Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
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
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 89 | 90 | theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neumann bou... | Mathlib/Analysis/Convex/GaugeRescale.lean | 63 | 67 | theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gauge t (gaugeRescale s t x) = gauge s x := by |
rcases eq_or_ne x 0 with rfl | hx
· simp
· exact gauge_gaugeRescale' s ((gauge_pos hta htb).2 hx).ne'
|
/-
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.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 77 | 90 | theorem IsTopologicalBasis.insert_empty {s : Set (Set α)} (h : IsTopologicalBasis s) :
IsTopologicalBasis (insert ∅ s) := by |
refine ⟨?_, by rw [sUnion_insert, empty_union, h.sUnion_eq], ?_⟩
· rintro t₁ (rfl | h₁) t₂ (rfl | h₂) x ⟨hx₁, hx₂⟩
· cases hx₁
· cases hx₁
· cases hx₂
· obtain ⟨t₃, h₃, hs⟩ := h.exists_subset_inter _ h₁ _ h₂ x ⟨hx₁, hx₂⟩
exact ⟨t₃, .inr h₃, hs⟩
· rw [h.eq_generateFrom]
refine le_antisym... |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 86 | 89 | theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
∃ b, 0 < b ∧ b < a ∧ x ∈ b • s := by |
obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
exact ⟨b, hb, hba, hx⟩
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 83 | 87 | theorem permutationsAux2_comp_append {t : α} {ts ys : List α} {r : List β} (f : List α → β) :
((permutationsAux2 t [] r ys) fun x => f (x ++ ts)).2 = (permutationsAux2 t ts r ys f).2 := by |
induction' ys with ys_hd _ ys_ih generalizing f
· simp
· simp [ys_ih fun xs => f (ys_hd :: xs)]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
/-!
# Direct... | Mathlib/Order/Directed.lean | 66 | 67 | theorem directedOn_range {f : ι → α} : Directed r f ↔ DirectedOn r (Set.range f) := by |
simp_rw [Directed, DirectedOn, Set.forall_mem_range, Set.exists_range_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 60 | 60 | theorem mem_support_map_iff : b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b := 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, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 76 | 78 | theorem range_eq_map [RingHomSurjective τ₁₂] (f : F) : range f = map f ⊤ := by |
ext
simp
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
/-!
# "Gauge rescale" homeomorphism between convex sets
Given two convex von Neumann bou... | Mathlib/Analysis/Convex/GaugeRescale.lean | 69 | 73 | theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by |
by_cases hx : gauge t x = 0
· simp [gaugeRescale, hx, gauge_nonneg]
· exact (gauge_gaugeRescale' s hx).le
|
/-
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 | 92 | 100 | theorem IsCut.congr [IsCut cmp cut] [TransCmp cmp] (H : cmp x y = .eq) : cut x = cut y := by |
cases ey : cut y
· exact IsCut.le_lt_trans (fun h => nomatch H.symm.trans <| OrientedCmp.cmp_eq_gt.1 h) ey
· cases ex : cut x
· exact IsCut.le_lt_trans (fun h => nomatch H.symm.trans h) ex |>.symm.trans ey
· rfl
· refine IsCut.le_gt_trans (cmp := cmp) (fun h => ?_) ex |>.symm.trans ey
cases H.s... |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
impor... | Mathlib/CategoryTheory/GlueData.lean | 93 | 95 | theorem t'_iji (i j : D.J) : D.t' i j i = pullback.fst ≫ D.t i j ≫ inv pullback.snd := by |
rw [← Category.assoc, ← D.t_fac]
simp
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 117 | 118 | theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by | simp only [Neg.neg, RatFunc.neg]
|
/-
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 | 126 | 129 | theorem relindex_mul_relindex (hHK : H ≤ K) (hKL : K ≤ L) :
H.relindex K * K.relindex L = H.relindex L := by |
rw [← relindex_subgroupOf hKL]
exact relindex_mul_index fun x hx => hHK hx
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
/-!
