pos h.2.1, ← abs_mul, ← ... | Mathlib/NumberTheory/Pell.lean | 542 | 560 |
/-
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... |
variable [CompleteLattice α] [InfConvergenceClass α] {f : ι → α}
| Mathlib/Topology/Order/MonotoneConvergence.lean | 147 | 148 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | convert rfl
@[simp]
theorem rootSet_C [CommRing S] [IsDomain S] [Algebra T S] (a : T) : (C a).rootSet S = ∅ := by
| Mathlib/Algebra/Polynomial/Roots.lean | 493 | 496 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | theorem normalizerCondition_of_isNilpotent [h : IsNilpotent G] : NormalizerCondition G := by
-- roughly based on https://groupprops.subwiki.org/wiki/Nilpotent_implies_normalizer_condition
rw [normalizerCondition_iff_only_full_group_self_normalizing]
apply @nilpotent_center_quotient_ind _ G _ _ <;> clear! G
· in... | Mathlib/GroupTheory/Nilpotent.lean | 778 | 789 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Lattice
/-!
# Intervals in ℕ
This file defines intervals of naturals. `List.Ico m n... | exact chain_succ_range' _ _ 1
· rw [eq_nil_of_le (le_of_not_gt h)]
trivial
| Mathlib/Data/List/Intervals.lean | 120 | 122 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | (∀ᶠ x in f ⊓ 𝓟 s, p x) ↔ ∀ᶠ x in f, x ∈ s → p x :=
mem_inf_principal
| Mathlib/Order/Filter/Basic.lean | 686 | 687 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | upperSemicontinuous_iff_isClosed_preimage.1 hf y
| Mathlib/Topology/Semicontinuous.lean | 784 | 785 |
/-
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... |
theorem union_def {s₁ s₂ : Set α} : s₁ ∪ s₂ = { a | a ∈ s₁ ∨ a ∈ s₂ } :=
| Mathlib/Data/Set/Basic.lean | 601 | 602 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,034 | 1,044 | |
/-
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.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | isMin_bot.not_supIrred
| Mathlib/Order/Irreducible.lean | 84 | 85 |
/-
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.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | theorem succ_strictMono : StrictMono (succ : α → α) := fun _ _ => succ_lt_succ
theorem covBy_succ (a : α) : a ⋖ succ a :=
| Mathlib/Order/SuccPred/Basic.lean | 284 | 286 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | have : a.val ^ m < n := by
obtain rfl | ha := eq_or_ne a 0
| Mathlib/Data/ZMod/Basic.lean | 1,027 | 1,028 |
/-
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)` ... | have : ContinuousAt f ((g ∘ e.symm) (e x)) := by
simp_rw [Function.comp, e.left_inv xe, f.continuousAt xf]
exact this.comp_continuousWithinAt h1
theorem liftPropOn_indep_chart [HasGroupoid M G] [HasGroupoid M' G'] (he : e ∈ G.maximalAtlas M)
(hf : f ∈ G'.maximalAtlas M') (h : LiftPropOn P g s) {y : H}
... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 319 | 327 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.Adjunctions
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Al... | (trivial k G V).ρ g = LinearMap.id :=
rfl
| Mathlib/RepresentationTheory/Rep.lean | 112 | 114 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... | lemma gaussianReal_apply_eq_integral (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) :
gaussianReal μ v s = ENNReal.ofReal (∫ x in s, gaussianPDFReal μ v x) := by
rw [gaussianReal_apply _ hv s, ofReal_integral_eq_lintegral_ofReal]
· rfl
| Mathlib/Probability/Distributions/Gaussian.lean | 212 | 215 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.TensorProduct.Associator
/-!
