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) 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.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 110 | 121 | theorem intDegree_add_le {x y : RatFunc K} (hy : y ≠ 0) (hxy : x + y ≠ 0) :
intDegree (x + y) ≤ max (intDegree x) (intDegree y) := by |
by_cases hx : x = 0
· simp only [hx, zero_add, ne_eq] at hxy
simp [hx, hxy]
rw [intDegree_add hxy, ←
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hx y.denom_ne_zero,
mul_comm y.denom, ←
natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hy x.denom_ne_zero,
... |
/-
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.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 92 | 94 | theorem Separable.of_mul_right {f g : R[X]} (h : (f * g).Separable) : g.Separable := by |
rw [mul_comm] at h
exact h.of_mul_left
|
/-
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.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 113 | 115 | theorem iteratedDerivWithin_zero : iteratedDerivWithin 0 f s = f := by |
ext x
simp [iteratedDerivWithin]
|
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.D... | Mathlib/CategoryTheory/CofilteredSystem.lean | 82 | 101 | theorem nonempty_sections_of_finite_cofiltered_system {J : Type u} [Category.{w} J]
[IsCofilteredOrEmpty J] (F : J ⥤ Type v) [∀ j : J, Finite (F.obj j)]
[∀ j : J, Nonempty (F.obj j)] : F.sections.Nonempty := by |
-- Step 1: lift everything to the `max u v w` universe.
let J' : Type max w v u := AsSmall.{max w v} J
let down : J' ⥤ J := AsSmall.down
let F' : J' ⥤ Type max u v w := down ⋙ F ⋙ uliftFunctor.{max u w, v}
haveI : ∀ i, Nonempty (F'.obj i) := fun i => ⟨⟨Classical.arbitrary (F.obj (down.obj i))⟩⟩
haveI : ∀ i... |
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Probability/ConditionalProbability.lean | 110 | 115 | theorem cond_toMeasurable_eq :
μ[|(toMeasurable μ s)] = μ[|s] := by |
unfold cond
by_cases hnt : μ s = ∞
· simp [hnt]
· simp [Measure.restrict_toMeasurable hnt]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlg... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 105 | 105 | theorem compl_apply_diag [Zero α] [One α] (i : V) : A.compl i i = 0 := by | simp [compl]
|
/-
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, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... | Mathlib/Algebra/GroupWithZero/Divisibility.lean | 71 | 78 | theorem dvdNotUnit_of_dvd_of_not_dvd {a b : α} (hd : a ∣ b) (hnd : ¬b ∣ a) : DvdNotUnit a b := by |
constructor
· rintro rfl
exact hnd (dvd_zero _)
· rcases hd with ⟨c, rfl⟩
refine ⟨c, ?_, rfl⟩
rintro ⟨u, rfl⟩
simp at hnd
|
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.EffectiveEpi.Basic
/-!
# Effective epimorphic sieves
We define the notion of effective epimorphic (... | Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean | 132 | 142 | theorem Sieve.effectiveEpimorphic_singleton {X Y : C} (f : Y ⟶ X) :
(Presieve.singleton f).EffectiveEpimorphic ↔ (EffectiveEpi f) := by |
constructor
· intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h
· rintro ⟨h⟩
show Nonempty _
rw [Sieve.generateSingleton_eq]
apply Nonempty.map (isColimitOfEffectiveEpiStruct _) h
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd2... | Mathlib/Topology/MetricSpace/Isometry.lean | 46 | 48 | theorem isometry_iff_dist_eq [PseudoMetricSpace α] [PseudoMetricSpace β] {f : α → β} :
Isometry f ↔ ∀ x y, dist (f x) (f y) = dist x y := by |
simp only [isometry_iff_nndist_eq, ← coe_nndist, NNReal.coe_inj]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 69 | 70 | theorem cast_to_int [AddGroupWithOne α] (n : PosNum) : ((n : ℤ) : α) = n := by |
rw [← to_nat_to_int, Int.cast_natCast, cast_to_nat]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.GroupTheory.Submonoid.Center
#align_import group_theory.subgroup.basic from "leanprover-community/mathlib"@"4be5... | Mathlib/GroupTheory/Subgroup/Center.lean | 130 | 131 | theorem eq_of_left_mem_center {g h : M} (H : IsConj g h) (Hg : g ∈ Set.center M) : g = h := by |
rcases H with ⟨u, hu⟩; rwa [← u.mul_left_inj, Hg.comm u]
|
/-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Option.Basic
import Mathlib.SetTheory.Cardinal.Basic
#al... | Mathlib/Computability/Encoding.lean | 143 | 149 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by |
intro n
cases' n with n <;> unfold encodeNum decodeNum
· rfl
rw [decode_encodePosNum n]
rw [PosNum.cast_to_num]
exact if_neg (encodePosNum_nonempty n)
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 94 | 94 | theorem tsub_le_iff_left : a - b ≤ c ↔ a ≤ b + c := by | rw [tsub_le_iff_right, add_comm]
|
/-
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 | 48 | 54 | theorem Orientation.measure_eq_volume (o : Orientation ℝ F (Fin n)) :
o.volumeForm.measure = volume := by |
have A : o.volumeForm.measure (stdOrthonormalBasis ℝ F).toBasis.parallelepiped = 1 :=
Orientation.measure_orthonormalBasis o (stdOrthonormalBasis ℝ F)
rw [addHaarMeasure_unique o.volumeForm.measure
(stdOrthonormalBasis ℝ F).toBasis.parallelepiped, A, one_smul]
simp only [volume, Basis.addHaar]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 22 | 31 | theorem volume_regionBetween_eq_integral' [SigmaFinite μ] (f_int : IntegrableOn f s μ)
(g_int : IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : f ≤ᵐ[μ.restrict s] g) :
μ.prod volume (regionBetween f g s) = ENNReal.ofReal (∫ y in s, (g - f) y ∂μ) := by |
have h : g - f =ᵐ[μ.restrict s] fun x => Real.toNNReal (g x - f x) :=
hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm
rw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs,
integral_congr_ae h, lintegral_congr_ae,
lintegral_coe_eq_integral _ ((integrable_cong... |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow fr... | Mathlib/Algebra/Regular/Pow.lean | 47 | 50 | theorem IsLeftRegular.pow_iff {n : ℕ} (n0 : 0 < n) : IsLeftRegular (a ^ n) ↔ IsLeftRegular a := by |
refine ⟨?_, IsLeftRegular.pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ]
exact IsLeftRegular.of_mul
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 123 | 125 | theorem NodupKeys.eq_of_mk_mem {a : α} {b b' : β a} {l : List (Sigma β)} (nd : NodupKeys l)
(h : Sigma.mk a b ∈ l) (h' : Sigma.mk a b' ∈ l) : b = b' := by |
cases nd.eq_of_fst_eq h h' rfl; 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.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fi... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 96 | 119 | theorem nodup_antidiagonalTuple (k n : ℕ) : List.Nodup (antidiagonalTuple k n) := by |
induction' k with k ih generalizing n
· cases n
· simp
· simp [eq_comm]
simp_rw [antidiagonalTuple, List.nodup_bind]
constructor
· intro i _
exact (ih i.snd).map (Fin.cons_right_injective (α := fun _ => ℕ) i.fst)
induction' n with n n_ih
· exact List.pairwise_singleton _ _
· rw [List.Nat.an... |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 105 | 105 | theorem box_zero : box (n + 1) 0 = ∅ := by | simp [box]
|
/-
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 | 137 | 139 | theorem rank'_lt (self : UnionFind) (i : Fin self.size) : (self.arr.get i).parent ≠ i →
self.rank i < self.rank (self.arr.get i).parent := by |
simpa only [← parentD_eq] using self.rankD_lt
|
/-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
impo... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 96 | 100 | theorem isAlgClosure_of_transcendence_basis [IsAlgClosed K] (hv : IsTranscendenceBasis R v) :
IsAlgClosure (Algebra.adjoin R (Set.range v)) K :=
letI := RingHom.domain_nontrivial (algebraMap R K)
{ alg_closed := by | infer_instance
algebraic := hv.isAlgebraic }
|
/-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "lean... | Mathlib/Topology/Sequences.lean | 113 | 116 | theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :
a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) := by |
rw [← seqClosure_eq_closure]
rfl
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_t... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 71 | 73 | theorem rightLim_eq (f : StieltjesFunction) (x : ℝ) : Function.rightLim f x = f x := by |
rw [← f.mono.continuousWithinAt_Ioi_iff_rightLim_eq, continuousWithinAt_Ioi_iff_Ici]
exact f.right_continuous' x
|
/-
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.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
im... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 108 | 109 | theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by |
simp_rw [mem_idealOfSet]; push_neg; rfl
|
/-
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 | 89 | 91 | theorem relindex_comap {G' : Type*} [Group G'] (f : G' →* G) (K : Subgroup G') :
relindex (comap f H) K = relindex H (map f K) := by |
rw [relindex, subgroupOf, comap_comap, index_comap, ← f.map_range, K.subtype_range]
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 73 | 74 | theorem isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x := by |
simp_rw [isOrtho_def, flip_apply]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 90 | 93 | theorem associator_inv_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv := by |
dsimp
simp
|
/-
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.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 46 | 49 | theorem uniqueMDiffWithinAt_univ : UniqueMDiffWithinAt I univ x := by |
unfold UniqueMDiffWithinAt
simp only [preimage_univ, univ_inter]
exact I.unique_diff _ (mem_range_self _)
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 90 | 93 | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by |
refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
convert hM.2 (star x) using 1
rw [mulVec_transpose, Matrix.dotProduct_mulVec, star_star, dotProduct_comm]
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathl... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 46 | 57 | theorem energy_le_one : P.energy G ≤ 1 :=
div_le_of_nonneg_of_le_mul (sq_nonneg _) zero_le_one <|
calc
∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2 ≤ P.parts.offDiag.card • (1 : ℚ) :=
sum_le_card_nsmul _ _ 1 fun uv _ =>
(sq_le_one_iff <| G.edgeDensity_nonneg _ _).2 <| G.edgeDensity_... |
rw [offDiag_card, one_mul]
norm_cast
rw [sq]
exact tsub_le_self
|
/-
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.Basic
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Neighborhoods to the left and to the right on an ... | Mathlib/Topology/Order/LeftRightNhds.lean | 82 | 87 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by |
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covBy_iff_Ioo_eq]
|
/-
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, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 108 | 110 | theorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n := by |
simp only [padicValNat, dite_eq_right_iff, PartENat.get_eq_iff_eq_coe, Nat.cast_zero,
multiplicity_eq_zero, and_imp, pos_iff_ne_zero, Ne, ← or_iff_not_imp_left]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
/-!