# Cycles of a li... | Mathlib/Data/List/Cycle.lean | 76 | 84 | theorem mem_of_nextOr_ne {xs : List α} {x d : α} (h : nextOr xs x d ≠ d) : x ∈ xs := by |
induction' xs with y ys IH
· simp at h
cases' ys with z zs
· simp at h
· by_cases hx : x = y
· simp [hx]
· rw [nextOr_cons_of_ne _ _ _ _ hx] at h
simpa [hx] using IH h
|
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e962... | Mathlib/Order/Height.lean | 93 | 106 | theorem exists_chain_of_le_chainHeight {n : ℕ} (hn : ↑n ≤ s.chainHeight) :
∃ l ∈ s.subchain, length l = n := by |
rcases (le_top : s.chainHeight ≤ ⊤).eq_or_lt with ha | ha <;>
rw [chainHeight_eq_iSup_subtype] at ha
· obtain ⟨_, ⟨⟨l, h₁, h₂⟩, rfl⟩, h₃⟩ :=
not_bddAbove_iff'.mp (WithTop.iSup_coe_eq_top.1 ha) n
exact ⟨l.take n, ⟨h₁.take _, fun x h ↦ h₂ _ <| take_subset _ _ h⟩,
(l.length_take n).trans <| min_eq... |
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 149 | 150 | theorem repr_self_apply (j) [Decidable (i = j)] : b.repr (b i) j = if i = j then 1 else 0 := by |
rw [repr_self, Finsupp.single_apply]
|
/-
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 | 70 | 73 | theorem noncommFoldr_eq_foldr (s : Multiset α) (h : LeftCommutative f) (b : β) :
noncommFoldr f s (fun x _ y _ _ => h x y) b = foldr f h b s := by |
induction s using Quotient.inductionOn
simp
|
/-
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 | 54 | 56 | theorem pythagoreanTriple_comm {x y z : ℤ} : PythagoreanTriple x y z ↔ PythagoreanTriple y x z := by |
delta PythagoreanTriple
rw [add_comm]
|
/-
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.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import... | Mathlib/RingTheory/Nilpotent/Lemmas.lean | 94 | 97 | theorem isNilpotent_mulRight_iff (a : A) : IsNilpotent (mulRight R a) ↔ IsNilpotent a := by |
constructor <;> rintro ⟨n, hn⟩ <;> use n <;>
simp only [mulRight_eq_zero_iff, pow_mulRight] at hn ⊢ <;>
exact hn
|
/-
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 | 60 | 61 | theorem PythagoreanTriple.zero : PythagoreanTriple 0 0 0 := by |
simp only [PythagoreanTriple, zero_mul, zero_add]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 102 | 110 | theorem range_vecMulLinear (M : Matrix m n R) :
LinearMap.range M.vecMulLinear = span R (range M) := by |
letI := Classical.decEq m
simp_rw [range_eq_map, ← iSup_range_stdBasis, Submodule.map_iSup, range_eq_map, ←
Ideal.span_singleton_one, Ideal.span, Submodule.map_span, image_image, image_singleton,
Matrix.vecMulLinear_apply, iSup_span, range_eq_iUnion, iUnion_singleton_eq_range,
LinearMap.stdBasis, coe_s... |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 116 | 117 | theorem zero_mem_iff {a : A} : (0 : R) ∈ σ a ↔ ¬IsUnit a := by |
rw [mem_iff, map_zero, zero_sub, IsUnit.neg_iff]
|
/-
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.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_prod... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 61 | 66 | theorem OrthonormalBasis.volume_parallelepiped (b : OrthonormalBasis ι ℝ F) :
volume (parallelepiped b) = 1 := by |
haveI : Fact (finrank ℝ F = finrank ℝ F) := ⟨rfl⟩
let o := (stdOrthonormalBasis ℝ F).toBasis.orientation
rw [← o.measure_eq_volume]
exact o.measure_orthonormalBasis b
|
/-
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.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 96 | 100 | theorem measure_diff_null (ht : μ t = 0) : μ (s \ t) = μ s :=
(measure_mono diff_subset).antisymm <| calc
μ s ≤ μ (s ∩ t) + μ (s \ t) := measure_le_inter_add_diff _ _ _
_ ≤ μ t + μ (s \ t) := by | gcongr; apply inter_subset_right
_ = μ (s \ t) := by simp [ht]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 100 | 102 | theorem Filter.Eventually.intCast_atBot [StrictOrderedRing R] [Archimedean R] {p : R → Prop}
(h : ∀ᶠ (x:R) in atBot, p x) : ∀ᶠ (n:ℤ) in atBot, p n := by |
rw [← Int.comap_cast_atBot (R := R)]; exact h.comap _
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 95 | 97 | theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by |
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
#align_import measure_theory.integral.layercake from "leanprover-commu... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 86 | 94 | theorem lintegral_rpow_eq_lintegral_meas_lt_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) := by |
rw [lintegral_rpow_eq_lintegral_meas_le_mul μ f_nn f_mble p_pos]
apply congr_arg fun z => ENNReal.ofReal p * z
apply lintegral_congr_ae
filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
with t ht
rw [ht]
|
/-
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.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
/-!
