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/-
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... | Mathlib/Topology/Connected/PathConnected.lean | 1,208 | 1,209 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,455 | 1,458 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... |
theorem contDiffOn_infty_iff_fderiv_of_isOpen (hs : IsOpen s) :
ContDiffOn 𝕜 ∞ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ∞ (fderiv 𝕜 f) s := by
rw [← ENat.coe_top_add_one, contDiffOn_succ_iff_fderiv_of_isOpen hs]
simp
| Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 885 | 890 |
/-
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... |
theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i ∈ range n, x ^ i / i ! ≤ exp x :=
| Mathlib/Data/Complex/Exponential.lean | 235 | 236 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | refine Submodule.eq_top_iff'.2 fun x => ?_
rw [mem_interior_iff_mem_nhds] at hy
have : Tendsto (fun c : R => y + c • x) (𝓝[{ x : R | IsUnit x }] 0) (𝓝 (y + (0 : R) • x)) :=
tendsto_const_nhds.add ((tendsto_nhdsWithin_of_tendsto_nhds tendsto_id).smul tendsto_const_nhds)
rw [zero_smul, add_zero] at this
o... | Mathlib/Topology/Algebra/Module/Basic.lean | 63 | 74 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,527 | 1,531 | |
/-
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 | 614 | 622 | |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Finite
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Regular
import Mathlib... |
theorem EventuallyLE.cardinal_bInter {S : Set ι} (hS : #S < c)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋂ i ∈ S, s i ‹_› ≤ᶠ[l] ⋂ i ∈ S, t i ‹_› := by
simp only [biInter_eq_iInter]
| Mathlib/Order/Filter/CardinalInter.lean | 145 | 149 |
/-
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.Data.Nat.ModEq
/-!
# Congruences modulo an integer
This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how
`Data.N... |
theorem mod_mul_left_mod (a b c : ℤ) : a % (b * c) % c = a % c :=
(mod_modEq _ _).of_mul_left _
end Int
| Mathlib/Data/Int/ModEq.lean | 263 | 271 |
/-
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
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 969 | 972 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Independence.Kernel
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Independence of sets of sets and measure spaces (σ-algebras)
* A fami... |
lemma iIndep_iff (m : ι → MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω} (μ : Measure Ω) :
iIndep m μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → MeasurableSet[m i] (f i)),
| Mathlib/Probability/Independence/Basic.lean | 170 | 172 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
/-!
# Construction of projections for the Dold-Kan correspondence
In this file... | simp only [P_zero, id_comp, comp_id]
· simp only [P_succ, add_comp, comp_add, assoc, comp_id, hq, reassoc_of% hq]
-- `erw` is needed to see through `natTransHσ q).app = Hσ q`
erw [(natTransHσ q).naturality f]
| Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 152 | 155 |
/-
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.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | _ = n * (n - 1) := by rw [sum_const, card_range, Nat.nsmul_eq_mul]
/-- Gauss' summation formula -/
theorem sum_range_id (n : ℕ) : ∑ i ∈ range n, i = n * (n - 1) / 2 := by
rw [← sum_range_id_mul_two n, Nat.mul_div_cancel _ zero_lt_two]
| Mathlib/Algebra/BigOperators/Intervals.lean | 258 | 262 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | /-- Any set is contained in the complement of the δ-thickening of the complement of its
δ-thickening. -/
lemma subset_compl_thickening_compl_thickening_self (δ : ℝ) (E : Set α) :
E ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ := by
intro x x_in_E
simp only [thickening, mem_compl_iff, mem_setOf_eq, not_lt]
