Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e... | Mathlib/Data/DList/Defs.lean | 80 | 81 | theorem toList_cons (x : α) (l : DList α) : toList (cons x l) = x :: toList l := by |
cases l; simp
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Limits.Shapes.Countable
import Mathlib.Topology.Category.Profinite.AsLimit
import Mathlib.Topology.Category.Profinite.CofilteredLimi... | Mathlib/Topology/Category/LightProfinite/Basic.lean | 322 | 335 | theorem mono_iff_injective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
Mono f ↔ Function.Injective f := by |
constructor
· intro hf x₁ x₂ h
let g₁ : of PUnit.{u+1} ⟶ X := ⟨fun _ => x₁, continuous_const⟩
let g₂ : of PUnit.{u+1} ⟶ X := ⟨fun _ => x₂, continuous_const⟩
have : g₁ ≫ f = g₂ ≫ f := by
ext
exact h
rw [cancel_mono] at this
apply_fun fun e => e PUnit.unit at this
exact this
· r... |
/-
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.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.wi... | Mathlib/RingTheory/WittVector/IsPoly.lean | 125 | 133 | theorem poly_eq_of_wittPolynomial_bind_eq [Fact p.Prime] (f g : ℕ → MvPolynomial ℕ ℤ)
(h : ∀ n, bind₁ f (wittPolynomial p _ n) = bind₁ g (wittPolynomial p _ n)) : f = g := by |
ext1 n
apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective
rw [← Function.funext_iff] at h
replace h :=
congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
bind₁_wi... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 132 | 134 | theorem isHomogeneous_C (r : R) : IsHomogeneous (C r : MvPolynomial σ R) 0 := by |
apply isHomogeneous_monomial
simp only [degree, Finsupp.zero_apply, Finset.sum_const_zero]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysi... | Mathlib/Analysis/Convex/Caratheodory.lean | 129 | 143 | theorem affineIndependent_minCardFinsetOfMemConvexHull :
AffineIndependent 𝕜 ((↑) : minCardFinsetOfMemConvexHull hx → E) := by |
let k := (minCardFinsetOfMemConvexHull hx).card - 1
have hk : (minCardFinsetOfMemConvexHull hx).card = k + 1 :=
(Nat.succ_pred_eq_of_pos (Finset.card_pos.mpr (minCardFinsetOfMemConvexHull_nonempty hx))).symm
classical
by_contra h
obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull ... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import... | Mathlib/Topology/List.lean | 70 | 71 | theorem nhds_nil : 𝓝 ([] : List α) = pure [] := by |
rw [nhds_list, List.traverse_nil _]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,463 | 2,464 | theorem mem_symmDiff : a ∈ s ∆ t ↔ a ∈ s ∧ a ∉ t ∨ a ∈ t ∧ a ∉ s := by |
simp_rw [symmDiff, sup_eq_union, mem_union, mem_sdiff]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Ring.Basic... | Mathlib/Algebra/Ring/Hom/Defs.lean | 573 | 576 | theorem map_ite_one_zero {F : Type*} [FunLike F α β] [RingHomClass F α β] (f : F)
(p : Prop) [Decidable p] :
f (ite p 1 0) = ite p 1 0 := by |
split_ifs with h <;> simp [h]
|
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Topology.Algebra.Order.Liminf... | Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 220 | 236 | theorem superpolynomialDecay_iff_abs_isBoundedUnder (hk : Tendsto k l atTop) :
SuperpolynomialDecay l k f ↔
∀ z : ℕ, IsBoundedUnder (· ≤ ·) l fun a : α => |k a ^ z * f a| := by |
refine
⟨fun h z => Tendsto.isBoundedUnder_le (Tendsto.abs (h z)), fun h =>
(superpolynomialDecay_iff_abs_tendsto_zero l k f).2 fun z => ?_⟩
obtain ⟨m, hm⟩ := h (z + 1)
have h1 : Tendsto (fun _ : α => (0 : β)) l (𝓝 0) := tendsto_const_nhds
have h2 : Tendsto (fun a : α => |(k a)⁻¹| * m) l (𝓝 0) :=
... |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 96 | 119 | theorem charpoly_degree_eq_dim [Nontrivial R] (M : Matrix n n R) :
M.charpoly.degree = Fintype.card n := by |
by_cases h : Fintype.card n = 0
· rw [h]
unfold charpoly
rw [det_of_card_zero]
· simp
· assumption
rw [← sub_add_cancel M.charpoly (∏ i : n, (X - C (M i i)))]
-- Porting note: added `↑` in front of `Fintype.card n`
have h1 : (∏ i : n, (X - C (M i i))).degree = ↑(Fintype.card n) := by
rw [... |
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 191 | 194 | theorem mem_treesOfNumNodesEq {x : Tree Unit} {n : ℕ} :
x ∈ treesOfNumNodesEq n ↔ x.numNodes = n := by |
induction x using Tree.unitRecOn generalizing n <;> cases n <;>
simp [treesOfNumNodesEq_succ, Nat.succ_eq_add_one, *]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
Coinductive formalization of unbounded computations.