# Convex join
... | Mathlib/Analysis/Convex/Join.lean | 36 | 37 | theorem mem_convexJoin : x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b := by |
simp [convexJoin]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
/-!
# Lem... | Mathlib/Algebra/CharZero/Quotient.lean | 42 | 47 | theorem nsmul_mem_zmultiples_iff_exists_sub_div {r : R} {n : ℕ} (hn : n ≠ 0) :
n • r ∈ AddSubgroup.zmultiples p ↔
∃ k : Fin n, r - (k : ℕ) • (p / n : R) ∈ AddSubgroup.zmultiples p := by |
rw [← natCast_zsmul r, zsmul_mem_zmultiples_iff_exists_sub_div (Int.natCast_ne_zero.mpr hn),
Int.cast_natCast]
rfl
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu, Anne Baanen
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Lo... | Mathlib/RingTheory/Localization/Module.lean | 56 | 71 | theorem LinearIndependent.of_isLocalizedModule {ι : Type*} {v : ι → M}
(hv : LinearIndependent R v) : LinearIndependent Rₛ (f ∘ v) := by |
rw [linearIndependent_iff'] at hv ⊢
intro t g hg i hi
choose! a g' hg' using IsLocalization.exist_integer_multiples S t g
have h0 : f (∑ i ∈ t, g' i • v i) = 0 := by
apply_fun ((a : R) • ·) at hg
rw [smul_zero, Finset.smul_sum] at hg
rw [map_sum, ← hg]
refine Finset.sum_congr rfl fun i hi => ?_... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#al... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 85 | 94 | theorem spanEval_ne_top : spanEval k ≠ ⊤ := by |
rw [Ideal.ne_top_iff_one, spanEval, Ideal.span, ← Set.image_univ,
Finsupp.mem_span_image_iff_total]
rintro ⟨v, _, hv⟩
replace hv := congr_arg (toSplittingField k v.support) hv
rw [AlgHom.map_one, Finsupp.total_apply, Finsupp.sum, AlgHom.map_sum, Finset.sum_eq_zero] at hv
· exact zero_ne_one hv
intro j ... |
/-
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.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
import Mathlib.Analysis.Complex.CauchyIntegral
#align_impo... | Mathlib/Analysis/Complex/RemovableSingularity.lean | 34 | 43 | theorem analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt {f : ℂ → E} {c : ℂ}
(hd : ∀ᶠ z in 𝓝[≠] c, DifferentiableAt ℂ f z) (hc : ContinuousAt f c) : AnalyticAt ℂ f c := by |
rcases (nhdsWithin_hasBasis nhds_basis_closedBall _).mem_iff.1 hd with ⟨R, hR0, hRs⟩
lift R to ℝ≥0 using hR0.le
replace hc : ContinuousOn f (closedBall c R) := by
refine fun z hz => ContinuousAt.continuousWithinAt ?_
rcases eq_or_ne z c with (rfl | hne)
exacts [hc, (hRs ⟨hz, hne⟩).continuousAt]
exa... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.Spec... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 111 | 114 | theorem HasStrictDerivAt.log (hf : HasStrictDerivAt f f' x) (hx : f x ≠ 0) :
HasStrictDerivAt (fun y => log (f y)) (f' / f x) x := by |
rw [div_eq_inv_mul]
exact (hasStrictDerivAt_log hx).comp x hf
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 100 | 101 | theorem Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _[0] ⟶ X _[0]) = 0 := by |
simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, Q, P_f_0_eq, sub_self]
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 77 | 78 | theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty := by | aesop_cat
|
/-
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.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 90 | 101 | theorem cos_eq_cos_iff {x y : ℂ} : cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x :=
calc
cos x = cos y ↔ cos x - cos y = 0 := sub_eq_zero.symm
_ ↔ -2 * sin ((x + y) / 2) * sin ((x - y) / 2) = 0 := by | rw [cos_sub_cos]
_ ↔ sin ((x + y) / 2) = 0 ∨ sin ((x - y) / 2) = 0 := by simp [(by norm_num : (2 : ℂ) ≠ 0)]
_ ↔ sin ((x - y) / 2) = 0 ∨ sin ((x + y) / 2) = 0 := or_comm
_ ↔ (∃ k : ℤ, y = 2 * k * π + x) ∨ ∃ k : ℤ, y = 2 * k * π - x := by
apply or_congr <;>
field_simp [sin_eq_zero_iff, (by norm... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 119 | 122 | theorem cramer_row_self (i : n) (h : ∀ j, b j = A j i) : A.cramer b = Pi.single i A.det := by |
rw [← transpose_transpose A, det_transpose]
convert cramer_transpose_row_self Aᵀ i