apply EMetri... | Mathlib/Topology/MetricSpace/Thickening.lean | 114 | 122 |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | SupClosed (ofDual ⁻¹' s) ↔ InfClosed s := Iff.rfl
| Mathlib/Order/SupClosed.lean | 242 | 243 |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... | Ultrafilter.compl_mem_iff_not_mem]
| Mathlib/Topology/Compactification/OnePoint.lean | 350 | 351 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... | (forall_mem_insertNone.2 ⟨hq, hmem⟩)
simpa using this
/-- Concave **Jensen's inequality** where an element plays a distinguished role. -/
lemma ConcaveOn.map_add_sum_le (hf : ConcaveOn 𝕜 s f) (h₀ : ∀ i ∈ t, 0 ≤ w i)
(h₁ : v + ∑ i ∈ t, w i = 1) (hmem : ∀ i ∈ t, p i ∈ s) (hv : 0 ≤ v) (hq : q ∈ s) :
v • ... | Mathlib/Analysis/Convex/Jensen.lean | 82 | 91 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... |
theorem degree_mono [Semiring S] {f : R[X]} {g : S[X]} (h : f.support ⊆ g.support) :
| Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 123 | 124 |
/-
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... |
-- TODO: Theory of order of an element under an action
theorem IsCycleOn.pow_apply_eq {s : Finset α} (hf : f.IsCycleOn s) (ha : a ∈ s) {n : ℕ} :
(f ^ n) a = a ↔ #s ∣ n := by
obtain rfl | hs := Finset.eq_singleton_or_nontrivial ha
· rw [coe_singleton, isCycleOn_singleton] at hf
simpa using IsFixedPt.iterate... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 756 | 766 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | refine le_ciSup (f := fun y : Iio b => a ^ (y : Cardinal)) ?_ ⟨c, h⟩
rw [← image_eq_range]
exact bddAbove_image.{u, u} _ bddAbove_Iio
| Mathlib/SetTheory/Cardinal/Basic.lean | 956 | 959 |
/-
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.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... | aeval f (bind₂ g φ) = eval₂Hom ((↑(aeval f : _ →ₐ[S] _) : _ →+* _).comp g) f φ :=
eval₂Hom_bind₂ _ _ _ _
| Mathlib/Algebra/MvPolynomial/Monad.lean | 268 | 269 |
/-
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... |
theorem reverse_foldl {l : List α} : reverse (foldl (fun t h => h :: t) [] l) = l := by
| Mathlib/Data/List/Basic.lean | 848 | 849 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
import Mathlib.Order.Directed
/-!
# Monotone functions on intervals
This file sho... | lemma AntitoneOn.mapsTo_Iic (h : AntitoneOn f (Iic b)) : MapsTo f (Iic b) (Ici (f b)) :=
fun _ _ ↦ by aesop
| Mathlib/Order/Interval/Set/Image.lean | 37 | 38 |
/-
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... | rfl
/-- `shiftLeft' b m n` performs a left shift of `m` `n` times
and adds the bit `b` as the least significant bit each time.
Returns the corresponding natural number -/
def shiftLeft' (b : Bool) (m : ℕ) : ℕ → ℕ
| 0 => m
| n + 1 => bit b (shiftLeft' b m n)
| Mathlib/Data/Nat/Bits.lean | 121 | 128 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | section LimFunctor
variable [HasLimitsOfShape J C]
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 450 | 452 |
/-
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.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... | variable [Preorder β] {s t : Finset α} {f : Finset α → β}
| Mathlib/Data/Finset/Interval.lean | 120 | 121 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
/-!
# Construction of projections for the Dold-Kan correspondence
In this file... | simp only [P_succ]
exact (of_P q n).induction
| Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 92 | 94 |
/-
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.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
... | theorem pos_of_pos {x : ℝ} (hx : 0 < x) : 0 < expNegInvGlue x := by
| Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 46 | 46 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | Mathlib/Data/List/Cycle.lean | 937 | 938 | |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Deepro Choudhury, Scott Carnahan
-/
import Mathlib.LinearAlgebra.RootSystem.Defs
import Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate
/-!