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "le... | Mathlib/Data/Seq/Computation.lean | 1,163 | 1,164 | theorem liftRel_pure_right (R : α → β → Prop) (ca : Computation α) (b : β) :
LiftRel R ca (pure b) ↔ ∃ a, a ∈ ca ∧ R a b := by | rw [LiftRel.swap, liftRel_pure_left]
|
/-
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.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 814 | 822 | theorem mem_nhds_induced [T : TopologicalSpace α] (f : β → α) (a : β) (s : Set β) :
s ∈ @nhds β (TopologicalSpace.induced f T) a ↔ ∃ u ∈ 𝓝 (f a), f ⁻¹' u ⊆ s := by |
letI := T.induced f
simp_rw [mem_nhds_iff, isOpen_induced_iff]
constructor
· rintro ⟨u, usub, ⟨v, openv, rfl⟩, au⟩
exact ⟨v, ⟨v, Subset.rfl, openv, au⟩, usub⟩
· rintro ⟨u, ⟨v, vsubu, openv, amem⟩, finvsub⟩
exact ⟨f ⁻¹' v, (Set.preimage_mono vsubu).trans finvsub, ⟨⟨v, openv, rfl⟩, amem⟩⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,544 | 1,545 | theorem Perm.extendDomain_refl : Perm.extendDomain (Equiv.refl _) f = Equiv.refl _ := by |
simp [Perm.extendDomain]
|
/-
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.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 120 | 120 | theorem nsmul (n : ℕ) (x : WittVector p R) : mapFun f (n • x) = n • mapFun f x := by | map_fun_tac
|
/-
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.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,325 | 2,333 | theorem prod_subtype_mul_prod_subtype {α β : Type*} [Fintype α] [CommMonoid β] (p : α → Prop)
(f : α → β) [DecidablePred p] :
(∏ i : { x // p x }, f i) * ∏ i : { x // ¬p x }, f i = ∏ i, f i := by |
classical
let s := { x | p x }.toFinset
rw [← Finset.prod_subtype s, ← Finset.prod_subtype sᶜ]
· exact Finset.prod_mul_prod_compl _ _
· simp [s]
· simp [s]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Compactness.LocallyCompact
/-!
# Sigma-compactness in topological spaces
## Main definitions
* `IsSigmaCompa... | Mathlib/Topology/Compactness/SigmaCompact.lean | 205 | 207 | theorem iUnion_compactCovering : ⋃ n, compactCovering X n = univ := by |
rw [compactCovering, iUnion_accumulate]
exact (Classical.choose_spec SigmaCompactSpace.exists_compact_covering).2
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 140 | 155 | theorem mellin_comp_mul_left (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (fun t => f (a * t)) s = (a : ℂ) ^ (-s) • mellin f s := by |
simp_rw [mellin]
have : EqOn (fun t : ℝ => (t : ℂ) ^ (s - 1) • f (a * t))
(fun t : ℝ => (a : ℂ) ^ (1 - s) • (fun u : ℝ => (u : ℂ) ^ (s - 1) • f u) (a * t))
(Ioi 0) := fun t ht ↦ by
dsimp only
rw [ofReal_mul, mul_cpow_ofReal_nonneg ha.le (le_of_lt ht), ← mul_smul,
(by ring : 1 - s = -(s ... |
/-
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.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
#align_impor... | Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 238 | 241 | theorem summable_sum_mul_range_of_summable_mul (h : Summable fun x : ℕ × ℕ ↦ f x.1 * g x.2) :
Summable fun n ↦ ∑ k ∈ range (n + 1), f k * g (n - k) := by |
simp_rw [← Nat.sum_antidiagonal_eq_sum_range_succ fun k l ↦ f k * g l]
exact summable_sum_mul_antidiagonal_of_summable_mul h
|
/-
Copyright (c) 2023 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.AdicValuation
import Mathlib.RingTheory.DedekindDomain.Factorization
import Mathlib.Algebra.O... | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | 221 | 231 | theorem zero : (0 : K_hat R K).IsFiniteAdele := by |
rw [IsFiniteAdele, Filter.eventually_cofinite]
have h_empty :
{v : HeightOneSpectrum R | ¬(0 : v.adicCompletion K) ∈ v.adicCompletionIntegers K} = ∅ := by
ext v; rw [mem_empty_iff_false, iff_false_iff]; intro hv
rw [mem_setOf] at hv; apply hv; rw [mem_adicCompletionIntegers]
have h_zero : (Valued.v... |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.CategoryTheory.Category.Basic
import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Tactic.NthRewrite
imp... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 174 | 186 | theorem lift_unique (φ : V ⥤q V') (Φ : FreeGroupoid V ⥤ V') (hΦ : of V ⋙q Φ.toPrefunctor = φ) :
Φ = lift φ := by |
apply Quotient.lift_unique
apply Paths.lift_unique
fapply @Quiver.Symmetrify.lift_unique _ _ _ _ _ _ _ _ _
· rw [← Functor.toPrefunctor_comp]
exact hΦ
· rintro X Y f
simp only [← Functor.toPrefunctor_comp, Prefunctor.comp_map, Paths.of_map, inv_eq_inv]
change Φ.map (inv ((Quotient.functor redStep... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Bases
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Co... | Mathlib/Topology/Sets/Opens.lean | 316 | 329 | theorem isBasis_iff_cover {B : Set (Opens α)} :
IsBasis B ↔ ∀ U : Opens α, ∃ Us, Us ⊆ B ∧ U = sSup Us := by |
constructor
· intro hB U
refine ⟨{ V : Opens α | V ∈ B ∧ V ≤ U }, fun U hU => hU.left, ext ?_⟩
rw [coe_sSup, hB.open_eq_sUnion' U.isOpen]
simp_rw [sUnion_eq_biUnion, iUnion, mem_setOf_eq, iSup_and, iSup_image]
rfl
· intro h
rw [isBasis_iff_nbhd]
intro U x hx
rcases h U with ⟨Us, hUs, ... |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 79 | 80 | theorem one_sub_gold : 1 - ψ = φ := by |
linarith [gold_add_goldConj]
|
/-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.GroupTheory.OrderOfElement
/-!