exact funext 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.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 30 | 32 | theorem preimage_Ici (e : α ≃o β) (b : β) : e ⁻¹' Ici b = Ici (e.symm b) := by |
ext x
simp [← e.le_iff_le]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 107 | 107 | theorem sup_sdiff_inf (x y : α) : x \ y ⊔ x ⊓ y = x := by | rw [sup_comm, sup_inf_sdiff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 111 | 112 | theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by |
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
|
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 41 | 44 | theorem lipschitzWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{f : α → β} : LipschitzWith K f ↔ ∀ x y, dist (f x) (f y) ≤ K * dist x y := by |
simp only [LipschitzWith, edist_nndist, dist_nndist]
norm_cast
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 97 | 98 | theorem sqrt_mul (x y : ℝ≥0) : sqrt (x * y) = sqrt x * sqrt y := by |
rw [sqrt_eq_iff_eq_sq, mul_pow, sq_sqrt, sq_sqrt]
|
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Ashvni Narayanan, Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.ZMod.Basic
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Rin... | Mathlib/Data/ZMod/Units.lean | 38 | 63 | theorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :
Function.Surjective (unitsMap h) := by |
suffices ∀ x : ℕ, x.Coprime n → ∃ k : ℕ, (x + k * n).Coprime m by
intro x
have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)
refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩
have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩
simp [unitsMap_def]
intro x hx
let ps := m.primeFacto... |
/-
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.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 122 | 130 | theorem Subpresheaf.eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι := by |
constructor
· rintro rfl
infer_instance
· intro H
ext U x
apply iff_true_iff.mpr
rw [← IsIso.inv_hom_id_apply (G.ι.app U) x]
exact ((inv (G.ι.app U)) x).2
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 78 | 80 | theorem encard_eq_coe_toFinset_card (s : Set α) [Fintype s] : encard s = s.toFinset.card := by |
have h := toFinite s
rw [h.encard_eq_coe_toFinset_card, toFinite_toFinset]
|
/-
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.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This module defin... | Mathlib/GroupTheory/GroupAction/Period.lean | 117 | 120 | theorem period_bounded_of_exponent_pos (exp_pos : 0 < Monoid.exponent M) (m : M) :
BddAbove (Set.range (fun a : α => period m a)) := by |
use Monoid.exponent M
simpa [upperBounds] using period_le_exponent exp_pos _
|
/-
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.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.Basic
#align_import algebra.star.pointwise from "leanprover-communit... | Mathlib/Algebra/Star/Pointwise.lean | 115 | 117 | theorem star_singleton {β : Type*} [InvolutiveStar β] (x : β) : ({x} : Set β)⋆ = {x⋆} := by |
ext1 y
rw [mem_star, mem_singleton_iff, mem_singleton_iff, star_eq_iff_star_eq, eq_comm]
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# The multiplicative and additive convolution of measures
In... | Mathlib/MeasureTheory/Group/Convolution.lean | 65 | 67 | theorem zero_mconv (μ : Measure M) : μ ∗ (0 : Measure M) = (0 : Measure M) := by |
unfold mconv
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, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.seto... | Mathlib/Data/Setoid/Partition.lean | 67 | 71 | theorem classes_ker_subset_fiber_set {β : Type*} (f : α → β) :
(Setoid.ker f).classes ⊆ Set.range fun y => { x | f x = y } := by |
rintro s ⟨x, rfl⟩
rw [Set.mem_range]
exact ⟨f x, rfl⟩
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 118 | 118 | theorem cgf_zero_fun : cgf 0 μ t = log (μ Set.univ).toReal := by | simp only [cgf, mgf_zero_fun]
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprove... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 116 | 117 | theorem exists_conts_b_of_denom {B : K} (nth_denom_eq : g.denominators n = B) :
