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 |
|---|---|---|---|---|---|
/-
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.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 176 | 176 | theorem prime_five : Prime 5 := by | decide
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 599 | 608 | theorem trans {a o o' : Ordinal.{u}} {f : ∀ b < o, Ordinal.{u}} (hf : IsFundamentalSequence a o f)
{g : ∀ b < o', Ordinal.{u}} (hg : IsFundamentalSequence o o' g) :
IsFundamentalSequence a o' fun i hi =>
f (g i hi) (by rw [← hg.2.2]; apply lt_blsub) := by |
refine ⟨?_, @fun i j _ _ h => hf.2.1 _ _ (hg.2.1 _ _ h), ?_⟩
· rw [hf.cof_eq]
exact hg.1.trans (ord_cof_le o)
· rw [@blsub_comp.{u, u, u} o _ f (@IsFundamentalSequence.monotone _ _ f hf)]
· exact hf.2.2
· exact hg.2.2
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 699 | 706 | theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by |
by_cases hab : a ≤ b
· rw [← Ioc_union_left hab, integrableOn_union,
eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
· rw [Icc_eq_empty hab, Ioc_eq_empty]
contrapose! hab
exact hab.le
|
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
#align_import category_theory.limits.shapes.wide_equalizers from "l... | Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean | 223 | 224 | theorem Cotrident.app_one (s : Cotrident f) (j : J) : f j ≫ s.ι.app one = s.ι.app zero := by |
rw [← s.w (line j), parallelFamily_map_left]
|
/-
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 | 436 | 437 | theorem isLowerSet_iff_Iio_subset : IsLowerSet s ↔ ∀ ⦃a⦄, a ∈ s → Iio a ⊆ s := by |
simp [isLowerSet_iff_forall_lt, subset_def, @forall_swap (_ ∈ s)]
|
/-
Copyright (c) 2020 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb... | Mathlib/Analysis/Convex/Hull.lean | 104 | 106 | theorem convexHull_nonempty_iff : (convexHull 𝕜 s).Nonempty ↔ s.Nonempty := by |
rw [nonempty_iff_ne_empty, nonempty_iff_ne_empty, Ne, Ne]
exact not_congr convexHull_empty_iff
|
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 269 | 271 | theorem gcdB_zero_left {s : R} : gcdB 0 s = 1 := by |
unfold gcdB
rw [xgcd, xgcd_zero_left]
|
/-
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.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 255 | 256 | theorem max?_mem {t : RBNode α} (h : t.max? = some a) : a ∈ t := by |
simpa using min?_mem ((min?_reverse _).trans h)
|
/-
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 | 287 | 334 | theorem finStronglyMeasurable_of_set_sigmaFinite [TopologicalSpace β] [Zero β]
{m : MeasurableSpace α} {μ : Measure α} (hf_meas : StronglyMeasurable f) {t : Set α}
(ht : MeasurableSet t) (hft_zero : ∀ x ∈ tᶜ, f x = 0) (htμ : SigmaFinite (μ.restrict t)) :
FinStronglyMeasurable f μ := by |
haveI : SigmaFinite (μ.restrict t) := htμ
let S := spanningSets (μ.restrict t)
have hS_meas : ∀ n, MeasurableSet (S n) := measurable_spanningSets (μ.restrict t)
let f_approx := hf_meas.approx
let fs n := SimpleFunc.restrict (f_approx n) (S n ∩ t)
have h_fs_t_compl : ∀ n, ∀ x, x ∉ t → fs n x = 0 := by
i... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Al... | Mathlib/Tactic/NormNum/Basic.lean | 481 | 482 | theorem ble_eq_false {x y : ℕ} : x.ble y = false ↔ y < x := by |
rw [← Nat.not_le, ← Bool.not_eq_true, Nat.ble_eq]
|
/-
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.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis
#align_import algebra.direct_sum.module from "leanprover-commun... | Mathlib/Algebra/DirectSum/Module.lean | 386 | 387 | theorem IsInternal.collectedBasis_mem (h : IsInternal A) {α : ι → Type*}
(v : ∀ i, Basis (α i) R (A i)) (a : Σi, α i) : h.collectedBasis v a ∈ A a.1 := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 568 | 569 | theorem tail_sum (L : List ℕ) : L.tail.sum = L.sum - L.headI := by |
rw [← headI_add_tail_sum L, add_comm, Nat.add_sub_cancel_right]
|
/-
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.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mat... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 63 | 70 | theorem condexp_indicator_aux (hs : MeasurableSet[m] s) (hf : f =ᵐ[μ.restrict sᶜ] 0) :
μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m]) := by |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm, Set.indicator_zero']; rfl
have hsf_zero : ∀ g : α → E, g =ᵐ[μ.restrict sᶜ] 0 → s.indicator g =ᵐ[μ] g := fun g =>
indicator_ae_eq_of_restrict_compl_ae_eq_zero (hm _ hs)
refine ((hsf_zero (μ[f|m]) (condexp_ae_eq_restrict_zero hs.compl hf)).trans ?_)... |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d8... | Mathlib/Data/Bool/Count.lean | 120 | 123 | theorem length_sub_one_le_two_mul_count_bool (hl : Chain' (· ≠ ·) l) (b : Bool) :
length l - 1 ≤ 2 * count b l := by |
rw [hl.two_mul_count_bool_eq_ite]
split_ifs <;> simp [le_tsub_add, Nat.le_succ_of_le]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mat... | Mathlib/Algebra/Ring/Equiv.lean | 785 | 788 | theorem symm_toRingHom_comp_toRingHom (e : R ≃+* S) :
e.symm.toRingHom.comp e.toRingHom = RingHom.id _ := by |
ext
simp
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 240 | 244 | theorem fromBlocks_add [Add α] (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) (A' : Matrix n l α) (B' : Matrix n m α) (C' : Matrix o l α)
(D' : Matrix o m α) : fromBlocks A B C D + fromBlocks A' B' C' D' =