# Root data and root system... | its roots. -/
@[ext]
protected lemma ext [CharZero R] [NoZeroSMulDivisors R M]
{P₁ P₂ : RootSystem ι R M N}
(he : P₁.toPerfectPairing = P₂.toPerfectPairing)
(hr : P₁.root = P₂.root) :
P₁ = P₂ := by
suffices ∀ P₁ P₂ : RootSystem ι R M N, P₁.toPerfectPairing = P₂.toPerfectPairing →
P₁.root = P₂.ro... | Mathlib/LinearAlgebra/RootSystem/Basic.lean | 188 | 212 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... | | base_cycles σ hσ => simp [hσ.cycleType, hσ.orderOf]
| induction_disjoint σ τ hd _ hσ hτ => simp [hd.cycleType, hd.orderOf, lcm_eq_nat_lcm, hσ, hτ]
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 179 | 181 |
/-
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, Mario Carneiro
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp.L... | rintro h
rw [h] at max
rcases max with ⟨⟨_h1, h2⟩⟩
obtain ⟨I, hIbot, hItop⟩ := not_isField_iff_exists_ideal_bot_lt_and_lt_top.mp not_field
exact ne_of_lt hItop (h2 I hIbot)
end Ring
| Mathlib/RingTheory/Ideal/Basic.lean | 266 | 272 |
/-
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.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | theorem image₂_subset_iff : image₂ f s t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, f x y ∈ u :=
forall_mem_image₂
| Mathlib/Data/Finset/NAry.lean | 98 | 100 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 548 | 571 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,140 | 2,142 | |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | · rw [← WithZero.coe_one, ← ofAdd_zero, WithZero.coe_lt_coe, ofAdd_lt, neg_lt_zero, ←
Int.ofNat_zero, Int.ofNat_lt, zero_lt_iff]
have h : (Ideal.span {r} : Ideal R) ≠ 0 := by
rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]
exact hr
apply Associates.count_ne_zero_iff_dvd h (by apply v... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 124 | 134 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | rw [he, Set.Infinite.Nat.sSup_eq_zero <| (exponent_eq_zero_iff_range_orderOf_infinite h).1 he]
case pos he =>
rw [csSup_eq_of_forall_le_of_forall_lt_exists_gt (Set.range_nonempty _)]
· simp_rw [Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff]
exact orderOf_le_exponent he
intro x hx
... | Mathlib/GroupTheory/Exponent.lean | 460 | 497 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Minpoly.Field
/-!
# Characteris... | have hPmonic : P.Monic := by
suffices (minpoly R f).Monic by
rwa [Monic.def, hP, mul_comm, leadingCoeff_mul_X, ← Monic.def] at this
exact minpoly.monic (isIntegral f)
have hzero : aeval f (minpoly R f) = 0 := minpoly.aeval _ _
simp only [hP, Module.End.mul_eq_comp, LinearMap.ext_iff, hf, aeval_X, ma... | Mathlib/LinearAlgebra/Charpoly/Basic.lean | 101 | 117 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 729 | 730 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.List.Lemmas
import Mathlib.Algebra.Group.Action.Hom
import Mathlib.Algebra.Group.Submonoid.Defs
import Mathlib.Data.List.FinRange
... |
instance SetLike.gMul {S : Type*} [SetLike S R] [Mul R] [Add ι] (A : ι → S)
[SetLike.GradedMul A] : GradedMonoid.GMul fun i => A i where
mul := fun a b => ⟨(a * b : R), SetLike.mul_mem_graded a.prop b.prop⟩
@[simp]
theorem SetLike.coe_gMul {S : Type*} [SetLike S R] [Mul R] [Add ι] (A : ι → S)
[SetLike.Grade... | Mathlib/Algebra/GradedMonoid.lean | 511 | 519 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 945 | 954 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 627 | 629 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | (s.map ↑f).SOppSide (f x) (f y) ↔ s.SOppSide x y :=
(show Function.Injective f.toAffineMap from f.injective).sOppSide_map_iff
theorem WSameSide.nonempty {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) :
| Mathlib/Analysis/Convex/Side.lean | 122 | 125 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... |
end Polynomial
| Mathlib/Algebra/Polynomial/Mirror.lean | 219 | 244 |
/-
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... | theorem norm_zpow_abs (a : E) (n : ℤ) : ‖a ^ |n|‖ = ‖a ^ n‖ := by
rcases le_total 0 n with hn | hn <;> simp [hn, abs_of_nonneg, abs_of_nonpos]
| Mathlib/Analysis/Normed/Group/Basic.lean | 424 | 425 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... |
end SemilatticeInf
| Mathlib/Order/Cover.lean | 183 | 184 |
/-
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
/-!
