Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... | else (1 / 2 : ℂ) * rexp (-π * (n + a) ^ 2 * t))
((evenKernel a t - (if (a : UnitAddCircle) = 0 then 1 else 0 : ℝ)) / 2) := by
refine (ofReal_sub .. ▸ (hasSum_ofReal.mpr (hasSum_int_evenKernel₀ a ht)).div_const
2).congr_fun fun n ↦ ?_
split_ifs
· rw [ofReal_zero, zero_div]
· rw [mul_com... | Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 528 | 543 |
/-
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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | theorem unbounded_of_unbounded_sUnion (r : α → α → Prop) [wo : IsWellOrder α r] {s : Set (Set α)}
(h₁ : Unbounded r <| ⋃₀ s) (h₂ : #s < Order.cof (swap rᶜ)) : ∃ x ∈ s, Unbounded r x := by
by_contra! h
simp_rw [not_unbounded_iff] at h
let f : s → α := fun x : s => wo.wf.sup x (h x.1 x.2)
refine h₂.not_le (le... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 704 | 719 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Transfer
/-!
# The Schur-Zassenhaus Theorem
In this file we prove the Schur-Zassenhaus theorem.
## Main results
- `Subgroup.exists_ri... |
end SchurZassenhausInduction
variable {n : ℕ} {G : Type u} [Group G]
/-- Do not use this lemma: It is made obsolete by `exists_right_complement'_of_coprime` -/
| Mathlib/GroupTheory/SchurZassenhaus.lean | 252 | 257 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
/-!
## Identities between ... | theorem p_nonzero [Nontrivial R] [CharP R p] : (p : 𝕎 R) ≠ 0 := by
intro h
simpa only [h, zero_coeff, zero_ne_one] using coeff_p_one p R
| Mathlib/RingTheory/WittVector/Identities.lean | 81 | 83 |
/-
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... | comp_v z₁ z₂ (zero_add n) p p q (add_zero p) hpq
/-- The associativity of the composition of cochains. -/
lemma comp_assoc {n₁ n₂ n₃ n₁₂ n₂₃ n₁₂₃ : ℤ}
(z₁ : Cochain F G n₁) (z₂ : Cochain G K n₂) (z₃ : Cochain K L n₃)
(h₁₂ : n₁ + n₂ = n₁₂) (h₂₃ : n₂ + n₃ = n₂₃) (h₁₂₃ : n₁ + n₂ + n₃ = n₁₂₃) :
(z₁.comp z₂ h... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 252 | 263 |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | @[simp] theorem derivative_inv' {R} [Field R] (f : R⟦X⟧) : d⁄dX R f⁻¹ = -f⁻¹ ^ 2 * d⁄dX R f := by
by_cases h : constantCoeff R f = 0
· suffices f⁻¹ = 0 by
rw [this, pow_two, zero_mul, neg_zero, zero_mul, map_zero]
rwa [MvPowerSeries.inv_eq_zero]
apply Derivation.leibniz_of_mul_eq_one
exact PowerSeries... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 166 | 172 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... | ext α l : 2
trans l.foldr sublistsAux [[]]
· rw [sublistsAux_eq_flatMap, sublists]
· simp only [sublistsFast, sublistsAux_eq_array_foldl, Array.foldr_toList]
rw [← foldr_hom Array.toList]
· intros _ _; congr
theorem sublists_append (l₁ l₂ : List α) :
sublists (l₁ ++ l₂) = (sublists l₂) >>= (fun x =... | Mathlib/Data/List/Sublists.lean | 120 | 129 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | #(interedges r s t) = ∑ ab ∈ P.parts ×ˢ Q.parts, #(interedges r ab.1 ab.2) := by
rw [card_interedges_finpartition_left _ P, sum_product]
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 159 | 160 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Monoid.Units
/-!
# Ordered monoid an... |
@[to_additive (attr := simp)]
| Mathlib/Algebra/Order/Hom/Monoid.lean | 586 | 587 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | /-- Definition of the natural algebra induced by the localization of an algebra.
