Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 509 | 517 | theorem pellZd_succ_succ (n) :
pellZd a1 (n + 2) + pellZd a1 n = (2 * a : ℕ) * pellZd a1 (n + 1) := by |
have : (1 : ℤ√(d a1)) + ⟨a, 1⟩ * ⟨a, 1⟩ = ⟨a, 1⟩ * (2 * a) := by
rw [Zsqrtd.natCast_val]
change (⟨_, _⟩ : ℤ√(d a1)) = ⟨_, _⟩
rw [dz_val]
dsimp [az]
ext <;> dsimp <;> ring_nf
simpa [mul_add, mul_comm, mul_left_comm, add_comm] using congr_arg (· * pellZd a1 n) this
|
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import field_theory.normal from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
noncomputable section
open scoped Classical Polynomial
open Polynomial ... | Mathlib/FieldTheory/Normal.lean | 66 | 81 | theorem Normal.exists_isSplittingField [h : Normal F K] [FiniteDimensional F K] :
∃ p : F[X], IsSplittingField F K p := by |
let s := Basis.ofVectorSpace F K
refine
⟨∏ x, minpoly F (s x), splits_prod _ fun x _ => h.splits (s x),
Subalgebra.toSubmodule.injective ?_⟩
rw [Algebra.top_toSubmodule, eq_top_iff, ← s.span_eq, Submodule.span_le, Set.range_subset_iff]
refine fun x =>
Algebra.subset_adjoin
(Multiset.mem_toF... |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 180 | 182 | theorem Memℒp.sub {f g : α → E} (hf : Memℒp f p μ) (hg : Memℒp g p μ) : Memℒp (f - g) p μ := by |
rw [sub_eq_add_neg]
exact hf.add hg.neg
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 73 | 76 | theorem LpAddConst_of_one_le {p : ℝ≥0∞} (hp : 1 ≤ p) : LpAddConst p = 1 := by |
rw [LpAddConst, if_neg]
intro h
exact lt_irrefl _ (h.2.trans_le hp)
|
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 207 | 209 | theorem hom_refl (C : G.ComponentCompl L) : C.hom (subset_refl L) = C := by |
change C.map _ = C
erw [induceHom_id G Lᶜ, ConnectedComponent.map_id]
|
import Mathlib.MeasureTheory.Covering.DensityTheorem
#align_import measure_theory.covering.liminf_limsup from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
open Set Filter Metric MeasureTheory TopologicalSpace
open scoped NNReal ENNReal Topology
variable {α : Type*} [MetricSpace α] [... | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | 157 | 178 | theorem blimsup_cthickening_ae_le_of_eventually_mul_le (p : ℕ → Prop) {s : ℕ → Set α} {M : ℝ}
(hM : 0 < M) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0))
(hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
(blimsup (fun i => cthickening (r₂ i... |
let R₁ i := max 0 (r₁ i)
let R₂ i := max 0 (r₂ i)
have hRp : 0 ≤ R₁ := fun i => le_max_left 0 (r₁ i)
replace hMr : ∀ᶠ i in atTop, M * R₁ i ≤ R₂ i := by
refine hMr.mono fun i hi ↦ ?_
rw [mul_max_of_nonneg _ _ hM.le, mul_zero]
exact max_le_max (le_refl 0) hi
simp_rw [← cthickening_max_zero (r₁ _), ... |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 51 | 55 | theorem sign_apply_eq (r : ℝ) : sign r = -1 ∨ sign r = 0 ∨ sign r = 1 := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· exact Or.inl <| sign_of_neg hn
· exact Or.inr <| Or.inl <| sign_zero
· exact Or.inr <| Or.inr <| sign_of_pos hp
|
import Mathlib.Topology.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
#align_import topology.algebra.order.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
set_option autoImplicit true
open Filter Set TopologicalSpace
open scoped Classical
open Topology
theor... | Mathlib/Topology/Order/ExtendFrom.lean | 36 | 42 | theorem eq_lim_at_left_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {la : β} (hab : a < b)
(ha : Tendsto f (𝓝[>] a) (𝓝 la)) : extendFrom (Ioo a b) f a = la := by |
apply extendFrom_eq
· rw [closure_Ioo hab.ne]
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc]
· simpa [hab]
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 139 | 140 | theorem Monic.add_of_right (hq : Monic q) (hpq : degree p < degree q) : Monic (p + q) := by |
rwa [Monic, leadingCoeff_add_of_degree_lt hpq]
|
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 203 | 203 | theorem add_ne_top : a + b ≠ ∞ ↔ a ≠ ∞ ∧ b ≠ ∞ := by | simpa only [lt_top_iff_ne_top] using add_lt_top
|
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 482 | 497 | theorem Perm.inter_append {l t₁ t₂ : List α} (h : Disjoint t₁ t₂) :
l ∩ (t₁ ++ t₂) ~ l ∩ t₁ ++ l ∩ t₂ := by |
induction l with
| nil => simp
| cons x xs l_ih =>
by_cases h₁ : x ∈ t₁
· have h₂ : x ∉ t₂ := h h₁
simp [*]
by_cases h₂ : x ∈ t₂
· simp only [*, inter_cons_of_not_mem, false_or_iff, mem_append, inter_cons_of_mem,
not_false_iff]
refine Perm.trans (Perm.cons _ l_ih) ?_
cha... |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 520 | 521 | theorem bind_of_mem {o : Part α} {a : α} (h : a ∈ o) (f : α → Part β) : o.bind f = f a := by |
rw [eq_some_iff.2 h, bind_some]
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 39 | 39 | theorem neg_pi_lt_log_im (x : ℂ) : -π < (log x).im := by | simp only [log_im, neg_pi_lt_arg]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 940 | 947 | theorem translationNumber_eq_int_iff (hf : Continuous f) {m : ℤ} :
τ f = m ↔ ∃ x : ℝ, f x = x + m := by |
constructor
· intro h
simp only [← h]
exact f.exists_eq_add_translationNumber hf
· rintro ⟨x, hx⟩
exact f.translationNumber_of_eq_add_int hx
|
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDomain
#align_import data.polynomial.field_division from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Polynomial
namespace Polynomial
u... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 78 | 89 | theorem lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors
{p : R[X]} {t : R} {n : ℕ} (h : p ≠ 0)
(hroot : ∀ m ≤ n, (derivative^[m] p).IsRoot t)
(hnzd : (n.factorial : R) ∈ nonZeroDivisors R) :
n < p.rootMultiplicity t := by |
by_contra! h'
replace hroot := hroot _ h'
simp only [IsRoot, eval_iterate_derivative_rootMultiplicity] at hroot
obtain ⟨q, hq⟩ := Nat.cast_dvd_cast (α := R) <| Nat.factorial_dvd_factorial h'
rw [hq, mul_mem_nonZeroDivisors] at hnzd
rw [nsmul_eq_mul, mul_left_mem_nonZeroDivisors_eq_zero_iff hnzd.1] at hroot... |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 797 | 802 | theorem dist_div_tan_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.tan (∡ p₃ p₁ p₂) = dist p₁ p₂ := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.tan_coe,
dist_div_tan_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inl (right_ne_of_oangle_eq_pi_div_two... |
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 349 | 356 | theorem cast_add (h : m ∣ n) (a b : ZMod n) : (cast (a + b : ZMod n) : R) = cast a + cast b := by |
cases n
· apply Int.cast_add
symm
dsimp [ZMod, ZMod.cast]
erw [← Nat.cast_add, ← sub_eq_zero, ← Nat.cast_sub (Nat.mod_le _ _),
@CharP.cast_eq_zero_iff R _ m]
exact h.trans (Nat.dvd_sub_mod _)
|
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.Perm.Support
import Mathlib.Logic.Equiv.Fin
import Mathlib.Tactic.NormNum.Ineq
#align_import group_theory.perm.sign from "leanprover-community/math... | Mathlib/GroupTheory/Perm/Sign.lean | 131 | 143 | theorem isConj_swap {w x y z : α} (hwx : w ≠ x) (hyz : y ≠ z) : IsConj (swap w x) (swap y z) :=
isConj_iff.2
(have h :
∀ {y z : α},
y ≠ z → w ≠ z → swap w y * swap x z * swap w x * (swap w y * swap x z)⁻¹ = swap y z :=
fun {y z} hyz hwz => by
rw [mul_inv_rev, swap_inv, swap_inv, mul_asso... | rw [hwz]; exact hyz.symm
⟨swap w z * swap x y, by rw [swap_comm y z, h hyz.symm hwy]⟩
else ⟨swap w y * swap x z, h hyz hwz⟩)
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 658 | 678 | theorem dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint {p₁ p₂ p : P} (h : p₁ ≠ p₂) :
dist p₁ p = dist p₂ p ↔
∃ r : ℝ, r • o.rotation (π / 2 : ℝ) (p₂ -ᵥ p₁) +ᵥ midpoint ℝ p₁ p₂ = p := by |
refine ⟨fun hd => ?_, fun hr => ?_⟩
· have hi : ⟪p₂ -ᵥ p₁, p -ᵥ midpoint ℝ p₁ p₂⟫ = 0 := by
rw [@dist_eq_norm_vsub' V, @dist_eq_norm_vsub' V, ←
mul_self_inj (norm_nonneg _) (norm_nonneg _), ← real_inner_self_eq_norm_mul_norm, ←
real_inner_self_eq_norm_mul_norm] at hd
simp_rw [vsub_midpo... |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 247 | 250 | theorem TendstoUniformly.comp (h : TendstoUniformly F f p) (g : γ → α) :
TendstoUniformly (fun n => F n ∘ g) (f ∘ g) p := by |
rw [tendstoUniformly_iff_tendstoUniformlyOnFilter] at h ⊢
simpa [principal_univ, comap_principal] using h.comp g
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 663 | 678 | theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by |
have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by
have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre)
filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }),
mem_nhdsWithin_of_mem_nhds this] with _ him hre
rw [arg, if_neg ... |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 157 | 161 | theorem condCount_union (hs : s.Finite) (htu : Disjoint t u) :
condCount s (t ∪ u) = condCount s t + condCount s u := by |
rw [condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,
cond_apply _ hs.measurableSet, Set.inter_union_distrib_left, measure_union, mul_add]
exacts [htu.mono inf_le_right inf_le_right, (hs.inter_of_left _).measurableSet]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 469 | 473 | theorem arcsin_eq_arccos {x : ℝ} (h : 0 ≤ x) : arcsin x = arccos (√(1 - x ^ 2)) := by |
rw [eq_comm, ← cos_arcsin]
exact
arccos_cos (arcsin_nonneg.2 h)
((arcsin_le_pi_div_two _).trans (div_le_self pi_pos.le one_le_two))
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 64 | 66 | theorem iteratedDerivWithin_const_mul (c : 𝕜) {f : 𝕜 → 𝕜} (hf : ContDiffOn 𝕜 n f s) :
iteratedDerivWithin n (fun z => c * f z) s x = c * iteratedDerivWithin n f s x := by |
simpa using iteratedDerivWithin_const_smul (F := 𝕜) hx h c hf
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 134 | 135 | theorem setLaverage_eq (f : α → ℝ≥0∞) (s : Set α) :
⨍⁻ x in s, f x ∂μ = (∫⁻ x in s, f x ∂μ) / μ s := by | rw [laverage_eq, restrict_apply_univ]
|
import Mathlib.Geometry.Manifold.MFDeriv.Atlas
noncomputable section
open scoped Manifold
open Set
section UniqueMDiff
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type*}
[Topolog... | Mathlib/Geometry/Manifold/MFDeriv/UniqueDifferential.lean | 120 | 131 | theorem UniqueMDiffWithinAt.smooth_bundle_preimage {p : TotalSpace F Z}
(hs : UniqueMDiffWithinAt I s p.proj) :
UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (π F Z ⁻¹' s) p := by |
set e := trivializationAt F Z p.proj