fromBlocks (A + A') (B + B') (C + C') (D + D') := by |
ext i j; rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl
|
/-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 307 | 308 | theorem noncommPiCoprod_range : (noncommPiCoprod hcomm).range = ⨆ i : ι, H i := by |
simp [noncommPiCoprod, MonoidHom.noncommPiCoprod_range]
|
/-
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.SpecificLimits.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.specific_limits.floor_pow from "leanprove... | Mathlib/Analysis/SpecificLimits/FloorPow.lean | 271 | 280 | theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ := by |
have cpos : 0 < c := zero_lt_one.trans hc
rcases eq_or_ne i 0 with (rfl | hi)
· simp only [pow_zero, Nat.floor_one, Nat.cast_one, mul_one, sub_le_self_iff, inv_nonneg, cpos.le]
calc
(1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
_ ≤ c ^ i - 1 := by
gcongr
simpa only [← div_eq_mul_inv, o... |
/-
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 | 115 | 119 | theorem hausdorffMeasure_of_lt_dimH {s : Set X} {d : ℝ≥0} (h : ↑d < dimH s) : μH[d] s = ∞ := by |
simp only [dimH_def, lt_iSup_iff] at h
rcases h with ⟨d', hsd', hdd'⟩
rw [ENNReal.coe_lt_coe, ← NNReal.coe_lt_coe] at hdd'
exact top_unique (hsd' ▸ hausdorffMeasure_mono hdd'.le _)
|
/-
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, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 237 | 240 | theorem induction_on_prime {P : α → Prop} (a : α) (h₁ : P 0) (h₂ : ∀ x : α, IsUnit x → P x)
(h₃ : ∀ a p : α, a ≠ 0 → Prime p → P a → P (p * a)) : P a := by |
simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime] at h₃
exact WfDvdMonoid.induction_on_irreducible a h₁ h₂ h₃
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520... | Mathlib/Data/ENNReal/Inv.lean | 495 | 497 | theorem sub_half (h : a ≠ ∞) : a - a / 2 = a / 2 := by |
lift a to ℝ≥0 using h
exact sub_eq_of_add_eq (mul_ne_top coe_ne_top <| by simp) (ENNReal.add_halves a)
|
/-
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, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#a... | Mathlib/Data/Rat/Floor.lean | 128 | 163 | theorem fract_inv_num_lt_num_of_pos {q : ℚ} (q_pos : 0 < q) : (fract q⁻¹).num < q.num := by |
-- we know that the numerator must be positive
have q_num_pos : 0 < q.num := Rat.num_pos.mpr q_pos
-- we will work with the absolute value of the numerator, which is equal to the numerator
have q_num_abs_eq_q_num : (q.num.natAbs : ℤ) = q.num := Int.natAbs_of_nonneg q_num_pos.le
set q_inv : ℚ := q.den / q.num... |
/-
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.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 976 | 978 | theorem MeasurableSet.map_coe_volume {s : Set α} (hs : MeasurableSet s) :
volume.map ((↑) : s → α) = restrict volume s := by |
rw [volume_set_coe_def, (MeasurableEmbedding.subtype_coe hs).map_comap volume, Subtype.range_coe]
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,606 | 1,610 | theorem surjOn_iInter [Nonempty ι] {s : ι → Set α} {t : Set β} {f : α → β}
(H : ∀ i, SurjOn f (s i) t) (Hinj : InjOn f (⋃ i, s i)) : SurjOn f (⋂ i, s i) t := by |
intro y hy
rw [Hinj.image_iInter_eq, mem_iInter]
exact fun i => H i hy
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#al... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 179 | 182 | theorem dist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄ := by |
rw [dist_eq_norm, vsub_sub_vsub_comm, dist_eq_norm_vsub V, dist_eq_norm_vsub V]
exact norm_sub_le _ _
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 283 | 294 | theorem ker_toZMod : RingHom.ker (toZMod : ℤ_[p] →+* ZMod p) = maximalIdeal ℤ_[p] := by |
ext x
rw [RingHom.mem_ker]
constructor
· intro h
simpa only [h, ZMod.cast_zero, sub_zero] using toZMod_spec x
· intro h
rw [← sub_zero x] at h
dsimp [toZMod, toZModHom]
convert zmod_congr_of_sub_mem_max_ideal x _ 0 _ h
· norm_cast
· apply sub_zmodRepr_mem
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 323 | 326 | theorem tendsto_prod_filter_iff {c : β} :
Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun _ => c) p p' := by |
simp_rw [nhds_eq_comap_uniformity, tendsto_comap_iff]
rfl
|
/-
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.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... | Mathlib/RingTheory/LocalProperties.lean | 608 | 626 | theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
[IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
(A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
(hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Alg... |
let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
let y := IsLocalization.commonDenomOfFinset M s
have hx₁ : (y : S) • (s : Set S') = g '' _ :=
(IsLocalization.finsetIntegerMultiple_image _ s).symm
obtain ⟨n, hn⟩ :=
Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
... |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Spe... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 117 | 156 | theorem MeasureTheory.measure_le_eq_lt [Nontrivial E] (r : ℝ) :
μ {x : E | g x ≤ r} = μ {x : E | g x < r} := by |
-- We copy `E` to a new type `F` on which we will put the norm defined by `g`
letI F : Type _ := E
letI : NormedAddCommGroup F :=
{ norm := g
dist := fun x y => g (x - y)
dist_self := by simp only [_root_.sub_self, h1, forall_const]
dist_comm := fun _ _ => by dsimp [dist]; rw [← h2, neg_sub]