# Games d... | Mathlib/SetTheory/Game/State.lean | 50 | 54 | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by |
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
|
/-
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.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import meas... | Mathlib/MeasureTheory/Function/L2Space.lean | 46 | 51 | theorem memℒp_two_iff_integrable_sq_norm {f : α → F} (hf : AEStronglyMeasurable f μ) :
Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ := by |
rw [← memℒp_one_iff_integrable]
convert (memℒp_norm_rpow_iff hf two_ne_zero ENNReal.two_ne_top).symm
· simp
· rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.two_ne_top]
|
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 135 | 139 | theorem eval₂_natCast (n : ℕ) : (n : R[X]).eval₂ f x = n := by |
induction' n with n ih
-- Porting note: `Nat.zero_eq` is required.
· simp only [eval₂_zero, Nat.cast_zero, Nat.zero_eq]
· rw [n.cast_succ, eval₂_add, ih, eval₂_one, n.cast_succ]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
/-!
# Power ... | Mathlib/RingTheory/PowerBasis.lean | 89 | 102 | theorem mem_span_pow' {x y : S} {d : ℕ} :
y ∈ Submodule.span R (Set.range fun i : Fin d => x ^ (i : ℕ)) ↔
∃ f : R[X], f.degree < d ∧ y = aeval x f := by |
have : (Set.range fun i : Fin d => x ^ (i : ℕ)) = (fun i : ℕ => x ^ i) '' ↑(Finset.range d) := by
ext n
simp_rw [Set.mem_range, Set.mem_image, Finset.mem_coe, Finset.mem_range]
exact ⟨fun ⟨⟨i, hi⟩, hy⟩ => ⟨i, hi, hy⟩, fun ⟨i, hi, hy⟩ => ⟨⟨i, hi⟩, hy⟩⟩
simp only [this, Finsupp.mem_span_image_iff_total, ... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.RingTheory.SimpleModule
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.simple from "leanprover-community/mathl... | Mathlib/Topology/Algebra/Module/Simple.lean | 28 | 34 | theorem LinearMap.isClosed_or_dense_ker (l : M →ₗ[R] N) :
IsClosed (LinearMap.ker l : Set M) ∨ Dense (LinearMap.ker l : Set M) := by |
rcases l.surjective_or_eq_zero with (hl | rfl)
· exact l.ker.isClosed_or_dense_of_isCoatom (LinearMap.isCoatom_ker_of_surjective hl)
· rw [LinearMap.ker_zero]
left
exact isClosed_univ
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 101 | 103 | theorem bind_eq_some' {x : Option α} {f : α → Option β} {b : β} :
x.bind f = some b ↔ ∃ a, x = some a ∧ f a = some b := by |
cases x <;> simp
|
/-
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.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
/-!