# Fixed-point-free automorphi... | Mathlib/GroupTheory/FixedPointFree.lean | 69 | 71 | theorem commute_all_of_involutive (g h : G) : Commute g h := by |
have key := map_mul φ g h
rwa [hφ.coe_eq_inv_of_involutive h2, inv_eq_iff_eq_inv, mul_inv_rev, inv_inv, inv_inv] at key
|
/-
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.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 865 | 866 | theorem exp_sub : exp (x - y) = exp x / exp y := by |
simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]
|
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Mo... | Mathlib/Data/Nat/Nth.lean | 137 | 138 | theorem nth_apply_eq_orderIsoOfNat (hf : (setOf p).Infinite) (n : ℕ) :
nth p n = @Nat.Subtype.orderIsoOfNat (setOf p) hf.to_subtype n := by | rw [nth, dif_neg hf]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87... | Mathlib/Algebra/Group/Indicator.lean | 232 | 234 | theorem mulIndicator_inter_mulSupport (s : Set α) (f : α → M) :
mulIndicator (s ∩ mulSupport f) f = mulIndicator s f := by |
rw [← mulIndicator_mulIndicator, mulIndicator_mulSupport]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "lean... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 264 | 266 | theorem toMeasure_apply_singleton (a : α) (h : MeasurableSet ({a} : Set α)) :
p.toMeasure {a} = p a := by |
simp [toMeasure_apply_eq_toOuterMeasure_apply _ _ h, toOuterMeasure_apply_singleton]
|
/-
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
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 413 | 421 | theorem diam_thickening_le {α : Type*} [PseudoMetricSpace α] (s : Set α) (hε : 0 ≤ ε) :
diam (thickening ε s) ≤ diam s + 2 * ε := by |
by_cases hs : IsBounded s
· exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans
(diam_cthickening_le _ hε)
obtain rfl | hε := hε.eq_or_lt
· simp [thickening_of_nonpos, diam_nonneg]
· rw [diam_eq_zero_of_unbounded (mt (IsBounded.subset · <| self_subset_thickening hε _) hs)]
po... |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint f... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 498 | 501 | theorem im_inner_adjoint_mul_self_eq_zero (T : E →ₗ[𝕜] E) (x : E) :
im ⟪x, LinearMap.adjoint T (T x)⟫ = 0 := by |
simp only [mul_apply, adjoint_inner_right, inner_self_eq_norm_sq_to_K]
norm_cast
|
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 139 | 144 | theorem hausdorffMeasure_of_dimH_lt {s : Set X} {d : ℝ≥0} (h : dimH s < d) : μH[d] s = 0 := by |
rw [dimH_def] at h
rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with ⟨d', hsd', hd'd⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at hd'd
exact (hausdorffMeasure_zero_or_top hd'd s).resolve_right fun h₂ => hsd'.not_le <|
le_iSup₂ (α := ℝ≥0∞) d' h₂
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Init.Align
import Mathlib.Data.Fintype.Order
import Mathlib.Algebra.DirectLimit
import Mathlib.ModelTheory.Quotients
import Mathlib.ModelT... | Mathlib/ModelTheory/DirectLimit.lean | 514 | 519 | theorem Equiv_isup_symm_inclusion_apply {i : ι} (x : S i) :
(Equiv_iSup S).symm (Substructure.inclusion (le_iSup _ _) x)
= of L ι _ (fun _ _ h ↦ Substructure.inclusion (S.monotone h)) i x := by |
apply (Equiv_iSup S).injective
simp only [Equiv.apply_symm_apply]
rfl
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 1,078 | 1,080 | theorem two_lt_ncard (hs : s.Finite := by | toFinite_tac) :
2 < s.ncard ↔ ∃ a ∈ s, ∃ b ∈ s, ∃ c ∈ s, a ≠ b ∧ a ≠ c ∧ b ≠ c := by
simp only [two_lt_ncard_iff hs, exists_and_left, exists_prop]
|
/-
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.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,603 | 1,607 | theorem abs_sub_round (x : α) : |x - round x| ≤ 1 / 2 := by |
rw [round_eq, abs_sub_le_iff]
have := floor_le (x + 1 / 2)
have := lt_floor_add_one (x + 1 / 2)
constructor <;> linarith
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 370 | 375 | theorem measure_le_le_exp_cgf [IsFiniteMeasure μ] (ε : ℝ) (ht : t ≤ 0)
(h_int : Integrable (fun ω => exp (t * X ω)) μ) :
(μ {ω | X ω ≤ ε}).toReal ≤ exp (-t * ε + cgf X μ t) := by |
refine (measure_le_le_exp_mul_mgf ε ht h_int).trans ?_
rw [exp_add]
exact mul_le_mul le_rfl (le_exp_log _) mgf_nonneg (exp_pos _).le
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_imp... | Mathlib/Data/Finset/Image.lean | 861 | 863 | theorem finsetCongr_refl : (Equiv.refl α).finsetCongr = Equiv.refl _ := by |
ext
simp
|
/-
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... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 571 | 601 | theorem Gamma_mul_Gamma_add_half (s : ℂ) :
Gamma s * Gamma (s + 1 / 2) = Gamma (2 * s) * (2 : ℂ) ^ (1 - 2 * s) * ↑(√π) := by |
suffices (fun z => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) = fun z =>
(Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(√π) by
convert congr_arg Inv.inv (congr_fun this s) using 1
· rw [mul_inv, inv_inv, inv_inv]
· rw [div_eq_mul_inv, mul_inv, mul_inv, inv_inv, inv_inv, ← cpow_neg, neg_sub]
have h1 : An... |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Jujian Zhang
-/
import Mathlib.Algebra.Polynomial.DenomsClearable
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/Liouville/Basic.lean | 123 | 173 | theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : ℤ[X]} (f0 : f ≠ 0)
(fa : eval α (map (algebraMap ℤ ℝ) f) = 0) :
∃ A : ℝ, 0 < A ∧ ∀ a : ℤ, ∀ b : ℕ,
(1 : ℝ) ≤ ((b : ℝ) + 1) ^ f.natDegree * (|α - a / (b + 1)| * A) := by |
-- `fR` is `f` viewed as a polynomial with `ℝ` coefficients.
set fR : ℝ[X] := map (algebraMap ℤ ℝ) f
-- `fR` is non-zero, since `f` is non-zero.
obtain fR0 : fR ≠ 0 := fun fR0 =>
(map_injective (algebraMap ℤ ℝ) fun _ _ A => Int.cast_inj.mp A).ne f0
(fR0.trans (Polynomial.map_zero _).symm)
-- reform... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
/-!