di... |
/-
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.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 276 | 281 | theorem iSup_induction {ι : Sort*} (S : ι → Submonoid M) {C : M → Prop} {x : M} (hx : x ∈ ⨆ i, S i)
(mem : ∀ (i), ∀ x ∈ S i, C x) (one : C 1) (mul : ∀ x y, C x → C y → C (x * y)) : C x := by |
rw [iSup_eq_closure] at hx
refine closure_induction hx (fun x hx => ?_) one mul
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hx
exact mem _ _ hi
|
/-
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.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3a... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 82 | 84 | theorem mem_sup_left {S T : Subsemigroup M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by |
have : S ≤ S ⊔ T := le_sup_left
tauto
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mat... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 466 | 476 | theorem count_finprod_coprime (exps : HeightOneSpectrum R → ℤ) :
count K v (∏ᶠ (w : HeightOneSpectrum R) (_ : w ≠ v),
(w.asIdeal : (FractionalIdeal R⁰ K)) ^ exps w) = 0 := by |
apply finprod_mem_induction fun I => count K v I = 0
· exact count_one K v
· intro I I' hI hI'
by_cases h : I ≠ 0 ∧ I' ≠ 0
· rw [count_mul' K v, if_pos h, hI, hI', add_zero]
· rw [count_mul' K v, if_neg h]
· intro w hw
rw [count_zpow, count_maximal_coprime K v hw, MulZeroClass.mul_zero]
|
/-
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.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf973... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 221 | 225 | theorem coe_eq_of_mem_split_of_lt_mem {y : ι → ℝ} (h₁ : J ∈ split I i x) (h₂ : y ∈ J)
(h₃ : x < y i) : (J : Set (ι → ℝ)) = ↑I ∩ { y | x < y i } := by |
refine (mem_split_iff'.1 h₁).resolve_left fun H => ?_
rw [← Box.mem_coe, H] at h₂
exact h₃.not_le h₂.2
|
/-
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 | 964 | 989 | theorem realize_toFormula (φ : L.BoundedFormula α n) (v : Sum α (Fin n) → M) :
φ.toFormula.Realize v ↔ φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr) := by |
induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3 a8 a9 a0
· rfl
· simp [BoundedFormula.Realize]
· simp [BoundedFormula.Realize]
· rw [toFormula, Formula.Realize, realize_imp, ← Formula.Realize, ih1, ← Formula.Realize, ih2,
realize_imp]
· rw [toFormula, Formula.Realize, realize_all, realize_all... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 140 | 142 | theorem fderiv_inner_apply (hf : DifferentiableAt ℝ f x) (hg : DifferentiableAt ℝ g x) (y : G) :
fderiv ℝ (fun t => ⟪f t, g t⟫) x y = ⟪f x, fderiv ℝ g x y⟫ + ⟪fderiv ℝ f x y, g x⟫ := by |
rw [(hf.hasFDerivAt.inner 𝕜 hg.hasFDerivAt).fderiv]; rfl
|
/-
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.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 471 | 479 | theorem setIntegral_condexpL2_indicator (hs : MeasurableSet[m] s) (ht : MeasurableSet t)
(hμs : μ s ≠ ∞) (hμt : μ t ≠ ∞) :
∫ x in s, (condexpL2 ℝ ℝ hm (indicatorConstLp 2 ht hμt 1) : α → ℝ) x ∂μ = (μ (t ∩ s)).toReal :=
calc
∫ x in s, (condexpL2 ℝ ℝ hm (indicatorConstLp 2 ht hμt 1) : α → ℝ) x ∂μ =
... | rw [smul_eq_mul, mul_one]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
... | Mathlib/Algebra/Polynomial/Coeff.lean | 138 | 138 | theorem mul_coeff_zero (p q : R[X]) : coeff (p * q) 0 = coeff p 0 * coeff q 0 := by | simp [coeff_mul]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 73 | 80 | theorem RespectsIso.basicOpen_iff_localization (hP : RespectsIso @P) {X Y : Scheme.{u}} [IsAffine X]
[IsAffine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (Opposite.op ⊤)) :
P (Scheme.Γ.map (f ∣_ Y.basicOpen r).op) ↔ P (Localization.awayMap (Scheme.Γ.map f.op) r) := by |
refine (hP.basicOpen_iff _ _).trans ?_
-- Porting note: was a one line term mode proof, but this `dsimp` is vital so the term mode
-- one liner is not possible
dsimp
rw [← hP.is_localization_away_iff]
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 148 | 151 | theorem diag_col_mul_row [Mul α] [AddCommMonoid α] (a b : n → α) :
diag (col a * row b) = a * b := by |
ext
simp [Matrix.mul_apply, col, row]
|
/-
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 | 1,147 | 1,148 | theorem indexOf_cons_self (a : α) (l : List α) : indexOf a (a :: l) = 0 := by |
rw [indexOf, findIdx_cons, beq_self_eq_true, cond]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sebastien Gouezel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Ba... | Mathlib/Topology/VectorBundle/Basic.lean | 808 | 812 | theorem localTriv_symmL {b : B} (hb : b ∈ Z.baseSet i) :
(Z.localTriv i).symmL R b = Z.coordChange i (Z.indexAt b) b := by |
ext1 v
rw [(Z.localTriv i).symmL_apply R, (Z.localTriv i).symm_apply]
exacts [rfl, hb]
|
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 718 | 724 | theorem affineIndependent_of_ne_of_mem_of_not_mem_of_mem {s : AffineSubspace k P} {p₁ p₂ p₃ : P}
(hp₁p₃ : p₁ ≠ p₃) (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∉ s) (hp₃ : p₃ ∈ s) :
AffineIndependent k ![p₁, p₂, p₃] := by |
rw [← affineIndependent_equiv (Equiv.swap (1 : Fin 3) 2)]
convert affineIndependent_of_ne_of_mem_of_mem_of_not_mem hp₁p₃ hp₁ hp₃ hp₂ using 1
ext x
fin_cases x <;> rfl
|
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanp... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 46 | 53 | theorem LinearMap.BilinForm.isSkewAdjoint_bracket {f g : Module.End R M}