# Spaces where indicators of closed sets have decreasing approx... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 75 | 85 | theorem measure_of_cont_bdd_of_tendsto_filter_indicator {ι : Type*} {L : Filter ι}
[L.IsCountablyGenerated] [TopologicalSpace Ω] [OpensMeasurableSpace Ω] (μ : Measure Ω)
[IsFiniteMeasure μ] {c : ℝ≥0} {E : Set Ω} (E_mble : MeasurableSet E) (fs : ι → Ω →ᵇ ℝ≥0)
(fs_bdd : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c)
... |
convert tendsto_lintegral_nn_filter_of_le_const μ fs_bdd fs_lim
have aux : ∀ ω, indicator E (fun _ ↦ (1 : ℝ≥0∞)) ω = ↑(indicator E (fun _ ↦ (1 : ℝ≥0)) ω) :=
fun ω ↦ by simp only [ENNReal.coe_indicator, ENNReal.coe_one]
simp_rw [← aux, lintegral_indicator _ E_mble]
simp only [lintegral_one, Measure.restrict... |
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functorial constructions
Lemmas on the interaction... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 68 | 70 | theorem HasProd.prod_mk {f : β → α} {g : β → γ} {a : α} {b : γ}
(hf : HasProd f a) (hg : HasProd g b) : HasProd (fun x ↦ (⟨f x, g x⟩ : α × γ)) ⟨a, b⟩ := by |
simp [HasProd, ← prod_mk_prod, Filter.Tendsto.prod_mk_nhds hf hg]
|
/-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... | Mathlib/Data/Nat/MaxPowDiv.lean | 68 | 75 | theorem base_mul_eq_succ {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv (p*n) = p.maxPowDiv n + 1 := by |
have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
rw [maxPowDiv.go, if_pos, mul_div_right _ this]
· apply go_succ
· refine ⟨hp, ?_, by simp⟩
apply Nat.mul_pos this hn
|
/-
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.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d... | Mathlib/Data/Finset/PImage.lean | 120 | 121 | theorem pimage_inter [DecidableEq α] : (s ∩ t).pimage f ⊆ s.pimage f ∩ t.pimage f := by |
simp only [← coe_subset, coe_pimage, coe_inter, PFun.image_inter]
|
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 48 | 49 | theorem eval₂_eq_sum {f : R →+* S} {x : S} : p.eval₂ f x = p.sum fun e a => f a * x ^ e := by |
rw [eval₂_def]
|
/-
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.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-com... | Mathlib/Algebra/Lie/Normalizer.lean | 64 | 67 | theorem le_normalizer : N ≤ N.normalizer := by |
intro m hm
rw [mem_normalizer]
exact fun x => N.lie_mem hm
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecf... | Mathlib/Data/PNat/Interval.lean | 67 | 72 | theorem card_Icc : (Icc a b).card = b + 1 - a := by |
rw [← Nat.card_Icc]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Icc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
|
/-
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.Order.Interval.Set.OrdConnectedComponent
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.t5 from "leanprover-community/math... | Mathlib/Topology/Order/T5.lean | 33 | 63 | theorem compl_section_ordSeparatingSet_mem_nhdsWithin_Ici (hd : Disjoint s (closure t))
(ha : a ∈ s) : (ordConnectedSection (ordSeparatingSet s t))ᶜ ∈ 𝓝[≥] a := by |
have hmem : tᶜ ∈ 𝓝[≥] a := by
refine mem_nhdsWithin_of_mem_nhds ?_
rw [← mem_interior_iff_mem_nhds, interior_compl]
exact disjoint_left.1 hd ha
rcases exists_Icc_mem_subset_of_mem_nhdsWithin_Ici hmem with ⟨b, hab, hmem', hsub⟩
by_cases H : Disjoint (Icc a b) (ordConnectedSection <| ordSeparatingSet ... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 96 | 97 | theorem lintegral_nnnorm_neg {f : α → β} : (∫⁻ a, ‖(-f) a‖₊ ∂μ) = ∫⁻ a, ‖f a‖₊ ∂μ := by |
simp only [Pi.neg_apply, nnnorm_neg]
|
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 131 | 132 | theorem transvection_mul_apply_of_ne (a b : n) (ha : a ≠ i) (c : R) (M : Matrix n n R) :
(transvection i j c * M) a b = M a b := by | simp [transvection, Matrix.add_mul, ha]
|
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
#align_import data.finmap from "leanprover-community/mathlib"@"c... | Mathlib/Data/Finmap.lean | 118 | 120 | theorem liftOn_toFinmap {γ} (s : AList β) (f : AList β → γ) (H) : liftOn ⟦s⟧ f H = f s := by |
cases s
rfl
|
/-
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 | 101 | 105 | theorem destutter'_of_chain (h : l.Chain R a) : l.destutter' R a = a :: l := by |
induction' l with b l hb generalizing a
· simp
obtain ⟨h, hc⟩ := chain_cons.mp h
rw [l.destutter'_cons_pos h, hb hc]
|
/-
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 | 44 | 44 | theorem join_concat (L : List (List α)) (l : List α) : join (L.concat l) = join L ++ l := by | simp
|
/-
Copyright (c) 2024 Jiecheng Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiecheng Zhao
-/
/-!