# Lemmas a... | Mathlib/Data/Int/Order/Units.lean | 49 | 49 | theorem neg_one_pow_ne_zero {n : ℕ} : (-1 : ℤ) ^ n ≠ 0 := by | simp
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 174 | 182 | theorem ofEven_ι (x y : M × R) :
ofEven Q ((even.ι (Q' Q)).bilin x y) =
(ι Q x.1 + algebraMap R _ x.2) * (ι Q y.1 - algebraMap R _ y.2) := by |
-- Porting note: entire proof was the term-mode `even.lift_ι (Q' Q) _ x y`
unfold ofEven
lift_lets
intro f
-- TODO: replacing `?_` with `_` takes way longer?
exact @even.lift_ι R (M × R) _ _ _ (Q' Q) _ _ _ ⟨f, ?_, ?_⟩ x y
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,834 | 1,839 | theorem aestronglyMeasurable_uIoc_iff [LinearOrder α] [PseudoMetrizableSpace β] {f : α → β}
{a b : α} :
AEStronglyMeasurable f (μ.restrict <| uIoc a b) ↔
AEStronglyMeasurable f (μ.restrict <| Ioc a b) ∧
AEStronglyMeasurable f (μ.restrict <| Ioc b a) := by |
rw [uIoc_eq_union, aestronglyMeasurable_union_iff]
|
/-
Copyright (c) 2023 Kim Liesinger. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Liesinger
-/
import Mathlib.Algebra.Group.Defs
/-!
# Levenshtein distances
We define the Levenshtein edit distance `levenshtein C xy ys` between two `List α`,
with a customizable ... | Mathlib/Data/List/EditDistance/Defs.lean | 226 | 239 | theorem suffixLevenshtein_cons₁
(x : α) (xs ys) :
suffixLevenshtein C (x :: xs) ys =
⟨levenshtein C (x :: xs) ys ::
(suffixLevenshtein C xs ys).1, by simp⟩ := by |
induction ys with
| nil =>
dsimp [levenshtein, suffixLevenshtein]
| cons y ys ih =>
apply suffixLevenshtein_cons₁_aux
· rfl
· rw [suffixLevenshtein_cons₂ (x :: xs), ih, impl_cons]
· rfl
· simp [suffixLevenshtein_length]
|
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd29... | Mathlib/CategoryTheory/EqToHom.lean | 276 | 279 | theorem postcomp_map_heq (H : D ⥤ E) (hx : F.obj X = G.obj X) (hy : F.obj Y = G.obj Y)
(hmap : HEq (F.map f) (G.map f)) : HEq ((F ⋙ H).map f) ((G ⋙ H).map f) := by |
dsimp
congr
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 385 | 387 | theorem tendstoUniformly_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι} :
TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x, edist (f x) (F n x) < ε := by |
simp only [← tendstoUniformlyOn_univ, tendstoUniformlyOn_iff, mem_univ, forall_const]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... | Mathlib/Data/Nat/Fib/Basic.lean | 204 | 205 | theorem fib_bit1 (n : ℕ) : fib (bit1 n) = fib (n + 1) ^ 2 + fib n ^ 2 := by |
rw [Nat.bit1_eq_succ_bit0, bit0_eq_two_mul, fib_two_mul_add_one]
|
/-
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.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 172 | 175 | theorem _root_.Commute.exp_right [T2Space 𝔸] {x y : 𝔸} (h : Commute x y) :
Commute x (exp 𝕂 y) := by |
rw [exp_eq_tsum]
exact Commute.tsum_right x fun n => (h.pow_right n).smul_right _
|
/-
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.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb... | Mathlib/Algebra/CharP/Two.lean | 33 | 33 | theorem add_self_eq_zero (x : R) : x + x = 0 := by | rw [← two_smul R x, two_eq_zero, zero_smul]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 210 | 212 | theorem iterate_succ_toZ (i : ι) (hi : i0 ≤ i) : succ^[(toZ i0 i).toNat] i0 = i := by |
rw [toZ_of_ge hi, Int.toNat_natCast]
exact Nat.find_spec (exists_succ_iterate_of_le hi)
|
/-
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.Data.List.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 235 | 241 | theorem map_permutationsAux (f : α → β) :
∀ ts is :
List α, map (map f) (permutationsAux ts is) = permutationsAux (map f ts) (map f is) := by |
refine permutationsAux.rec (by simp) ?_
introv IH1 IH2; rw [map] at IH2
simp only [foldr_permutationsAux2, map_append, map, map_map_permutationsAux2, permutations,
bind_map, IH1, append_assoc, permutationsAux_cons, cons_bind, ← IH2, map_bind]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure fr... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 211 | 214 | theorem Integrable.withDensityᵥ_trim_eq_integral {m m0 : MeasurableSpace α} {μ : Measure α}
(hm : m ≤ m0) {f : α → ℝ} (hf : Integrable f μ) {i : Set α} (hi : MeasurableSet[m] i) :
(μ.withDensityᵥ f).trim hm i = ∫ x in i, f x ∂μ := by |
rw [VectorMeasure.trim_measurableSet_eq hm hi, withDensityᵥ_apply hf (hm _ hi)]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Si... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 370 | 373 | theorem IsCyclic.image_range_orderOf (ha : ∀ x : α, x ∈ zpowers a) :
Finset.image (fun i => a ^ i) (range (orderOf a)) = univ := by |
simp_rw [← SetLike.mem_coe] at ha
simp only [_root_.image_range_orderOf, Set.eq_univ_iff_forall.mpr ha, Set.toFinset_univ]
|
/-
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.LinearAlgebra.Dual
/-!