(hf : f ∈ B.skewAdjointSubmodule) (hg : g ∈ B.skewAdjointSubmodule) :
⁅f, g⁆ ∈ B.skewAdjointSubmodule := by |
rw [mem_skewAdjointSubmodule] at *
have hfg : IsAdjointPair B B (f * g) (g * f) := by rw [← neg_mul_neg g f]; exact hf.mul hg
have hgf : IsAdjointPair B B (g * f) (f * g) := by rw [← neg_mul_neg f g]; exact hg.mul hf
change IsAdjointPair B B (f * g - g * f) (-(f * g - g * f)); rw [neg_sub]
exact hfg.sub hgf
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENa... | Mathlib/RingTheory/Multiplicity.lean | 617 | 622 | theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by |
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 284 | 293 | theorem eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : HasEigenvalue T μ)
(hnn : ∀ x : E, 0 ≤ RCLike.re ⟪x, T x⟫) : 0 ≤ μ := by |
obtain ⟨v, hv⟩ := hμ.exists_hasEigenvector
have hpos : (0 : ℝ) < ‖v‖ ^ 2 := by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2
have : RCLike.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2 := by
have := congr_arg RCLike.re (inner_product_apply_eigenvector hv.1)
-- Porting note: why can't `exact_mod_cast` do this? These l... |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.Data.Set.Pointwise.SMul
/-!
... | Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 82 | 87 | theorem fixedBy_subset_fixedBy_zpow (g : G) (j : ℤ) :
fixedBy α g ⊆ fixedBy α (g ^ j) := by |
intro a a_in_fixedBy
rw [mem_fixedBy, zpow_smul_eq_iff_minimalPeriod_dvd,
minimalPeriod_eq_one_iff_fixedBy.mpr a_in_fixedBy, Nat.cast_one]
exact one_dvd j
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 183 | 185 | theorem map_zero [Nonempty ι] : f 0 = 0 := by |
obtain ⟨i, _⟩ : ∃ i : ι, i ∈ Set.univ := Set.exists_mem_of_nonempty ι
exact map_coord_zero f i rfl
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 886 | 902 | theorem ae_tendsto_lintegral_nnnorm_sub_div {f : α → E} (hf : LocallyIntegrable f μ) :
∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by |
rcases hf.exists_nat_integrableOn with ⟨u, u_open, u_univ, hu⟩
have : ∀ n, ∀ᵐ x ∂μ,
Tendsto (fun a => (∫⁻ y in a, ‖(u n).indicator f y - (u n).indicator f x‖₊ ∂μ) / μ a)
(v.filterAt x) (𝓝 0) := by
intro n
apply ae_tendsto_lintegral_nnnorm_sub_div_of_integrable
exact (integrable_indicator_i... |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Ba... | Mathlib/RingTheory/AlgebraicIndependent.lean | 156 | 160 | theorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :
AlgebraicIndependent R x := by |
have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp
rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv
exact hfv.of_comp
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Algebra.Order.Monoid.Prod
import Mathlib... | Mathlib/Algebra/Order/Module/OrderedSMul.lean | 118 | 128 | theorem OrderedSMul.mk' (h : ∀ ⦃a b : M⦄ ⦃c : 𝕜⦄, a < b → 0 < c → c • a ≤ c • b) :
OrderedSMul 𝕜 M := by |
have hlt' : ∀ (a b : M) (c : 𝕜), a < b → 0 < c → c • a < c • b := by
refine fun a b c hab hc => (h hab hc).lt_of_ne ?_
rw [Ne, hc.ne'.isUnit.smul_left_cancel]
exact hab.ne
refine ⟨fun {a b c} => hlt' a b c, fun {a b c hab hc} => ?_⟩
obtain ⟨c, rfl⟩ := hc.ne'.isUnit
rw [← inv_smul_smul c a, ← inv_s... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 162 | 166 | theorem push_parentD (arr : Array UFNode) : parentD (arr.push ⟨arr.size, 0⟩) i = parentD arr i := by |
simp [parentD, Array.get_eq_getElem, Array.get_push]
split <;> split <;> try simp
· exact Nat.le_antisymm (Nat.ge_of_not_lt ‹_›) (Nat.le_of_lt_succ ‹_›)
· cases ‹¬_› (Nat.lt_succ_of_lt ‹_›)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Floris van Doorn
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.GroupTheory.GroupAc... | Mathlib/Data/Set/Pointwise/SMul.lean | 798 | 798 | theorem smul_zero_subset (s : Set α) : s • (0 : Set β) ⊆ 0 := by | simp [subset_def, mem_smul]
|
/-
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.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.Calculus.AffineMap
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.Affin... | Mathlib/Analysis/NormedSpace/AddTorsorBases.lean | 137 | 141 | theorem AffineBasis.centroid_mem_interior_convexHull {ι} [Fintype ι] (b : AffineBasis ι ℝ V) :
Finset.univ.centroid ℝ b ∈ interior (convexHull ℝ (range b)) := by |
haveI := b.nonempty
simp only [b.interior_convexHull, mem_setOf_eq, b.coord_apply_centroid (Finset.mem_univ _),
inv_pos, Nat.cast_pos, Finset.card_pos, Finset.univ_nonempty, forall_true_iff]
|
/-
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.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanpro... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 261 | 279 | theorem Fix.ind_rec {β : Type u} (g₁ g₂ : Fix F α → β)
(h :
∀ x : F (append1 α (Fix F α)),
appendFun id g₁ <$$> x = appendFun id g₂ <$$> x → g₁ (Fix.mk x) = g₂ (Fix.mk x)) :
∀ x, g₁ x = g₂ x := by |
apply Quot.ind
intro x
apply q.P.w_ind _ x
intro a f' f ih
show g₁ ⟦q.P.wMk a f' f⟧ = g₂ ⟦q.P.wMk a f' f⟧
rw [← Fix.ind_aux a f' f]
apply h
rw [← abs_map, ← abs_map, MvPFunctor.map_eq, MvPFunctor.map_eq]
congr 2
rw [MvPFunctor.appendContents, appendFun, appendFun, ← splitFun_comp, ← splitFun_comp]
... |
/-
Copyright (c) 2024 Miyahara Kō. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Miyahara Kō
-/
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
/-!