# Lemmas about `Array.extract`
Some useful lemmas about Array.extract
-/
set_option autoImplicit true
namespace Array
@[simp]
theorem extract_eq_nil_of_start_eq_... | Mathlib/Data/Array/ExtractLemmas.lean | 44 | 50 | theorem extract_extract {a : Array α} (h : s1 + e2 ≤ e1) :
(a.extract s1 e1).extract s2 e2 = a.extract (s1 + s2) (s1 + e2) := by |
apply ext
· simp only [size_extract]
omega
· intro i h1 h2
simp only [get_extract, Nat.add_assoc]
|
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere... | Mathlib/Geometry/Euclidean/Sphere/Power.lean | 84 | 96 | theorem mul_dist_eq_abs_sub_sq_dist {a b p q : P} (hp : ∃ k : ℝ, k ≠ 1 ∧ b -ᵥ p = k • (a -ᵥ p))
(hq : dist a q = dist b q) : dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2| := by |
let m : P := midpoint ℝ a b
have h1 := vsub_sub_vsub_cancel_left a p m
have h2 := vsub_sub_vsub_cancel_left p q m
have h3 := vsub_sub_vsub_cancel_left a q m
have h : ∀ r, b -ᵥ r = m -ᵥ r + (m -ᵥ a) := fun r => by
rw [midpoint_vsub_left, ← right_vsub_midpoint, add_comm, vsub_add_vsub_cancel]
iterate 4 r... |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Module.Card
import Mathlib.SetTheory.Cardinal.CountableCover
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.Analysis.Specific... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 97 | 106 | theorem cardinal_eq_of_mem_nhds
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E]
{s : Set E} {x : E} (hs : s ∈ 𝓝 x) : #s = #E := by |
let g := Homeomorph.addLeft x
let t := g ⁻¹' s
have : t ∈ 𝓝 0 := g.continuous.continuousAt.preimage_mem_nhds (by simpa [g] using hs)
have A : #t = #E := cardinal_eq_of_mem_nhds_zero 𝕜 this
have B : #t = #s := Cardinal.mk_subtype_of_equiv s g.toEquiv
rwa [B] at A
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 96 | 107 | theorem one_le_succ_nth_stream_b {ifp_succ_n : IntFractPair K}
(succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_succ_n) : 1 ≤ ifp_succ_n.b := by |
obtain ⟨ifp_n, nth_stream_eq, stream_nth_fr_ne_zero, ⟨-⟩⟩ :
∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr ≠ 0
∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n :=
succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq
suffices 1 ≤ ifp_n.fr⁻¹ by rwa [IntFractPair.of, le_floor, cast_one]
suffices if... |
/-
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.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 45 | 50 | theorem Finsupp.toFreeAbelianGroup_comp_singleAddHom (x : X) :
Finsupp.toFreeAbelianGroup.comp (Finsupp.singleAddHom x) =
(smulAddHom ℤ (FreeAbelianGroup X)).flip (of x) := by |
ext
simp only [AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply, one_smul,
toFreeAbelianGroup, Finsupp.liftAddHom_apply_single]
|
/-
Copyright (c) 2022 John Nicol. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: John Nicol
-/
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.wilson from "leanprover-community/mathlib"@"c471da714c044131b90c133701e51b877c246677"
/-!
# Wilson's the... | Mathlib/NumberTheory/Wilson.lean | 101 | 104 | theorem prime_iff_fac_equiv_neg_one (h : n ≠ 1) : Prime n ↔ ((n - 1)! : ZMod n) = -1 := by |
refine ⟨fun h1 => ?_, fun h2 => prime_of_fac_equiv_neg_one h2 h⟩
haveI := Fact.mk h1
exact ZMod.wilsons_lemma n
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 93 | 94 | theorem log_mul_ofReal (r : ℝ) (hr : 0 < r) (x : ℂ) (hx : x ≠ 0) :
log (x * r) = Real.log r + log x := by | rw [mul_comm, log_ofReal_mul hr 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.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 62 | 70 | theorem coeff_det_X_add_C_zero (A B : Matrix n n α) :
coeff (det ((X : α[X]) • A.map C + B.map C)) 0 = det B := by |
rw [det_apply, finset_sum_coeff, det_apply]
refine Finset.sum_congr rfl ?_
rintro g -
convert coeff_smul (R := α) (sign g) _ 0
rw [coeff_zero_prod]
refine Finset.prod_congr rfl ?_
simp
|
/-
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 | 101 | 103 | theorem kernel.ι_op :
(kernel.ι f.op).unop = eqToHom (Opposite.unop_op _) ≫ cokernel.π f ≫ (kernelOpUnop f).inv := by |
simp [kernelOpUnop]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 127 | 131 | theorem monomial_def [DecidableEq σ] (n : σ →₀ ℕ) :
(monomial R n) = LinearMap.stdBasis R (fun _ ↦ R) n := by |
rw [monomial]
-- unify the `Decidable` arguments
convert rfl
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510e... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 62 | 65 | theorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}
(h : Function.Injective (algebraMap A B)) :
aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by |
rw [aeval_algebraMap_apply, ← (algebraMap A B).map_zero, h.eq_iff]
|
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polyno... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 56 | 59 | theorem integralNormalization_support {f : R[X]} :