have hp : p ∈ e.source := FiberBundle.mem_trivializationAt_proj_source
have : UniqueMDiffWithinAt (I.prod 𝓘(𝕜, F)) (s ×ˢ univ) (e p) := by
rw [← Prod.mk.eta (p := e p), FiberBundle.trivializationAt_proj_fst]
exact hs.prod (uniqueMDiffWithinAt_univ _)
rw [← e.left_... |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 120 | 121 | theorem liftOn_mk {α : Type*} {f : M × S → α} (wd : ∀ (p p' : M × S), p ≈ p' → f p = f p')
(m : M) (s : S) : liftOn (mk m s) f wd = f ⟨m, s⟩ := by | convert Quotient.liftOn_mk f wd ⟨m, s⟩
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 499 | 502 | theorem A_mono (L : F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : A f L r ε ⊆ A f L r δ := by |
rintro x ⟨r', r'r, hr'⟩
refine ⟨r', r'r, fun y hy z hz => (hr' y hy z hz).trans (mul_le_mul_of_nonneg_right h ?_)⟩
linarith [hy.1, hy.2, r'r.2]
|
import Mathlib.Order.Filter.Prod
#align_import order.filter.n_ary from "leanprover-community/mathlib"@"78f647f8517f021d839a7553d5dc97e79b508dea"
open Function Set
open Filter
namespace Filter
variable {α α' β β' γ γ' δ δ' ε ε' : Type*} {m : α → β → γ} {f f₁ f₂ : Filter α}
{g g₁ g₂ : Filter β} {h h₁ h₂ : Filt... | Mathlib/Order/Filter/NAry.lean | 124 | 125 | theorem map₂_sup_right : map₂ m f (g₁ ⊔ g₂) = map₂ m f g₁ ⊔ map₂ m f g₂ := by |
simp_rw [← map_prod_eq_map₂, prod_sup, map_sup]
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.calculus.fderiv.linear from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topolo... | Mathlib/Analysis/Calculus/FDeriv/Linear.lean | 136 | 139 | theorem IsBoundedLinearMap.fderivWithin (h : IsBoundedLinearMap 𝕜 f)
(hxs : UniqueDiffWithinAt 𝕜 s x) : fderivWithin 𝕜 f s x = h.toContinuousLinearMap := by |
rw [DifferentiableAt.fderivWithin h.differentiableAt hxs]
exact h.fderiv
|
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import ring_theory.polynomial.hermite.gaussian from "leanprover-communit... | Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 58 | 64 | theorem hermite_eq_deriv_gaussian (n : ℕ) (x : ℝ) : aeval x (hermite n) =
(-1 : ℝ) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2)) := by |
rw [deriv_gaussian_eq_hermite_mul_gaussian]
field_simp [Real.exp_ne_zero]
rw [← @smul_eq_mul ℝ _ ((-1) ^ n), ← inv_smul_eq_iff₀, mul_assoc, smul_eq_mul, ← inv_pow, ←
neg_inv, inv_one]
exact pow_ne_zero _ (by norm_num)
|
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 124 | 139 | theorem rank_submodule_le_one_iff (s : Submodule K V) [Module.Free K s] :
Module.rank K s ≤ 1 ↔ ∃ v₀ ∈ s, s ≤ K ∙ v₀ := by |
simp_rw [rank_le_one_iff, le_span_singleton_iff]
constructor
· rintro ⟨⟨v₀, hv₀⟩, h⟩
use v₀, hv₀
intro v hv
obtain ⟨r, hr⟩ := h ⟨v, hv⟩
use r
rwa [Subtype.ext_iff, coe_smul] at hr
· rintro ⟨v₀, hv₀, h⟩
use ⟨v₀, hv₀⟩
rintro ⟨v, hv⟩
obtain ⟨r, hr⟩ := h v hv
use r
rwa [Subt... |
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β γ m n : Type*}
variable {R : Type*}
namespace Matrix
open Matrix
def hadamard [Mul α] (A : Matrix m n α) (B : Matrix m n α) : Matrix m n α :... | Mathlib/Data/Matrix/Hadamard.lean | 148 | 151 | theorem dotProduct_vecMul_hadamard [DecidableEq m] [DecidableEq n] (v : m → α) (w : n → α) :
dotProduct (v ᵥ* (A ⊙ B)) w = trace (diagonal v * A * (B * diagonal w)ᵀ) := by |
rw [← sum_hadamard_eq, Finset.sum_comm]
simp [dotProduct, vecMul, Finset.sum_mul, mul_assoc]
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Logic.Unique
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Lift
#align_import algebra.group.units from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
assert_not_exists Multiplicative
a... | Mathlib/Algebra/Group/Units.lean | 276 | 276 | theorem inv_mul_of_eq {a : α} (h : ↑u = a) : ↑u⁻¹ * a = 1 := by | rw [← h, u.inv_mul]
|
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 223 | 234 | theorem isTheta_cpow_rpow (hl_im : IsBoundedUnder (· ≤ ·) l fun x => |(g x).im|)
(hl : ∀ᶠ x in l, f x = 0 → re (g x) = 0 → g x = 0) :
(fun x => f x ^ g x) =Θ[l] fun x => abs (f x) ^ (g x).re :=
calc
(fun x => f x ^ g x) =Θ[l]
(show α → ℝ from fun x => abs (f x) ^ (g x).re / Real.exp (arg (f x) * i... |
simp only [ofReal_one, div_one]
rfl
|
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Ring.Subring.Basic
... | Mathlib/RingTheory/Perfection.lean | 129 | 130 | theorem coeff_pow_p (f : Ring.Perfection R p) (n : ℕ) :
coeff R p (n + 1) (f ^ p) = coeff R p n f := by | rw [RingHom.map_pow]; exact f.2 n
|
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 185 | 192 | theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by |
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 843 | 846 | theorem NormedCommGroup.nhds_basis_norm_lt (x : E) :
(𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { y | ‖y / x‖ < ε } := by |
simp_rw [← ball_eq']
exact Metric.nhds_basis_ball
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 866 | 868 | theorem get_eq_get? (l : List α) (i : Fin l.length) :
l.get i = (l.get? i).get (by simp [get?_eq_get]) := by |
simp [get_eq_iff]
|
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.Topology.QuasiSeparated
