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 | 373 | 377 | theorem yn_add (m n) : yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n := by |
injection pellZd_add a1 m n with _ h
zify
rw [h]
simp [pellZd]
|
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
∀ x ∈ s, convexHull ℝ (connectedComponentIn s ... | Mathlib/Analysis/Convex/AmpleSet.lean | 79 | 86 | theorem image {s : Set E} (h : AmpleSet s) (L : E ≃ᵃL[ℝ] F) :
AmpleSet (L '' s) := forall_mem_image.mpr fun x hx ↦
calc (convexHull ℝ) (connectedComponentIn (L '' s) (L x))
_ = (convexHull ℝ) (L '' (connectedComponentIn s x)) :=
.symm <| congrArg _ <| L.toHomeomorph.image_connectedComponentIn hx
... | rw [h x hx, image_univ, L.surjective.range_eq]
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 84 | 85 | theorem degrees_def [DecidableEq σ] (p : MvPolynomial σ R) :
p.degrees = p.support.sup fun s : σ →₀ ℕ => Finsupp.toMultiset s := by | rw [degrees]; convert rfl
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,506 | 1,528 | theorem merge_loop (s : α → α → Bool) (l r t) :
merge.loop s l r t = reverseAux t (merge s l r) := by |
rw [merge]; generalize hn : l.length + r.length = n
induction n using Nat.recAux generalizing l r t with
| zero =>
rw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_add_eq_zero_left hn)]
rw [eq_nil_of_length_eq_zero (Nat.eq_zero_of_add_eq_zero_right hn)]
simp only [merge.loop, reverseAux]
| succ n ih =>... |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 121 | 131 | theorem liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by |
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom... |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 312 | 317 | theorem ofReal_sub (p : ℝ) {q : ℝ} (hq : 0 ≤ q) :
ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q := by |
obtain h | h := le_total p q
· rw [ofReal_of_nonpos (sub_nonpos_of_le h), tsub_eq_zero_of_le (ofReal_le_ofReal h)]
refine ENNReal.eq_sub_of_add_eq ofReal_ne_top ?_
rw [← ofReal_add (sub_nonneg_of_le h) hq, sub_add_cancel]
|
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 105 | 108 | theorem rdropWhile_concat (x : α) :
rdropWhile p (l ++ [x]) = if p x then rdropWhile p l else l ++ [x] := by |
simp only [rdropWhile, dropWhile, reverse_append, reverse_singleton, singleton_append]
split_ifs with h <;> simp [h]
|
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Group.Pointwise
#align_import measure_theory.group.fundamental_domain from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
open scoped ENNReal Pointwise Topology NNRea... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 121 | 137 | theorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)
(h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
(h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)
(h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ :=
... |
replace h_meas : ∀ g : G, NullMeasurableSet (g • s) μ := fun g => by
rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹)
have h_meas' : NullMeasurableSet {a | ∃ g : G, g • a ∈ s} μ := by
rw [← iUnion_smul_eq_setOf_exists]; exact .iUnion h_meas
rw [ae_iff_measure_eq h_... |
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 376 | 381 | theorem integral_nonneg {g : ℝⁿ → ℝ} (hg : ∀ x ∈ Box.Icc I, 0 ≤ g x) (μ : Measure ℝⁿ)
[IsLocallyFiniteMeasure μ] : 0 ≤ integral I l g μ.toBoxAdditive.toSMul := by |
by_cases hgi : Integrable I l g μ.toBoxAdditive.toSMul
· refine ge_of_tendsto' hgi.hasIntegral fun π => sum_nonneg fun J _ => ?_
exact mul_nonneg ENNReal.toReal_nonneg (hg _ <| π.tag_mem_Icc _)
· rw [integral, dif_neg hgi]
|
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 | 217 | 217 | theorem arg_I : arg I = π / 2 := by | simp [arg, le_refl]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 71 | 73 | theorem integralNormalization_coeff_ne_degree {f : R[X]} {i : ℕ} (hi : f.degree ≠ i) :
coeff (integralNormalization f) i = coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) := by |
rw [integralNormalization_coeff, if_neg hi]
|
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 423 | 428 | theorem nullHomotopicMap_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (hom : ∀ i j, C.X i ⟶ D.X j) :
(nullHomotopicMap hom).f k₁ = C.d k₁ k₀ ≫ hom k₀ k₁ := by |
dsimp only [nullHomotopicMap]
rw [dNext_eq hom r₁₀, prevD, AddMonoidHom.mk'_apply, D.shape, comp_zero, add_zero]
exact hk₁ _
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 196 | 197 | theorem T_pow (m : ℤ) (n : ℕ) : (T m ^ n : R[T;T⁻¹]) = T (n * m) := by |
rw [T, T, single_pow n, one_pow, nsmul_eq_mul]
|
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 142 | 144 | theorem mem_prehaar_empty {K₀ : Set G} {f : Compacts G → ℝ} :