# ... | variable [CommSemiring F] [Algebra F K] [IsScalarTower F E K] (L : Subalgebra F K) {F}
/-- If `K / E / F` is a ring extension tower, `L` is a subalgebra of `K / F`,
then `E[L]` is generated by any basis of `L / F` as an `E`-module. -/
theorem Subalgebra.adjoin_eq_span_basis {ι : Type*} (bL : Basis ι F L) :
toSubmo... | Mathlib/RingTheory/Adjoin/Basic.lean | 173 | 202 |
/-
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... | exp_injective.eq_iff
| Mathlib/Data/Complex/Exponential.lean | 316 | 317 |
/-
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... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 911 | 919 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... |
end ScalarTower
| Mathlib/RingTheory/Coprime/Basic.lean | 243 | 245 |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... | replace hxy : x - y ∈ f ⁻¹' {0} := by simp [hxy]
rw [← closure_eq s, ← closure_union, ← add_sub_cancel y x]
| Mathlib/Algebra/Ring/Subring/Basic.lean | 1,082 | 1,083 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | ext; simp
@[simp]
| Mathlib/FieldTheory/Perfect.lean | 151 | 153 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | (hζ : IsPrimitiveRoot ζ ↑p) (hodd : p ≠ 2) :
Prime (Algebra.norm ℤ (hζ.toInteger - 1)) := by
have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by simpa using hcycl
replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simpa using hζ
exact hζ.prime_norm_toInteger_sub_one_of_prime_ne_two hodd
/-- In a `p ^ (k... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 430 | 438 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.SimplicialObject.Split
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Construction of the inverse functor of the Dold-Kan equivalence
... | simpa only [Γ₀.Obj.map_on_summand K (Splitting.IndexSet.id (op Δ)) e.op
(rfl : e ≫ 𝟙 Δ = e ≫ 𝟙 Δ),
Γ₀.Obj.Termwise.mapMono_id] using id_comp _
/-- The functor `Γ₀ : ChainComplex C ℕ ⥤ SimplicialObject C`, on morphisms. -/
@[simps]
| Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 266 | 271 |
/-
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
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Sub.Basic
impo... | Mathlib/Algebra/Order/Ring/Canonical.lean | 47 | 47 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | refine ⟨?_, ?_⟩
· intro h
refine Nat.prime_def.mpr ⟨?_, fun m hdvd => ?_⟩
· match n with
| 0 => contradiction
| 1 => contradiction
| Nat.succ (Nat.succ n) => simp [succ_le_succ]
· rw [← mem_singleton, ← h, mem_properDivisors]
have := Nat.le_of_dvd ?_ hdvd
· simpa [hdvd, thi... | Mathlib/NumberTheory/Divisors.lean | 464 | 476 |
/-
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
/-... | @[simp]
theorem integral_sub_center_inv (c : ℂ) {R : ℝ} (hR : R ≠ 0) :
(∮ z in C(c, R), (z - c)⁻¹) = 2 * π * I := by
simp [circleIntegral, ← div_eq_mul_inv, mul_div_cancel_left₀ _ (circleMap_ne_center hR)]
/-- If `f' : ℂ → E` is a derivative of a complex differentiable function on the circle
| Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 403 | 408 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... |
instance (α : ι → Type*) [∀ i, AddCommMonoid (α i)] [∀ i, PartialOrder (α i)]
| Mathlib/Data/DFinsupp/Order.lean | 124 | 125 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 672 | 673 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | variable [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F]
variable (𝕜 E F)
instance instTopologicalSpace : TopologicalSpace 𝓢(E, F) :=
| Mathlib/Analysis/Distribution/SchwartzSpace.lean | 482 | 485 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Action.Pointwise.Finset
import Mathlib.GroupTheory.QuotientGroup.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Basic
/-!
# Stabiliz... | stabilizer_inf_stabilizer_le_stabilizer_apply₂ fun _ ↦ smul_set_sdiff
@[to_additive]
lemma stabilizer_union_eq_left (hdisj : Disjoint s t) (hstab : stabilizer G s ≤ stabilizer G t)
(hstab_union : stabilizer G (s ∪ t) ≤ stabilizer G t) :
stabilizer G (s ∪ t) = stabilizer G s := by
| Mathlib/Algebra/Pointwise/Stabilizer.lean | 81 | 86 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | theorem whiskerLeft_rightUnitor (f : a ⟶ b) (g : b ⟶ c) :
f ◁ (ρ_ g).hom = (α_ f g (𝟙 c)).inv ≫ (ρ_ (f ≫ g)).hom := by
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 374 | 375 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,841 | 1,844 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | instance commSemigroup : CommSemigroup ℚ := by infer_instance
| Mathlib/Data/Rat/Defs.lean | 334 | 334 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | instance : OrderBot (Part α) where
bot := none
bot_le := by rintro x _ ⟨⟨_⟩, _⟩
theorem le_total_of_le_of_le {x y : Part α} (z : Part α) (hx : x ≤ z) (hy : y ≤ z) :
| Mathlib/Data/Part.lean | 342 | 346 |
/-
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.NNReal
/-!