∃ conts, g.continuants n = conts ∧ conts.b = B := by | simpa
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Topology.Algebra.Polynomial
... | Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean | 91 | 94 | theorem card_rootSet_le_derivative {F : Type*} [CommRing F] [Algebra F ℝ] (p : F[X]) :
Fintype.card (p.rootSet ℝ) ≤ Fintype.card (p.derivative.rootSet ℝ) + 1 := by |
simpa only [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe, derivative_map] using
card_roots_toFinset_le_derivative (p.map (algebraMap F ℝ))
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8e... | Mathlib/Algebra/Polynomial/Expand.lean | 48 | 49 | theorem expand_eq_sum {f : R[X]} : expand R p f = f.sum fun e a => C a * (X ^ p) ^ e := by |
simp [expand, eval₂]
|
/-
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 | 101 | 103 | theorem inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ := by |
simp_rw [cthickening, ← infEdist_inv]
rfl
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 77 | 79 | theorem divOf_zero (x : k[G]) : x /ᵒᶠ 0 = x := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
simp only [AddMonoidAlgebra.divOf_apply, zero_add]
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | Mathlib/Data/Vector/Snoc.lean | 154 | 159 | theorem mapAccumr₂_snoc (f : α → β → σ → σ × φ) (x : α) (y : β) :
mapAccumr₂ f (xs.snoc x) (ys.snoc y) c
= let q := f x y c
let r := mapAccumr₂ f xs ys q.1
(r.1, r.2.snoc q.2) := by |
induction xs, ys using Vector.inductionOn₂ <;> simp_all
|
/-
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 | 62 | 63 | theorem sub_smul_slope_vadd (f : k → PE) (a b : k) : (b - a) • slope f a b +ᵥ f a = f b := by |
rw [sub_smul_slope, vsub_vadd]
|
/-
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.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 58 | 59 | theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) := by |
simp [← Ioi_inter_Iic]
|
/-
Copyright (c) 2023 Jonas van der Schaaf. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Christian Merten, Jonas van der Schaaf
-/
import Mathlib.AlgebraicGeometry.OpenImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib... | Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean | 79 | 89 | theorem spec_of_surjective {R S : CommRingCat} (f : R ⟶ S) (h : Function.Surjective f) :
IsClosedImmersion (Scheme.specMap f) where
base_closed := PrimeSpectrum.closedEmbedding_comap_of_surjective _ _ h
surj_on_stalks x := by |
erw [← localRingHom_comp_stalkIso, CommRingCat.coe_comp, CommRingCat.coe_comp]
apply Function.Surjective.comp (Function.Surjective.comp _ _) _
· exact (ConcreteCategory.bijective_of_isIso (StructureSheaf.stalkIso S x).inv).2
· exact surjective_localRingHom_of_surjective f h x.asIdeal
· let g := (St... |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 118 | 121 | theorem PowerBasis.norm_gen_eq_coeff_zero_minpoly (pb : PowerBasis R S) :
norm R pb.gen = (-1) ^ pb.dim * coeff (minpoly R pb.gen) 0 := by |
rw [norm_eq_matrix_det pb.basis, det_eq_sign_charpoly_coeff, charpoly_leftMulMatrix,
Fintype.card_fin]
|
/-
Copyright (c) 2019 mathlib community. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Wojciech Nawrocki
-/
import Batteries.Data.RBMap.Basic
import Mathlib.Init.Data.Nat.Notation
import Mathlib.Mathport.Rename
import Mathlib.Tactic.TypeStar
import Mat... | Mathlib/Data/Tree/Basic.lean | 94 | 96 | theorem numLeaves_pos (x : Tree α) : 0 < x.numLeaves := by |
rw [numLeaves_eq_numNodes_succ]
exact x.numNodes.zero_lt_succ
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 87 | 89 | theorem HasDerivWithinAt.smul (hc : HasDerivWithinAt c c' s x) (hf : HasDerivWithinAt f f' s x) :
HasDerivWithinAt (fun y => c y • f y) (c x • f' + c' • f x) s x := by |
simpa using (HasFDerivWithinAt.smul hc hf).hasDerivWithinAt
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Lattice
#align_import data.set.accumulate from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
/-!