(integralNormalization f).support ⊆ f.support := by |
intro
simp (config := { contextual := true }) [integralNormalization, coeff_monomial, mem_support_iff]
|
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 99 | 103 | theorem Path.cast_cast {u v u' v' u'' v'' : U} (p : Path u v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(p.cast hu hv).cast hu' hv' = p.cast (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
|
/-
Copyright (c) 2019 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Paul Lezeau, Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 114 | 118 | theorem IsIntegrallyClosed.degree_le_of_ne_zero {s : S} (hs : IsIntegral R s) {p : R[X]}
(hp0 : p ≠ 0) (hp : Polynomial.aeval s p = 0) : degree (minpoly R s) ≤ degree p := by |
rw [degree_eq_natDegree (minpoly.ne_zero hs), degree_eq_natDegree hp0]
norm_cast
exact natDegree_le_of_dvd ((isIntegrallyClosed_dvd_iff hs _).mp hp) hp0
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 119 | 124 | theorem le_closure (f : E →ₗ.[R] F) : f ≤ f.closure := by |
by_cases hf : f.IsClosable
· refine le_of_le_graph ?_
rw [← hf.graph_closure_eq_closure_graph]
exact (graph f).le_topologicalClosure
rw [closure_def' hf]
|
/-
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.OuterMeasure.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set α` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 129 | 133 | theorem extend_mono' ⦃s₁ s₂ : Set α⦄ (h₁ : P s₁) (hs : s₁ ⊆ s₂) : extend m s₁ ≤ extend m s₂ := by |
refine le_iInf ?_
intro h₂
rw [extend_eq m h₁]
exact m_mono h₁ h₂ hs
|
/-
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.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 33 | 42 | theorem exp_bound_sq (x z : ℂ) (hz : ‖z‖ ≤ 1) :
‖exp (x + z) - exp x - z • exp x‖ ≤ ‖exp x‖ * ‖z‖ ^ 2 :=
calc
‖exp (x + z) - exp x - z * exp x‖ = ‖exp x * (exp z - 1 - z)‖ := by |
congr
rw [exp_add]
ring
_ = ‖exp x‖ * ‖exp z - 1 - z‖ := norm_mul _ _
_ ≤ ‖exp x‖ * ‖z‖ ^ 2 :=
mul_le_mul_of_nonneg_left (abs_exp_sub_one_sub_id_le hz) (norm_nonneg _)
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import algebra.order.sub.with_top from "leanprover-community/mathlib"@"a... | Mathlib/Algebra/Order/Sub/WithTop.lean | 55 | 55 | theorem sub_top {a : WithTop α} : a - ⊤ = (⊥ : α) := by | cases a <;> rfl
|
/-
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.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "l... | Mathlib/Algebra/Group/UniqueProds.lean | 95 | 101 | theorem set_subsingleton (h : UniqueMul A B a0 b0) :
Set.Subsingleton { ab : G × G | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0 } := by |
rintro ⟨x1, y1⟩ (hx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0) ⟨x2, y2⟩
(hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0)
rcases h hx.1 hx.2.1 hx.2.2 with ⟨rfl, rfl⟩
rcases h hy.1 hy.2.1 hy.2.2 with ⟨rfl, rfl⟩
rfl
|
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal approximation theory o... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 92 | 112 | theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
lfpApprox f x (a+1) = f (lfpApprox f x a) := by |
apply le_antisymm
· conv => left; unfold lfpApprox
apply sSup_le
simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop, Set.union_singleton,
Set.mem_insert_iff, Set.mem_setOf_eq, forall_eq_or_imp, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
apply And.intro
· apply le_... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Reid Barton
-/
import Mathlib.Data.TypeMax
import Mathlib.Logic.UnivLE
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import category_theory.limits.types f... | Mathlib/CategoryTheory/Limits/Types.lean | 83 | 87 | theorem isLimitEquivSections_symm_apply {c : Cone F} (t : IsLimit c)
(x : F.sections) (j : J) :
c.π.app j ((isLimitEquivSections t).symm x) = (x : ∀ j, F.obj j) j := by |
conv_rhs => rw [← (isLimitEquivSections t).right_inv x]
rfl
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Basic
/-!
# Compact subsets of products as limits in `Profinite`
This file exhibits a compact subset `C` of a product... | Mathlib/Topology/Category/Profinite/Product.lean | 58 | 62 | theorem surjective_π_app :
Function.Surjective (π_app C J) := by |
intro x
obtain ⟨y, hy⟩ := x.prop
exact ⟨⟨y, hy.1⟩, Subtype.ext hy.2⟩
|
/-
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.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 95 | 101 | theorem smul_sphere' {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) :
c • sphere x r = sphere (c • x) (‖c‖ * r) := by |
ext y
rw [mem_smul_set_iff_inv_smul_mem₀ hc]
conv_lhs => rw [← inv_smul_smul₀ hc x]
simp only [mem_sphere, dist_smul₀, norm_inv, ← div_eq_inv_mul, div_eq_iff (norm_pos_iff.2 hc).ne',
mul_comm r]
|
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
/-!
