Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Set.Card
import Mathlib.GroupTheory.Subgroup.Center
open MulAction ConjClasses
variable (G : Type*) [Group G]
theorem sum_conjClasses_card_eq_card [Fintype <| Conj... | Mathlib/GroupTheory/ClassEquation.lean | 47 | 70 | theorem Group.nat_card_center_add_sum_card_noncenter_eq_card [Finite G] :
Nat.card (Subgroup.center G) + ∑ᶠ x ∈ noncenter G, Nat.card x.carrier = Nat.card G := by |
classical
cases nonempty_fintype G
rw [@Nat.card_eq_fintype_card G, ← sum_conjClasses_card_eq_card, ←
Finset.sum_sdiff (ConjClasses.noncenter G).toFinset.subset_univ]
simp only [Nat.card_eq_fintype_card, Set.toFinset_card]
congr 1
swap
· convert finsum_cond_eq_sum_of_cond_iff _ _
simp [Set.mem_to... | 0 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 112 | 133 | theorem coeff_add_of_disjoint (x y : 𝕎 R) (h : ∀ n, x.coeff n = 0 ∨ y.coeff n = 0) :
(x + y).coeff n = x.coeff n + y.coeff n := by |
let P : ℕ → Prop := fun n => y.coeff n = 0
haveI : DecidablePred P := Classical.decPred P
set z := mk p fun n => if P n then x.coeff n else y.coeff n
have hx : select P z = x := by
ext1 n; rw [select, coeff_mk, coeff_mk]
split_ifs with hn
· rfl
· rw [(h n).resolve_right hn]
have hy : select (... | 0 |
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 | 150 | 156 | theorem isSpecial_iff : u.IsSpecial ↔ u.IsSpecial' := by |
dsimp [IsSpecial, IsSpecial']
let ⟨wp, x, y, zp, ap, bp⟩ := u
constructor <;> intro h <;> simp [w, z, succPNat] at * <;>
simp only [← coe_inj, mul_coe, mk_coe] at *
· simp_all [← h, Nat.mul, Nat.succ_eq_add_one]; ring
· simp [Nat.succ_eq_add_one, Nat.mul_add, Nat.add_mul, ← Nat.add_assoc] at h; rw [← h];... | 0 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {σ R : Type*} [CommSemiring R]
namespace MvPolynomial
theorem monomial_dvd_monomial {r s : ... | Mathlib/Algebra/MvPolynomial/Division.lean | 260 | 263 | theorem X_dvd_monomial {i : σ} {j : σ →₀ ℕ} {r : R} :
(X i : MvPolynomial σ R) ∣ monomial j r ↔ r = 0 ∨ j i ≠ 0 := by |
refine monomial_dvd_monomial.trans ?_
simp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
| 0 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Order.OmegaCompletePartialOrder
namespace SimpleGraph
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <|... | Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean | 43 | 49 | theorem chromaticNumber_pathGraph (n : ℕ) (h : 2 ≤ n) :
(pathGraph n).chromaticNumber = 2 := by |
have hc := (pathGraph.bicoloring n).colorable
apply le_antisymm
· exact hc.chromaticNumber_le
· simpa only [pathGraph_two_eq_top, chromaticNumber_top] using
chromaticNumber_mono_of_embedding (pathGraph_two_embedding n h)
| 0 |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Typ... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 82 | 85 | theorem congr_set {s t : Set H} {x : H} {f : H → H'} (hu : s =ᶠ[𝓝 x] t) : P f s x ↔ P f t x := by |
obtain ⟨o, host, ho, hxo⟩ := mem_nhds_iff.mp hu.mem_iff
simp_rw [subset_def, mem_setOf, ← and_congr_left_iff, ← mem_inter_iff, ← Set.ext_iff] at host
rw [hG.is_local ho hxo, host, ← hG.is_local ho hxo]
| 0 |
import Mathlib.RingTheory.HahnSeries.Addition
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 152 | 161 | theorem smul_coeff_right [SMulZeroClass R W] {x : HahnSeries Γ R}
{y : HahnModule Γ R W} {a : Γ} {s : Set Γ} (hs : s.IsPWO) (hys : ((of R).symm y).support ⊆ s) :
((of R).symm <| x • y).coeff a =
∑ ij ∈ addAntidiagonal x.isPWO_support hs a,
x.coeff ij.fst • ((of R).symm y).coeff ij.snd := by |
rw [smul_coeff]
apply sum_subset_zero_on_sdiff (addAntidiagonal_mono_right hys) _ fun _ _ => rfl
