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) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Sym.Sym2
import Mathlib.Logic.Relation
#align_import order.game_add from "leanprover-community/mathlib"@"fee218fb033b2fd390c447f8be27754bc9093be9"
/-!
# G... | Mathlib/Order/GameAdd.lean | 201 | 215 | theorem Acc.sym2_gameAdd {a b} (ha : Acc rα a) (hb : Acc rα b) :
Acc (Sym2.GameAdd rα) s(a, b) := by |
induction' ha with a _ iha generalizing b
induction' hb with b hb ihb
refine Acc.intro _ fun s => ?_
induction' s using Sym2.inductionOn with c d
rw [Sym2.GameAdd]
dsimp
rintro ((rc | rd) | (rd | rc))
· exact iha c rc ⟨b, hb⟩
· exact ihb d rd
· rw [Sym2.eq_swap]
exact iha d rd ⟨b, hb⟩
· rw [S... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,683 | 1,685 | theorem norm_setToFun_le_mul_norm' (hT : DominatedFinMeasAdditive μ T C) (f : α →₁[μ] E) :
‖setToFun μ T hT f‖ ≤ max C 0 * ‖f‖ := by |
rw [L1.setToFun_eq_setToL1]; exact L1.norm_setToL1_le_mul_norm' hT f
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#... | Mathlib/Data/Finset/Fold.lean | 88 | 96 | theorem fold_const [hd : Decidable (s = ∅)] (c : β) (h : op c (op b c) = op b c) :
Finset.fold op b (fun _ => c) s = if s = ∅ then b else op b c := by |
classical
induction' s using Finset.induction_on with x s hx IH generalizing hd
· simp
· simp only [Finset.fold_insert hx, IH, if_false, Finset.insert_ne_empty]
split_ifs
· rw [hc.comm]
· exact h
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "lean... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 735 | 736 | theorem Fork.π_comp_hom {s t : Cofork f g} (f : s ⟶ t) : s.π ≫ f.hom = t.π := by |
cases s; cases t; cases f; aesop
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.ConcreteCategory.Bun... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean | 258 | 266 | theorem iso_isometry {V W : SemiNormedGroupCat₁} (i : V ≅ W) : Isometry i.hom := by |
change Isometry (⟨⟨i.hom, map_zero _⟩, fun _ _ => map_add _ _ _⟩ : V →+ W)
refine AddMonoidHomClass.isometry_of_norm _ ?_
intro v
apply le_antisymm (i.hom.2 v)
calc
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
‖v‖ = ‖i.inv (i.hom v)‖ := by erw [Iso.hom_inv_id_apply]
_ ≤ ... |
/-
Copyright (c) 2020 Bhavik Mehta, E. W. Ayers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.Mul... | Mathlib/CategoryTheory/Sites/Grothendieck.lean | 286 | 290 | theorem isGLB_sInf (s : Set (GrothendieckTopology C)) : IsGLB s (sInf s) := by |
refine @IsGLB.of_image _ _ _ _ sieves ?_ _ _ ?_
· intros
rfl
· exact _root_.isGLB_sInf _
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 521 | 529 | theorem Set.Finite.convexHull_eq_image {s : Set E} (hs : s.Finite) : convexHull R s =
haveI := hs.fintype
(⇑(∑ x : s, (@LinearMap.proj R s _ (fun _ => R) _ _ x).smulRight x.1)) '' stdSimplex R s := by |
letI := hs.fintype
rw [← convexHull_basis_eq_stdSimplex, LinearMap.image_convexHull, ← Set.range_comp]
apply congr_arg
simp_rw [Function.comp]
convert Subtype.range_coe.symm
simp [LinearMap.sum_apply, ite_smul, Finset.filter_eq, Finset.mem_univ]
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.CategoryTheory.Sites.CoverPreserving
import Mathlib.CategoryTheory.Sites.Sheafification
#align_import cate... | Mathlib/CategoryTheory/Sites/CoverLifting.lean | 263 | 274 | theorem gluedSection_is_unique (y) (hy : x.IsAmalgamation y) : y = gluedSection ℱ hS hx := by |
unfold gluedSection limit.lift
-- Porting note: next line was `ext W`
-- Now `ext` can't see that `ran` is defined as a limit.