f ∈ haarProduct K₀ ↔ ∀ K : Compacts G, f K ∈ Icc (0 : ℝ) (index (K : Set G) K₀) := by |
simp only [haarProduct, Set.pi, forall_prop_of_true, mem_univ, mem_setOf_eq]
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 306 | 319 | theorem hasFiniteIntegral_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
(bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
(h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : HasFiniteIntegral f μ := by |
/- `‖F n a‖ ≤ bound a` and `‖F n a‖ --> ‖f a‖` implies `‖f a‖ ≤ bound a`,
and so `∫ ‖f‖ ≤ ∫ bound < ∞` since `bound` is has_finite_integral -/
rw [hasFiniteIntegral_iff_norm]
calc
(∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a, ENNReal.ofReal (bound a) ∂μ :=
lintegral_mono_ae <| all_ae_ofReal_f_le_bound h_... |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 186 | 187 | theorem coprime_add_mul_right_right (m n k : ℕ) : Coprime m (n + k * m) ↔ Coprime m n := by |
rw [Coprime, Coprime, gcd_add_mul_right_right]
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 295 | 295 | theorem none_orElse' (x : Option α) : none.orElse (fun _ ↦ x) = x := by | cases x <;> rfl
|
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 227 | 243 | theorem MonotoneOn.ae_differentiableWithinAt_of_mem {f : ℝ → ℝ} {s : Set ℝ} (hf : MonotoneOn f s) :
∀ᵐ x, x ∈ s → DifferentiableWithinAt ℝ f s x := by |
/- We use a global monotone extension of `f`, and argue that this extension is differentiable
almost everywhere. Such an extension need not exist (think of `1/x` on `(0, +∞)`), but it exists
if one restricts first the function to a compact interval `[a, b]`. -/
apply ae_of_mem_of_ae_of_mem_inter_Ioo
intr... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.trigonometric.inverse_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classic... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 108 | 114 | theorem deriv_arcsin : deriv arcsin = fun x => 1 / √(1 - x ^ 2) := by |
funext x
by_cases h : x ≠ -1 ∧ x ≠ 1
· exact (hasDerivAt_arcsin h.1 h.2).deriv
· rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_arcsin.1 h)]
simp only [not_and_or, Ne, Classical.not_not] at h
rcases h with (rfl | rfl) <;> simp
|
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 | 69 | 76 | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by |
induction' n with n hn
· simp
· have : leadingCoeff (X + 1 : S[X]) = 1 := leadingCoeff_X_add_C 1
rw [ascPochhammer_succ_left, Monic.def, leadingCoeff_mul,
leadingCoeff_comp (ne_zero_of_eq_one <| natDegree_X_add_C 1 : natDegree (X + 1) ≠ 0), hn,
monic_X, one_mul, one_mul, this, one_pow]
|
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 68 | 72 | theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by |
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 427 | 430 | theorem exists_mem_of_measure_ne_zero_of_ae (hs : μ s ≠ 0) {p : α → Prop}
(hp : ∀ᵐ x ∂μ.restrict s, p x) : ∃ x, x ∈ s ∧ p x := by |
rw [← μ.restrict_apply_self, ← frequently_ae_mem_iff] at hs
exact (hs.and_eventually hp).exists
|
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 109 | 111 | theorem ae_not_liouvilleWith : ∀ᵐ x, ∀ p > (2 : ℝ), ¬LiouvilleWith p x := by |
simpa only [ae_iff, not_forall, Classical.not_not, setOf_exists] using
volume_iUnion_setOf_liouvilleWith
|
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
open Polynomial Algebra FiniteD... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 387 | 399 | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by |
refine le_antisymm (adjoin_le fun x hx => ?_) (adjoin_mono fun x hx => ?_)
· suffices hx : x ^ n.1 = 1 by
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine (isRoot_of_unity_iff n.pos B).2 ?_
refine ⟨... |
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Filter Germ Topology
def Hyperreal : Type :=
Germ (hyperfilter ℕ : Filter ℕ) ℝ deri... | Mathlib/Data/Real/Hyperreal.lean | 794 | 801 | theorem st_inv (x : ℝ*) : st x⁻¹ = (st x)⁻¹ := by |
by_cases h0 : x = 0
· rw [h0, inv_zero, ← coe_zero, st_id_real, inv_zero]
by_cases h1 : Infinitesimal x
· rw [((infinitesimal_iff_infinite_inv h0).mp h1).st_eq, h1.st_eq, inv_zero]
by_cases h2 : Infinite x
· rw [(infinitesimal_inv_of_infinite h2).st_eq, h2.st_eq, inv_zero]
exact ((isSt_st' h2).inv h1).st... |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 340 | 343 | theorem cthickening_ball (hε : 0 ≤ ε) (hδ : 0 < δ) (x : E) :
cthickening ε (ball x δ) = closedBall x (ε + δ) := by |
rw [← thickening_singleton, cthickening_thickening hε hδ,
cthickening_singleton _ (add_nonneg hε hδ.le)]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 483 | 485 | theorem zero_convolution : 0 ⋆[L, μ] g = 0 := by |
ext
simp_rw [convolution_def, Pi.zero_apply, L.map_zero₂, integral_zero]