# Operator norm as an `NNNorm`
Operator norm as an `NNNorm`, i.e. takin... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 49 | 53 | theorem nnnorm_def (f : E →SL[σ₁₂] F) : ‖f‖₊ = sInf { c | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊ } := by |
ext
rw [NNReal.coe_sInf, coe_nnnorm, norm_def, NNReal.coe_image]
simp_rw [← NNReal.coe_le_coe, NNReal.coe_mul, coe_nnnorm, mem_setOf_eq, NNReal.coe_mk,
exists_prop]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 49 | 50 | theorem normed_sub (x : E) : f.normed μ (c - x) = f.normed μ (c + x) := by |
simp_rw [f.normed_def, f.sub]
|
/-
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.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0... | Mathlib/Data/Int/Log.lean | 87 | 90 | theorem log_of_left_le_one {b : ℕ} (hb : b ≤ 1) (r : R) : log b r = 0 := by |
rcases le_total 1 r with h | h
· rw [log_of_one_le_right _ h, Nat.log_of_left_le_one hb, Int.ofNat_zero]
· rw [log_of_right_le_one _ h, Nat.clog_of_left_le_one hb, Int.ofNat_zero, neg_zero]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Perm/Sign.lean | 99 | 110 | theorem swap_induction_on [Finite α] {P : Perm α → Prop} (f : Perm α) :
P 1 → (∀ f x y, x ≠ y → P f → P (swap x y * f)) → P f := by |
cases nonempty_fintype α
cases' (truncSwapFactors f).out with l hl
induction' l with g l ih generalizing f
· simp (config := { contextual := true }) only [hl.left.symm, List.prod_nil, forall_true_iff]
· intro h1 hmul_swap
rcases hl.2 g (by simp) with ⟨x, y, hxy⟩
rw [← hl.1, List.prod_cons, hxy.2]
... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27dd... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 48 | 51 | theorem essSup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
essSup f (μ.trim hm) = essSup f μ := by |
simp_rw [essSup]
exact limsup_trim hm hf
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.le... | Mathlib/Data/Int/Lemmas.lean | 75 | 77 | theorem natAbs_inj_of_nonpos_of_nonneg {a b : ℤ} (ha : a ≤ 0) (hb : 0 ≤ b) :
natAbs a = natAbs b ↔ -a = b := by |
simpa only [Int.natAbs_neg] using natAbs_inj_of_nonneg_of_nonneg (neg_nonneg_of_nonpos ha) hb
|
/-
Copyright (c) 2021 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.Nat.Nth
import Mathlib.NumberTheory.SmoothNumbers
#align_import number_theory.prime_counting from "leanprover-commu... | Mathlib/NumberTheory/PrimeCounting.lean | 83 | 102 | theorem primeCounting'_add_le {a k : ℕ} (h0 : 0 < a) (h1 : a < k) (n : ℕ) :
π' (k + n) ≤ π' k + Nat.totient a * (n / a + 1) :=
calc
π' (k + n) ≤ ((range k).filter Prime).card + ((Ico k (k + n)).filter Prime).card := by |
rw [primeCounting', count_eq_card_filter_range, range_eq_Ico, ←
Ico_union_Ico_eq_Ico (zero_le k) le_self_add, filter_union]
apply card_union_le
_ ≤ π' k + ((Ico k (k + n)).filter Prime).card := by
rw [primeCounting', count_eq_card_filter_range]
_ ≤ π' k + ((Ico k (k + n)).filter (Copr... |
/-
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, Yaël Dillies
-/
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Data.Set.MulAntidiagonal
#align_import data.finset.mul_antidiagonal from "leanprover-communi... | Mathlib/Data/Finset/MulAntidiagonal.lean | 72 | 73 | theorem mem_mulAntidiagonal : x ∈ mulAntidiagonal hs ht a ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ x.1 * x.2 = a := by |
simp only [mulAntidiagonal, Set.Finite.mem_toFinset, Set.mem_mulAntidiagonal]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
... | Mathlib/Analysis/MellinTransform.lean | 64 | 75 | theorem MellinConvergent.comp_mul_left {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : 0 < a) :
MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s := by |
have := integrableOn_Ioi_comp_mul_left_iff (fun t : ℝ => (t : ℂ) ^ (s - 1) • f t) 0 ha
rw [mul_zero] at this
have h1 : EqOn (fun t : ℝ => (↑(a * t) : ℂ) ^ (s - 1) • f (a * t))
((a : ℂ) ^ (s - 1) • fun t : ℝ => (t : ℂ) ^ (s - 1) • f (a * t)) (Ioi 0) := fun t ht ↦ by
simp only [ofReal_mul, mul_cpow_ofRea... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.PowerSeries.Basic
import Mathlib.Data.Finsupp.PWO
#align_import ring_theory.hahn_series... | Mathlib/RingTheory/HahnSeries/PowerSeries.lean | 132 | 142 | theorem ofPowerSeries_X : ofPowerSeries Γ R PowerSeries.X = single 1 1 := by |
ext n
simp only [single_coeff, ofPowerSeries_apply, RingHom.coe_mk]
split_ifs with hn
· rw [hn]
convert @embDomain_coeff ℕ R _ _ Γ _ _ _ 1 <;> simp
· rw [embDomain_notin_image_support]
simp only [not_exists, Set.mem_image, toPowerSeries_symm_apply_coeff, mem_support,
PowerSeries.coeff_X]
in... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic.Abel
import Mathlib.Tactic.GCongr.Core
#align_import data.int.modeq from "leanprover-community/mathlib"@"47a1a73351de... | Mathlib/Data/Int/ModEq.lean | 112 | 114 | theorem neg_modEq_neg : -a ≡ -b [ZMOD n] ↔ a ≡ b [ZMOD n] := by |
-- Porting note: Restore old proof once #3309 is through