-- See https://github.com/leanprover-community/mathlib4/issues/5229
refine limit.hom_ext (fun (W : StructuredArrow (op U) G.op) ↦ ?_)
erw [limit.lift_π]
convert helper ℱ hS hx (�... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,501 | 1,504 | theorem IsClosed.mul_right_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
IsClosed (t * s) := by |
rw [← image_op_smul]
exact IsClosed.smul_left_of_isCompact ht (hs.image continuous_op)
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 630 | 631 | theorem nrComplexPlaces_eq_zero_of_finrank_eq_one (h : finrank ℚ K = 1) :
NrComplexPlaces K = 0 := by | linarith [card_add_two_mul_card_eq_rank K]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,599 | 1,601 | theorem lt_lsub_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} :
a < lsub.{_, v} f ↔ ∃ i, a ≤ f i := by |
simpa only [not_forall, not_lt, not_le] using not_congr (@lsub_le_iff.{_, v} _ f a)
|
/-
Copyright (c) 2022 Cuma Kökmen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Cuma Kökmen, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 176 | 178 | theorem torusIntegral_sub (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) :
(∯ x in T(c, R), f x - g x) = (∯ x in T(c, R), f x) - ∯ x in T(c, R), g x := by |
simpa only [sub_eq_add_neg, ← torusIntegral_neg] using torusIntegral_add hf hg.neg
|
/-
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 | 345 | 359 | theorem isCompactElement_iff (s : Opens α) :
CompleteLattice.IsCompactElement s ↔ IsCompact (s : Set α) := by |
rw [isCompact_iff_finite_subcover, CompleteLattice.isCompactElement_iff]
refine ⟨?_, fun H ι U hU => ?_⟩
· introv H hU hU'
obtain ⟨t, ht⟩ := H ι (fun i => ⟨U i, hU i⟩) (by simpa)
refine ⟨t, Set.Subset.trans ht ?_⟩
rw [coe_finset_sup, Finset.sup_eq_iSup]
rfl
· obtain ⟨t, ht⟩ :=
H (fun i =>... |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.C... | Mathlib/Tactic/CancelDenoms/Core.lean | 81 | 86 | theorem cancel_factors_le {α} [LinearOrderedField α] {a b ad bd a' b' gcd : α}
(ha : ad * a = a') (hb : bd * b = b') (had : 0 < ad) (hbd : 0 < bd) (hgcd : 0 < gcd) :
(a ≤ b) = (1 / gcd * (bd * a') ≤ 1 / gcd * (ad * b')) := by |
rw [mul_le_mul_left, ← ha, ← hb, ← mul_assoc, ← mul_assoc, mul_comm bd, mul_le_mul_left]
· exact mul_pos had hbd
· exact one_div_pos.2 hgcd
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,335 | 1,337 | theorem mk_compl_eq_mk_compl_infinite {α : Type*} [Infinite α] {s t : Set α} (hs : #s < #α)
(ht : #t < #α) : #(sᶜ : Set α) = #(tᶜ : Set α) := by |
rw [mk_compl_of_infinite s hs, mk_compl_of_infinite t ht]
|
/-
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.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.MeasureTheory.Integral.DivergenceTheorem
import Mathlib.MeasureTheory.Integral.CircleIntegral
i... | Mathlib/Analysis/Complex/CauchyIntegral.lean | 295 | 324 | theorem circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable {c : ℂ}
{r R : ℝ} (h0 : 0 < r) (hle : r ≤ R) {f : ℂ → E} {s : Set ℂ} (hs : s.Countable)
(hc : ContinuousOn f (closedBall c R \ ball c r))
(hd : ∀ z ∈ (ball c R \ closedBall c r) \ s, DifferentiableAt ℂ f z) :
(∮ z i... |
/- We apply the previous lemma to `fun z ↦ f (c + exp z)` on the rectangle
`[log r, log R] × [0, 2 * π]`. -/
set A := closedBall c R \ ball c r
obtain ⟨a, rfl⟩ : ∃ a, Real.exp a = r := ⟨Real.log r, Real.exp_log h0⟩
obtain ⟨b, rfl⟩ : ∃ b, Real.exp b = R := ⟨Real.log R, Real.exp_log (h0.trans_le hle)⟩
rw [... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 164 | 166 | theorem union_ae_eq_right : (s ∪ t : Set α) =ᵐ[μ] t ↔ μ (s \ t) = 0 := by |
simp [eventuallyLE_antisymm_iff, ae_le_set, union_diff_right,
diff_eq_empty.2 Set.subset_union_right]
|
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 79 | 82 | theorem HasStrictFDerivAt.cpow (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x)
(h0 : f x ∈ slitPlane) : HasStrictFDerivAt (fun x => f x ^ g x)
((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x := by |
convert (@hasStrictFDerivAt_cpow ((fun x => (f x, g x)) x) h0).comp x (hf.prod hg)
|
/-
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, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 530 | 533 | theorem extraction_of_frequently_atTop {P : ℕ → Prop} (h : ∃ᶠ n in atTop, P n) :
∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, P (φ n) := by |
rw [frequently_atTop'] at h
exact extraction_of_frequently_atTop' h
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "le... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 165 | 168 | theorem _root_.IsPrimitiveRoot.minpoly_dvd_cyclotomic {n : ℕ} {K : Type*} [Field K] {μ : K}
(h : IsPrimitiveRoot μ n) (hpos : 0 < n) [CharZero K] : minpoly ℤ μ ∣ cyclotomic n ℤ := by |
apply minpoly.isIntegrallyClosed_dvd (h.isIntegral hpos)
simpa [aeval_def, eval₂_eq_eval_map, IsRoot.def] using h.isRoot_cyclotomic hpos
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 121 | 122 | theorem BEqCmp.cmp_iff_eq [BEq α] [LawfulBEq α] [BEqCmp (α := α) cmp] : cmp x y = .eq ↔ x = y := by |
simp [BEqCmp.cmp_iff_beq]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 377 | 398 | theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) :