# Limits and a... | Tendsto (fun x : ℝ => exp (b * x) / x ^ s) atTop atTop := by
refine ((tendsto_rpow_atTop hb).comp (tendsto_exp_div_rpow_atTop (s / b))).congr' ?_
filter_upwards [eventually_ge_atTop (0 : ℝ)] with x hx₀
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 126 | 128 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | Mathlib/Data/Num/Lemmas.lean | 1,296 | 1,309 | |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | ext1 x
simp_rw [convolution_def]
rw [← integral_sub_left_eq_self _ μ x]
simp_rw [sub_sub_self, flip_apply]
/-- The symmetric definition of convolution. -/
theorem convolution_eq_swap : (f ⋆[L, μ] g) x = ∫ t, L (f (x - t)) (g t) ∂μ := by
rw [← convolution_flip]; rfl
| Mathlib/Analysis/Convolution.lean | 657 | 664 |
/-
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... | the `GeneralizedBooleanAlgebra` ones -/
section CogeneralizedBooleanAlgebra
| Mathlib/Order/SymmDiff.lean | 479 | 480 |
/-
Copyright (c) 2024 Jana Göken. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Artur Szafarczyk, Suraj Krishna M S, Jean-Baptiste Stiegler, Isabelle Dubois,
Tomáš Jakl, Lorenzo Zanichelli, Alina Yan, Emilie Uthaiwat, Jana Göken,
Filippo A. E. Nuccio
-/
import Mathlib... | lemma isClosed_preCantorSet (n : ℕ) : IsClosed (preCantorSet n) := by
let f := Homeomorph.mulLeft₀ (1 / 3 : ℝ) (by norm_num)
let g := (Homeomorph.addLeft (2 : ℝ)).trans f
induction n with
| zero => exact isClosed_Icc
| succ n ih =>
refine IsClosed.union ?_ ?_
· simpa [f, div_eq_inv_mul] using f.isClos... | Mathlib/Topology/Instances/CantorSet.lean | 81 | 90 |
/-
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.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | h₁.homologyIso.inv ≫ homologyMap φ =
rightHomologyMap' φ h₁ h₂ ≫ h₂.homologyIso.inv := by
simp only [← cancel_mono h₂.homologyIso.hom, assoc, Iso.inv_hom_id_assoc, comp_id,
← RightHomologyData.rightHomologyIso_hom_naturality φ h₁ h₂, Iso.inv_hom_id]
@[reassoc]
lemma rightHomologyIso_hom_naturality ... | Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 756 | 762 |
/-
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.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... |
theorem NatTrans.equifibered_of_discrete {ι : Type*} {F G : Discrete ι ⥤ C}
(α : F ⟶ G) : NatTrans.Equifibered α := by
rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
simp only [Discrete.functor_map_id]
exact IsPullback.of_horiz_isIso ⟨by rw [Category.id_comp, Category.comp_id]⟩
| Mathlib/CategoryTheory/Limits/VanKampen.lean | 75 | 80 |
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
/-!
# Basics on bilinear maps
This file provides basics on bilin... | obtain ⟨x', hx⟩ := hₗ x
subst hx
obtain ⟨y', hy⟩ := hᵣ y
| Mathlib/LinearAlgebra/BilinearMap.lean | 343 | 345 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Subsemigroup.Defs
import M... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 460 | 462 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... | simp only [toFinsupp_monomial, toFinsupp_mul, AddMonoidAlgebra.single_mul_single]
@[simp]
| Mathlib/Algebra/Polynomial/Basic.lean | 422 | 424 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | refine
iSup_le fun n =>
le_trans ?_
(add_le_add_left
(mul_le_mul_of_nonneg_left (ENNReal.sum_le_tsum <| Finset.Ico (u 0 + 1) (u n + 1)) ?_) _)
· simpa using Finset.sum_schlomilch_le hf h_pos h_nonneg hu h_succ_diff n
· exact zero_le _
theorem tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m... | Mathlib/Analysis/PSeries.lean | 161 | 171 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... |
variable [Monoid R] [AddCommMonoid M] [DistribMulAction R M]
theorem isTorsion'_powers_iff (p : R) :
IsTorsion' M (Submonoid.powers p) ↔ ∀ x : M, ∃ n : ℕ, p ^ n • x = 0 := by
constructor
· intro h x
| Mathlib/Algebra/Module/Torsion.lean | 810 | 816 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | protected lemma Filter.HasBasis.isUniformInducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
IsUniformInducing f ↔
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 49 | 51 |
/-
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.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
impor... | theorem t'_iji (i j : D.J) : D.t' i j i = pullback.fst _ _ ≫ D.t i j ≫ inv (pullback.snd _ _) := by
rw [← Category.assoc, ← D.t_fac]
simp
| Mathlib/CategoryTheory/GlueData.lean | 93 | 95 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Equiv.Opposite
import Mathlib.Algebra.... | rw [← Nat.sub_add_cancel hk, pow_add, mul_apply, hm, map_zero]
theorem commute_pow_left_of_commute
[Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₁₂ : R →+* R₂}
| Mathlib/Algebra/Module/LinearMap/End.lean | 135 | 138 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Logic.Small.Basic
import Mathlib.RingTheory.Ideal.Defs
/-!