# iterate
Proves various lemmas about `List.iterate`.
-/
variable {α : Type*}
namespace List
... | Mathlib/Data/List/Iterate.lean | 54 | 56 | theorem take_iterate (f : α → α) (a : α) (m n : ℕ) :
take m (iterate f a n) = iterate f a (min m n) := by |
rw [← range_map_iterate, ← range_map_iterate, ← map_take, take_range]
|
/-
Copyright (c) 2022 Violeta Hernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández
-/
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.List.AList
#align_import data.finsupp.alist from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/AList.lean | 41 | 44 | theorem toAList_keys_toFinset [DecidableEq α] (f : α →₀ M) :
f.toAList.keys.toFinset = f.support := by |
ext
simp [toAList, AList.mem_keys, AList.keys, List.keys]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,724 | 1,727 | theorem aevalTower_id :
aevalTower (AlgHom.id S S) = (aeval : (σ → S) → MvPolynomial σ S →ₐ[S] S) := by |
ext
simp only [aevalTower_X, aeval_X]
|
/-
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 | 1,906 | 1,907 | theorem foldr_eta : ∀ l : List α, foldr cons [] l = l := by |
simp only [foldr_self_append, append_nil, forall_const]
|
/-
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.BumpFunction.FiniteDimension
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpac... | Mathlib/Geometry/Manifold/BumpFunction.lean | 226 | 233 | theorem exists_r_pos_lt_subset_ball {s : Set M} (hsc : IsClosed s) (hs : s ⊆ support f) :
∃ r ∈ Ioo 0 f.rOut,
s ⊆ (chartAt H c).source ∩ extChartAt I c ⁻¹' ball (extChartAt I c c) r := by |
set e := extChartAt I c
have : IsClosed (e '' s) := f.isClosed_image_of_isClosed hsc hs
rw [support_eq_inter_preimage, subset_inter_iff, ← image_subset_iff] at hs
rcases exists_pos_lt_subset_ball f.rOut_pos this hs.2 with ⟨r, hrR, hr⟩
exact ⟨r, hrR, subset_inter hs.1 (image_subset_iff.1 hr)⟩
|
/-
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.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 338 | 341 | theorem diagonalObjPullbackFstIso_inv_snd_fst {X Y Z : C} (f : X ⟶ Z) (g : Y ⟶ Z) :
(diagonalObjPullbackFstIso f g).inv ≫ pullback.snd ≫ pullback.fst = pullback.snd := by |
delta diagonalObjPullbackFstIso
simp
|
/-
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 | 818 | 818 | theorem ofSet_univ_eq_refl : ofSet univ isOpen_univ = PartialHomeomorph.refl X := by | ext <;> simp
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen
-/
import Mathlib.Logic.Relation
import Mathlib.Order.GaloisConnection
#align_import data.setoid.basic from "leanprover-community/mathlib"@"bbe... | Mathlib/Data/Setoid/Basic.lean | 257 | 260 | theorem sSup_def {s : Set (Setoid α)} : sSup s = EqvGen.Setoid (sSup (Rel '' s)) := by |
rw [sSup_eq_eqvGen, sSup_image]
congr with (x y)
simp only [iSup_apply, iSup_Prop_eq, exists_prop]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 314 | 316 | theorem mem_primitiveRoots {ζ : R} (h0 : 0 < k) : ζ ∈ primitiveRoots k R ↔ IsPrimitiveRoot ζ k := by |
rw [primitiveRoots, mem_filter, Multiset.mem_toFinset, mem_nthRoots h0, and_iff_right_iff_imp]
exact IsPrimitiveRoot.pow_eq_one
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 259 | 269 | theorem le_add_order_smul_norm_of_isOfFinAddOrder {u : AddCircle p} (hu : IsOfFinAddOrder u)
(hu' : u ≠ 0) : p ≤ addOrderOf u • ‖u‖ := by |
obtain ⟨n, hn⟩ := exists_norm_eq_of_isOfFinAddOrder hu
replace hu : (addOrderOf u : ℝ) ≠ 0 := by
norm_cast
exact (addOrderOf_pos_iff.mpr hu).ne'
conv_lhs => rw [← mul_one p]
rw [hn, nsmul_eq_mul, ← mul_assoc, mul_comm _ p, mul_assoc, mul_div_cancel₀ _ hu,
mul_le_mul_left hp.out, Nat.one_le_cast, Na... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 516 | 518 | theorem _root_.Basis.toBasis_toOrthonormalBasis (v : Basis ι 𝕜 E) (hv : Orthonormal 𝕜 v) :
(v.toOrthonormalBasis hv).toBasis = v := by |
simp [Basis.toOrthonormalBasis, OrthonormalBasis.toBasis]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "lea... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 746 | 751 | theorem UniformIntegrable.ae_eq {g : ι → α → β} (hf : UniformIntegrable f p μ)
(hfg : ∀ n, f n =ᵐ[μ] g n) : UniformIntegrable g p μ := by |
obtain ⟨hfm, hunif, C, hC⟩ := hf
refine ⟨fun i => (hfm i).congr (hfg i), (unifIntegrable_congr_ae hfg).1 hunif, C, fun i => ?_⟩
rw [← snorm_congr_ae (hfg i)]
exact hC 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.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 461 | 474 | theorem orthogonalProjection_eq_circumcenter_of_exists_dist_eq {n : ℕ} (s : Simplex ℝ P n) {p : P}
(hr : ∃ r, ∀ i, dist (s.points i) p = r) :
↑(s.orthogonalProjectionSpan p) = s.circumcenter := by |
change ∃ r : ℝ, ∀ i, (fun x => dist x p = r) (s.points i) at hr
have hr : ∃ (r : ℝ), ∀ (a : P),