Given an algebra `R → S`, a submonoid `R` of `M`, and a localization `Rₘ` for `M`,
let `Sₘ` be the localization of `S` to the image of `M` under `algebraMap R S`.
| Mathlib/RingTheory/Localization/Basic.lean | 445 | 447 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Mario Carneiro
-/
import Mathlib.Computability.Halting
/-!
# Strong reducibility and degrees.
This file defines the notions of computable many-one reduction and one-one
reduc... | Mathlib/Computability/Reduce.lean | 495 | 496 | |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Comb... | _ ≤ 1 + ε ^ 5 * 49⁻¹ := by gcongr; norm_num
_ = 1 + ε ^ 5 / 49 := by rw [div_eq_mul_inv]
private theorem density_sub_eps_le_sum_density_div_card [Nonempty α]
(hPα : #P.parts * 16 ^ #P.parts ≤ card α) (hPε : ↑100 ≤ ↑4 ^ #P.parts * ε ^ 5)
{hU : U ∈ P.parts} {hV : V ∈ P.parts} {A B : Finset (Finset α)}
... | Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 240 | 254 |
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Order.RelClasses
/-!
# Euclidean domains
This file introduces Euclidean domains and provides the extende... |
end GCD
| Mathlib/Algebra/EuclideanDomain/Defs.lean | 233 | 235 |
/-
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... | end
lemma iterate_two_mul (hf : Involutive f) (n : ℕ) : f^[2 * n] = id := by
| Mathlib/Algebra/Ring/Parity.lean | 319 | 321 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... |
/-- Lagrange basis polynomials indexed by `s : Finset ι`, defined at nodes `v i` for a
| Mathlib/LinearAlgebra/Lagrange.lean | 191 | 192 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.Comma.Arrow
/-!
# Commutative squares
This file provide an API for commutative squares in categories.
If `top`, `left`, `right`... | CommSq i.unop h.unop g.unop f.unop :=
⟨by simp only [← unop_comp, p.w]⟩
| Mathlib/CategoryTheory/CommSq.lean | 64 | 65 |
/-
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.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... |
@[simp]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 198 | 199 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | intro h
letI := h.toField
simpa using congr_arg natDegree (monic_X.eq_one_of_isUnit <| monic_X (R := R).ne_zero.isUnit)
section multiplicity
/-- An algorithm for deciding polynomial divisibility.
The algorithm is "compute `p %ₘ q` and compare to `0`".
See `Polynomial.modByMonic` for the algorithm that computes ... | Mathlib/Algebra/Polynomial/Div.lean | 472 | 491 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 477 | 479 | |
/-
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.... |
@[simp] lemma natCast_eq_zero {a n : ℕ} [NeZero n] : (a : Fin n) = 0 ↔ n ∣ a := by
| Mathlib/Data/Fin/Basic.lean | 381 | 382 |
/-
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.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | | succ n ih => rw [succ_nsmul, prod_add, ih, pow_succ]
end PrimeMultiset
| Mathlib/Data/PNat/Factors.lean | 195 | 198 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... |
@[simp]
theorem bihimp_top : a ⇔ ⊤ = a := by rw [bihimp, himp_top, top_himp, inf_top_eq]
| Mathlib/Order/SymmDiff.lean | 213 | 215 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Measure theory in the product of groups
In this file we show propertie... | Mathlib/MeasureTheory/Group/Prod.lean | 489 | 492 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | simpa [SqLe, mul_left_comm, mul_assoc] using Nat.mul_le_mul_left (n * n) xy
theorem sqLe_mul {d x y z w : ℕ} :
(SqLe x 1 y d → SqLe z 1 w d → SqLe (x * w + y * z) d (x * z + d * y * w) 1) ∧
(SqLe x 1 y d → SqLe w d z 1 → SqLe (x * z + d * y * w) 1 (x * w + y * z) d) ∧
(SqLe y d x 1 → SqLe z 1 w d →... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 378 | 392 |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | rw [← Nat.not_le, ← Bool.not_eq_true, Nat.ble_eq]