# Accumulat... | Mathlib/Data/Set/Accumulate.lean | 56 | 61 | theorem iUnion_accumulate [Preorder α] : ⋃ x, Accumulate s x = ⋃ x, s x := by |
apply Subset.antisymm
· simp only [subset_def, mem_iUnion, exists_imp, mem_accumulate]
intro z x x' ⟨_, hz⟩
exact ⟨x', hz⟩
· exact iUnion_mono fun i => subset_accumulate
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Eric Wieser
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathl... | Mathlib/RingTheory/MatrixAlgebra.lean | 113 | 121 | theorem right_inv (M : Matrix n n A) : (toFunAlgHom R A n) (invFun R A n M) = M := by |
simp only [invFun, AlgHom.map_sum, stdBasisMatrix, apply_ite ↑(algebraMap R A), smul_eq_mul,
mul_boole, toFunAlgHom_apply, RingHom.map_zero, RingHom.map_one, Matrix.map_apply,
Pi.smul_def]
convert Finset.sum_product (β := Matrix n n A)
conv_lhs => rw [matrix_eq_sum_std_basis M]
refine Finset.sum_congr ... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformFilterBasis
import Mathlib.RingTheory.Valuation.ValuationSubring
#align_i... | Mathlib/Topology/Algebra/Valuation.lean | 37 | 81 | theorem subgroups_basis : RingSubgroupsBasis fun γ : Γ₀ˣ => (v.ltAddSubgroup γ : AddSubgroup R) :=
{ inter := by |
rintro γ₀ γ₁
use min γ₀ γ₁
simp only [ltAddSubgroup, ge_iff_le, Units.min_val, Units.val_le_val, lt_min_iff,
AddSubgroup.mk_le_mk, setOf_subset_setOf, le_inf_iff, and_imp, imp_self, implies_true,
forall_const, and_true]
tauto
mul := by
rintro γ
cases' exists_squa... |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import... | Mathlib/Order/Filter/Cocardinal.lean | 70 | 72 | theorem frequently_cocardinal {p : α → Prop} :
(∃ᶠ x in cocardinal α hreg, p x) ↔ c ≤ # { x | p x } := by |
simp only [Filter.Frequently, eventually_cocardinal, not_not,coe_setOf, not_lt]
|
/-
Copyright (c) 2023 François G. Dorais. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: François G. Dorais
-/
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem ge... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 84 | 87 | theorem get_append_right {a b : ByteArray} (hle : a.size ≤ i) (h : i < (a ++ b).size)
(h' : i - a.size < b.size := Nat.sub_lt_left_of_lt_add hle (size_append .. ▸ h)) :
(a ++ b)[i] = b[i - a.size] := by |
simp [getElem_eq_data_getElem]; exact Array.get_append_right hle
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Valuation
import Mathlib.Topology.Algebra.WithZeroTopology
import Mathlib.Topology.Algebra.UniformField
#align_import topology.algebr... | Mathlib/Topology/Algebra/ValuedField.lean | 117 | 127 | theorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) := by |
rw [continuous_iff_continuousAt]
intro x
rcases eq_or_ne x 0 with (rfl | h)
· rw [ContinuousAt, map_zero, WithZeroTopology.tendsto_zero]
intro γ hγ
rw [Filter.Eventually, Valued.mem_nhds_zero]
use Units.mk0 γ hγ; rfl
· have v_ne : (v x : Γ₀) ≠ 0 := (Valuation.ne_zero_iff _).mpr h
rw [Continuo... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Type... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 112 | 129 | theorem left_unitality (X : Type u) :
(λ_ ((free R).obj X)).hom =
(ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [ε, μ]
let q : X →₀ R := ((λ_ (of R (X →₀ R))).hom) (1 ⊗ₜ[R] Finsupp.single x 1)
cha... |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
/-!
# D... | Mathlib/Data/List/Destutter.lean | 64 | 70 | theorem destutter'_sublist (a) : l.destutter' R a <+ a :: l := by |
induction' l with b l hl generalizing a
· simp
rw [destutter']
split_ifs
· exact Sublist.cons₂ a (hl b)
· exact (hl a).trans ((l.sublist_cons b).cons_cons a)
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
/-!
# Birkhoff average in a normed space
In this file we prove some lemmas about ... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 42 | 44 | theorem dist_birkhoffSum_apply_birkhoffSum (f : α → α) (g : α → E) (n : ℕ) (x : α) :
dist (birkhoffSum f g n (f x)) (birkhoffSum f g n x) = dist (g (f^[n] x)) (g x) := by |
simp only [dist_eq_norm, birkhoffSum_apply_sub_birkhoffSum]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | Mathlib/RingTheory/Localization/InvSubmonoid.lean | 87 | 91 | theorem surj'' (z : S) : ∃ (r : R) (m : M), z = r • (toInvSubmonoid M S m : S) := by |
rcases IsLocalization.surj M z with ⟨⟨r, m⟩, e : z * _ = algebraMap R S r⟩
refine ⟨r, m, ?_⟩
rw [Algebra.smul_def, ← e, mul_assoc]
simp
|
/-
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 | 131 | 133 | theorem rootsOfUnity.coe_pow [CommMonoid R] (ζ : rootsOfUnity k R) (m : ℕ) :