a ∈ Set.range (fun (i : Fin (n + 1)) => s.points i) → dist a p = r := by
cases' hr with r hr
use r
refine Set.forall_mem_range.mpr ?_
exact hr
rw [exists_dist_eq_iff_exists_dist_orthogonalProject... |
/-
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
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "lea... | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 190 | 200 | theorem injective_toCircle (hT : T ≠ 0) : Function.Injective (@toCircle T) := by |
intro a b h
induction a using QuotientAddGroup.induction_on'
induction b using QuotientAddGroup.induction_on'
simp_rw [toCircle_apply_mk] at h
obtain ⟨m, hm⟩ := expMapCircle_eq_expMapCircle.mp h.symm
rw [QuotientAddGroup.eq]; simp_rw [AddSubgroup.mem_zmultiples_iff, zsmul_eq_mul]
use m
field_simp at hm... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.legendre_sym... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 301 | 308 | theorem quadraticChar_neg_one [DecidableEq F] (hF : ringChar F ≠ 2) :
quadraticChar F (-1) = χ₄ (Fintype.card F) := by |
have h := quadraticChar_eq_pow_of_char_ne_two hF (neg_ne_zero.mpr one_ne_zero)
rw [h, χ₄_eq_neg_one_pow (FiniteField.odd_card_of_char_ne_two hF)]
set n := Fintype.card F / 2
cases' Nat.even_or_odd n with h₂ h₂
· simp only [Even.neg_one_pow h₂, eq_self_iff_true, if_true]
· simp only [Odd.neg_one_pow h₂, Rin... |
/-
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.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf... | Mathlib/Data/Nat/WithBot.lean | 43 | 49 | theorem add_eq_two_iff {n m : WithBot ℕ} :
n + m = 2 ↔ n = 0 ∧ m = 2 ∨ n = 1 ∧ m = 1 ∨ n = 2 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_two_iff
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 747 | 749 | theorem map_fst_darts_append {u v : V} (p : G.Walk u v) :
p.darts.map (·.fst) ++ [v] = p.support := by |
induction p <;> simp! [*]
|
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#a... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 237 | 241 | theorem TendstoInMeasure.exists_seq_tendstoInMeasure_atTop {u : Filter ι} [NeBot u]
[IsCountablyGenerated u] {f : ι → α → E} {g : α → E} (hfg : TendstoInMeasure μ f u g) :
∃ ns : ℕ → ι, TendstoInMeasure μ (fun n => f (ns n)) atTop g := by |
obtain ⟨ns, h_tendsto_ns⟩ : ∃ ns : ℕ → ι, Tendsto ns atTop u := exists_seq_tendsto u
exact ⟨ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@... | Mathlib/Topology/Basic.lean | 742 | 748 | theorem interior_frontier (h : IsClosed s) : interior (frontier s) = ∅ := by |
have A : frontier s = s \ interior s := h.frontier_eq
have B : interior (frontier s) ⊆ interior s := by rw [A]; exact interior_mono diff_subset
have C : interior (frontier s) ⊆ frontier s := interior_subset
have : interior (frontier s) ⊆ interior s ∩ (s \ interior s) :=
subset_inter B (by simpa [A] using C... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 55 | 60 | theorem hexagon_forward (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).hom ≫ (braiding X _).hom ≫ (α_ Y Z X).hom =
(braiding X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (braiding X Z).hom := by |
apply TensorProduct.ext_threefold
intro x y z
rfl
|
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 411 | 424 | theorem noncommProd_union_of_disjoint [DecidableEq α] {s t : Finset α} (h : Disjoint s t)
(f : α → β) (comm : { x | x ∈ s ∪ t }.Pairwise fun a b => Commute (f a) (f b)) :
noncommProd (s ∪ t) f comm =
noncommProd s f (comm.mono <| coe_subset.2 subset_union_left) *
noncommProd t f (comm.mono <| coe_... |
obtain ⟨sl, sl', rfl⟩ := exists_list_nodup_eq s
obtain ⟨tl, tl', rfl⟩ := exists_list_nodup_eq t
rw [List.disjoint_toFinset_iff_disjoint] at h
calc noncommProd (List.toFinset sl ∪ List.toFinset tl) f comm
= noncommProd ⟨↑(sl ++ tl), Multiset.coe_nodup.2 (sl'.append tl' h)⟩ f
(by convert comm; simp... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 105 | 123 | theorem StableUnderBaseChange.Γ_pullback_fst (hP : StableUnderBaseChange @P) (hP' : RespectsIso @P)
{X Y S : Scheme} [IsAffine X] [IsAffine Y] [IsAffine S] (f : X ⟶ S) (g : Y ⟶ S)
(H : P (Scheme.Γ.map g.op)) : P (Scheme.Γ.map (pullback.fst : pullback f g ⟶ _).op) := by |
-- Porting note (#11224): change `rw` to `erw`
erw [← PreservesPullback.iso_inv_fst AffineScheme.forgetToScheme (AffineScheme.ofHom f)
(AffineScheme.ofHom g)]
rw [op_comp, Functor.map_comp, hP'.cancel_right_isIso, AffineScheme.forgetToScheme_map]
have :=
_root_.congr_arg Quiver.Hom.unop
(Preser... |
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprov... | Mathlib/Algebra/CharP/MixedCharZero.lean | 178 | 189 | theorem PNat.isUnit_natCast [h : Fact (∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I))]
(n : ℕ+) : IsUnit (n : R) := by |
-- `n : R` is a unit iff `(n)` is not a proper ideal in `R`.