#align_import algebraic_geometry.morphisms.quasi_separated from "leanprover-community/mathlib"@"1a51edf13debfcbe223fa06b1cb353b9ed9751cc"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite Topolog... | Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 121 | 123 | theorem quasi_compact_affineProperty_diagonal_eq :
QuasiCompact.affineProperty.diagonal = QuasiSeparated.affineProperty := by |
funext; rw [quasi_compact_affineProperty_iff_quasiSeparatedSpace]; rfl
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 149 | 151 | theorem IsCoprime.of_mul_right_left (H : IsCoprime x (y * z)) : IsCoprime x y := by |
rw [isCoprime_comm] at H ⊢
exact H.of_mul_left_left
|
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 46 | 47 | theorem rank_zero [Nontrivial K] : rank (0 : V →ₗ[K] V') = 0 := by |
rw [rank, LinearMap.range_zero, rank_bot]
|
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Set
open scoped RealInnerProductSpace
variable {V P : Type*} [NormedAddCommGroup V] [InnerP... | Mathlib/Geometry/Euclidean/PerpBisector.lean | 97 | 98 | theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by |
simp only [mem_perpBisector_iff_dist_eq, dist_comm]
|
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open sc... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 456 | 529 | theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
{ε : ℝ} (εpos : 0 < ε) :
∃ g : α → EReal,
(∀ x, (f x : EReal) < g x) ∧
LowerSemicontinuous g ∧
Integrable (fun x => EReal.toReal (g x)) μ ∧
(∀ᵐ x ∂μ, g x < ⊤) ∧ (∫ x, EReal.toReal (g x) ∂μ... |
let δ : ℝ≥0 := ⟨ε / 2, (half_pos εpos).le⟩
have δpos : 0 < δ := half_pos εpos
let fp : α → ℝ≥0 := fun x => Real.toNNReal (f x)
have int_fp : Integrable (fun x => (fp x : ℝ)) μ := hf.real_toNNReal
rcases exists_lt_lowerSemicontinuous_integral_gt_nnreal fp int_fp δpos with
⟨gp, fp_lt_gp, gpcont, gp_lt_top,... |
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.MetricSpace.PseudoMetric
import Mathlib.Topology.Order.LocalExtr
#align_import analysis.convex.extrema from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E β : Type*} [AddCo... | Mathlib/Analysis/Convex/Extrema.lean | 54 | 69 | theorem IsMinOn.of_isLocalMinOn_of_convexOn {f : E → β} {a : E} (a_in_s : a ∈ s)
(h_localmin : IsLocalMinOn f s a) (h_conv : ConvexOn ℝ s f) : IsMinOn f s a := by |
intro x x_in_s
let g : ℝ →ᵃ[ℝ] E := AffineMap.lineMap a x
have hg0 : g 0 = a := AffineMap.lineMap_apply_zero a x
have hg1 : g 1 = x := AffineMap.lineMap_apply_one a x
have hgc : Continuous g := AffineMap.lineMap_continuous
have h_maps : MapsTo g (Icc 0 1) s := by
simpa only [g, mapsTo', ← segment_eq_im... |
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 113 | 116 | theorem transvection_mul_transvection_same (h : i ≠ j) (c d : R) :
transvection i j c * transvection i j d = transvection i j (c + d) := by |
simp [transvection, Matrix.add_mul, Matrix.mul_add, h, h.symm, add_smul, add_assoc,
stdBasisMatrix_add]
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 342 | 344 | theorem hasDerivWithinAt_Iio_iff_Iic [PartialOrder 𝕜] :
HasDerivWithinAt f f' (Iio x) x ↔ HasDerivWithinAt f f' (Iic x) x := by |
rw [← Iic_diff_right, hasDerivWithinAt_diff_singleton]
|
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.normed_space.enorm from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
attribute [local instance] Classical.propDecidable
open ENNReal
structure ENorm (𝕜 : Type*) (V : Type*) [NormedField 𝕜] [Ad... | Mathlib/Analysis/NormedSpace/ENorm.lean | 107 | 110 | theorem map_neg (x : V) : e (-x) = e x :=
calc
e (-x) = ‖(-1 : 𝕜)‖₊ * e x := by | rw [← map_smul, neg_one_smul]
_ = e x := by simp
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 156 | 156 | theorem diag_mem_sym2_iff : Sym2.diag a ∈ s.sym2 ↔ a ∈ s := by | simp [diag_mem_sym2_mem_iff]
|
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.AdjoinRoot
#align_import ring_theory.adjoin.field from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
noncomputable section
open Polynomial
variable {R K L M : Type*} [CommRing R]... | Mathlib/RingTheory/Adjoin/Field.lean | 106 | 110 | theorem IsIntegral.minpoly_splits_tower_top [Algebra K L] [IsScalarTower R K L]
(h : Splits (algebraMap R L) (minpoly R x)) :
Splits (algebraMap K L) (minpoly K x) := by |
rw [IsScalarTower.algebraMap_eq R K L] at h
exact int.minpoly_splits_tower_top' h
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 652 | 658 | theorem prod_id : ∀ {n} {α β : TypeVec.{u} n}, (id ⊗' id) = (id : α ⊗ β ⟹ _) := by |
intros
ext i a
induction' i with _ _ _ i_ih
· cases a
rfl
· apply i_ih
|
import Mathlib.Data.List.Sigma
#align_import data.list.alist from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v w
open List
variable {α : Type u} {β : α → Type v}
structure AList (β : α → Type v) : Type max u v where
entries : List (Sigma β)
nodupKeys : entri... | Mathlib/Data/List/AList.lean | 207 | 208 | theorem mem_replace {a a' : α} {b : β a} {s : AList β} : a' ∈ replace a b s ↔ a' ∈ s := by |
rw [mem_keys, keys_replace, ← mem_keys]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 171 | 174 | theorem weightedVSubOfPoint_map (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P) (b : P) :
(s₂.map e).weightedVSubOfPoint p b w = s₂.weightedVSubOfPoint (p ∘ e) b (w ∘ e) := by |