(((ζ ^ m :) : Rˣ) : R) = ((ζ : Rˣ) : R) ^ m := by |
rw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Mario Carneiro, Martin Dvorak
-/
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac... | Mathlib/Data/List/Join.lean | 105 | 109 | theorem take_sum_join' (L : List (List α)) (i : ℕ) :
L.join.take (Nat.sum ((L.map length).take i)) = (L.take i).join := by |
induction L generalizing i
· simp
· cases i <;> simp [take_append, *]
|
/-
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 | 82 | 83 | theorem ringExpChar.eq_one (R : Type*) [NonAssocSemiring R] [CharZero R] : ringExpChar R = 1 := by |
rw [ringExpChar, ringChar.eq_zero, max_eq_right zero_le_one]
|
/-
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.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-commun... | Mathlib/SetTheory/Game/Ordinal.lean | 58 | 59 | theorem toPGame_rightMoves (o : Ordinal) : o.toPGame.RightMoves = PEmpty := by |
rw [toPGame, RightMoves]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import dat... | Mathlib/Data/Nat/Lattice.lean | 59 | 62 | theorem sInf_empty : sInf ∅ = 0 := by |
rw [sInf_eq_zero]
right
rfl
|
/-
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, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-co... | Mathlib/Order/Interval/Set/ProjIcc.lean | 109 | 110 | theorem projIcc_eq_right (h : a < b) : projIcc a b h.le x = ⟨b, right_mem_Icc.2 h.le⟩ ↔ b ≤ x := by |
simp [projIcc, Subtype.ext_iff, max_min_distrib_left, h.le, h.not_le]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 94 | 94 | theorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by | simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int
#align_import data.int.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b... | Mathlib/Data/Int/Cast/Field.lean | 38 | 42 | theorem cast_div [DivisionRing α] {m n : ℤ} (n_dvd : n ∣ m) (hn : (n : α) ≠ 0) :
((m / n : ℤ) : α) = m / n := by |
rcases n_dvd with ⟨k, rfl⟩
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Int.mul_ediv_cancel_left _ this, mul_comm n, Int.cast_mul, mul_div_cancel_right₀ _ hn]
|
/-
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.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community... | Mathlib/Probability/Process/Adapted.lean | 99 | 105 | theorem Filtration.adapted_natural [MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β]
{u : ι → Ω → β} (hum : ∀ i, StronglyMeasurable[m] (u i)) :
Adapted (Filtration.natural u hum) u := by |
intro i
refine StronglyMeasurable.mono ?_ (le_iSup₂_of_le i (le_refl i) le_rfl)
rw [stronglyMeasurable_iff_measurable_separable]
exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).isSeparable_range⟩
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
#align_import group_theory.subgroup.saturated from "leanprover-community/mathlib"@"f7f... | Mathlib/GroupTheory/Subgroup/Saturated.lean | 42 | 56 | theorem saturated_iff_zpow {H : Subgroup G} :
Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := by |
constructor
· intros hH n g hgn
induction' n with n n
· simp only [Int.natCast_eq_zero, Int.ofNat_eq_coe, zpow_natCast] at hgn ⊢
exact hH hgn
· suffices g ^ (n + 1) ∈ H by
refine (hH this).imp ?_ id
simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]
simpa only [inv_mem_iff, zp... |
/-
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.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib... | Mathlib/Data/Set/Pairwise/Basic.lean | 41 | 42 | theorem pairwise_on_bool (hr : Symmetric r) {a b : α} :
Pairwise (r on fun c => cond c a b) ↔ r a b := by | simpa [Pairwise, Function.onFun] using @hr a b
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!... | Mathlib/InformationTheory/Hamming.lean | 122 | 123 | theorem hammingDist_lt_one {x y : ∀ i, β i} : hammingDist x y < 1 ↔ x = y := by |
rw [Nat.lt_one_iff, hammingDist_eq_zero]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 123 | 123 | theorem zsmul (z : ℤ) (x : WittVector p R) : mapFun f (z • x) = z • mapFun f x := by | map_fun_tac
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Combinatorics.Young.YoungDiagram
#align_import combinatorics.young.semistandard_tableau from "leanprover-community/mathlib"@"b363547b3113d350d053abdf288... | Mathlib/Combinatorics/Young/SemistandardTableau.lean | 129 | 133 | theorem row_weak_of_le {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ}
(hj : j1 ≤ j2) (cell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 := by |
cases' eq_or_lt_of_le hj with h h