|
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#align_import measure_theory.integral.peak_function from "leanprover-community/mathlib"@"13b0d72fd8533ba459ac66e9a885e35ffabb32b2"
open Set Filter MeasureTheory MeasureTheory.Measure TopologicalSpace Metric
open scoped Topology ENNReal
open Set
variable... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 99 | 182 | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux
(hs : MeasurableSet s) (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂... |
refine Metric.tendsto_nhds.2 fun ε εpos => ?_
obtain ⟨δ, hδ, δpos, δone⟩ : ∃ δ, (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ < ε ∧ 0 < δ ∧ δ < 1:= by
have A :
Tendsto (fun δ => (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ) (𝓝[>] 0)
(𝓝 ((0 * ∫ x in s, ‖g x‖ ∂μ) + 2 * 0)) := by
apply Tendsto.mono_left _ nhdsWithin... |
import Mathlib.CategoryTheory.Adjunction.Opposites
import Mathlib.CategoryTheory.Comma.Presheaf
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.CategoryTheory.Limits.Over... | Mathlib/CategoryTheory/Limits/Presheaf.lean | 486 | 503 | theorem final_toCostructuredArrow_comp_pre {c : Cocone (F ⋙ yoneda)} (hc : IsColimit c) :
Functor.Final (c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) := by |
apply Functor.cofinal_of_isTerminal_colimit_comp_yoneda
suffices IsTerminal (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙
CostructuredArrow.toOver yoneda c.pt)) by
apply IsTerminal.isTerminalOfObj (overEquivPresheafCostructuredArrow c.pt).inverse
apply IsTerminal.ofIso th... |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 125 | 134 | theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) := by |
refine
⟨fun hf u hu => hf.comp hu, fun h =>
((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 ?_⟩
rintro s ⟨hbs, hsc⟩
refine ⟨closure (f ⁻¹' s), ⟨mt ?_ hbs, isClosed_closure⟩, fun x => mt fun hx => subset_closure hx⟩
rw [← seqClosure_eq_closure]
rintro ⟨u, hus, hu⟩
exact hsc.mem_of_te... |
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 172 | 186 | theorem smul_apply (f : R[X]) (g : PolynomialModule R M) (n : ℕ) :
(f • g) n = ∑ x ∈ Finset.antidiagonal n, f.coeff x.1 • g x.2 := by |
induction' f using Polynomial.induction_on' with p q hp hq f_n f_a
· rw [add_smul, Finsupp.add_apply, hp, hq, ← Finset.sum_add_distrib]
congr
ext
rw [coeff_add, add_smul]
· rw [Finset.Nat.sum_antidiagonal_eq_sum_range_succ fun i j => (monomial f_n f_a).coeff i • g j,
monomial_smul_apply]
si... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 141 | 142 | theorem two_nsmul_coe_div_two (θ : ℝ) : (2 : ℕ) • (↑(θ / 2) : Angle) = θ := by |
rw [← coe_nsmul, two_nsmul, add_halves]
|
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 396 | 414 | theorem eq_properDivisors_of_subset_of_sum_eq_sum {s : Finset ℕ} (hsub : s ⊆ n.properDivisors) :
((∑ x ∈ s, x) = ∑ x ∈ n.properDivisors, x) → s = n.properDivisors := by |
cases n
· rw [properDivisors_zero, subset_empty] at hsub
simp [hsub]
classical
rw [← sum_sdiff hsub]
intro h
apply Subset.antisymm hsub
rw [← sdiff_eq_empty_iff_subset]
contrapose h
rw [← Ne, ← nonempty_iff_ne_empty] at h
apply ne_of_lt
rw [← zero_add (∑ x ∈ s, x), ← add_assoc... |
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 358 | 359 | theorem mfderivWithin_of_mem_nhds (h : s ∈ 𝓝 x) : mfderivWithin I I' f s x = mfderiv I I' f x := by |
rw [← mfderivWithin_univ, ← univ_inter s, mfderivWithin_inter h]
|
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 102 | 110 | theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
(ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by |
rw [← kernel.kernel_sum_seq κ]
have : ∀ a, kernel.sum (kernel.seq κ) a (Prod.mk a ⁻¹' t) =
∑' n, kernel.seq κ n a (Prod.mk a ⁻¹' t) := fun a =>
kernel.sum_apply' _ _ (measurable_prod_mk_left ht)
simp_rw [this]
refine Measurable.ennreal_tsum fun n => ?_
exact measurable_kernel_prod_mk_left_of_finite... |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {α β F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 70 | 76 | theorem continuousOn_tsum [TopologicalSpace β] {f : α → β → F} {s : Set β}
(hf : ∀ i, ContinuousOn (f i) s) (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
ContinuousOn (fun x => ∑' n, f n x) s := by |
classical
refine (tendstoUniformlyOn_tsum hu hfu).continuousOn (eventually_of_forall ?_)
intro t
exact continuousOn_finset_sum _ fun i _ => hf i
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 102 | 104 | theorem binomial_eq [DecidableEq α] (h : a ≠ b) :
multinomial {a, b} f = (f a + f b)! / ((f a)! * (f b)!) := by |
simp [multinomial, Finset.sum_pair h, Finset.prod_pair h]
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 299 | 310 | theorem iIndepFun.mgf_sum [IsProbabilityMeasure μ] {X : ι → Ω → ℝ}