simp_rw [weightedVSubOfPoint_apply]
exact Finset.sum_map _ _ _
|
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 248 | 250 | theorem image₂_inter_singleton [DecidableEq α] (s₁ s₂ : Finset α) (hf : Injective fun a => f a b) :
image₂ f (s₁ ∩ s₂) {b} = image₂ f s₁ {b} ∩ image₂ f s₂ {b} := by |
simp_rw [image₂_singleton_right, image_inter _ _ hf]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Finsupp.Fin
import Mathlib.Logic.Equiv.Fin
#align_import data.mv_polynomial.equiv from "leanprover-community/mathlib"@"2f5b500... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 466 | 477 | theorem finSuccEquiv_support (f : MvPolynomial (Fin (n + 1)) R) :
(finSuccEquiv R n f).support = Finset.image (fun m : Fin (n + 1) →₀ ℕ => m 0) f.support := by |
ext i
rw [Polynomial.mem_support_iff, Finset.mem_image, Finsupp.ne_iff]
constructor
· rintro ⟨m, hm⟩
refine ⟨cons i m, ?_, cons_zero _ _⟩
rw [← support_coeff_finSuccEquiv]
simpa using hm
· rintro ⟨m, h, rfl⟩
refine ⟨tail m, ?_⟩
rwa [← coeff, zero_apply, ← mem_support_iff, support_coeff_fi... |
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Algebra.Algebra.Defs
import Mathlib.LinearAlgebra.Projection
import Mat... | Mathlib/Topology/Algebra/Module/Basic.lean | 198 | 201 | theorem Submodule.dense_iff_topologicalClosure_eq_top {s : Submodule R M} :
Dense (s : Set M) ↔ s.topologicalClosure = ⊤ := by |
rw [← SetLike.coe_set_eq, dense_iff_closure_eq]
simp
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 609 | 610 | theorem zero_lt_one_add_norm_sq' (x : E) : 0 < 1 + ‖x‖ ^ 2 := by |
positivity
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 133 | 136 | theorem Wbtw.dist_add_dist {x y z : P} (h : Wbtw ℝ x y z) :
dist x y + dist y z = dist x z := by |
obtain ⟨a, ⟨ha₀, ha₁⟩, rfl⟩ := h
simp [abs_of_nonneg, ha₀, ha₁, sub_mul]
|
import Mathlib.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Order.Monoid.Defs
#align_import algebra.order.monoid.canonical.defs from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
universe u
variable {α : Type u}
class ExistsMulOf... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 273 | 285 | theorem lt_iff_exists_mul [CovariantClass α α (· * ·) (· < ·)] : a < b ↔ ∃ c > 1, b = a * c := by |
rw [lt_iff_le_and_ne, le_iff_exists_mul, ← exists_and_right]
apply exists_congr
intro c
rw [and_comm, and_congr_left_iff, gt_iff_lt]
rintro rfl
constructor
· rw [one_lt_iff_ne_one]
apply mt
rintro rfl
rw [mul_one]
· rw [← (self_le_mul_right a c).lt_iff_ne]
apply lt_mul_of_one_lt_right'
|
import Mathlib.RingTheory.Derivation.Lie
import Mathlib.Geometry.Manifold.DerivationBundle
#align_import geometry.manifold.algebra.left_invariant_derivation from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
noncomputable section
open scoped LieGroup Manifold Derivation
variable {𝕜 ... | Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean | 239 | 246 | theorem evalAt_mul : evalAt (g * h) X = 𝒅ₕ (L_apply I g h) (evalAt h X) := by |
ext f
rw [← left_invariant, apply_hfdifferential, apply_hfdifferential, L_mul, fdifferential_comp,
apply_fdifferential]
-- Porting note: more agressive here
erw [LinearMap.comp_apply]
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [apply_fdifferential, ← apply_hfdifferential... |
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 365 | 381 | theorem y_eq_one_of_mem_closedBall {D : ℝ} {x : E} (Dpos : 0 < D)
(hx : x ∈ closedBall (0 : E) (1 - D)) : y D x = 1 := by |
change (w D ⋆[lsmul ℝ ℝ, μ] φ) x = 1
have B : ∀ y : E, y ∈ ball x D → φ y = 1 := by
have C : ball x D ⊆ ball 0 1 := by
apply ball_subset_ball'
simp only [mem_closedBall] at hx
linarith only [hx]
intro y hy
simp only [φ, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]
... |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 443 | 445 | theorem aroots_X [CommRing S] [IsDomain S] [Algebra T S] :
aroots (X : T[X]) S = {0} := by |
rw [aroots_def, map_X, roots_X]
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform RealInnerProductSpace
open Complex hiding exp continuous_exp abs_of_nonneg sq_abs
noncomp... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 79 | 83 | theorem isLittleO_exp_neg_mul_sq_cocompact {a : ℂ} (ha : 0 < a.re) (s : ℝ) :
(fun x : ℝ => Complex.exp (-a * x ^ 2)) =o[cocompact ℝ] fun x : ℝ => |x| ^ s := by |
convert cexp_neg_quadratic_isLittleO_abs_rpow_cocompact (?_ : (-a).re < 0) 0 s using 1
· simp_rw [zero_mul, add_zero]
· rwa [neg_re, neg_lt_zero]
|
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
#align_import category_theory.sites.closed from "leanprover-community/mathlib"@"4cfc30e317caad46858393f1a7a33f609296cc30"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable (J₁ J₂ : GrothendieckTopol... | Mathlib/CategoryTheory/Sites/Closed.lean | 124 | 132 | theorem pullback_close {X Y : C} (f : Y ⟶ X) (S : Sieve X) :
J₁.close (S.pullback f) = (J₁.close S).pullback f := by |
apply le_antisymm
· refine J₁.le_close_of_isClosed (Sieve.pullback_monotone _ (J₁.le_close S)) ?_
apply J₁.isClosed_pullback _ _ (J₁.close_isClosed _)
· intro Z g hg
change _ ∈ J₁ _
rw [← Sieve.pullback_comp]
apply hg