| Mathlib/Tactic/NormNum/Basic.lean | 481 | 482 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.DedekindD... | rw [Polynomial.eval₂_mul] at this
cases mul_eq_zero.mp this <;> contradiction
rw [div_add_div _ _ hx hy, eq_div_iff (mul_ne_zero hx hy), div_eq_mul_inv, mul_right_comm, ←
| Mathlib/FieldTheory/RatFunc/AsPolynomial.lean | 147 | 149 |
/-
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.unique (hf : HasFPowerSeriesOnBall f p x r)
(hg : HasFPowerSeriesOnBall g p x r) : (EMetric.ball x r).EqOn f g := fun _ hy ↦
| Mathlib/Analysis/Analytic/Basic.lean | 586 | 588 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 445 | 446 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | have : v.limRatioMeas hρ y = ∞ := hy.1
simp only [this, ENNReal.coe_lt_top, mem_setOf_eq]
· simp only [(zero_lt_one.trans_le hq).ne', true_or, ENNReal.coe_eq_zero, Ne,
not_false_iff]
have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s)) := by
apply ENNReal.Tendsto.mul... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 484 | 505 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 242 | 386 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... |
alias pow_two_sub_pow_two := sq_sub_sq
| Mathlib/Algebra/Ring/Commute.lean | 185 | 186 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,180 | 1,183 | |
/-
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 | 3,050 | 3,051 | |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# The plus construction for presheaves.
This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D`
where `C` i... | J.toPlus P ≫ J.plusLift η hQ = η := by
dsimp [plusLift]
rw [← Category.assoc]
rw [Iso.comp_inv_eq]
dsimp only [isoToPlus, asIso]
rw [toPlus_naturality]
theorem plusLift_unique {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (hQ : Presheaf.IsSheaf J Q)
(γ : J.plusObj P ⟶ Q) (hγ : J.toPlus P ≫ γ = η) : γ = J.plusLift η hQ... | Mathlib/CategoryTheory/Sites/Plus.lean | 264 | 278 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... |
variable (R)
@[simp]
| Mathlib/RingTheory/WittVector/WittPolynomial.lean | 116 | 119 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | theorem snoc_injective2 : Function.Injective2 (@snoc n α) := fun x y xₙ yₙ h ↦
⟨funext fun i ↦ by simpa using congr_fun h (castSucc i), by simpa using congr_fun h (last n)⟩
@[simp]
theorem snoc_inj {x y : ∀ i : Fin n, α i.castSucc} {xₙ yₙ : α (last n)} :
| Mathlib/Data/Fin/Tuple/Basic.lean | 559 | 563 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.ExactFunctor
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Preadditive.Biproducts
imp... | zero := ⟨F.obj 0, by rw [IsZero.iff_id_eq_zero, ← F.map_id, id_zero, F.map_zero]⟩
end
section InducedCategory
| Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean | 128 | 132 |
/-
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.FieldTheory.Finiteness
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 141 | 148 | |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... |
/-- The path component of `x` is the set of points that can be joined to `x`. -/
def pathComponent (x : X) :=
{ y | Joined x y }
theorem mem_pathComponent_iff : x ∈ pathComponent y ↔ Joined y x := .rfl
| Mathlib/Topology/Connected/PathConnected.lean | 194 | 200 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
have : ω = [] := eq_nil_of_length_eq_zero (hω.trans h)
rw [this, wordProd_nil]
· rintro rfl
exact cs.length_one
@[simp]
theorem length_inv (w : W) : ℓ (w⁻¹) = ℓ w := by
| Mathlib/GroupTheory/Coxeter/Length.lean | 91 | 98 |
/-
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 ... | Iff.rfl
| Mathlib/Order/Irreducible.lean | 204 | 204 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 1,686 | 1,688 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
/-!