ApproximatesLinearOn (hf.toPartialEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
(N * (N⁻¹ - c)⁻¹ * c) := fun x hx y hy ↦ by
set A := hf.toPartialEquiv hc
have Af : ∀ z, A z = f z := fun z => rfl
rcases ... |
congr 2
simp only [ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.map_sub]
abel
_ ≤ N * (c * ‖y' - x'‖) := mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _)
_ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) := by
gcongr
rw [← dist_eq_norm, ← dist_eq... |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 265 | 269 | theorem ExpChar.neg_one_pow_expChar_pow [Ring R] (q n : ℕ) [hR : ExpChar R q] :
(-1 : R) ^ q ^ n = -1 := by |
cases' hR with _ _ hprime _
· simp only [one_pow, pow_one]
haveI := Fact.mk hprime; exact CharP.neg_one_pow_char_pow R q n
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 624 | 632 | theorem cyclicPermutations_rotate (l : List α) (k : ℕ) :
(l.rotate k).cyclicPermutations = l.cyclicPermutations.rotate k := by |
have : (l.rotate k).cyclicPermutations.length = length (l.cyclicPermutations.rotate k) := by
cases l
· simp
· rw [length_cyclicPermutations_of_ne_nil] <;> simp
refine ext_get this fun n hn hn' => ?_
rw [get_rotate, get_cyclicPermutations, rotate_rotate, ← rotate_mod, Nat.add_comm]
cases l <;> simp
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c89... | Mathlib/Algebra/Regular/SMul.lean | 164 | 167 | theorem pow_iff {n : ℕ} (n0 : 0 < n) : IsSMulRegular M (a ^ n) ↔ IsSMulRegular M a := by |
refine ⟨?_, pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ, ← smul_eq_mul]
exact of_smul _
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 644 | 646 | theorem fromEdgeSet_edgeSet : fromEdgeSet G.edgeSet = G := by |
ext v w
exact ⟨fun h => h.1, fun h => ⟨h, G.ne_of_adj h⟩⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,903 | 1,904 | theorem Ioc_union_Ioc_left : Ioc a c ∪ Ioc b c = Ioc (min a b) c := by |
rw [Ioc_union_Ioc, max_self] <;> exact (min_le_right _ _).trans (le_max_right _ _)
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 621 | 624 | theorem _root_.MeasurableEmbedding.integrable_map_iff {f : α → δ} (hf : MeasurableEmbedding f)
{g : δ → β} : Integrable g (Measure.map f μ) ↔ Integrable (g ∘ f) μ := by |
simp_rw [← memℒp_one_iff_integrable]
exact hf.memℒp_map_measure_iff
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_imp... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 212 | 214 | theorem toComplex_div_re (x y : ℤ[i]) : ((x / y : ℤ[i]) : ℂ).re = round (x / y : ℂ).re := by |
rw [div_def, ← @Rat.round_cast ℝ _ _]
simp [-Rat.round_cast, mul_assoc, div_eq_mul_inv, mul_add, add_mul]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 185 | 207 | theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / Int.gcd x y) := by |
by_cases h0 : Int.gcd x y = 0
· have hx : x = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_left h0
have hy : y = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_right h0
have hz : z = 0 := by
simpa only [PythagoreanTriple, hx, hy, a... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 152 | 154 | theorem pi_div_two_le_two : π / 2 ≤ 2 := by |
rw [Real.pi, mul_div_cancel_left₀ _ (two_ne_zero' ℝ)]
exact (Classical.choose_spec exists_cos_eq_zero).1.2
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 127 | 129 | theorem WithTop.coe_iSup [SupSet α] (f : ι → α) (h : BddAbove (Set.range f)) :
↑(⨆ i, f i) = (⨆ i, f i : WithTop α) := by |
rw [iSup, iSup, WithTop.coe_sSup' h, ← range_comp]; rfl
|
/-
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.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Prefixes, suffixes... | Mathlib/Data/List/Infix.lean | 73 | 76 | theorem prefix_concat_iff {l₁ l₂ : List α} {a : α} :
l₁ <+: l₂ ++ [a] ↔ l₁ = l₂ ++ [a] ∨ l₁ <+: l₂ := by |
simpa only [← reverse_concat', reverse_inj, reverse_suffix] using
suffix_cons_iff (l₁ := l₁.reverse) (l₂ := l₂.reverse)
|
/-
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 Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 223 | 228 | theorem den_div_intCast_eq_one_iff (m n : ℤ) (hn : n ≠ 0) : ((m : ℚ) / n).den = 1 ↔ n ∣ m := by |
replace hn : (n : ℚ) ≠ 0 := num_ne_zero.mp hn
constructor
· rw [Rat.den_eq_one_iff, eq_div_iff hn]
exact mod_cast (Dvd.intro_left _)
· exact (intCast_div _ _ · ▸ rfl)
|
/-
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.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 42 | 46 | theorem bound_of_shell [RingHomIsometric σ₁₂] (f : E →ₛₗ[σ₁₂] F) {ε C : ℝ} (ε_pos : 0 < ε) {c : 𝕜}
(hc : 1 < ‖c‖) (hf : ∀ x, ε / ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖) (x : E) :
‖f x‖ ≤ C * ‖x‖ := by |
by_cases hx : x = 0; · simp [hx]
exact SemilinearMapClass.bound_of_shell_semi_normed f ε_pos hc hf (norm_ne_zero_iff.2 hx)
|
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 103 | 107 | theorem inf_orthogonal_eq_bot : K ⊓ Kᗮ = ⊥ := by |
rw [eq_bot_iff]
intro x
rw [mem_inf]
exact fun ⟨hx, ho⟩ => inner_self_eq_zero.1 (ho x hx)
|
/-
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.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/... | Mathlib/Probability/IdentDistrib.lean | 209 | 221 | theorem snorm_eq [NormedAddCommGroup γ] [OpensMeasurableSpace γ] (h : IdentDistrib f g μ ν)