# Perfect pairings of modules
A perfect pairing of two (left) modules may be defined either as:
1. A bilinear map `M × N → R` such ... | Mathlib/LinearAlgebra/PerfectPairing.lean | 71 | 74 | theorem apply_toDualLeft_symm_apply (f : Dual R N) (x : N) : p (p.toDualLeft.symm f) x = f x := by |
have h := LinearEquiv.apply_symm_apply p.toDualLeft f
rw [toDualLeft_apply] at h
exact congrFun (congrArg DFunLike.coe h) x
|
/-
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.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 437 | 439 | theorem Nat.sq_add_sq_modEq (p : ℕ) [Fact p.Prime] (x : ℕ) :
∃ a b : ℕ, a ≤ p / 2 ∧ b ≤ p / 2 ∧ a ^ 2 + b ^ 2 ≡ x [MOD p] := by |
simpa only [← Int.natCast_modEq_iff] using Nat.sq_add_sq_zmodEq p x
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | Mathlib/Topology/UniformSpace/Ascoli.lean | 85 | 125 | theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
(UniformFun.uniformSpace X α).comap F =
(Pi.uniformSpace _).comap F := by |
-- The `≤` inequality is trivial
refine le_antisymm (UniformSpace.comap_mono UniformFun.uniformContinuous_toFun) ?_
-- A bit of rewriting to get a nice intermediate statement.
change comap _ _ ≤ comap _ _
simp_rw [Pi.uniformity, Filter.comap_iInf, comap_comap, Function.comp]
refine ((UniformFun.hasBasis_un... |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 348 | 354 | theorem zero_eq [Nontrivial A] : σ (0 : A) = {0} := by |
refine Set.Subset.antisymm ?_ (by simp [Algebra.algebraMap_eq_smul_one, mem_iff])
rw [spectrum, Set.compl_subset_comm]
intro k hk
rw [Set.mem_compl_singleton_iff] at hk
have : IsUnit (Units.mk0 k hk • (1 : A)) := IsUnit.smul (Units.mk0 k hk) isUnit_one
simpa [mem_resolventSet_iff, Algebra.algebraMap_eq_smu... |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 67 | 78 | theorem trinomial_natDegree (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).natDegree = n := by |
refine
natDegree_eq_of_degree_eq_some
((Finset.sup_le fun i h => ?_).antisymm <|
le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])
replace h := support_trinomial' k m n u v w h
rw [mem_insert, mem_insert, mem_singleton] at h
rcases h with (rfl | rfl | rfl)
· exact WithBot.c... |
/-
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
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 96 | 96 | theorem Bool.symmDiff_eq_xor : ∀ p q : Bool, p ∆ q = xor p q := by | decide
|
/-
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.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/m... | Mathlib/Topology/Order/LeftRightLim.lean | 163 | 174 | theorem rightLim_le_leftLim (h : x < y) : rightLim f x ≤ leftLim f y := by |
letI : TopologicalSpace α := Preorder.topology α
haveI : OrderTopology α := ⟨rfl⟩
rcases eq_or_ne (𝓝[<] y) ⊥ with (h' | h')
· simp [leftLim, h']
exact rightLim_le hf h
obtain ⟨a, ⟨xa, ay⟩⟩ : (Ioo x y).Nonempty :=
forall_mem_nonempty_iff_neBot.2 (neBot_iff.2 h') (Ioo x y)
(Ioo_mem_nhdsWithin_Ii... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 206 | 207 | theorem nhdsWithin_eq_nhdsWithin' {a : α} {s t u : Set α} (hs : s ∈ 𝓝 a) (h₂ : t ∩ s = u ∩ s) :
𝓝[t] a = 𝓝[u] a := by | rw [nhdsWithin_restrict' t hs, nhdsWithin_restrict' u hs, h₂]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 766 | 770 | theorem eleven_dvd_iff :
11 ∣ n ↔ (11 : ℤ) ∣ ((digits 10 n).map fun n : ℕ => (n : ℤ)).alternatingSum := by |
have t := dvd_iff_dvd_ofDigits 11 10 (-1 : ℤ) (by norm_num) n
rw [ofDigits_neg_one] at t
exact t
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 760 | 761 | theorem withTopEquiv_zero : withTopEquiv 0 = 0 := by |
simp
|
/-
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.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Order.Ring.Finset
import Mat... | Mathlib/Analysis/Normed/Field/Basic.lean | 422 | 423 | theorem norm_pow_le' (a : α) {n : ℕ} (h : 0 < n) : ‖a ^ n‖ ≤ ‖a‖ ^ n := by |
simpa only [NNReal.coe_pow, coe_nnnorm] using NNReal.coe_mono (nnnorm_pow_le' a h)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 249 | 252 | theorem LinearEquiv.dualMap_refl :
(LinearEquiv.refl R M₁).dualMap = LinearEquiv.refl R (Dual R M₁) := by |
ext
rfl
|
/-
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.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import pr... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 242 | 245 | theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
MeasurableSet {x | Integrable (f x) (κ x)} := by |
simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]
exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.ennnorm) measurable_const
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.List.Sigma
import Mathlib.Data.Int.Range