(h_indep : iIndepFun (fun i => inferInstance) X μ) (h_meas : ∀ i, Measurable (X i))
(s : Finset ι) : mgf (∑ i ∈ s, X i) μ t = ∏ i ∈ s, mgf (X i) μ t := by |
classical
induction' s using Finset.induction_on with i s hi_notin_s h_rec h_int
· simp only [sum_empty, mgf_zero_fun, measure_univ, ENNReal.one_toReal, prod_empty]
· have h_int' : ∀ i : ι, AEStronglyMeasurable (fun ω : Ω => exp (t * X i ω)) μ := fun i =>
((h_meas i).const_mul t).exp.aestronglyMeasurable... |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 132 | 132 | theorem div_neg (a : K) : a / -b = -(a / b) := by | rw [← div_neg_eq_neg_div]
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 479 | 480 | theorem image_sub_const_Ico : (fun x => x - a) '' Ico b c = Ico (b - a) (c - a) := by |
simp [sub_eq_neg_add]
|
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 91 | 92 | theorem smul_unitBall {c : 𝕜} (hc : c ≠ 0) : c • ball (0 : E) (1 : ℝ) = ball (0 : E) ‖c‖ := by |
rw [_root_.smul_ball hc, smul_zero, mul_one]
|
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 | 552 | 573 | theorem perm_lookmap (f : α → Option α) {l₁ l₂ : List α}
(H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁) (p : l₁ ~ l₂) :
lookmap f l₁ ~ lookmap f l₂ := by |
induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ p₁ _ IH₁ IH₂; · simp
· cases h : f a
· simp [h]
exact IH (pairwise_cons.1 H).2
· simp [lookmap_cons_some _ _ h, p]
· cases' h₁ : f a with c <;> cases' h₂ : f b with d
· simp [h₁, h₂]
apply swap
· simp [h₁, lookmap_cons_some _ _ h₂]
a... |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 1,733 | 1,734 | theorem preimage_sUnion {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋃₀s = ⋃ t ∈ s, f ⁻¹' t := by |
rw [sUnion_eq_biUnion, preimage_iUnion₂]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,029 | 1,045 | theorem lintegral_abs_det_fderiv_le_addHaar_image_aux2 (hs : MeasurableSet s) (h's : μ s ≠ ∞)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
(∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) := by |
-- We just need to let the error tend to `0` in the previous lemma.
have :
Tendsto (fun ε : ℝ≥0 => μ (f '' s) + 2 * ε * μ s) (𝓝[>] 0)
(𝓝 (μ (f '' s) + 2 * (0 : ℝ≥0) * μ s)) := by
apply Tendsto.mono_left _ nhdsWithin_le_nhds
refine tendsto_const_nhds.add ?_
refine ENNReal.Tendsto.mul_const ?... |
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 | 93 | 114 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by |
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [re_ofReal_mul, im_ofReal_mul, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left₀ _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_... |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | Mathlib/Algebra/Lie/Solvable.lean | 82 | 85 | theorem derivedSeriesOfIdeal_add (k l : ℕ) : D (k + l) I = D k (D l I) := by |
induction' k with k ih
· rw [Nat.zero_add, derivedSeriesOfIdeal_zero]
· rw [Nat.succ_add k l, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ, ih]
|
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacobi_theta.basic from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
open Complex Real Asymptotics Filter Topology
open scope... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 75 | 86 | theorem hasSum_nat_jacobiTheta {τ : ℂ} (hτ : 0 < im τ) :
HasSum (fun n : ℕ => cexp (π * I * ((n : ℂ) + 1) ^ 2 * τ)) ((jacobiTheta τ - 1) / 2) := by |
have := hasSum_jacobiTheta₂_term 0 hτ
simp_rw [jacobiTheta₂_term, mul_zero, zero_add, ← jacobiTheta_eq_jacobiTheta₂] at this
have := this.nat_add_neg
rw [← hasSum_nat_add_iff' 1] at this
simp_rw [Finset.sum_range_one, Int.cast_neg, Int.cast_natCast, Nat.cast_zero, neg_zero,
Int.cast_zero, sq (0 : ℂ), mul... |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 52 | 55 | theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by |
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
|
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 286 | 296 | theorem finish_v (hr : u.r = 0) : u.finish.v = u.v := by |
let ha : u.r + u.b * u.q = u.a := u.rq_eq
rw [hr, zero_add] at ha
ext
· change (u.wp + 1) * u.b + ((u.wp + 1) * u.qp + u.x) * u.b = u.w * u.a + u.x * u.b
have : u.wp + 1 = u.w := rfl
rw [this, ← ha, u.qp_eq hr]
ring
· change u.y * u.b + (u.y * u.qp + u.z) * u.b = u.y * u.a + u.z * u.b
rw [← h... |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 102 | 104 | theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by |
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 717 | 720 | theorem setIntegral_trim {X} {m m0 : MeasurableSpace X} {μ : Measure X} (hm : m ≤ m0) {f : X → E}
(hf_meas : StronglyMeasurable[m] f) {s : Set X} (hs : MeasurableSet[m] s) :