(p : ℝ≥0∞) : snorm f p μ = snorm g p ν := by |
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp only [h_top, snorm, snormEssSup, ENNReal.top_ne_zero, eq_self_iff_true, if_true, if_false]
apply essSup_eq
exact h.comp (measurable_coe_nnreal_ennreal.comp measurable_nnnorm)
simp only [snorm_eq_snorm' h0 h_top, snorm', one_div]
congr 1
... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 260 | 277 | theorem adjMatrix_pow_apply_eq_card_walk [DecidableEq V] [Semiring α] (n : ℕ) (u v : V) :
(G.adjMatrix α ^ n) u v = Fintype.card { p : G.Walk u v | p.length = n } := by |
rw [card_set_walk_length_eq]
induction' n with n ih generalizing u v
· obtain rfl | h := eq_or_ne u v <;> simp [finsetWalkLength, *]
· simp only [pow_succ', finsetWalkLength, ih, adjMatrix_mul_apply]
rw [Finset.card_biUnion]
· norm_cast
simp only [Nat.cast_sum, card_map, neighborFinset_def]
... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 287 | 287 | theorem lift_root : lift i a h (root f) = a := by | rw [root, lift_mk, eval₂_X]
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 785 | 786 | theorem add_eq_left {a b : Cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) : a + b = a := by |
rw [add_eq_max ha, max_eq_left hb]
|
/-
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.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Ta... | Mathlib/Logic/Relation.lean | 296 | 299 | theorem trans (hab : ReflTransGen r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by |
induction hbc with
| refl => assumption
| tail _ hcd hac => exact hac.tail hcd
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,012 | 1,017 | theorem integral_Iic_of_hasDerivAt_of_tendsto'
(hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x) (f'int : IntegrableOn f' (Iic a))
(hf : Tendsto f atBot (𝓝 m)) : ∫ x in Iic a, f' x = f a - m := by |
refine integral_Iic_of_hasDerivAt_of_tendsto ?_ (fun x hx => hderiv x hx.out.le)
f'int hf
exact (hderiv a right_mem_Iic).continuousAt.continuousWithinAt
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 50 | 64 | theorem norm_cderiv_le (hr : 0 < r) (hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M) :
‖cderiv r f z‖ ≤ M / r := by |
have hM : 0 ≤ M := by
obtain ⟨w, hw⟩ : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
exact (norm_nonneg _).trans (hf w hw)
have h1 : ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2 := by
intro w hw
simp only [mem_sphere_iff_norm, norm_eq_abs] at hw
simp only [norm_smul, i... |
/-
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.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.adhesive from "leanprover-community/mathlib"@"... | Mathlib/CategoryTheory/Adhesive.lean | 59 | 63 | theorem IsPushout.IsVanKampen.flip {H : IsPushout f g h i} (H' : H.IsVanKampen) :
H.flip.IsVanKampen := by |
introv W' hf hg hh hi w
simpa only [IsPushout.flip_iff, IsPullback.flip_iff, and_comm] using
H' g' f' i' h' αW αY αX αZ hg hf hi hh w.flip
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 78 | 79 | theorem eq_of_slope_eq_zero {f : k → PE} {a b : k} (h : slope f a b = (0 : E)) : f a = f b := by |
rw [← sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
#... | Mathlib/Tactic/Ring/Basic.lean | 598 | 598 | theorem pow_atom (a : R) (b) : a ^ b = a ^ b * (nat_lit 1).rawCast + 0 := by | simp
|
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Algebra.Periodic
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.periodi... | Mathlib/Data/Nat/Periodic.lean | 48 | 54 | theorem filter_multiset_Ico_card_eq_of_periodic (n a : ℕ) (p : ℕ → Prop) [DecidablePred p]
(pp : Periodic p a) : card (filter p (Ico n (n + a))) = a.count p := by |
rw [count_eq_card_filter_range, Finset.card, Finset.filter_val, Finset.range_val, ←
multiset_Ico_map_mod n, ← map_count_True_eq_filter_card, ← map_count_True_eq_filter_card,
map_map]
congr; funext n
exact (Function.Periodic.map_mod_nat pp n).symm
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 266 | 267 | theorem comp_coeff_zero'' (q : FormalMultilinearSeries 𝕜 E F) (p : FormalMultilinearSeries 𝕜 E E) :
(q.comp p) 0 = q 0 := by | ext v; exact q.comp_coeff_zero p _ _
|
/-
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.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 213 | 217 | theorem eq_of_forall_toMeasure_apply_eq (μ ν : FiniteMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by |
apply Subtype.ext
ext1 s s_mble
exact h s s_mble
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,063 | 1,063 | theorem Icc_bot : Icc ⊥ a = Iic a := by | simp [← Ici_inter_Iic]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,856 | 1,863 | theorem limsup_preimage_iterate_ae_eq {f : α → α} (hf : QuasiMeasurePreserving f μ μ)
(hs : f ⁻¹' s =ᵐ[μ] s) : limsup (α := Set α) (fun n => (preimage f)^[n] s) atTop =ᵐ[μ] s :=
haveI : ∀ n, (preimage f)^[n] s =ᵐ[μ] s := by |
intro n
induction' n with n ih
· rfl
simpa only [iterate_succ', comp_apply] using ae_eq_trans (hf.ae_eq ih) hs