# Derivatives of aff... | theorem hasStrictDerivAt_lineMap : HasStrictDerivAt (lineMap a b) (b - a) x := by
simpa using (lineMap a b : 𝕜 →ᵃ[𝕜] E).hasStrictDerivAt
| Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 64 | 65 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vanderm... | | n + 1 => by
rw [prod_range_succ, prod_range_succ_factorial n, mul_comm, superFactorial]
variable {R : Type*} [CommRing R]
| Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 68 | 71 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
/-!... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 436 | 436 | |
/-
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 | 947 | 952 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Log
import Mathlib.Data.Nat.Pri... | #{i ∈ Ico 1 b | p ^ i ≤ k % p ^ i + (n - k) % p ^ i} :=
calc
∑ i ∈ Finset.Ico 1 b, n / p ^ i = ∑ i ∈ Finset.Ico 1 b, (k + (n - k)) / p ^ i := by
simp only [add_tsub_cancel_of_le hkn]
| Mathlib/Data/Nat/Multiplicity.lean | 179 | 182 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | ⇑(frobeniusAlgEquivOfAlgebraic K L) = (· ^ q) := rfl
variable [Finite L]
open Polynomial in
theorem orderOf_frobeniusAlgHom : orderOf (frobeniusAlgHom K L) = Module.finrank K L :=
(orderOf_eq_iff Module.finrank_pos).mpr <| by
have := Fintype.ofFinite L
refine ⟨DFunLike.ext _ _ fun x ↦ ?_, fun m lt pos e... | Mathlib/FieldTheory/Finite/Basic.lean | 356 | 369 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 1,549 | 1,552 | |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Normed.Algebra.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Data.Complex.Exponential
import Ma... | hasFDerivAt_exp_smul_const_of_mem_ball 𝕂 x t htx
deriv₁.hasFDerivAt.unique deriv₂ ▸ deriv₁
theorem hasStrictFDerivAt_exp_smul_const_of_mem_ball' (x : 𝔸) (t : 𝕊)
(htx : t • x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
HasStrictFDerivAt (fun u : 𝕊 => exp 𝕂 (u • x))
(((1 : 𝕊 →L[𝕂] 𝕊).... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 301 | 308 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | theorem smul_eq_C_mul (a : R) : a • p = C a * p := by simp [ext_iff]
theorem update_eq_add_sub_coeff {R : Type*} [Ring R] (p : R[X]) (n : ℕ) (a : R) :
p.update n a = p + Polynomial.C (a - p.coeff n) * Polynomial.X ^ n := by
ext
rw [coeff_update_apply, coeff_add, coeff_C_mul_X_pow]
| Mathlib/Algebra/Polynomial/Coeff.lean | 325 | 330 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Partition.Equipartition
/-!
# Turán's theorem
In this file we prove Turán's theorem, the first importan... | by_contra h
rw [not_isEmpty_iff] at h
obtain ⟨f, ha⟩ := h
simp only [turanGraph, top_adj] at ha
obtain ⟨x, y, d, c⟩ := Fintype.exists_ne_map_eq_of_card_lt (fun x ↦
(⟨(f x).1 % r, Nat.mod_lt _ hr⟩ : Fin r)) (by simp)
simp only [Fin.mk.injEq] at c
exact absurd c ((@ha x y).mpr d)
/-- An `r + 1`-cliquef... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 86 | 95 |
/-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | · refine intervalIntegral.intervalIntegrable_cpow' ?_
rwa [sub_re, one_re, ← zero_sub, sub_lt_sub_iff_right]
· apply Continuous.continuousOn
continuity
-- pointwise limit of f
have f_tends : ∀ x : ℝ, x ∈ Ioi (0 : ℝ) →
Tendsto (fun n : ℕ => f n x) atTop (𝓝 <| ↑(Real.exp (-x)) * (x : ℂ) ^ (... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 291 | 350 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | if hx : n / b = 0 then
simp only [hx, if_true, List.length, congrArg (fun l ↦ l + 1) hlen]
| Mathlib/Data/Nat/Digits.lean | 751 | 752 |
/-
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.Subsemigroup.Operations
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Order.Filter.Extr
/-!