∫ x in s, f x ∂μ = ∫ x in s, f x ∂μ.trim hm := by |
rwa [integral_trim hm hf_meas, restrict_trim hm μ]
|
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 182 | 184 | theorem toRightMovesNim_one_symm (i) :
(@toRightMovesNim 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by |
simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 120 | 124 | theorem differentiableAt_iff_restrictScalars (hf : DifferentiableAt 𝕜 f x) :
DifferentiableAt 𝕜' f x ↔ ∃ g' : E →L[𝕜'] F, g'.restrictScalars 𝕜 = fderiv 𝕜 f x := by |
rw [← differentiableWithinAt_univ, ← fderivWithin_univ]
exact
differentiableWithinAt_iff_restrictScalars 𝕜 hf.differentiableWithinAt uniqueDiffWithinAt_univ
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 274 | 281 | theorem deriv_mul_const_field (v : 𝕜') : deriv (fun y => u y * v) x = deriv u x * v := by |
by_cases hu : DifferentiableAt 𝕜 u x
· exact deriv_mul_const hu v
· rw [deriv_zero_of_not_differentiableAt hu, zero_mul]
rcases eq_or_ne v 0 with (rfl | hd)
· simp only [mul_zero, deriv_const]
· refine deriv_zero_of_not_differentiableAt (mt (fun H => ?_) hu)
simpa only [mul_inv_cancel_right₀ h... |
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 263 | 265 | theorem bodd_bit (b n) : bodd (bit b n) = b := by |
rw [bit_val]
cases b <;> cases bodd n <;> simp [(show bodd 2 = false by rfl)]
|
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 54 | 54 | theorem id : IsSemiringHom (@id α) := by | constructor <;> intros <;> rfl
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 182 | 189 | theorem _root_.Submodule.adjoint_subtypeL (U : Submodule 𝕜 E) [CompleteSpace U] :
U.subtypeL† = orthogonalProjection U := by |
symm
rw [eq_adjoint_iff]
intro x u
rw [U.coe_inner, inner_orthogonalProjection_left_eq_right,
orthogonalProjection_mem_subspace_eq_self]
rfl
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 1,306 | 1,321 | theorem le_iff_sign {x y : EReal} :
x ≤ y ↔ sign x < sign y ∨
sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
sign x = SignType.zero ∧ sign y = SignType.zero ∨
sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by |
constructor
· intro h
refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
cases hy : sign y <;> rw [hs, hy] at h ⊢
· simp
· left; simpa using h
· right; right; simpa using h
· rintro (h | h | h | h)
· exact (sign.monotone.reflect_lt ... |
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 | 667 | 667 | theorem transnumAuxSeq_zero : f.transnumAuxSeq 0 = f 0 := by | simp [transnumAuxSeq]
|
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Completion
import Mathlib.Topology.MetricSpace.GromovHausdorffRealized
import Mathlib.Topology.MetricSpace.Kuratowski
#align_import topology.metric_space.gromov_hausdorff from "leanprover-community/... | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 150 | 173 | theorem toGHSpace_eq_toGHSpace_iff_isometryEquiv {X : Type u} [MetricSpace X] [CompactSpace X]
[Nonempty X] {Y : Type v} [MetricSpace Y] [CompactSpace Y] [Nonempty Y] :
toGHSpace X = toGHSpace Y ↔ Nonempty (X ≃ᵢ Y) :=
⟨by
simp only [toGHSpace]
rw [Quotient.eq]
rintro ⟨e⟩
have I :
(Nonemp... |
dsimp only [NonemptyCompacts.kuratowskiEmbedding]; rfl
have f := (kuratowskiEmbedding.isometry X).isometryEquivOnRange
have g := (kuratowskiEmbedding.isometry Y).isometryEquivOnRange.symm
exact ⟨f.trans <| (cast I e).trans g⟩, by
rintro ⟨e⟩
simp only [toGHSpace, Quotient.eq']
have f := (k... |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 249 | 252 | theorem eq_one_of_x_eq_one (h₀ : d ≠ 0) {a : Solution₁ d} (ha : a.x = 1) : a = 1 := by |
have prop := a.prop_y
rw [ha, one_pow, sub_self, mul_eq_zero, or_iff_right h₀, sq_eq_zero_iff] at prop
exact ext ha prop
|
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 220 | 223 | theorem sum_measure_preimage_singleton (s : Finset β) {f : α → β}
(hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : (∑ b ∈ s, μ (f ⁻¹' {b})) = μ (f ⁻¹' ↑s) := by |
simp only [← measure_biUnion_finset (pairwiseDisjoint_fiber f s) hf,
Finset.set_biUnion_preimage_singleton]
|
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.centering from "leanprover-community/mathlib"@"bea6c853b6edbd15e9d0941825abd04d77933ed0"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
variable {Ω E : Type*} {m0 : ... | Mathlib/Probability/Martingale/Centering.lean | 135 | 153 | theorem martingalePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
(hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
(hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : martingalePart (f + g) ℱ μ n =ᵐ[μ] f n := by |
set h := f - martingalePart (f + g) ℱ μ with hhdef
have hh : h = predictablePart (f + g) ℱ μ - g := by
rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) ℱ μ),
martingalePart_add_predictablePart]
have hhpred : Adapted ℱ fun n => h (n + 1) := by
rw [hh]
exact adapted_predict... |
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 577 | 580 | theorem _root_.MeasurableEmbedding.setIntegral_map {Y} {_ : MeasurableSpace Y} {f : X → Y}
(hf : MeasurableEmbedding f) (g : Y → E) (s : Set Y) :
∫ y in s, g y ∂Measure.map f μ = ∫ x in f ⁻¹' s, g (f x) ∂μ := by |
rw [hf.restrict_map, hf.integral_map]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 329 | 331 | theorem map_volume_mul_right {a : ℝ} (h : a ≠ 0) :
Measure.map (· * a) volume = ENNReal.ofReal |a⁻¹| • volume := by |
simpa only [mul_comm] using Real.map_volume_mul_left h
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.TensorProduct.Basic
#align_import data.matrix.kronecker from "leanpr... | Mathlib/Data/Matrix/Kronecker.lean | 225 | 230 | theorem trace_kroneckerMapBilinear [CommSemiring R] [Fintype m] [Fintype n] [AddCommMonoid α]
[AddCommMonoid β] [AddCommMonoid γ] [Module R α] [Module R β] [Module R γ]
(f : α →ₗ[R] β →ₗ[R] γ) (A : Matrix m m α) (B : Matrix n n β) :
trace (kroneckerMapBilinear f A B) = f (trace A) (trace B) := by |
simp_rw [Matrix.trace, Matrix.diag, kroneckerMapBilinear_apply_apply, LinearMap.map_sum₂,
map_sum, ← Finset.univ_product_univ, Finset.sum_product, kroneckerMap_apply]
|
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | Mathlib/Topology/Germ.lean | 94 | 102 | theorem Filter.Eventually.germ_congr_set
{P : ∀ x : X, Germ (𝓝 x) Y → Prop} (hf : ∀ᶠ x in 𝓝ˢ A, P x f)
(h : ∀ᶠ z in 𝓝ˢ A, g z = f z) : ∀ᶠ x in 𝓝ˢ A, P x g := by |
rw [eventually_nhdsSet_iff_forall] at *
intro x hx
apply ((hf x hx).and (h x hx).eventually_nhds).mono
intro y hy
convert hy.1 using 1
exact Germ.coe_eq.mpr hy.2
|
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 79 | 92 | theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
p.removeZero.leftInv i = p.leftInv i := by |
ext1 n
induction' n using Nat.strongRec' with n IH
match n with
| 0 => simp -- if one replaces `simp` with `refl`, the proof times out in the kernel.
| 1 => simp -- TODO: why?
| n + 2 =>
simp only [leftInv, neg_inj]
refine Finset.sum_congr rfl fun c cuniv => ?_
rcases c with ⟨c, hc⟩
ext v
... |
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 469 | 472 | theorem Antiperiodic.zsmul_sub_eq [AddCommGroup α] [AddGroup β] (h : Antiperiodic f c) (n : ℤ) :
f (n • c - x) = (n.negOnePow : ℤ) • f (-x) := by |
rw [sub_eq_add_neg, add_comm]
exact h.add_zsmul_eq n
|
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 166 | 170 | theorem toOuterMeasure_caratheodory : p.toOuterMeasure.caratheodory = ⊤ := by |
refine eq_top_iff.2 <| le_trans (le_sInf fun x hx => ?_) (le_sum_caratheodory _)
have ⟨y, hy⟩ := hx
exact
((le_of_eq (dirac_caratheodory y).symm).trans (le_smul_caratheodory _ _)).trans (le_of_eq hy)
|
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 220 | 223 | theorem _root_.Ideal.fg_of_isUnit (inj : Function.Injective (algebraMap R P)) (I : Ideal R)
(h : IsUnit (I : FractionalIdeal S P)) : I.FG := by |
rw [← coeIdeal_fg S inj I]
exact FractionalIdeal.fg_of_isUnit I h
|
import Mathlib.Init.Classical
import Mathlib.Order.FixedPoints
import Mathlib.Order.Zorn
#align_import set_theory.cardinal.schroeder_bernstein from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Set Function
open scoped Classical
universe u v
namespace Function
namespace Embedd... | Mathlib/SetTheory/Cardinal/SchroederBernstein.lean | 100 | 131 | theorem min_injective [I : Nonempty ι] : ∃ i, Nonempty (∀ j, β i ↪ β j) :=
let ⟨s, hs, ms⟩ :=
show ∃ s ∈ sets β, ∀ a ∈ sets β, s ⊆ a → a = s from
zorn_subset (sets β) fun c hc hcc =>
⟨⋃₀c, fun x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi =>
(hcc.total hpc hqc).elim (fun h => hc hqc x (h hxp) y hyq... |
simpa only [ne_eq, not_exists, not_forall, not_and] using h
let ⟨f, hf⟩ := Classical.axiom_of_choice h
have : f ∈ s :=
have : insert f s ∈ sets β := fun x hx y hy => by
cases' hx with hx hx <;> cases' hy with hy hy; · simp [hx, hy]
· subst x
exa... |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 378 | 380 | theorem IsCyclic.image_range_card (ha : ∀ x : α, x ∈ zpowers a) :
Finset.image (fun i => a ^ i) (range (Fintype.card α)) = univ := by |
rw [← orderOf_eq_card_of_forall_mem_zpowers ha, IsCyclic.image_range_orderOf ha]
|
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
universe v v₁ u₁ u
open CategoryTheory LinearMap Opposite
variable {C : Type u₁} [Category.{v₁} C]
structure PresheafOfModules (R : Cᵒᵖ ⥤ RingCat.{u}) where
presheaf : Cᵒᵖ ⥤ AddCommGroupCat.{v}
module : ∀ X : ... | Mathlib/Algebra/Category/ModuleCat/Presheaf.lean | 85 | 88 | theorem map_comp (P : PresheafOfModules R) {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