# The `A`-module structure on `M ⊗[R] N`
Whe... | (TensorProduct.lift <| lift <| LinearMap.flip <|
(TensorProduct.lcurry A M P ((M ⊗[A] P) ⊗[R] Q)).restrictScalars R
| Mathlib/LinearAlgebra/TensorProduct/Tower.lean | 477 | 478 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | have heq := stationaryPoint_spec hf le_rfl hj
simpa [h, heq]
· intro h
simp [norm, h]
end
section Embedding
open CauSeq
variable {p : ℕ} [Fact p.Prime]
theorem equiv_zero_of_val_eq_of_equiv_zero {f g : PadicSeq p}
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 121 | 134 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Determinants in free (finite) modules
Quite a lot of our... | letI : Nonempty (Module.Free.ChooseBasisIndex R M) := (Module.Free.chooseBasis R M).index_nonempty
nontriviality R
exact LinearMap.det_zero' (Module.Free.chooseBasis R M)
| Mathlib/LinearAlgebra/FreeModule/Determinant.lean | 25 | 29 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,254 | 1,255 | |
/-
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.Commute.Defs
import Mathlib.Algebra.Group.H... | Mathlib/Algebra/Group/Commute/Hom.lean | 41 | 43 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheory... |
variable (hf₁ : f 1 = 1) (hmul : ∀ {m n}, Nat.Coprime m n → f (m * n) = f m * f n)
include hf₁ hmul in
/-- We relate a finite product over primes in `s` to an infinite sum over `s`-factored numbers. -/
lemma summable_and_hasSum_factoredNumbers_prod_filter_prime_tsum
(hsum : ∀ {p : ℕ}, p.Prime → Summable (fun n : ... | Mathlib/NumberTheory/EulerProduct/Basic.lean | 75 | 101 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | rfl
theorem modNat_rev (i : Fin (m * n)) : i.rev.modNat = i.modNat.rev := by
| Mathlib/Data/Fin/Basic.lean | 1,338 | 1,340 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... |
@[simp]
| Mathlib/CategoryTheory/ComposableArrows.lean | 318 | 319 |
/-
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.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... |
variable {N : ℕ} [NeZero N]
/-- The `AddMonoidHom` from `ZMod N` to `ℝ / ℤ` sending `j mod N` to `j / N mod 1`. -/
noncomputable def toAddCircle : ZMod N →+ UnitAddCircle :=
lift N ⟨AddMonoidHom.mk' (fun j ↦ ↑(j / N : ℝ)) (by simp [add_div]),
by simp [div_self (NeZero.ne _)]⟩
| Mathlib/Topology/Instances/AddCircle.lean | 662 | 669 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.Over
/-!
# Restriction of Schemes and Morphisms
## Main definition
- `AlgebraicGeometry.Schem... | @[reassoc (attr := simp)]
lemma Scheme.isoOfEq_inv_ι (X : Scheme.{u}) {U V : X.Opens} (e : U = V) :
(X.isoOfEq e).inv ≫ U.ι = V.ι :=
IsOpenImmersion.isoOfRangeEq_inv_fac _ _ _
@[simp]
| Mathlib/AlgebraicGeometry/Restrict.lean | 367 | 372 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | map p'.subtype (range <| inclusion h) = p := by simp [range_inclusion, map_comap_eq, h]
| Mathlib/Algebra/Module/Submodule/Range.lean | 308 | 309 |
/-
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.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | induction toList l <;> intro h
· rfl
· rcases cons_subset.1 h with ⟨⟨_, ⟨⟩, _⟩, _⟩
theorem mem_of_subset' {a} : ∀ {l₁ l₂ : Lists' α true} (_ : l₁ ⊆ l₂) (_ : a ∈ l₁.toList), a ∈ l₂
| nil, _, Lists'.Subset.nil, h => by cases h
| cons' a0 l0, l₂, s, h => by
| Mathlib/SetTheory/Lists.lean | 163 | 169 |
/-
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... | @[simp]
theorem laverage_mul_measure_univ (μ : Measure α) [IsFiniteMeasure μ] (f : α → ℝ≥0∞) :
(⨍⁻ (a : α), f a ∂μ) * μ univ = ∫⁻ x, f x ∂μ := by
| Mathlib/MeasureTheory/Integral/Average.lean | 232 | 234 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,183 | 1,184 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 418 | 421 | |
/-
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 Batteries.Data.List.Perm
import Mathlib.Tactic.Common
/-!
# Counting in lists
This fi... | theorem count_map_of_injective [DecidableEq β] (l : List α) (f : α → β)
(hf : Function.Injective f) (x : α) : count (f x) (map f l) = count x l := by
simp only [count, countP_map]
unfold Function.comp
| Mathlib/Data/List/Count.lean | 54 | 57 |
/-
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.Lattice
/-!
# Formal concept analysis
This file defines concept lattices. A concept of a relation `r : α → β → Prop` is a pair of sets
`s : Set ... | closure_snd := rfl }⟩
| Mathlib/Order/Concept.lean | 244 | 245 |
/-
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.Data.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... | @[to_additive]
lemma prod_lt_one (hf : f < 1) : ∏ i, f i < 1 :=
Finset.prod_lt_one' (fun _ _ ↦ hf.le _) <| by simpa using (Pi.lt_def.1 hf).2
@[to_additive sum_pos_iff_of_nonneg]
| Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 547 | 551 |
/-
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... | exact fun h ↦ IsLocalization.exists_of_eq ((toFractionRingRingEquiv K).symm.injective h)
surj' := by
rintro ⟨z⟩
convert IsLocalization.surj K[X]⁰ z
simp only [← ofFractionRing_algebraMap, Function.comp_apply, ← ofFractionRing_mul,
| Mathlib/FieldTheory/RatFunc/Basic.lean | 659 | 663 |
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