# Lemmas about the `sup` and `in... | (∏ i ∈ s, f i).support.sup degb ≤ ∑ i ∈ s, (f i).support.sup degb :=
(sup_support_multiset_prod_le degb0 degbm _).trans_eq <| congr_arg _ <| Multiset.map_map _ _ _
theorem le_inf_support_finset_prod (degt0 : 0 ≤ degt 0)
(degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) (s : Finset ι) (f : ι → R[A]) :
(∑ i ... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 183 | 190 |
/-
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... | alias ⟨SameCycle.of_pow_right, SameCycle.pow_right⟩ := sameCycle_pow_right
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 152 | 153 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | case inr => simp
case inr => simp
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 615 | 617 |
/-
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... | IntegrableOn f (s ∪ t) μ :=
(hs.add_measure ht).mono_measure <| Measure.restrict_union_le _ _
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 160 | 162 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... |
theorem symmDiff_sdiff : a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) := by
| Mathlib/Order/SymmDiff.lean | 141 | 142 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... |
@[to_additive]
theorem one_div_mul_eq_div : 1 / a * b = b / a := by simp
| Mathlib/Algebra/Group/Basic.lean | 585 | 588 |
/-
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... | theorem restrict_eq_zero : μ.restrict s = 0 ↔ μ s = 0 := by
rw [← measure_univ_eq_zero, restrict_apply_univ]
| Mathlib/MeasureTheory/Measure/Restrict.lean | 203 | 205 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | Mathlib/Data/Real/Sqrt.lean | 56 | 56 | |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.BoundedOrder.Lattice
import Mathlib.Data.List.Induction
import Mathlib.Order.MinMax
import Ma... |
@[simp]
theorem foldr_min_of_ne_nil (h : l ≠ []) : ↑(l.foldr min ⊤) = l.minimum :=
@foldr_max_of_ne_nil αᵒᵈ _ _ _ h
theorem le_min_of_forall_le (l : List α) (a : α) (h : ∀ x ∈ l, a ≤ x) : a ≤ l.foldr min ⊤ :=
@max_le_of_forall_le αᵒᵈ _ _ _ _ h
| Mathlib/Data/List/MinMax.lean | 514 | 520 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | theorem coe_lt_coe {p q : Seminorm 𝕜 E} : (p : E → ℝ) < q ↔ p < q :=
Iff.rfl
theorem le_def {p q : Seminorm 𝕜 E} : p ≤ q ↔ ∀ x, p x ≤ q x :=
Iff.rfl
| Mathlib/Analysis/Seminorm.lean | 245 | 250 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | rwa [fst_image_prod _ st.2, fst_image_prod _ (h.mono H).snd] at this
· have := image_subset (Prod.snd : α × β → β) H
rwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this
| Mathlib/Data/Set/Prod.lean | 340 | 342 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.ShortComplex.Retract
import Mathlib.CategoryTheory.MorphismProperty.Composition
/-!
#... | constructor
· intro
exact quasiIsoAt_of_comp_left φ φ' i
· intro
infer_instance
| Mathlib/Algebra/Homology/QuasiIso.lean | 174 | 179 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | first point (reversed) is zero. -/
theorem _root_.Wbtw.oangle₃₁₂_eq_zero {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) : ∡ p₃ p₁ p₂ = 0 := by
rw [oangle_eq_zero_iff_oangle_rev_eq_zero, h.oangle₂₁₃_eq_zero]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 435 | 438 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
/-!
# Cartan subalgebras
Cartan subalgebras are one of the most important concepts in Lie theor... | fun y hy => I.lie_mem hy
open LieIdeal
/-- A nilpotent Lie algebra is its own Cartan subalgebra. -/
| Mathlib/Algebra/Lie/CartanSubalgebra.lean | 114 | 118 |
/-
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... | (comap g V W hVW).comp (comap f U V hUV) :=
RingHom.ext fun s =>
Subtype.eq <|
funext fun p => by
rw [comap_apply, Localization.localRingHom_comp _ (PrimeSpectrum.comap g p.1).asIdeal] <;>
simp
@[elementwise, reassoc]
theorem toOpen_comp_comap (f : R →+* S) (U : Opens (PrimeSpectrum.T... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 1,080 | 1,097 |
/-
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... | (hf : IsInducing f) (h : IsTopologicalBasis T) : IsTopologicalBasis ((preimage f) '' T) :=
.of_hasBasis_nhds fun a ↦ by
convert (hf.basis_nhds (h.nhds_hasBasis (a := f a))).to_image_id with s
aesop
| Mathlib/Topology/Bases.lean | 230 | 234 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Monoid.Units
/-!