P.map (f ≫ g) = (P.map g).comp (P.map f) := by |
ext
simp [map_apply]
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 350 | 355 | theorem IsLindelof.countable_of_discrete [DiscreteTopology X] (hs : IsLindelof s) :
s.Countable := by |
have : ∀ x : X, ({x} : Set X) ∈ 𝓝 x := by simp [nhds_discrete]
rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, ht, _, hssubt⟩
rw [biUnion_of_singleton] at hssubt
exact ht.mono hssubt
|
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 145 | 148 | theorem exponent_eq_zero_iff_forall : exponent G = 0 ↔ ∀ n > 0, ∃ g : G, g ^ n ≠ 1 := by |
rw [exponent_eq_zero_iff, ExponentExists]
push_neg
rfl
|
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 465 | 482 | theorem exists_nontrivial_relation_sum_zero_of_not_affine_ind {t : Finset V}
(h : ¬AffineIndependent k ((↑) : t → V)) :
∃ f : V → k, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 := by |
classical
rw [affineIndependent_iff_of_fintype] at h
simp only [exists_prop, not_forall] at h
obtain ⟨w, hw, hwt, i, hi⟩ := h
simp only [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w ((↑) : t → V) hw 0,
vsub_eq_sub, Finset.weightedVSubOfPoint_apply, sub_zero] at hwt
let f : ... |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 400 | 404 | theorem orElse_eq_some (o o' : Option α) (x : α) :
(o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by |
cases o
· simp only [true_and, false_or, eq_self_iff_true, none_orElse]
· simp only [some_orElse, or_false, false_and]
|
import Mathlib.RingTheory.Valuation.Integers
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.Bezout
import Mathlib.Tactic.FieldSimp
#align_import... | Mathlib/RingTheory/Valuation/ValuationRing.lean | 406 | 413 | theorem of_integers : ValuationRing 𝒪 := by |
constructor
intro a b
rcases le_total (v (algebraMap 𝒪 K a)) (v (algebraMap 𝒪 K b)) with h | h
· obtain ⟨c, hc⟩ := Valuation.Integers.dvd_of_le hh h
use c; exact Or.inr hc.symm
· obtain ⟨c, hc⟩ := Valuation.Integers.dvd_of_le hh h
use c; exact Or.inl hc.symm
|
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 40 | 60 | theorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[>] a,
s ∈ 𝓝[Ioc a b] a,
s ∈ 𝓝[Ioo a b] a,
∃ u ∈ Ioc a b, Ioo a u ⊆ s,
∃ u ∈ Ioi a, Ioo a u ⊆ s] := by |
tfae_have 1 ↔ 2
· rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]
tfae_have 1 ↔ 3
· rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
tfae_have 4 → 5
· exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
tfae_have 5 → 1
· rintro ⟨u, hau, hu⟩
exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu
tfae_have 1... |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 117 | 117 | theorem lift_one : q.lift (1 : ℍ[R,c₁,c₂]) = 1 := by | simp [lift]
|
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 31 | 33 | theorem tsub_add_cancel_of_le (h : a ≤ b) : b - a + a = b := by |
rw [add_comm]
exact add_tsub_cancel_of_le h
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 916 | 924 | theorem Integrable.real_toNNReal {f : α → ℝ} (hf : Integrable f μ) :
Integrable (fun x => ((f x).toNNReal : ℝ)) μ := by |
refine
⟨hf.aestronglyMeasurable.aemeasurable.real_toNNReal.coe_nnreal_real.aestronglyMeasurable, ?_⟩
rw [hasFiniteIntegral_iff_norm]
refine lt_of_le_of_lt ?_ ((hasFiniteIntegral_iff_norm _).1 hf.hasFiniteIntegral)
apply lintegral_mono
intro x
simp [ENNReal.ofReal_le_ofReal, abs_le, le_abs_self]
|
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Integral.DominatedConve... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,279 | 1,310 | theorem integrableOn_deriv_right_of_nonneg (hcont : ContinuousOn g (Icc a b))
(hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x)
(g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x) : IntegrableOn g' (Ioc a b) := by |
by_cases hab : a < b; swap
· simp [Ioc_eq_empty hab]
rw [integrableOn_Ioc_iff_integrableOn_Ioo]
have meas_g' : AEMeasurable g' (volume.restrict (Ioo a b)) := by
apply (aemeasurable_derivWithin_Ioi g _).congr
refine (ae_restrict_mem measurableSet_Ioo).mono fun x hx => ?_
exact (hderiv x hx).derivWit... |
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 | 1,171 | 1,174 | theorem isOpen_iff_nhds : IsOpen s ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s :=
calc
IsOpen s ↔ s ⊆ interior s := subset_interior_iff_isOpen.symm
_ ↔ ∀ x ∈ s, 𝓝 x ≤ 𝓟 s := by | simp_rw [interior_eq_nhds, subset_def, mem_setOf]
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 132 | 138 | theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array β),
length (mapIdx.go f l arr) = length l + arr.size := by |
intro f l
induction' l with head tail ih
· intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add]
· intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons];
simp only [Nat.add_succ, add_zero, Nat.add_comm]
|
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 265 | 275 | theorem seq_norm_le (n : ℕ) :
Int.natAbs (Algebra.norm ℤ (seq K w₁ hB n : 𝓞 K)) ≤ B := by |
cases n with
| zero =>
have : 1 ≤ B := by
contrapose! hB
simp only [Nat.lt_one_iff.mp hB, CharP.cast_eq_zero, mul_zero, zero_le]
simp only [ne_eq, seq, map_one, Int.natAbs_one, this]
| succ n =>
rw [← Nat.cast_le (α := ℚ), Int.cast_natAbs, Int.cast_abs, Algebra.coe_norm_int]
... |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 360 | 369 | theorem total_of_right_unique (U : Relator.RightUnique r) (ab : ReflTransGen r a b)
(ac : ReflTransGen r a c) : ReflTransGen r b c ∨ ReflTransGen r c b := by |
induction' ab with b d _ bd IH
· exact Or.inl ac
· rcases IH with (IH | IH)
· rcases cases_head IH with (rfl | ⟨e, be, ec⟩)
· exact Or.inr (single bd)
· cases U bd be
exact Or.inl ec
· exact Or.inr (IH.tail bd)
|
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Set Filter Function Topology
noncomputable section
namespace TopologicalSpace
universe u
variable {α : Type u} {β : Type*} ... | Mathlib/Topology/Bases.lean | 454 | 456 | theorem IsSeparable.mono {s u : Set α} (hs : IsSeparable s) (hu : u ⊆ s) : IsSeparable u := by |
rcases hs with ⟨c, c_count, hs⟩
exact ⟨c, c_count, hu.trans hs⟩
|
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
| Mathlib/Algebra/Ring/Center.lean | 24 | 37 | theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by | rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul]
mid_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, mul_zero, zero_... |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 85 | 90 | theorem gradedCommAux_lof_tmul (i j : ι) (a : 𝒜 i) (b : ℬ j) :
gradedCommAux R 𝒜 ℬ (lof R _ 𝒜ℬ (i, j) (a ⊗ₜ b)) =
(-1 : ℤˣ)^(j * i) • lof R _ ℬ𝒜 (j, i) (b ⊗ₜ a) := by |
rw [gradedCommAux]
dsimp
simp [mul_comm i j]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 85 | 87 | theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by |
simp only [rpow_def_of_nonneg hx]
split_ifs <;> simp [*, exp_ne_zero]
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 246 | 246 | theorem get_eq_iff : List.get l n = x ↔ l.get? n.1 = some x := by | simp [get?_eq_some]
|
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 189 | 194 | theorem δ_comp_σ_of_gt' {n} {i : Fin (n + 3)} {j : Fin (n + 2)} (H : j.succ < i) :
X.σ j ≫ X.δ i =
X.δ (i.pred fun (hi : i = 0) => by simp only [Fin.not_lt_zero, hi] at H) ≫
X.σ (j.castLT ((add_lt_add_iff_right 1).mp (lt_of_lt_of_le H i.is_le))) := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_gt' H]
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 263 | 264 | theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ := by |
rw [← coe_subset, coe_Icc, coe_Ioo, Set.Icc_subset_Ioo_iff h₁]
|
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 150 | 152 | theorem t0Space_iff_uniformity :
T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by |
simp only [t0Space_iff_inseparable, inseparable_iff_ker_uniformity, mem_ker, id]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 203 | 207 | theorem eval₂_repr_eq_eval₂_of_map_eq (h : IsAdjoinRoot S f) (z : S) (w : R[X])
(hzw : h.map w = z) : (h.repr z).eval₂ i x = w.eval₂ i x := by |
rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw
obtain ⟨y, hy⟩ := hzw
rw [← sub_eq_zero, ← eval₂_sub, hy, eval₂_mul, hx, zero_mul]
|
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 124 | 130 | theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
prevD i f = f i (i - 1) ≫ D.d (i - 1) i := by |
dsimp [prevD]
cases i
· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
not_false_iff, comp_zero]
· congr <;> simp
|
import Mathlib.FieldTheory.Finite.Basic
#align_import field_theory.chevalley_warning from "leanprover-community/mathlib"@"e001509c11c4d0f549d91d89da95b4a0b43c714f"
universe u v
section FiniteField
open MvPolynomial
open Function hiding eval
open Finset FiniteField
variable {K σ ι : Type*} [Fintype K] [Field ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 168 | 171 | theorem char_dvd_card_solutions_of_fintype_sum_lt [Fintype ι] {f : ι → MvPolynomial σ K}
(h : (∑ i, (f i).totalDegree) < Fintype.card σ) :
p ∣ Fintype.card { x : σ → K // ∀ i, eval x (f i) = 0 } := by |
simpa using char_dvd_card_solutions_of_sum_lt p h
|
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 378 | 380 | theorem coeff_bit0_mul (P Q : R[X]) (n : ℕ) : coeff (bit0 P * Q) n = 2 * coeff (P * Q) n := by |
-- Porting note: `two_mul` is required.
simp [bit0, add_mul, two_mul]
|
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