· rw [h]
· exact T.row_weak h cell
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | Mathlib/Data/Int/CardIntervalMod.lean | 42 | 44 | theorem Ico_filter_dvd_card : ((Ico a b).filter (r ∣ ·)).card =
max (⌈b / (r : ℚ)⌉ - ⌈a / (r : ℚ)⌉) 0 := by |
rw [Ico_filter_dvd_eq _ _ hr, card_map, card_Ico, toNat_eq_max]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 25 | 30 | theorem support_mul [Mul G] [DecidableEq G] (a b : MonoidAlgebra k G) :
(a * b).support ⊆ a.support * b.support := by |
rw [MonoidAlgebra.mul_def]
exact support_sum.trans <| biUnion_subset.2 fun _x hx ↦
support_sum.trans <| biUnion_subset.2 fun _y hy ↦
support_single_subset.trans <| singleton_subset_iff.2 <| mem_image₂_of_mem hx hy
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce... | Mathlib/Algebra/MvPolynomial/Expand.lean | 59 | 61 | theorem expand_one : expand 1 = AlgHom.id R (MvPolynomial σ R) := by |
ext1 f
rw [expand_one_apply, AlgHom.id_apply]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 104 | 104 | theorem prod_univ {s : Set α} : s ×ˢ (univ : Set β) = Prod.fst ⁻¹' s := by | simp [prod_eq]
|
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 45 | 56 | theorem card_div_mul_le_card_div_mul_card_div (A B C : Finset α) :
(A / C).card * B.card ≤ (A / B).card * (B / C).card := by |
rw [← card_product (A / B), ← mul_one ((A / B) ×ˢ (B / C)).card]
refine card_mul_le_card_mul (fun b ac ↦ ac.1 * ac.2 = b) (fun x hx ↦ ?_)
fun x _ ↦ card_le_one_iff.2 fun hu hv ↦
((mem_bipartiteBelow _).1 hu).2.symm.trans ?_
obtain ⟨a, ha, c, hc, rfl⟩ := mem_div.1 hx
refine card_le_card_of_inj_on (fun... |
/-
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.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 78 | 79 | theorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a := by |
rw [← succ_zero, opow_succ]; simp only [opow_zero, one_mul]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathli... | Mathlib/NumberTheory/LucasLehmer.lean | 145 | 145 | theorem sMod_mod (p i : ℕ) : sMod p i % (2 ^ p - 1) = sMod p i := by | cases i <;> simp [sMod]
|
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | Mathlib/Data/List/Sym.lean | 46 | 47 | theorem sym2_eq_nil_iff {xs : List α} : xs.sym2 = [] ↔ xs = [] := by |
cases xs <;> simp [List.sym2]
|
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 35 | 41 | theorem Ioi_add_bij : BijOn (· + d) (Ioi a) (Ioi (a + d)) := by |
refine
⟨fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h =>
?_⟩
obtain ⟨c, rfl⟩ := exists_add_of_le (mem_Ioi.mp h).le
rw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h
exact ⟨a + c, h, by rw [add_right_comm]⟩
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 86 | 89 | theorem single_mem_gradeBy {R} [CommSemiring R] (f : M → ι) (m : M) (r : R) :
Finsupp.single m r ∈ gradeBy R f (f m) := by |
intro x hx
rw [Finset.mem_singleton.mp (Finsupp.support_single_subset hx)]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Order.SuccPred.LinearLocallyFinite
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_sampling from "leanprover-com... | Mathlib/Probability/Martingale/OptionalSampling.lean | 51 | 58 | theorem condexp_stopping_time_ae_eq_restrict_eq_const
[(Filter.atTop : Filter ι).IsCountablyGenerated] (h : Martingale f ℱ μ)
(hτ : IsStoppingTime ℱ τ) [SigmaFinite (μ.trim hτ.measurableSpace_le)] (hin : i ≤ n) :
μ[f n|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] f i := by |
refine Filter.EventuallyEq.trans ?_ (ae_restrict_of_ae (h.condexp_ae_eq hin))
refine condexp_ae_eq_restrict_of_measurableSpace_eq_on hτ.measurableSpace_le (ℱ.le i)
(hτ.measurableSet_eq' i) fun t => ?_
rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 90 | 92 | theorem multiplicity_eq_factorization {n p : ℕ} (pp : p.Prime) (hn : n ≠ 0) :
multiplicity p n = n.factorization p := by |
simp [factorization, pp, padicValNat_def' pp.ne_one hn.bot_lt]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.Opposite
import Mathlib.CategoryTheory.Limits.Opposites
#align_import category_... | Mathlib/CategoryTheory/Abelian/Opposite.lean | 124 | 126 | theorem kernel.ι_unop :
(kernel.ι g.unop).op = eqToHom (Opposite.op_unop _) ≫ cokernel.π g ≫ (kernelUnopOp g).inv := by |
simp
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 109 | 112 | theorem hasBasis_nhds_of_ne_zero {x : Γ₀} (h : x ≠ 0) :
HasBasis (𝓝 x) (fun _ : Unit => True) fun _ => {x} := by |
rw [nhds_of_ne_zero h]
exact hasBasis_pure _
|
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