(limsup_ae_eq_of_forall_ae_eq (fun n => (preimage f)^[n] s) this).trans (ae_eq_refl _)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances... | Mathlib/Topology/Instances/Int.lean | 41 | 43 | theorem pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n := by |
intro m n hne
rw [dist_eq]; norm_cast; rwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontin... | Mathlib/Topology/Semicontinuous.lean | 403 | 406 | theorem ContinuousAt.comp_lowerSemicontinuousAt {g : γ → δ} {f : α → γ} (hg : ContinuousAt g (f x))
(hf : LowerSemicontinuousAt f x) (gmon : Monotone g) : LowerSemicontinuousAt (g ∘ f) x := by |
simp only [← lowerSemicontinuousWithinAt_univ_iff] at hf ⊢
exact hg.comp_lowerSemicontinuousWithinAt hf gmon
|
/-
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.Category.Cat
import Mathlib.CategoryTheory.Elements
#align_import category_theory.grothendieck from "leanprover-community/mathlib"@"14b... | Mathlib/CategoryTheory/Grothendieck.lean | 132 | 136 | theorem congr {X Y : Grothendieck F} {f g : X ⟶ Y} (h : f = g) :
f.fiber = eqToHom (by subst h; rfl) ≫ g.fiber := by |
subst h
dsimp
simp
|
/-
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.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 787 | 792 | theorem UniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ}
(hu : UniformInducing u) :
UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S := by |
have := UniformFun.postcomp_uniformInducing (α := ι) hu
simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff]
rfl
|
/-
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 | 398 | 402 | theorem _root_.Function.Injective.mem_finset_image (hf : Injective f) :
f a ∈ s.image f ↔ a ∈ s := by |
refine ⟨fun h => ?_, Finset.mem_image_of_mem f⟩
obtain ⟨y, hy, heq⟩ := mem_image.1 h
exact hf heq ▸ hy
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 149 | 161 | theorem edist_le (f : α → E) {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s) :
edist (f x) (f y) ≤ eVariationOn f s := by |
wlog hxy : y ≤ x generalizing x y
· rw [edist_comm]
exact this hy hx (le_of_not_le hxy)
let u : ℕ → α := fun n => if n = 0 then y else x
have hu : Monotone u := monotone_nat_of_le_succ fun
| 0 => hxy
| (_ + 1) => le_rfl
have us : ∀ i, u i ∈ s := fun
| 0 => hy
| (_ + 1) => hx
simpa only [Finset.... |
/-
Copyright (c) 2016 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 Batteries.Tactic.Alias
import Batteries.Data.Nat.Basic
/-! # Basic lemmas about natural numbers
The primary purpose ... | .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean | 74 | 79 | theorem recDiag_zero_succ {motive : Nat → Nat → Sort _} (zero_zero : motive 0 0)
(zero_succ : ∀ n, motive 0 n → motive 0 (n+1)) (succ_zero : ∀ m, motive m 0 → motive (m+1) 0)
(succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (n) :
Nat.recDiag zero_zero zero_succ succ_zero succ_succ 0 (n+1)
= zero_s... |
simp [Nat.recDiag]; rfl
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 1,017 | 1,049 | theorem exists_maximal_independent (s : ι → M) :
∃ I : Set ι,
(LinearIndependent R fun x : I => s x) ∧
∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I) := by |
classical
rcases exists_maximal_independent' R s with ⟨I, hIlinind, hImaximal⟩
use I, hIlinind
intro i hi
specialize hImaximal (I ∪ {i}) (by simp)
set J := I ∪ {i} with hJ
have memJ : ∀ {x}, x ∈ J ↔ x = i ∨ x ∈ I := by simp [hJ]
have hiJ : i ∈ J := by simp [J]
have h := by
refin... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 128 | 149 | theorem lintegral_join {m : Measure (Measure α)} {f : α → ℝ≥0∞} (hf : Measurable f) :
∫⁻ x, f x ∂join m = ∫⁻ μ, ∫⁻ x, f x ∂μ ∂m := by |
simp_rw [lintegral_eq_iSup_eapprox_lintegral hf, SimpleFunc.lintegral,
join_apply (SimpleFunc.measurableSet_preimage _ _)]
suffices
∀ (s : ℕ → Finset ℝ≥0∞) (f : ℕ → ℝ≥0∞ → Measure α → ℝ≥0∞), (∀ n r, Measurable (f n r)) →
Monotone (fun n μ => ∑ r ∈ s n, r * f n r μ) →
⨆ n, ∑ r ∈ s n, r * ∫⁻ μ, f... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 118 | 120 | theorem HasStrictDerivAt.complexToReal_fderiv {f : ℂ → ℂ} {f' x : ℂ} (h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (f' • (1 : ℂ →L[ℝ] ℂ)) x := by |
simpa only [Complex.restrictScalars_one_smulRight] using h.hasStrictFDerivAt.restrictScalars ℝ
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanp... | Mathlib/Probability/Process/Stopping.lean | 612 | 615 | theorem measurableSet_min_const_iff (hτ : IsStoppingTime f τ) (s : Set Ω) {i : ι} :
MeasurableSet[(hτ.min_const i).measurableSpace] s ↔
MeasurableSet[hτ.measurableSpace] s ∧ MeasurableSet[f i] s := by |
rw [measurableSpace_min_const hτ]; apply MeasurableSpace.measurableSet_inf
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
/-!
# The colon ideal
This file defines `Submodule.colon N P` as the ideal of all elements `r... | Mathlib/RingTheory/Ideal/Colon.lean | 76 | 78 | theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} :
r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by |
simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 140 | 141 | theorem inf_relindex_left : (H ⊓ K).relindex H = K.relindex H := by |
rw [inf_comm, inf_relindex_right]
|
/-
Copyright (c) 2024 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.ContinuousFunction.CocompactMap
/-!