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 312 | 314 | theorem logb_lt_logb_of_base_lt_one (hx : 0 < x) (hxy : x < y) : logb b y < logb b x := by |
rw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]
exact log_lt_log hx hxy
|
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 314 | 320 | theorem gluedLift_p1 : gluedLift 𝒰 f g s ≫ p1 𝒰 f g = s.fst := by |
rw [← cancel_epi (𝒰.pullbackCover s.fst).fromGlued]
apply Multicoequalizer.hom_ext
intro b
simp_rw [OpenCover.fromGlued, Multicoequalizer.π_desc_assoc, gluedLift, ← Category.assoc]
simp_rw [(𝒰.pullbackCover s.fst).ι_glueMorphisms]
simp [p1, pullback.condition]
|
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 56 | 60 | theorem tendsto_pow_div_pow_atTop_atTop {p q : ℕ} (hpq : q < p) :
Tendsto (fun x : 𝕜 => x ^ p / x ^ q) atTop atTop := by |
rw [tendsto_congr' pow_div_pow_eventuallyEq_atTop]
apply tendsto_zpow_atTop_atTop
omega
|
import Mathlib.Geometry.RingedSpace.PresheafedSpace
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.Topology.Sheaves.Stalks
#align_import algebraic_geometry.stalks from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
universe v u v' u'
open Opposite Cate... | Mathlib/Geometry/RingedSpace/Stalks.lean | 229 | 246 | theorem stalkSpecializes_stalkMap {X Y : PresheafedSpace.{_, _, v} C}
(f : X ⟶ Y) {x y : X} (h : x ⤳ y) :
Y.presheaf.stalkSpecializes (f.base.map_specializes h) ≫ stalkMap f x =
stalkMap f y ≫ X.presheaf.stalkSpecializes h := by |
-- Porting note: the original one liner `dsimp [stalkMap]; simp [stalkMap]` doesn't work,
-- I had to uglify this
dsimp [stalkSpecializes, stalkMap, stalkFunctor, stalkPushforward]
-- We can't use `ext` here due to https://github.com/leanprover/std4/pull/159
refine colimit.hom_ext fun j => ?_
induction j w... |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 70 | 74 | theorem IsMatching.toEdge.surjective {M : Subgraph G} (h : M.IsMatching) :
Function.Surjective h.toEdge := by |
rintro ⟨e, he⟩
refine Sym2.ind (fun x y he => ?_) e he
exact ⟨⟨x, M.edge_vert he⟩, h.toEdge_eq_of_adj _ he⟩
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 373 | 373 | theorem div_lt_one (hb : 0 < b) : a / b < 1 ↔ a < b := by | rw [div_lt_iff hb, one_mul]
|
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 78 | 80 | theorem evalFrom_append_singleton (S : Set σ) (x : List α) (a : α) :
M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a := by |
simp only [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
|
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import Mathlib.Analysis.Complex.Polynomial
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 197 | 203 | theorem zmod_char_primitive_of_eq_one_only_at_zero (n : ℕ) (ψ : AddChar (ZMod n) C)
(hψ : ∀ a, ψ a = 1 → a = 0) : IsPrimitive ψ := by |
refine fun a ha => (isNontrivial_iff_ne_trivial _).mpr fun hf => ?_
have h : mulShift ψ a 1 = (1 : AddChar (ZMod n) C) (1 : ZMod n) :=
congr_fun (congr_arg (↑) hf) 1
rw [mulShift_apply, mul_one] at h; norm_cast at h
exact ha (hψ a h)
|
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 485 | 490 | theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k := by |
induction' k with k ih
· simpa
simp only [ucs_succ]
-- Porting note: `mono` makes no progress
apply monotone_normalizer ih
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 110 | 113 | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 := by |
cases n
· simp
· simp [X_mul, Nat.succ_ne_zero, ascPochhammer_succ_left]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 38 | 42 | theorem opRingEquiv_op_monomial (n : ℕ) (r : R) :
opRingEquiv R (op (monomial n r : R[X])) = monomial n (op r) := by |
simp only [opRingEquiv, RingEquiv.coe_trans, Function.comp_apply,
AddMonoidAlgebra.opRingEquiv_apply, RingEquiv.op_apply_apply, toFinsuppIso_apply, unop_op,
toFinsupp_monomial, Finsupp.mapRange_single, toFinsuppIso_symm_apply, ofFinsupp_single]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 151 | 154 | theorem isConnected_setOf_sameRay_and_ne_zero {x : E} (hx : x ≠ 0) :
IsConnected { y | SameRay ℝ x y ∧ y ≠ 0 } := by |
simp_rw [← exists_pos_left_iff_sameRay_and_ne_zero hx]
exact isConnected_Ioi.image _ (continuous_id.smul continuous_const).continuousOn
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
#align_import measure_theory.covering.vitali_family from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Metric Set Filter TopologicalSpace MeasureTheory.Measure
open Filter MeasureTheory Topology
variable {α : Type*}... | Mathlib/MeasureTheory/Covering/VitaliFamily.lean | 170 | 179 | theorem measure_le_tsum_of_absolutelyContinuous [SecondCountableTopology α] {ρ : Measure α}
(hρ : ρ ≪ μ) : ρ s ≤ ∑' p : h.index, ρ (h.covering p) :=
calc
ρ s ≤ ρ ((s \ ⋃ p ∈ h.index, h.covering p) ∪ ⋃ p ∈ h.index, h.covering p) :=
measure_mono (by simp only [subset_union_left, diff_union_self])
_ ≤ ... |
rw [hρ h.measure_diff_biUnion, zero_add,
measure_biUnion h.index_countable h.covering_disjoint fun x hx => h.measurableSet_u hx]
|
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 107 | 126 | theorem isSatisfiable_iff_isFinitelySatisfiable {T : L.Theory} :