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.Testing.SlimCheck.Sample... | Mathlib/Testing/SlimCheck/Functions.lean | 335 | 366 | theorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs) (h₁ : xs ~ ys)
(x : α) : List.applyId.{u} (xs.zip ys) x ∈ ys ↔ x ∈ xs := by |
simp only [List.applyId]
cases h₃ : List.dlookup x (List.map Prod.toSigma (xs.zip ys)) with
| none =>
dsimp [Option.getD]
rw [h₁.mem_iff]
| some val =>
have h₂ : ys.Nodup := h₁.nodup_iff.1 h₀
replace h₁ : xs.length = ys.length := h₁.length_eq
dsimp
induction xs generalizing ys with
... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 368 | 371 | theorem coeff_succ_mul_X (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (φ * X) = coeff R n φ := by |
simp only [coeff, Finsupp.single_add]
convert φ.coeff_add_mul_monomial (single () n) (single () 1) _
rw [mul_one]; rfl
|
/-
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.Polynomial.Expand
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
import Mathlib.RingTheory.Adjoin.... | Mathlib/RingTheory/IntegralClosure.lean | 425 | 428 | theorem IsIntegral.of_pow {x : B} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
IsIntegral R x := by |
rcases hx with ⟨p, hmonic, heval⟩
exact ⟨expand R n p, hmonic.expand hn, by rwa [← aeval_def, expand_aeval]⟩
|
/-
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.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.m... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 300 | 301 | theorem zpow_sub {A : M} (ha : IsUnit A.det) (z1 z2 : ℤ) : A ^ (z1 - z2) = A ^ z1 / A ^ z2 := by |
rw [sub_eq_add_neg, zpow_add ha, zpow_neg ha, div_eq_mul_inv]
|
/-
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.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,207 | 1,213 | theorem idealRange_eq_top_of_surjective (h : Function.Surjective f) : f.idealRange = ⊤ := by |
rw [← f.range_eq_top] at h
rw [idealRange_eq_lieSpan_range, h, ← LieSubalgebra.coe_to_submodule, ←
LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.top_coeSubmodule,
LieSubalgebra.top_coe_submodule, LieSubmodule.coe_lieSpan_submodule_eq_iff]
use ⊤
exact LieSubmodule.top_coeSubmodule
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | Mathlib/Control/Fold.lean | 396 | 404 | theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by |
unfold length
rw [foldl_toList]
generalize toList xs = ys
rw [← Nat.add_zero ys.length]
generalize 0 = n
induction' ys with _ _ ih generalizing n
· simp
· simp_arith [ih]
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Heather Macbeth
-/
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.algebra.order.monotone_continuity from... | Mathlib/Topology/Order/MonotoneContinuity.lean | 63 | 75 | theorem continuousWithinAt_right_of_monotoneOn_of_exists_between {f : α → β} {s : Set α} {a : α}
(h_mono : MonotoneOn f s) (hs : s ∈ 𝓝[≥] a) (hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b) :
ContinuousWithinAt f (Ici a) a := by |
have ha : a ∈ Ici a := left_mem_Ici
have has : a ∈ s := mem_of_mem_nhdsWithin ha hs
refine tendsto_order.2 ⟨fun b hb => ?_, fun b hb => ?_⟩
· filter_upwards [hs, @self_mem_nhdsWithin _ _ a (Ici a)] with _ hxs hxa using hb.trans_le
(h_mono has hxs hxa)
· rcases hfs b hb with ⟨c, hcs, hac, hcb⟩
have ... |
/-
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.BigOperators.Group.Finset
import Mathlib.Order.SupIndep
import Mathlib.Order.Atoms
#align_import order.partition.finpa... | Mathlib/Order/Partition/Finpartition.lean | 483 | 485 | theorem exists_mem (ha : a ∈ s) : ∃ t ∈ P.parts, a ∈ t := by |
simp_rw [← P.sup_parts] at ha
exact mem_sup.1 ha
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Stro... | Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean | 292 | 300 | theorem opNorm_le_of_shell' {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 < ε) (hC : 0 ≤ C) {c : 𝕜}
(hc : ‖c‖ < 1) (hf : ∀ x, ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) : ‖f‖ ≤ C := by |
by_cases h0 : c = 0
· refine opNorm_le_of_ball ε_pos hC fun x hx => hf x ?_ ?_
· simp [h0]
· rwa [ball_zero_eq] at hx
· rw [← inv_inv c, norm_inv, inv_lt_one_iff_of_pos (norm_pos_iff.2 <| inv_ne_zero h0)] at hc
refine opNorm_le_of_shell ε_pos hC hc ?_
rwa [norm_inv, div_eq_mul_inv, inv_inv]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 359 | 373 | theorem realize_mapTermRel_id [L'.Structure M]
{ft : ∀ n, L.Term (Sum α (Fin n)) → L'.Term (Sum β (Fin n))}
{fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n} {v : α → M}
{v' : β → M} {xs : Fin n → M}
(h1 :
∀ (n) (t : L.Term (Sum α (Fin n))) (xs : Fin n → M),
(ft n t... |
induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih
· rfl
· simp [mapTermRel, Realize, h1]
· simp [mapTermRel, Realize, h1, h2]
· simp [mapTermRel, Realize, ih1, ih2]
· simp only [mapTermRel, Realize, ih, id]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
/-!