# Injecti... | ExtensionOfMaxAdjoin.extensionToFun_wd i f h _ ⟨i m, _⟩ 0 _, map_zero, add_zero]
simp
theorem extensionOfMax_le (h : Module.Baer R Q) {y : N} :
extensionOfMax i f ≤ extensionOfMaxAdjoin i f h y :=
⟨le_sup_left, fun x x' EQ => by
| Mathlib/Algebra/Module/Injective.lean | 341 | 346 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | section Order
open scoped ComplexOrder
| Mathlib/Analysis/RCLike/Basic.lean | 748 | 750 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Tilted
/-!
# Log-likelihood Ratio
The likelihood ratio between two measures `μ` and `ν` is their Radon-Nikodym derivative
`μ.rnDeri... | filter_upwards [exp_llr μ ν, Measure.rnDeriv_pos' hμν] with x hx hx_pos
rwa [if_neg hx_pos.ne'] at hx
lemma neg_llr [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪ ν) :
- llr μ ν =ᵐ[μ] llr ν μ := by
| Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | 57 | 61 |
/-
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.Abelian
import Mathlib.LinearAlgebra.TensorProduct.Tower
/-!
# Tensor products of Lie modules
Tensor products of Lie modules carry natural Lie ... |
variable (R L M N P Q)
/-- The universal property for tensor product of modules of a Lie algebra: the `R`-linear
| Mathlib/Algebra/Lie/TensorProduct.lean | 80 | 83 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | simpa [tsub_self] using key
theorem thickenedIndicatorAux_mono {δ₁ δ₂ : ℝ} (hle : δ₁ ≤ δ₂) (E : Set α) :
| Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 89 | 91 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | end InnerRegularWRT
namespace InnerRegular
variable [TopologicalSpace α]
/-- The measure of a measurable set is the supremum of the measures of compact sets it contains. -/
theorem _root_.MeasurableSet.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : MeasurableSet U)
(μ : Measure α) [InnerRegular μ] :
μ U = ⨆ (K ... | Mathlib/MeasureTheory/Measure/Regular.lean | 638 | 651 |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... |
theorem contDiffOn_id {s} : ContDiffOn 𝕜 n (id : E → E) s :=
contDiff_id.contDiffOn
/-- Bilinear functions are `C^n`. -/
theorem IsBoundedBilinearMap.contDiff (hb : IsBoundedBilinearMap 𝕜 b) : ContDiff 𝕜 n b :=
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 157 | 162 |
/-
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... | rw [midpoint_comm, oangle_midpoint_left]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 509 | 510 |
/-
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.... | lemma predAbove_castSucc_of_le (p i : Fin n) (h : i ≤ p) : p.predAbove (castSucc i) = i := by
rw [predAbove_of_le_castSucc _ _ (castSucc_le_castSucc_iff.mpr h), castPred_castSucc]
| Mathlib/Data/Fin/Basic.lean | 1,208 | 1,210 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | exact this.extend_real
choose g hg using this
refine ⟨fun x i => g i x, LipschitzWith.of_dist_le_mul fun x y => ?_, fun x hx ↦ ?_⟩
· exact (dist_pi_le_iff (mul_nonneg K.2 dist_nonneg)).2 fun i => (hg i).1.dist_le_mul x y
· ext1 i
exact (hg i).2 hx
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 383 | 414 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | theorem reverse_reverse {v : Vector α n} : v.reverse.reverse = v := by
cases v
simp [Vector.reverse]
| Mathlib/Data/Vector/Basic.lean | 253 | 255 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Left Homology of short complexes
Given a short complex `S : Shor... | simp only [← h.cyclesIso_hom_comp_i, Iso.inv_hom_id_assoc]
@[reassoc (attr := simp)]
lemma leftHomologyπ_comp_leftHomologyIso_hom :