# Cocompact maps in normed groups
This file gives a characterizati... | Mathlib/Analysis/Normed/Group/CocompactMap.lean | 29 | 39 | theorem CocompactMapClass.norm_le [FunLike 𝓕 E F] [CocompactMapClass 𝓕 E F] (ε : ℝ) :
∃ r : ℝ, ∀ x : E, r < ‖x‖ → ε < ‖f x‖ := by |
have h := cocompact_tendsto f
rw [tendsto_def] at h
specialize h (Metric.closedBall 0 ε)ᶜ (mem_cocompact_of_closedBall_compl_subset 0 ⟨ε, rfl.subset⟩)
rcases closedBall_compl_subset_of_mem_cocompact h 0 with ⟨r, hr⟩
use r
intro x hx
suffices x ∈ f⁻¹' (Metric.closedBall 0 ε)ᶜ by aesop
apply hr
simp [h... |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 336 | 351 | theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0)
(hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) : (mkMetric m₁ : OuterMeasure X) ≤ c • mkMetric m₂ := by |
classical
rcases (mem_nhdsWithin_Ici_iff_exists_Ico_subset' zero_lt_one).1 hle with ⟨r, hr0, hr⟩
refine fun s =>
le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s)
(ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc))
(mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, hr0⟩) fun r' ... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontin... | Mathlib/Topology/Semicontinuous.lean | 1,116 | 1,120 | theorem upperSemicontinuousAt_sum {f : ι → α → γ} {a : Finset ι}
(ha : ∀ i ∈ a, UpperSemicontinuousAt (f i) x) :
UpperSemicontinuousAt (fun z => ∑ i ∈ a, f i z) x := by |
simp_rw [← upperSemicontinuousWithinAt_univ_iff] at *
exact upperSemicontinuousWithinAt_sum ha
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 669 | 677 | theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ)
(ha : Tendsto a l atBot) :
Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <| ∫ x in Iic b, f x ∂μ) := by |
let φ i := Ioi (a i)
have hφ : AECover (μ.restrict <| Iic b) l φ := aecover_Ioi ha
refine (hφ.integral_tendsto_of_countably_generated hfi).congr' ?_
filter_upwards [ha.eventually (eventually_le_atBot <| b)] with i hai
rw [intervalIntegral.integral_of_le hai, Measure.restrict_restrict (hφ.measurableSet i)]
... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,641 | 1,643 | theorem span_eq_top_of_surjective {s : Set P₁} (hf : Function.Surjective f)
(h : affineSpan k s = ⊤) : affineSpan k (f '' s) = ⊤ := by |
rw [← AffineSubspace.map_span, h, map_top_of_surjective f hf]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Order.UpperLower.Basic
#align_import combinatorics.set_family.intersecting from "leanprover-community/mathlib"@"d90e4e186... | Mathlib/Combinatorics/SetFamily/Intersecting.lean | 61 | 61 | theorem intersecting_singleton : ({a} : Set α).Intersecting ↔ a ≠ ⊥ := by | simp [Intersecting]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanp... | Mathlib/Probability/Process/Stopping.lean | 607 | 609 | theorem measurableSpace_min_const (hτ : IsStoppingTime f τ) {i : ι} :
(hτ.min_const i).measurableSpace = hτ.measurableSpace ⊓ f i := by |
rw [hτ.measurableSpace_min (isStoppingTime_const _ i), measurableSpace_const]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Logic.Function.Basic
#align_import group_theory.semidirect_product from "lean... | Mathlib/GroupTheory/SemidirectProduct.lean | 198 | 198 | theorem rightHom_inr (g : G) : rightHom (inr g : N ⋊[φ] G) = g := by | simp [rightHom]
|
/-
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 | 662 | 663 | theorem getLast_congr {l₁ l₂ : List α} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) :
getLast l₁ h₁ = getLast l₂ h₂ := by | subst l₁; rfl
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 693 | 694 | theorem differentiableOn_const_sub_iff (c : F) :
DifferentiableOn 𝕜 (fun y => c - f y) s ↔ DifferentiableOn 𝕜 f s := by | simp [sub_eq_add_neg]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 460 | 463 | theorem mapDomain_notin_range {f : α → β} (x : α →₀ M) (a : β) (h : a ∉ Set.range f) :
mapDomain f x a = 0 := by |
rw [mapDomain, sum_apply, sum]
exact Finset.sum_eq_zero fun a' _ => single_eq_of_ne fun eq => h <| eq ▸ Set.mem_range_self _
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 634 | 635 | theorem exists_integral_le (hf : Integrable f μ) : ∃ x, ∫ a, f a ∂μ ≤ f x := by |
simpa only [average_eq_integral] using exists_average_le (IsProbabilityMeasure.ne_zero μ) hf
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 827 | 831 | theorem insertNth_apply_above {i j : Fin (n + 1)} (h : i < j) (x : α i)
(p : ∀ k, α (i.succAbove k)) :
i.insertNth x p j = @Eq.recOn _ _ (fun x _ ↦ α x) _
(succAbove_pred_of_lt _ _ h) (p <| j.pred _) := by |
rw [insertNth, succAboveCases, dif_neg h.ne', dif_neg h.not_lt]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