# Counting in lists
T... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 81 | 82 | theorem countP_eq_length : countP p l = l.length ↔ ∀ a ∈ l, p a := by |
rw [countP_eq_length_filter, filter_length_eq_length]
|
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
impor... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 233 | 236 | theorem coeFn_compQuasiMeasurePreserving (g : β →ₘ[ν] γ) (hf : QuasiMeasurePreserving f μ ν) :
g.compQuasiMeasurePreserving f hf =ᵐ[μ] g ∘ f := by |
rw [compQuasiMeasurePreserving_eq_mk]
apply coeFn_mk
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 452 | 465 | theorem support_swap_mul_swap {x y z : α} (h : List.Nodup [x, y, z]) :
support (swap x y * swap y z) = {x, y, z} := by |
simp only [List.not_mem_nil, and_true_iff, List.mem_cons, not_false_iff, List.nodup_cons,
List.mem_singleton, and_self_iff, List.nodup_nil] at h
push_neg at h
apply le_antisymm
· convert support_mul_le (swap x y) (swap y z) using 1
rw [support_swap h.left.left, support_swap h.right.left]
simp [Fins... |
/-
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.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 118 | 128 | theorem convolution_integrand_bound_right_of_le_of_subset {C : ℝ} (hC : ∀ i, ‖g i‖ ≤ C) {x t : G}
{s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * C) t := by |
-- Porting note: had to add `f := _`
refine le_indicator (f := fun t ↦ ‖L (f t) (g (x - t))‖) (fun t _ => ?_) (fun t ht => ?_) t
· apply_rules [L.le_of_opNorm₂_le_of_le, le_rfl]
· have : x - t ∉ support g := by
refine mt (fun hxt => hu ?_) ht
refine ⟨_, Set.neg_mem_neg.mpr (subset_closure hxt), _, ... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable
#align_import measure_theory.function.cond... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | 185 | 211 | theorem integral_norm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
(hf : StronglyMeasurable f) (hfi : IntegrableOn f s μ) (hg : StronglyMeasurable[m] g)
(hgi : IntegrableOn g s μ)
(hgf : ∀ t, MeasurableSet[m] t → μ t < ∞ → ∫ x in t, g x ∂μ = ∫ x in t, f x ∂μ)
(hs : MeasurableSet[m] s) (... |
rw [integral_norm_eq_pos_sub_neg hgi, integral_norm_eq_pos_sub_neg hfi]
have h_meas_nonneg_g : MeasurableSet[m] {x | 0 ≤ g x} :=
(@stronglyMeasurable_const _ _ m _ _).measurableSet_le hg
have h_meas_nonneg_f : MeasurableSet {x | 0 ≤ f x} :=
stronglyMeasurable_const.measurableSet_le hf
have h_meas_nonpo... |
/-
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, Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
#align_import linear_algebra.direct_sum.tensor_product fr... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 173 | 176 | theorem directSumLeft_symm_lof_tmul (i : ι₁) (x : M₁ i) (y : M₂') :
(directSumLeft R M₁ M₂').symm (DirectSum.lof R _ _ i (x ⊗ₜ[R] y)) =
DirectSum.lof R _ _ i x ⊗ₜ[R] y := by |
rw [LinearEquiv.symm_apply_eq, directSumLeft_tmul_lof]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba5... | Mathlib/Order/Bounded.lean | 384 | 387 | theorem unbounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
Unbounded (· < ·) (s ∩ { b | a < b }) ↔ Unbounded (· < ·) s := by |
rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]
exact bounded_lt_inter_lt a
|
/-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 410 | 412 | theorem toIdeal_iSup₂ {κ : Sort*} {κ' : κ → Sort*} (s : ∀ i, κ' i → HomogeneousIdeal 𝒜) :
(⨆ (i) (j), s i j).toIdeal = ⨆ (i) (j), (s i j).toIdeal := by |
simp_rw [toIdeal_iSup]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 283 | 292 | theorem log_eq_zero {x : ℝ} : log x = 0 ↔ x = 0 ∨ x = 1 ∨ x = -1 := by |
constructor
· intro h
rcases lt_trichotomy x 0 with (x_lt_zero | rfl | x_gt_zero)
· refine Or.inr (Or.inr (neg_eq_iff_eq_neg.mp ?_))
rw [← log_neg_eq_log x] at h
exact eq_one_of_pos_of_log_eq_zero (neg_pos.mpr x_lt_zero) h
· exact Or.inl rfl
· exact Or.inr (Or.inl (eq_one_of_pos_of_log_... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 838 | 865 | theorem nodup_permutations'Aux_iff {s : List α} {x : α} : Nodup (permutations'Aux x s) ↔ x ∉ s := by |
refine ⟨fun h => ?_, nodup_permutations'Aux_of_not_mem _ _⟩
intro H
obtain ⟨k, hk, hk'⟩ := nthLe_of_mem H
rw [nodup_iff_nthLe_inj] at h
refine k.succ_ne_self.symm $ h k (k + 1) ?_ ?_ ?_
· simpa [Nat.lt_succ_iff] using hk.le
· simpa using hk
rw [nthLe_permutations'Aux, nthLe_permutations'Aux]
have hl ... |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 445 | 451 | theorem IsUpperSet.total (hs : IsUpperSet s) (ht : IsUpperSet t) : s ⊆ t ∨ t ⊆ s := by |
by_contra! h
simp_rw [Set.not_subset] at h
obtain ⟨⟨a, has, hat⟩, b, hbt, hbs⟩ := h
obtain hab | hba := le_total a b
· exact hbs (hs hab has)
· exact hat (ht hba hbt)
|
/-
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.Logic.Equiv.TransferInstance
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb... | Mathlib/Algebra/Group/AddChar.lean | 305 | 306 | theorem mulShift_spec' (ψ : AddChar R M) (n : ℕ) (x : R) : mulShift ψ n x = ψ x ^ n := by |
rw [mulShift_apply, ← nsmul_eq_mul, map_nsmul_eq_pow]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 530 | 533 | theorem isIso_of_source_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : X ≅ 0) (j : Y ≅ 0) :
IsIso f := by |
rw [zero_of_source_iso_zero f i]
exact (isIsoZeroEquivIsoZero _ _).invFun ⟨i, j⟩
|
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomi... | Mathlib/Algebra/Polynomial/Taylor.lean | 93 | 94 | theorem taylor_coeff_one : (taylor r f).coeff 1 = f.derivative.eval r := by |
rw [taylor_coeff, hasseDeriv_one]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
/-!