T.IsSatisfiable ↔ T.IsFinitelySatisfiable :=
⟨Theory.IsSatisfiable.isFinitelySatisfiable, fun h => by
classical
set M : Finset T → Type max u v := fun T0 : Finset T =>
(h (T0.map (Function.Embedding.subtype fun x => x ∈ T)) T0.map_... |
refine ⟨fun φ hφ => ?_⟩
rw [Ultraproduct.sentence_realize]
refine
Filter.Eventually.filter_mono (Ultrafilter.of_le _)
(Filter.eventually_atTop.2
⟨{⟨φ, hφ⟩}, fun s h' =>
Theory.realize_sentence_of_mem (s.map (Function.Embedding.subtype fun x =>... |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 747 | 752 | theorem dist_div_cos_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(π / 2)) : dist p₃ p₂ / Real.Angle.cos (∡ p₂ p₃ p₁) = dist p₁ p₃ := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
dist_div_cos_angle_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inr (right_ne_of_oangle_eq_pi_div_two h))]
|
import Mathlib.CategoryTheory.Sites.CompatiblePlus
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.compatible_sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryThe... | Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean | 110 | 114 | theorem sheafificationWhiskerRightIso_inv_app :
(J.sheafificationWhiskerRightIso F).inv.app P = (J.sheafifyCompIso F P).inv := by |
dsimp [sheafificationWhiskerRightIso, sheafifyCompIso]
simp only [Category.id_comp, Category.comp_id]
erw [Category.id_comp]
|
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 225 | 237 | theorem prefix_take_iff {x y : List α} {n : ℕ} : x <+: y.take n ↔ x <+: y ∧ x.length ≤ n := by |
constructor
· intro h
constructor
· exact List.IsPrefix.trans h <| List.take_prefix n y
· replace h := h.length_le
rw [length_take, Nat.le_min] at h
exact h.left
· intro ⟨hp, hl⟩
have hl' := hp.length_le
rw [List.prefix_iff_eq_take] at *
rw [hp, List.take_take]
simp [min_e... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 29 | 30 | theorem aeval_T (x : A) (n : ℤ) : aeval x (T R n) = (T A n).eval x := by |
rw [aeval_def, eval₂_eq_eval_map, map_T]
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 274 | 275 | theorem subset_interior_iff_isOpen : s ⊆ interior s ↔ IsOpen s := by |
simp only [interior_eq_iff_isOpen.symm, Subset.antisymm_iff, interior_subset, true_and]
|
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 250 | 254 | theorem get_rotate (l : List α) (n : ℕ) (k : Fin (l.rotate n).length) :
(l.rotate n).get k =
l.get ⟨(k + n) % l.length, mod_lt _ (length_rotate l n ▸ k.1.zero_le.trans_lt k.2)⟩ := by |
rw [← Option.some_inj, ← get?_eq_get, ← get?_eq_get, get?_rotate]
exact k.2.trans_eq (length_rotate _ _)
|
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import number_theory.padics.hensel from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/NumberTheory/Padics/Hensel.lean | 43 | 49 | theorem padic_polynomial_dist {p : ℕ} [Fact p.Prime] (F : Polynomial ℤ_[p]) (x y : ℤ_[p]) :
‖F.eval x - F.eval y‖ ≤ ‖x - y‖ :=
let ⟨z, hz⟩ := F.evalSubFactor x y
calc
‖F.eval x - F.eval y‖ = ‖z‖ * ‖x - y‖ := by | simp [hz]
_ ≤ 1 * ‖x - y‖ := by gcongr; apply PadicInt.norm_le_one
_ = ‖x - y‖ := by simp
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 350 | 351 | theorem setAverage_eq (f : α → E) (s : Set α) :
⨍ x in s, f x ∂μ = (μ s).toReal⁻¹ • ∫ x in s, f x ∂μ := by | rw [average_eq, restrict_apply_univ]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 204 | 206 | theorem mem_sublists {s t : List α} : s ∈ sublists t ↔ s <+ t := by |
rw [← reverse_sublist, ← mem_sublists', sublists'_reverse,
mem_map_of_injective reverse_injective]
|
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Multiset
section Pi
variable {α : Type*}
open Function
def Pi.empty (δ : α → Sort*) : ∀ a ∈ (0 : Multiset α), δ a :=
nofun
#align multiset.pi.empty Multi... | Mathlib/Data/Multiset/Pi.lean | 71 | 80 | theorem Pi.cons_injective {a : α} {b : δ a} {s : Multiset α} (hs : a ∉ s) :
Function.Injective (Pi.cons s a b) := fun f₁ f₂ eq =>
funext fun a' =>
funext fun h' =>
have ne : a ≠ a' := fun h => hs <| h.symm ▸ h'
have : a' ∈ a ::ₘ s := mem_cons_of_mem h'
calc
f₁ a' h' = Pi.cons s a b f... | rw [Pi.cons_ne this ne.symm]
_ = Pi.cons s a b f₂ a' this := by rw [eq]
_ = f₂ a' h' := by rw [Pi.cons_ne this ne.symm]
|
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.normed_space.hahn_banach.separation from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 128 | 146 | theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht₁ : Convex ℝ t)
(ht₃ : IsOpen t) (disj : Disjoint s t) :
∃ (f : E →L[ℝ] ℝ) (u : ℝ), (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u < f b := by |
obtain rfl | ⟨a₀, ha₀⟩ := s.eq_empty_or_nonempty
· exact ⟨0, -1, by simp, fun b _hb => by norm_num⟩
obtain rfl | ⟨b₀, hb₀⟩ := t.eq_empty_or_nonempty
· exact ⟨0, 1, fun a _ha => by norm_num, by simp⟩
obtain ⟨f, s, hf₁, hf₂⟩ := geometric_hahn_banach_open hs₁ hs₂ ht₁ disj
have hf : IsOpenMap f := by
refin... |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 91 | 92 | theorem WithTop.iInf_empty [IsEmpty ι] [InfSet α] (f : ι → WithTop α) :
⨅ i, f i = ⊤ := by | rw [iInf, range_eq_empty, WithTop.sInf_empty]
|
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