[`data.finset.sym`@`98e83c3d541c77cdb7da20d79611a780ff8e7d90`..`02ba8949f486ebecf93fe7460f1ed0564b5e442c`](https://leanprover-community.github.io/mathlib-port-status/file/d... | Mathlib/Data/Finset/Sym.lean | 152 | 154 | theorem diag_mem_sym2_mem_iff : (∀ b, b ∈ Sym2.diag a → b ∈ s) ↔ a ∈ s := by |
rw [← mem_sym2_iff]
exact mk_mem_sym2_iff.trans <| and_self_iff
|
/-
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 | 489 | 493 | theorem Equiv.induced_symm {α β : Type*} (e : α ≃ β) :
TopologicalSpace.induced e.symm = TopologicalSpace.coinduced e := by |
ext t U
rw [isOpen_induced_iff, isOpen_coinduced]
simp only [e.symm.preimage_eq_iff_eq_image, exists_eq_right, ← preimage_equiv_eq_image_symm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,477 | 1,483 | theorem Iio_union_Ioo' (h₁ : c < b) : Iio b ∪ Ioo c d = Iio (max b d) := by |
ext x
cases' lt_or_le x b with hba hba
· simp [hba, h₁]
· simp only [mem_Iio, mem_union, mem_Ioo, lt_max_iff]
refine or_congr Iff.rfl ⟨And.right, ?_⟩
exact fun h₂ => ⟨h₁.trans_le hba, h₂⟩
|
/-
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, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 765 | 772 | theorem ContinuousOn.strictMonoOn_of_injOn_Ioo {a b : α} {f : α → δ} (hab : a < b)
(hf_c : ContinuousOn f (Ioo a b)) (hf_i : InjOn f (Ioo a b)) :
StrictMonoOn f (Ioo a b) ∨ StrictAntiOn f (Ioo a b) := by |
haveI : Inhabited (Ioo a b) := Classical.inhabited_of_nonempty (nonempty_Ioo_subtype hab)
let g : Ioo a b → δ := Set.restrict (Ioo a b) f
have : StrictMono g ∨ StrictAnti g :=
Continuous.strictMono_of_inj hf_c.restrict hf_i.injective
exact this.imp strictMono_restrict.mp strictAntiOn_iff_strictAnti.mpr
|
/-
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.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Sets.Opens
#align_import topology.local_homeomorph from "leanprover-community/mathlib"@"431589b... | Mathlib/Topology/PartialHomeomorph.lean | 427 | 430 | theorem eventually_nhdsWithin {x : X} (p : Y → Prop) {s : Set X}
(hx : x ∈ e.source) : (∀ᶠ y in 𝓝[e.symm ⁻¹' s] e x, p y) ↔ ∀ᶠ x in 𝓝[s] x, p (e x) := by |
refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e31... | Mathlib/Data/QPF/Multivariate/Basic.lean | 141 | 157 | theorem liftR_iff {α : TypeVec n} (r : ∀ /- ⦃i⦄ -/ {i}, α i → α i → Prop) (x y : F α) :
LiftR r x y ↔ ∃ a f₀ f₁, x = abs ⟨a, f₀⟩ ∧ y = abs ⟨a, f₁⟩ ∧ ∀ i j, r (f₀ i j) (f₁ i j) := by |
constructor
· rintro ⟨u, xeq, yeq⟩
cases' h : repr u with a f
use a, fun i j => (f i j).val.fst, fun i j => (f i j).val.snd
constructor
· rw [← xeq, ← abs_repr u, h, ← abs_map]; rfl
constructor
· rw [← yeq, ← abs_repr u, h, ← abs_map]; rfl
intro i j
exact (f i j).property
rintro ⟨... |
/-
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 | 1,713 | 1,716 | theorem eq_prod_range_div' {M : Type*} [CommGroup M] (f : ℕ → M) (n : ℕ) :
f n = ∏ i ∈ range (n + 1), if i = 0 then f 0 else f i / f (i - 1) := by |
conv_lhs => rw [Finset.eq_prod_range_div f]
simp [Finset.prod_range_succ', mul_comm]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a... | Mathlib/Analysis/NormedSpace/ENorm.lean | 82 | 92 | theorem map_smul (c : 𝕜) (x : V) : e (c • x) = ‖c‖₊ * e x := by |
apply le_antisymm (e.map_smul_le' c x)
by_cases hc : c = 0
· simp [hc]
calc
(‖c‖₊ : ℝ≥0∞) * e x = ‖c‖₊ * e (c⁻¹ • c • x) := by rw [inv_smul_smul₀ hc]
_ ≤ ‖c‖₊ * (‖c⁻¹‖₊ * e (c • x)) := mul_le_mul_left' (e.map_smul_le' _ _) _
_ = e (c • x) := by
rw [← mul_assoc, nnnorm_inv, ENNReal.coe_inv, EN... |
/-
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 | 162 | 162 | theorem lt_floor_add_one (a : α) : a < ⌊a⌋₊ + 1 := by | simpa using lt_succ_floor a
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 371 | 373 | theorem image_id' (s : Set α) : (fun x => x) '' s = s := by |
ext
simp
|
/-
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 | 150 | 155 | theorem _root_.HasCompactSupport.convolution_integrand_bound_left (hcf : HasCompactSupport f)
(hf : Continuous f) {x t : G} {s : Set G} (hx : x ∈ s) :
‖L (f (x - t)) (g t)‖ ≤
(-tsupport f + s).indicator (fun t => (‖L‖ * ⨆ i, ‖f i‖) * ‖g t‖) t := by |
convert hcf.convolution_integrand_bound_right L.flip hf hx using 1
simp_rw [L.opNorm_flip, mul_right_comm]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Data.List.Chain
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.Category.ULift
... | Mathlib/CategoryTheory/IsConnected.lean | 183 | 187 | theorem induct_on_objects [IsPreconnected J] (p : Set J) {j₀ : J} (h0 : j₀ ∈ p)
(h1 : ∀ {j₁ j₂ : J} (_ : j₁ ⟶ j₂), j₁ ∈ p ↔ j₂ ∈ p) (j : J) : j ∈ p := by |
let aux (j₁ j₂ : J) (f : j₁ ⟶ j₂) := congrArg ULift.up <| (h1 f).eq