# Lemmas about booleans
These are the lemmas about booleans w... | Mathlib/Init/Data/Bool/Lemmas.lean | 97 | 97 | theorem coe_true : ↑true = True := by | simp
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
i... | Mathlib/MeasureTheory/Constructions/Polish.lean | 621 | 626 | theorem Continuous.map_eq_borel {X Y : Type*} [TopologicalSpace X] [PolishSpace X]
[MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [T0Space Y] [SecondCountableTopology Y]
{f : X → Y} (hf : Continuous f) (hsurj : Surjective f) :
MeasurableSpace.map f ‹MeasurableSpace X› = borel Y := by |
borelize Y
exact hf.measurable.map_measurableSpace_eq hsurj
|
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 1,519 | 1,523 | theorem map_eq_map_iff {f g : α → β} {l : List α} : map f l = map g l ↔ ∀ x ∈ l, f x = g x := by |
refine ⟨?_, map_congr⟩; intro h x hx
rw [mem_iff_get] at hx; rcases hx with ⟨n, hn, rfl⟩
rw [get_map_rev f, get_map_rev g]
congr!
|
/-
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.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 242 | 259 | theorem balancedCore_mem_nhds_zero (hU : U ∈ 𝓝 (0 : E)) : balancedCore 𝕜 U ∈ 𝓝 (0 : E) := by |
-- Getting neighborhoods of the origin for `0 : 𝕜` and `0 : E`
obtain ⟨r, V, hr, hV, hrVU⟩ : ∃ (r : ℝ) (V : Set E),
0 < r ∧ V ∈ 𝓝 (0 : E) ∧ ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U := by
have h : Filter.Tendsto (fun x : 𝕜 × E => x.fst • x.snd) (𝓝 (0, 0)) (𝓝 0) :=
continuous_smul.tendsto... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 1,260 | 1,264 | theorem isUnit_of_associated_mul [CancelCommMonoidWithZero α] {p b : α} (h : Associated (p * b) p)
(hp : p ≠ 0) : IsUnit b := by |
cases' h with a ha
refine isUnit_of_mul_eq_one b a ((mul_right_inj' hp).mp ?_)
rwa [← mul_assoc, mul_one]
|
/-
Copyright (c) 2023 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanp... | Mathlib/RingTheory/Localization/NormTrace.lean | 61 | 69 | theorem Algebra.norm_localization [Module.Free R S] [Module.Finite R S] (a : S) :
Algebra.norm Rₘ (algebraMap S Sₘ a) = algebraMap R Rₘ (Algebra.norm R a) := by |
cases subsingleton_or_nontrivial R
· haveI : Subsingleton Rₘ := Module.subsingleton R Rₘ
simp [eq_iff_true_of_subsingleton]
let b := Module.Free.chooseBasis R S
letI := Classical.decEq (Module.Free.ChooseBasisIndex R S)
rw [Algebra.norm_eq_matrix_det (b.localizationLocalization Rₘ M Sₘ),
Algebra.norm... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 187 | 190 | theorem hexagon_reverse_inv (X Y Z : C) :
(α_ Z X Y).hom ≫ (β_ (X ⊗ Y) Z).inv ≫ (α_ X Y Z).hom =
(β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫ X ◁ (β_ Y Z).inv := by |
simp
|
/-
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.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 701 | 703 | theorem LinearMap.toMatrix_mul (f g : M₁ →ₗ[R] M₁) :
LinearMap.toMatrix v₁ v₁ (f * g) = LinearMap.toMatrix v₁ v₁ f * LinearMap.toMatrix v₁ v₁ g := by |
rw [LinearMap.mul_eq_comp, LinearMap.toMatrix_comp v₁ v₁ v₁ f g]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,097 | 2,098 | theorem compl_mem_kernMap {s : Set β} : sᶜ ∈ kernMap m f ↔ ∃ t, tᶜ ∈ f ∧ m '' t = s := by |
simp_rw [mem_kernMap_iff_compl, compl_compl]
|
/-
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
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 119 | 128 | theorem eq_of_degree_le_of_eval_index_eq (hvs : Set.InjOn v s)
(h_deg_le : f.degree ≤ s.card)
(h_deg_eq : f.degree = g.degree)
(hlc : f.leadingCoeff = g.leadingCoeff)
(h_eval : ∀ i ∈ s, f.eval (v i) = g.eval (v i)) : f = g := by |
rcases eq_or_ne f 0 with rfl | hf
· rwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq
· exact eq_of_degree_sub_lt_of_eval_index_eq s hvs
(lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_le)
h_eval
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathli... | Mathlib/Data/ZMod/Basic.lean | 1,325 | 1,331 | theorem natAbs_valMinAbs_add_le {n : ℕ} (a b : ZMod n) :
(a + b).valMinAbs.natAbs ≤ (a.valMinAbs + b.valMinAbs).natAbs := by |
cases' n with n
· rfl
apply natAbs_min_of_le_div_two n.succ
· simp_rw [Int.cast_add, coe_valMinAbs]
· apply natAbs_valMinAbs_le
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Range
#align_import data.fin.vec_notation from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b... | Mathlib/Data/Fin/VecNotation.lean | 379 | 381 | theorem cons_vec_bit0_eq_alt0 (x : α) (u : Fin n → α) (i : Fin (n + 1)) :
vecCons x u (bit0 i) = vecAlt0 rfl (vecAppend rfl (vecCons x u) (vecCons x u)) i := by |
rw [vecAlt0_vecAppend]; rfl
|
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