rw [← Ideal.span_singleton_eq_top]
-- So by contrapositive, we should show the quotient does not have characteristic zero.
apply not_imp_comm.mp (h.elim (Ideal.span {↑n}))
intro h_char_zero
-- In particular, the image of `n` in the quotient shoul... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 1,112 | 1,116 | theorem mkContinuousMultilinear_norm_le' (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) (C : ℝ)
(H : ∀ m₁ m₂, ‖f m₁ m₂‖ ≤ (C * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖) :
‖mkContinuousMultilinear f C H‖ ≤ max C 0 := by |
dsimp only [mkContinuousMultilinear]
exact mkContinuous_norm_le _ (le_max_right _ _) _
|
/-
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 | 526 | 530 | theorem isRotated_iff_mem_map_range : l ~r l' ↔ l' ∈ (List.range (l.length + 1)).map l.rotate := by |
simp_rw [mem_map, mem_range, isRotated_iff_mod]
exact
⟨fun ⟨n, hn, h⟩ => ⟨n, Nat.lt_succ_of_le hn, h⟩,
fun ⟨n, hn, h⟩ => ⟨n, Nat.le_of_lt_succ hn, h⟩⟩
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
#align_import ring_theory.fractional_ideal from "lean... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 611 | 613 | theorem mul_le {I J K : FractionalIdeal S P} : I * J ≤ K ↔ ∀ i ∈ I, ∀ j ∈ J, i * j ∈ K := by |
simp only [mul_def]
exact Submodule.mul_le
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 506 | 507 | theorem dist_div_eq_dist_mul_left (a b c : E) : dist (a / b) c = dist a (c * b) := by |
rw [← dist_mul_right _ _ b, div_mul_cancel]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Atlas
/-!
# Unique derivative sets in manifolds
In this file, we prove various properties of unique... | Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean | 39 | 49 | theorem UniqueMDiffWithinAt.image_denseRange (hs : UniqueMDiffWithinAt I s x)
{f : M → M'} {f' : E →L[𝕜] E'} (hf : HasMFDerivWithinAt I I' f s x f')
(hd : DenseRange f') : UniqueMDiffWithinAt I' (f '' s) (f x) := by |
/- Rewrite in coordinates, apply `HasFDerivWithinAt.uniqueDiffWithinAt`. -/
have := hs.inter' <| hf.1 (extChartAt_source_mem_nhds I' (f x))
refine (((hf.2.mono ?sub1).uniqueDiffWithinAt this hd).mono ?sub2).congr_pt ?pt
case pt => simp only [mfld_simps]
case sub1 => mfld_set_tac
case sub2 =>
rintro _ ⟨... |
/-
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 | 2,616 | 2,617 | theorem filter_comm (s : Finset α) : (s.filter p).filter q = (s.filter q).filter p := by |
simp_rw [filter_filter, and_comm]
|
/-
Copyright (c) 2022 Joseph Hua. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Rob Lewis, Joseph Hua
-/
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.endofunctor.algebra ... | Mathlib/CategoryTheory/Endofunctor/Algebra.lean | 446 | 448 | theorem Algebra.homEquiv_naturality_str (adj : F ⊣ G) (A₁ A₂ : Algebra F) (f : A₁ ⟶ A₂) :
(adj.homEquiv A₁.a A₁.a) A₁.str ≫ G.map f.f = f.f ≫ (adj.homEquiv A₂.a A₂.a) A₂.str := by |
rw [← Adjunction.homEquiv_naturality_right, ← Adjunction.homEquiv_naturality_left, f.h]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,127 | 1,130 | theorem re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two (x y : E) :
re ⟪x, y⟫ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 := by |
rw [@norm_sub_mul_self 𝕜]
ring
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 660 | 663 | theorem apply_mem_span_image_iff_mem_span [RingHomSurjective σ₁₂] {f : F} {x : M}
{s : Set M} (hf : Function.Injective f) :
f x ∈ Submodule.span R₂ (f '' s) ↔ x ∈ Submodule.span R s := by |
rw [← Submodule.mem_comap, ← Submodule.map_span, Submodule.comap_map_eq_of_injective hf]
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENa... | Mathlib/RingTheory/Multiplicity.lean | 591 | 598 | theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x ∈ s, f x) = ∑ x ∈ s, multiplicity p (f x) := by |
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 137 | 138 | theorem ball_div_singleton : ball x δ / {y} = ball (x / y) δ := by |
simp_rw [div_eq_mul_inv, inv_singleton, ball_mul_singleton]
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,094 | 2,095 | theorem subset_pair_iff : s ⊆ {a, b} ↔ ∀ x ∈ s, x = a ∨ x = b := by |
simp [subset_def]
|
/-
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 | 321 | 322 | theorem zpow_neg_mul_zpow_self (n : ℤ) {A : M} (h : IsUnit A.det) : A ^ (-n) * A ^ n = 1 := by |
rw [zpow_neg h, nonsing_inv_mul _ (h.det_zpow _)]