S.leftHomologyπ ≫ h.leftHomologyIso.hom = h.cyclesIso.hom ≫ h.π := by
dsimp only [leftHomologyπ, leftHomologyIso, cyclesIso, leftHomologyMapIso',
| Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean | 714 | 719 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.ToIntervalMod
import Mathli... | protected lemma neg {f : ℝ → ℝ} (hf : GrowsPolynomially f) : GrowsPolynomially (-f) := by
intro b hb
obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf b hb
refine ⟨c₂, hc₂_mem, c₁, hc₁_mem, ?_⟩
filter_upwards [hf] with x hx
intro u hu
simp only [Pi.neg_apply, Set.neg_mem_Icc_iff, neg_mul_eq_mul_neg, neg_neg]
ex... | Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | 254 | 261 |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | /-- Given a ring homomorphism `f : R →+* S` and a unit `x` in `S`, the induced homomorphism
`R[T;T⁻¹] →+* S` sending `T` to `x` and `T⁻¹` to `x⁻¹`. -/
def eval₂ : R[T;T⁻¹] →+* S :=
| Mathlib/Algebra/Polynomial/Laurent.lean | 537 | 539 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... |
section Lattice
variable [DecidableEq α]
theorem card_union_add_card_inter (s t : Finset α) :
#(s ∪ t) + #(s ∩ t) = #s + #t :=
Finset.induction_on t (by simp) fun a r har h => by by_cases a ∈ s <;>
simp [*, ← Nat.add_assoc, Nat.add_right_comm _ 1]
theorem card_inter_add_card_union (s t : Finset α) :
#... | Mathlib/Data/Finset/Card.lean | 475 | 486 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | theorem det_toLin' (f : Matrix ι ι R) : LinearMap.det (Matrix.toLin' f) = Matrix.det f := by
simp only [← toLin_eq_toLin', det_toLin]
/-- To show `P (LinearMap.det f)` it suffices to consider `P (Matrix.det (toMatrix _ _ f))` and
`P 1`. -/
@[elab_as_elim]
theorem det_cases [DecidableEq M] {P : A → Prop} (f : M →ₗ[A]... | Mathlib/LinearAlgebra/Determinant.lean | 204 | 210 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Pointwise.Set.Card
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Topology.Algebr... | choose! g hg using A
set L : ℕ → Set G := fun n => (fun T => T ∪ g T • K)^[n] K
have Lcompact : ∀ n, IsCompact (L n) := by
intro n
induction' n with n IH
· exact hK
· simp_rw [L, iterate_succ']
apply IsCompact.union IH (hK.smul (g (L n)))
have Lclosed : ∀ n, IsClosed (L n) := by
intro ... | Mathlib/MeasureTheory/Group/Measure.lean | 604 | 619 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... |
theorem binaryRec_of_ne_zero {C : Nat → Sort*} (z : C 0) (f : ∀ b n, C n → C (bit b n)) {n}
(h : n ≠ 0) :
binaryRec z f n = bit_decomp n ▸ f (bodd n) (div2 n) (binaryRec z f (div2 n)) := by
cases n using bitCasesOn with
| h b n =>
rw [binaryRec_eq _ _ (by right; simpa [bit_eq_zero_iff] using h)]
| Mathlib/Data/Nat/Bitwise.lean | 67 | 73 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono... | let s' : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂) :=
PullbackCone.mk s.fst s.snd (s.condition_assoc _)
refine ⟨big_k.isLimit.lift s', big_k.isLimit.fac _ WalkingCospan.left,
big_k.isLimit.fac _ WalkingCospan.right, fun m₁ m₂ => ?_⟩
apply big_k.isLimit.hom_ext
refine (Pullback... | Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean | 110 | 123 |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... | ⟨fun v => g <| v ∘ f, Poly.induction (C := fun g => IsPoly (fun v => g (v ∘ f)))
(fun i => by simpa using IsPoly.proj _) (fun n => by simpa using IsPoly.const _)
(fun f g pf pg => by simpa using IsPoly.sub pf pg)
(fun f g pf pg => by simpa using IsPoly.mul pf pg) _⟩
| Mathlib/NumberTheory/Dioph.lean | 244 | 248 |
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