simp [-sub_neg_eq_add, neg_sub_neg, modEq_iff_dvd, dvd_sub_comm]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#a... | Mathlib/Topology/Instances/Complex.lean | 50 | 116 | theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ : K →+* ℂ}
(hc : UniformContinuous ψ) : ψ.toFun = K.subtype ∨ ψ.toFun = conj ∘ K.subtype := by |
letI : TopologicalDivisionRing ℂ := TopologicalDivisionRing.mk
letI : TopologicalRing K.topologicalClosure :=
Subring.instTopologicalRing K.topologicalClosure.toSubring
set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure)
have ui : UniformInducing ι :=
⟨by
erw [unifor... |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geo... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 65 | 65 | theorem oangle_self_right (p₁ p₂ : P) : ∡ p₁ p₂ p₂ = 0 := by | simp [oangle]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Yury Kudryashov
-/
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 111 | 112 | theorem pderiv_monomial_single {i : σ} {n : ℕ} : pderiv i (monomial (single i n) a) =
monomial (single i (n - 1)) (a * n) := by | simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 58 | 61 | theorem nonneg (x : ℝ) : 0 ≤ expNegInvGlue x := by |
cases le_or_gt x 0 with
| inl h => exact ge_of_eq (zero_of_nonpos h)
| inr h => exact le_of_lt (pos_of_pos h)
|
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.L... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 105 | 105 | theorem areaForm_to_volumeForm (x y : E) : ω x y = o.volumeForm ![x, y] := by | simp [areaForm]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptoti... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 84 | 88 | theorem abs_tendsto_atTop (hdeg : 0 < P.degree) :
Tendsto (fun x => abs <| eval x P) atTop atTop := by |
rcases le_total 0 P.leadingCoeff with hP | hP
· exact tendsto_abs_atTop_atTop.comp (P.tendsto_atTop_of_leadingCoeff_nonneg hdeg hP)
· exact tendsto_abs_atBot_atTop.comp (P.tendsto_atBot_of_leadingCoeff_nonpos hdeg hP)
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.convex.... | Mathlib/Analysis/Convex/StrictConvexBetween.lean | 46 | 51 | theorem Wbtw.dist_le_max_dist (p : P) {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) :
dist p₂ p ≤ max (dist p₁ p) (dist p₃ p) := by |
by_cases hp₁ : p₂ = p₁; · simp [hp₁]
by_cases hp₃ : p₂ = p₃; · simp [hp₃]
have hs : Sbtw ℝ p₁ p₂ p₃ := ⟨h, hp₁, hp₃⟩
exact (hs.dist_lt_max_dist _).le
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
/-!
# Fractional ide... | Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 87 | 90 | theorem mem_span_basisOfFractionalIdeal {I : (FractionalIdeal (𝓞 K)⁰ K)ˣ} {x : K} :
x ∈ Submodule.span ℤ (Set.range (basisOfFractionalIdeal K I)) ↔ x ∈ (I : Set K) := by |
rw [basisOfFractionalIdeal, (fractionalIdealBasis K I.1).ofIsLocalizedModule_span ℚ ℤ⁰ _]
simp
|
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"... | Mathlib/LinearAlgebra/Coevaluation.lean | 47 | 54 | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by |
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 67 | 68 | theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := by |
rw [← coe_nonempty, coe_Ioc, Set.nonempty_Ioc]
|
/-
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 | 119 | 120 | theorem singleton_mul_ball : {x} * ball y δ = ball (x * y) δ := by |
simp only [preimage_mul_ball, image_mul_left, singleton_mul, div_inv_eq_mul, mul_comm y x]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7c... | Mathlib/Probability/Kernel/Composition.lean | 93 | 96 | theorem compProdFun_empty (κ : kernel α β) (η : kernel (α × β) γ) (a : α) :
compProdFun κ η a ∅ = 0 := by |
simp only [compProdFun, Set.mem_empty_iff_false, Set.setOf_false, measure_empty,
MeasureTheory.lintegral_const, zero_mul]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Grothendieck
import Mathlib.CategoryTheory.Sites.Pretopology
import Mathlib.CategoryTheory.Limits.Lattice
import Mathlib.Topology.Sets... | Mathlib/CategoryTheory/Sites/Spaces.lean | 92 | 95 | theorem pretopology_toGrothendieck :
Pretopology.toGrothendieck _ (Opens.pretopology T) = Opens.grothendieckTopology T := by |
rw [← pretopology_ofGrothendieck]
apply (Pretopology.gi (Opens T)).l_u_eq
|
/-
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 | 33 | 35 | theorem range_coe : range (some : α → WithTop α) = Iio ⊤ := by |
ext x
rw [mem_Iio, WithTop.lt_top_iff_ne_top, mem_range, ne_top_iff_exists]
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-commu... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 46 | 51 | theorem ker_aeval_ring_hom'_unit_polynomial (f : End K V) (c : K[X]ˣ) :
LinearMap.ker (aeval f (c : K[X])) = ⊥ := by |
rw [Polynomial.eq_C_of_degree_eq_zero (degree_coe_units c)]
simp only [aeval_def, eval₂_C]
apply ker_algebraMap_end
apply coeff_coe_units_zero_ne_zero c
|
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