|
/-
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 | 57 | 57 | theorem eq_true_eq_not_eq_false (b : Bool) : (¬b = false) = (b = true) := by | simp
|
/-
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 | 1,660 | 1,661 | theorem EventuallyEq.sub_eq [AddGroup β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) :
f - g =ᶠ[l] 0 := by | simpa using ((EventuallyEq.refl l f).sub h).symm
|
/-
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.Control.Monad.Basic
import Mathlib.Data.Part
import Mathlib.Order.Chain
import Mathlib.Order.Hom.Order
import Mathlib.Algebra.Order.Ring.Nat
#align_import o... | Mathlib/Order/OmegaCompletePartialOrder.lean | 554 | 556 | theorem bot_continuous : Continuous (⊥ : α →o β) := by |
rw [← sSup_empty]
exact sSup_continuous _ fun f hf => hf.elim
|
/-
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.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b6093548394... | Mathlib/SetTheory/Ordinal/Principal.lean | 282 | 290 | theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o := by |
rcases lt_or_eq_of_le ho with (ho | rfl)
· have h₂ : succ (1 : Ordinal) = 2 := by simp
rw [← h₂, lt_succ_iff] at ho
rcases lt_or_eq_of_le ho with (ho | rfl)
· rw [lt_one_iff_zero.1 ho]
exact principal_zero
· exact principal_mul_one
· exact principal_mul_two
|
/-
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.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,116 | 1,129 | theorem prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : α → Cardinal.{v}) :
prod f = Cardinal.lift.{u} (∏ i, f i) := by |
revert f
refine Fintype.induction_empty_option ?_ ?_ ?_ α (h_fintype := h)
· intro α β hβ e h f
letI := Fintype.ofEquiv β e.symm
rw [← e.prod_comp f, ← h]
exact mk_congr (e.piCongrLeft _).symm
· intro f
rw [Fintype.univ_pempty, Finset.prod_empty, lift_one, Cardinal.prod, mk_eq_one]
· intro α ... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 389 | 392 | theorem cos_oangle_sub_left_mul_norm_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : Real.Angle.cos (o.oangle (x - y) x) * ‖x - y‖ = ‖x‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
exact (-o).cos_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two h
|
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import ... | Mathlib/Data/Nat/Choose/Multinomial.lean | 284 | 296 | theorem multinomial_coe_fill_of_not_mem {m : Fin (n + 1)} {s : Sym α (n - m)} {x : α} (hx : x ∉ s) :
(fill x m s : Multiset α).multinomial = n.choose m * (s : Multiset α).multinomial := by |
rw [Multiset.multinomial_filter_ne x]
rw [← mem_coe] at hx
refine congrArg₂ _ ?_ ?_
· rw [card_coe, count_coe_fill_self_of_not_mem hx]
· refine congrArg _ ?_
rw [coe_fill, coe_replicate, Multiset.filter_add]
rw [Multiset.filter_eq_self.mpr]
· rw [add_right_eq_self]
rw [Multiset.filter_eq_ni... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 124 | 126 | theorem rank_add_rank_le_rank_prod [Nontrivial R] :
Module.rank R M + Module.rank R M₁ ≤ Module.rank R (M × M₁) := by |
convert ← lift_rank_add_lift_rank_le_rank_prod R M M₁ <;> apply lift_id
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Rin... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 619 | 624 | theorem Zlattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[FiniteDimensional ℝ E] {L : AddSubgroup E} [DiscreteTopology L] [IsZlattice ℝ L] [Finite ι]
(b : Basis ι ℤ L) [MeasurableSpace E] [OpensMeasurableSpace E] (μ : MeasureTheory.Measure E) :
MeasureTheory.IsAddFundamen... |
convert Zspan.isAddFundamentalDomain (b.ofZlatticeBasis ℝ) μ
all_goals exact (b.ofZlatticeBasis_span ℝ).symm
|
/-
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, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,893 | 1,905 | theorem count_factors_eq_find_of_dvd_pow [DecidableEq (Associates α)] {a p : Associates α}
(hp : Irreducible p) [∀ n : ℕ, Decidable (a ∣ p ^ n)] {n : ℕ} (h : a ∣ p ^ n) :
@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩ = p.count a.factors := by |
apply le_antisymm
· refine Nat.find_le ⟨1, ?_⟩
rw [mul_one]
symm
exact eq_pow_count_factors_of_dvd_pow hp h
· have hph := pow_ne_zero (@Nat.find (fun n => a ∣ p ^ n) _ ⟨n, h⟩) hp.ne_zero
cases' subsingleton_or_nontrivial α with hα hα
· simp [eq_iff_true_of_subsingleton] at hph
convert cou... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.