intro b hb
simp only [not_and, mem_sdiff, mem_addAntidiagonal, HahnSeries.mem_support, not_imp_not] at hb
rw [hb.2 hb.1.1 hb.1.2.2, smul_zero]
| 0 |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.AlgebraicIndependent
#align_import field_theory.is_alg_closed.classification from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 41 | 59 | theorem cardinal_mk_le_sigma_polynomial :
#L ≤ #(Σ p : R[X], { x : L // x ∈ p.aroots L }) :=
@mk_le_of_injective L (Σ p : R[X], {x : L | x ∈ p.aroots L})
(fun x : L =>
let p := Classical.indefiniteDescription _ (Algebra.IsAlgebraic.isAlgebraic x)
⟨p.1, x, by
dsimp
have h : p.1.map ... |
rw [Ne, ← Polynomial.degree_eq_bot,
Polynomial.degree_map_eq_of_injective (NoZeroSMulDivisors.algebraMap_injective R L),
Polynomial.degree_eq_bot]
exact p.2.1
erw [Polynomial.mem_roots h, Polynomial.IsRoot, Polynomial.eval_map, ← Polynomial.aeval_def,
p.2.2... | 0 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 176 | 180 | theorem symmDiff_sdiff_eq_sup : a ∆ (b \ a) = a ⊔ b := by |
rw [symmDiff, sdiff_idem]
exact
le_antisymm (sup_le_sup sdiff_le sdiff_le)
(sup_le le_sdiff_sup <| le_sdiff_sup.trans <| sup_le le_sup_right le_sdiff_sup)
| 0 |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 174 | 180 | theorem mem_approxAddOrderOf_iff {δ : ℝ} {x : UnitAddCircle} {n : ℕ} (hn : 0 < n) :
x ∈ approxAddOrderOf UnitAddCircle n δ ↔ ∃ m < n, gcd m n = 1 ∧ ‖x - ↑((m : ℝ) / n)‖ < δ := by |
simp only [mem_approx_add_orderOf_iff, mem_setOf_eq, ball, exists_prop, dist_eq_norm,
AddCircle.addOrderOf_eq_pos_iff hn, mul_one]
constructor
· rintro ⟨y, ⟨m, hm₁, hm₂, rfl⟩, hx⟩; exact ⟨m, hm₁, hm₂, hx⟩
· rintro ⟨m, hm₁, hm₂, hx⟩; exact ⟨↑((m : ℝ) / n), ⟨m, hm₁, hm₂, rfl⟩, hx⟩
| 0 |
import Mathlib.ModelTheory.Algebra.Ring.Basic
import Mathlib.RingTheory.FreeCommRing
namespace FirstOrder
namespace Ring
open Language
variable {α : Type*}
section
attribute [local instance] compatibleRingOfRing
private theorem exists_term_realize_eq_freeCommRing (p : FreeCommRing α) :
∃ t : Language.rin... | Mathlib/ModelTheory/Algebra/Ring/FreeCommRing.lean | 54 | 63 | theorem realize_termOfFreeCommRing (p : FreeCommRing α) (v : α → R) :
(termOfFreeCommRing p).realize v = FreeCommRing.lift v p := by |
let _ := compatibleRingOfRing (FreeCommRing α)
rw [termOfFreeCommRing]
conv_rhs => rw [← Classical.choose_spec (exists_term_realize_eq_freeCommRing p)]
induction Classical.choose (exists_term_realize_eq_freeCommRing p) with
| var _ => simp
| func f a ih =>
cases f <;>
simp [ih]
| 0 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.ae_measurable_order from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open MeasureTheory Set TopologicalSpace
open scoped Classical
open ENNReal NNReal
| Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean | 34 | 107 | theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type*}
{m : MeasurableSpace α} (μ : Measure α) {β : Type*} [CompleteLinearOrder β] [DenselyOrdered β]
[TopologicalSpace β] [OrderTopology β] [SecondCountableTopology β] [MeasurableSpace β]
[BorelSpace β] (s : Set β) (s_count : s.Coun... |
haveI : Encodable s := s_count.toEncodable
have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
intro p q
by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
· rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv, h'u, h'v... | 0 |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 119 | 153 | theorem nonempty_hom_of_forall_finite_subgraph_hom [Finite W]
(h : ∀ G' : G.Subgraph, G'.verts.Finite → G'.coe →g F) : Nonempty (G →g F) := by |
-- Obtain a `Fintype` instance for `W`.