# Ordered monoid an... | @[to_additive "Any type satisfying `OrderAddMonoidIsoClass` can be cast into `OrderAddMonoidIso` via
`OrderAddMonoidIsoClass.toOrderAddMonoidIso`"]
instance [EquivLike F α β] [OrderIsoClass F α β] [MulEquivClass F α β] : CoeTC F (α ≃*o β) :=
| Mathlib/Algebra/Order/Hom/Monoid.lean | 177 | 179 |
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Principal Ideals
This file deals with the set of principal ideals of a `CommRing R`.
## Main definitions and res... | theorem associatesEquivIsPrincipal_apply (x : R) :
associatesEquivIsPrincipal R (.mk x) = span {x} := rfl
@[simp]
| Mathlib/RingTheory/Ideal/IsPrincipal.lean | 75 | 78 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 988 | 988 | |
/-
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.Analysis.Convex.StrictConvexBetween
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.Normed.Affine.Convex
import Mathlib.Geometry.E... | /-- A set of points is concyclic if it is cospherical and coplanar. (Most results are stated
directly in terms of `Cospherical` instead of using `Concyclic`.) -/
structure Concyclic (ps : Set P) : Prop where
Cospherical : Cospherical ps
Coplanar : Coplanar ℝ ps
| Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 200 | 204 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 588 | 595 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,286 | 1,294 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | exact inv_pos.2 (lt_of_lt_of_le zero_lt_one (one_le_exp (neg_nonneg.2 h)))
| Mathlib/Data/Complex/Exponential.lean | 275 | 275 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 343 | 351 | |
/-
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.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 970 | 971 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... | RingHom.liftOfRightInverse_comp_apply _ _ _ _ _
@[simp]
theorem truncate_truncate {n₁ n₂ n₃ : ℕ} (h1 : n₁ ≤ n₂) (h2 : n₂ ≤ n₃)
(x : TruncatedWittVector p n₃ R) :
| Mathlib/RingTheory/WittVector/Truncated.lean | 346 | 350 |
/-
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.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... | terms of `a ^ (Fintype.card F / 2)`. -/
theorem quadraticChar_eq_pow_of_char_ne_two (hF : ringChar F ≠ 2) {a : F} (ha : a ≠ 0) :
quadraticChar F a = if a ^ (Fintype.card F / 2) = 1 then 1 else -1 :=
quadraticCharFun_eq_pow_of_char_ne_two hF ha
| Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 179 | 182 |
/-
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... |
open Function in
protected theorem map_update_add [DecidableEq ι] (i : ι) (u v : s i →ₗ[R] t i) :
map (update f i (u + v)) = map (update f i u) + map (update f i v) := by
ext x
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 551 | 555 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | tauto
theorem map_edgeFinset_induce_of_support_subset (h : G.support ⊆ s) :
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 455 | 457 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Tactic.FinCases
import Mathlib.Topology.Connected.LocallyConnected
import Mathlib.Topology.Sets.Closeds
/-!
# L... | left_inv := by
intro
simp [comap_comap]
right_inv := by
intro
simp [comap_comap]
| Mathlib/Topology/LocallyConstant/Basic.lean | 484 | 489 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | lemma mul_left_injective_of_invertible [Invertible A] :
Function.Injective (fun (x : Matrix m n α) => x * A) :=
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 281 | 282 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | -- In addition, type class inference would not be able to infer `v`.
/-- A valuation gives a preorder on the underlying ring. -/
def toPreorder : Preorder R :=
Preorder.lift v
/-- If `v` is a valuation on a division ring then `v(x) = 0` iff `x = 0`. -/
theorem zero_iff [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : ... | Mathlib/RingTheory/Valuation/Basic.lean | 196 | 202 |
/-
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.Localization.Opposite
/-!