injection constant_of_preserves_morphisms (fun k => ULift.up.{u₁} (k ∈ p)) aux j j₀ with i
rwa [i]
|
/-
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.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 320 | 323 | theorem w : forkMap P X π ≫ firstMap P X π = forkMap P X π ≫ secondMap P X π := by |
ext x ij
simp only [firstMap, secondMap, forkMap, types_comp_apply, Types.pi_lift_π_apply,
← FunctorToTypes.map_comp_apply, ← op_comp, pullback.condition]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-... | Mathlib/Data/TypeVec.lean | 582 | 586 | theorem repeatEq_iff_eq {α : TypeVec n} {i x y} :
ofRepeat (repeatEq α i (prod.mk _ x y)) ↔ x = y := by |
induction' i with _ _ _ i_ih
· rfl
erw [repeatEq, i_ih]
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | Mathlib/Analysis/MeanInequalitiesPow.lean | 101 | 110 | theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
(hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) := by |
have : 0 < p := by positivity
rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one]
· exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp
all_goals
apply_rules [sum_nonneg, rpow_nonneg]
intro i hi
apply_rules [mul_nonneg, rpow_nonneg, hw i hi, hz i hi]
|
/-
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.Group.Hom.Defs
import Mathlib.Algebra.Group.Subse... | Mathlib/Algebra/Group/Submonoid/Basic.lean | 561 | 562 | theorem closure_singleton_le_iff_mem (m : M) (p : Submonoid M) : closure {m} ≤ p ↔ m ∈ p := by |
rw [closure_le, singleton_subset_iff, SetLike.mem_coe]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_imp... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 57 | 60 | theorem pentagon_inv_inv_hom (W X Y Z : C) :
(α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ X) Y Z).hom =
(𝟙 W ⊗ (α_ X Y Z).hom) ≫ (α_ W X (Y ⊗ Z)).inv := by |
coherence
|
/-
Copyright (c) 2022 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.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 247 | 268 | theorem induction_on {x y : A}
(hy : y ∈ elementalStarAlgebra R x) {P : (u : A) → u ∈ elementalStarAlgebra R x → Prop}
(self : P x (self_mem R x)) (star_self : P (star x) (star_self_mem R x))
(algebraMap : ∀ r, P (algebraMap R A r) (_root_.algebraMap_mem _ r))
(add : ∀ u hu v hv, P u hu → P v hv → P (u ... |
apply closure (adjoin R {x} : Set A) subset_closure (fun y hy ↦ ?_) y hy
rw [SetLike.mem_coe, ← mem_toSubalgebra, adjoin_toSubalgebra] at hy
induction hy using Algebra.adjoin_induction'' with
| mem u hu =>
obtain ((rfl : u = x) | (hu : star u = x)) := by simpa using hu
· exact self
· simp_rw [← hu,... |
/-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
#align_import algebra.star.order from "leanprover-community/mathlib"@"31c24... | Mathlib/Algebra/Star/Order.lean | 156 | 166 | theorem conjugate_nonneg {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ star c * a * c := by |
rw [StarOrderedRing.nonneg_iff] at ha
refine AddSubmonoid.closure_induction ha (fun x hx => ?_)
(by rw [mul_zero, zero_mul]) fun x y hx hy => ?_
· obtain ⟨x, rfl⟩ := hx
convert star_mul_self_nonneg (x * c) using 1
rw [star_mul, ← mul_assoc, mul_assoc _ _ c]
· calc
0 ≤ star c * x * c + 0 := by... |
/-
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.Data.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | Mathlib/Data/Matroid/Dual.lean | 237 | 240 | theorem coindep_iff_exists' : M.Coindep X ↔ (∃ B, M.Base B ∧ B ⊆ M.E \ X) ∧ X ⊆ M.E := by |
simp_rw [Coindep, dual_indep_iff_exists', and_comm (a := _ ⊆ _), and_congr_left_iff, subset_diff]
exact fun _ ↦ ⟨fun ⟨B, hB, hXB⟩ ↦ ⟨B, hB, hB.subset_ground, hXB.symm⟩,
fun ⟨B, hB, _, hBX⟩ ↦ ⟨B, hB, hBX.symm⟩⟩
|
/-
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.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7... | Mathlib/Algebra/RingQuot.lean | 419 | 420 | theorem mkRingHom_rel {r : R → R → Prop} {x y : R} (w : r x y) : mkRingHom r x = mkRingHom r y := by |
simp [mkRingHom_def, Quot.sound (Rel.of w)]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 544 | 546 | theorem eq_orthogonalProjection_of_eq_submodule {K' : Submodule 𝕜 E} [HasOrthogonalProjection K']
(h : K = K') (u : E) : (orthogonalProjection K u : E) = (orthogonalProjection K' u : E) := by |
subst h; rfl
|
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 200 | 204 | theorem contMDiffAt_totalSpace (f : M → TotalSpace F E) (x₀ : M) :
ContMDiffAt IM (IB.prod 𝓘(𝕜, F)) n f x₀ ↔
ContMDiffAt IM IB n (fun x => (f x).proj) x₀ ∧
ContMDiffAt IM 𝓘(𝕜, F) n (fun x => (trivializationAt F E (f x₀).proj (f x)).2) x₀ := by |
simp_rw [← contMDiffWithinAt_univ]; exact contMDiffWithinAt_totalSpace f
|
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