section Sigma
variable {α α₁ α₂ : Type*} {β : α → Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
namespace Sigma
instance inst... | Mathlib/Data/Sigma/Basic.lean | 75 | 80 | theorem _root_.Function.eq_of_sigmaMk_comp {γ : Type*} [Nonempty γ]
{a b : α} {f : γ → β a} {g : γ → β b} (h : Sigma.mk a ∘ f = Sigma.mk b ∘ g) :
a = b ∧ HEq f g := by |
rcases ‹Nonempty γ› with ⟨i⟩
obtain rfl : a = b := congr_arg Sigma.fst (congr_fun h i)
simpa [funext_iff] using h
|
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 184 | 195 | theorem coe_real_preimage_closedBall_eq_iUnion (x ε : ℝ) :
(↑) ⁻¹' closedBall (x : AddCircle p) ε = ⋃ z : ℤ, closedBall (x + z • p) ε := by |
rcases eq_or_ne p 0 with (rfl | hp)
· simp [iUnion_const]
ext y
simp only [dist_eq_norm, mem_preimage, mem_closedBall, zsmul_eq_mul, mem_iUnion, Real.norm_eq_abs,
← QuotientAddGroup.mk_sub, norm_eq, ← sub_sub]
refine ⟨fun h => ⟨round (p⁻¹ * (y - x)), h⟩, ?_⟩
rintro ⟨n, hn⟩
rw [← mul_le_mul_left (abs_... |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_import analysis.calculus.deriv.inverse from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
va... | Mathlib/Analysis/Calculus/Deriv/Inverse.lean | 112 | 117 | theorem not_differentiableWithinAt_of_local_left_inverse_hasDerivWithinAt_zero {f g : 𝕜 → 𝕜} {a : 𝕜}
{s t : Set 𝕜} (ha : a ∈ s) (hsu : UniqueDiffWithinAt 𝕜 s a) (hf : HasDerivWithinAt f 0 t (g a))
(hst : MapsTo g s t) (hfg : f ∘ g =ᶠ[𝓝[s] a] id) : ¬DifferentiableWithinAt 𝕜 g s a := by |
intro hg
have := (hf.comp a hg.hasDerivWithinAt hst).congr_of_eventuallyEq_of_mem hfg.symm ha
simpa using hsu.eq_deriv _ this (hasDerivWithinAt_id _ _)
|
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
noncomputable section
open scoped Manifold
open Bundle Set Topology
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H) {M : Type*} [To... | Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean | 200 | 210 | theorem symm_comp_deriv {x : M} (hx : x ∈ e.source) :
(mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) =
ContinuousLinearMap.id 𝕜 (TangentSpace I x) := by |
have : mfderiv I I (e.symm ∘ e) x = (mfderiv I' I e.symm (e x)).comp (mfderiv I I' e x) :=
mfderiv_comp x (he.mdifferentiableAt_symm (e.map_source hx)) (he.mdifferentiableAt hx)
rw [← this]
have : mfderiv I I (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id I
rw [← this]
apply Filter.Even... |
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071ab828a4ee6e26a52afe4"
section Lemmas
namespace Mathlib.Meta.NormNum
def jacobiSymNat (a b : ℕ) : ℤ :=
jacobiSym a b
#align norm_num.jacobi_sym_... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 68 | 69 | theorem jacobiSymNat.zero_right (a : ℕ) : jacobiSymNat a 0 = 1 := by |
rw [jacobiSymNat, jacobiSym.zero_right]
|
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Support
import Mathlib.Topology.Order.IsLUB
#align_import topology.algebra.order.compact from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafee5f9423"
open Filter OrderDual TopologicalSp... | Mathlib/Topology/Algebra/Order/Compact.lean | 178 | 188 | theorem IsCompact.exists_isLeast [ClosedIicTopology α] {s : Set α} (hs : IsCompact s)
(ne_s : s.Nonempty) : ∃ x, IsLeast s x := by |
haveI : Nonempty s := ne_s.to_subtype
suffices (s ∩ ⋂ x ∈ s, Iic x).Nonempty from
⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
rw [biInter_eq_iInter]
by_contra H
rw [not_nonempty_iff_eq_empty] at H
rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Ii... |
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 133 | 134 | theorem mul_invOf_mul_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * ⅟ b * b = a := by |
simp [mul_assoc]
|
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 194 | 204 | theorem hom_eq_iff_not_disjoint (C : G.ComponentCompl L) (h : K ⊆ L) (D : G.ComponentCompl K) :
C.hom h = D ↔ ¬Disjoint (C : Set V) (D : Set V) := by |
rw [Set.not_disjoint_iff]
constructor
· rintro rfl
refine C.ind fun x xnL => ?_
exact ⟨x, ⟨xnL, rfl⟩, ⟨fun xK => xnL (h xK), rfl⟩⟩
· refine C.ind fun x xnL => ?_
rintro ⟨x, ⟨_, e₁⟩, _, rfl⟩
rw [← e₁]
rfl
|
import Mathlib.Logic.Small.Defs
import Mathlib.Logic.Equiv.Set
#align_import logic.small.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
universe u w v v'
section
open scoped Classical
instance small_subtype (α : Type v) [Small.{w} α] (P : α → Prop) : Small.{w} { x // P x } ... | Mathlib/Logic/Small/Basic.lean | 46 | 54 | theorem small_of_injective_of_exists {α : Type v} {β : Type w} {γ : Type v'} [Small.{u} α]
(f : α → γ) {g : β → γ} (hg : Function.Injective g) (h : ∀ b : β, ∃ a : α, f a = g b) :
Small.{u} β := by |
by_cases hβ : Nonempty β
· refine small_of_surjective (f := Function.invFun g ∘ f) (fun b => ?_)
obtain ⟨a, ha⟩ := h b
exact ⟨a, by rw [Function.comp_apply, ha, Function.leftInverse_invFun hg]⟩
· simp only [not_nonempty_iff] at hβ
infer_instance
|
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 444 | 447 | theorem cast_toZModPow (m n : ℕ) (h : m ≤ n) (x : ℤ_[p]) :
ZMod.cast (toZModPow n x) = toZModPow m x := by |
rw [← zmod_cast_comp_toZModPow _ _ h]
rfl
|
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