# Calculus of fractions
Following the definitions by [Gabriel and Zisman][gabriel-zisman-1967],
given a morphism prope... | (h : LeftFractionRel z₁ z₂) : RightFractionRel z₁.unop z₂.unop := by
obtain ⟨Z, t₁, t₂, hs, hf, ht⟩ := h
exact ⟨Opposite.unop Z, t₁.unop, t₂.unop, Quiver.Hom.op_inj hs,
Quiver.Hom.op_inj hf, ht⟩
lemma leftFractionRel_op_iff
| Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean | 897 | 902 |
/-
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... | exact sub_right_injective this
| Mathlib/Probability/Martingale/BorelCantelli.lean | 287 | 287 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | MeasurableSet[hτ.measurableSpace] {ω | i ≤ τ ω} :=
hτ.measurableSet_ge_of_countable_range' (Set.to_countable _) i
protected theorem measurableSet_lt_of_countable_range' (hτ : IsStoppingTime f τ)
(h_countable : (Set.range τ).Countable) (i : ι) :
| Mathlib/Probability/Process/Stopping.lean | 484 | 488 |
/-
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.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... | /-- For `r`, instead of `p` -/
theorem one_div_nonneg' : 0 ≤ 1 / r := le_of_lt h.one_div_pos'
| Mathlib/Data/Real/ConjExponents.lean | 101 | 102 |
/-
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 | 1,219 | 1,222 | |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
import Mathlib.Topology.S... |
/-- The neighbourhood filter of a nonzero element consists of all sets containing that
| Mathlib/Topology/Algebra/WithZeroTopology.lean | 78 | 79 |
/-
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
/-... | ((differentiable_circleMap c R θ).hasDerivAt.tendsto_nhdsNE
(deriv_circleMap_ne_zero hR)).eventually this
filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (ball_mem_nhds _ zero_lt_one)]
simp_all [dist_eq, sub_eq_zero]
refine (((hasDerivAt_circleMap c R θ).isBigO_sub.mono... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 256 | 262 |
/-
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.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | (g : Spec B ⟶ Spec C) : HasPullback f g := by
rw [← Scheme.Spec.map_preimage f, ← Scheme.Spec.map_preimage g]
exact ⟨⟨⟨_, isLimitOfHasPullbackOfPreservesLimit
| Mathlib/AlgebraicGeometry/Pullbacks.lean | 425 | 427 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Tactic.Group
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
... |
lemma IsEverywherePos.smul_measure (hs : IsEverywherePos μ s) {c : ℝ≥0∞} (hc : c ≠ 0) :
IsEverywherePos (c • μ) s :=
| Mathlib/MeasureTheory/Measure/EverywherePos.lean | 165 | 167 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | refine fun h => ⟨fun i hi j hj hij => ?_, fun i hi j hj hij => ?_⟩ <;>
simp_rw [Finset.disjoint_sup_left, Finset.disjoint_sup_right] <;>
intro i' hi' j' hj'
· exact h (mk_mem_product hi hi') (mk_mem_product hj hj') (ne_of_apply_ne Prod.fst hij)
· exact h (mk_mem_product hi' hi) (mk_mem_product hj' hj) (... | Mathlib/Order/SupIndep.lean | 222 | 226 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... | rw [choose_succ_right _ _ hn, Nat.add_one_sub_one]
| Mathlib/Data/Nat/Choose/Basic.lean | 79 | 80 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | · left
rw [eq_div_iff_mul_eq (two_ne_zero' ℝ), eq_sub_iff_add_eq] at hn
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 206 | 207 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... | nonZeroDivisorsRight M₀ = nonZeroSMulDivisors M₀ M₀ := rfl
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 121 | 122 |
/-
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.LinearAlgebra.Quotient.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# Projection to a subspace
In this file we define
* `Submodule.linearProjOfIsCom... |
theorem IsProj.eq_conj_prodMap {f : E →ₗ[R] E} (h : IsProj p f) :
f = (p.prodEquivOfIsCompl (ker f) h.isCompl).conj (prodMap id 0) := by
| Mathlib/LinearAlgebra/Projection.lean | 440 | 442 |
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