cases nonempty_fintype W
-- Establish the required interface instances.
haveI : ∀ G' : G.Finsubgraphᵒᵖ, Nonempty ((finsubgraphHomFunctor G F).obj G') := fun G' =>
⟨h G'.unop G'.unop.property⟩
haveI : ∀ G' : G.Finsubgraphᵒᵖ, Fintype ((finsubgraphHomFunctor G F).obj ... | 0 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.preadditive.of_biproducts from "leanprover-community/mathlib"@"061ea99a5610cfc72c286aa930d3c1f47f74f3d0"
noncomputable section
universe v u
op... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 88 | 96 | theorem distrib (f g h k : X ⟶ Y) : (f +ᵣ g) +ₗ h +ᵣ k = (f +ₗ h) +ᵣ g +ₗ k := by |
let diag : X ⊞ X ⟶ Y ⊞ Y := biprod.lift (biprod.desc f g) (biprod.desc h k)
have hd₁ : biprod.inl ≫ diag = biprod.lift f h := by ext <;> simp [diag]
have hd₂ : biprod.inr ≫ diag = biprod.lift g k := by ext <;> simp [diag]
have h₁ : biprod.lift (f +ᵣ g) (h +ᵣ k) = biprod.lift (𝟙 X) (𝟙 X) ≫ diag := by
ext ... | 0 |
import Mathlib.Topology.PartitionOfUnity
import Mathlib.Analysis.Convex.Combination
#align_import analysis.convex.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function
open Topology
variable {ι X E : Type*} [TopologicalSpace X] [AddCommGroup E] [Modu... | Mathlib/Analysis/Convex/PartitionOfUnity.lean | 51 | 60 | theorem exists_continuous_forall_mem_convex_of_local (ht : ∀ x, Convex ℝ (t x))
(H : ∀ x : X, ∃ U ∈ 𝓝 x, ∃ g : X → E, ContinuousOn g U ∧ ∀ y ∈ U, g y ∈ t y) :
∃ g : C(X, E), ∀ x, g x ∈ t x := by |
choose U hU g hgc hgt using H
obtain ⟨f, hf⟩ := PartitionOfUnity.exists_isSubordinate isClosed_univ (fun x => interior (U x))
(fun x => isOpen_interior) fun x _ => mem_iUnion.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩
refine ⟨⟨fun x => ∑ᶠ i, f i x • g i x,
hf.continuous_finsum_smul (fun i => isOpen_interi... | 0 |
import Mathlib.Logic.Equiv.Defs
#align_import data.erased from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
universe u
def Erased (α : Sort u) : Sort max 1 u :=
Σ's : α → Prop, ∃ a, (fun b => a = b) = s
#align erased Erased
namespace Erased
@[inline]
def mk {α} (a : α) : Erased... | Mathlib/Data/Erased.lean | 56 | 59 | theorem out_mk {α} (a : α) : (mk a).out = a := by |
let h := (mk a).2; show Classical.choose h = a
have := Classical.choose_spec h
exact cast (congr_fun this a).symm rfl
| 0 |
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 | 162 | 181 | theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : E} (hx : DifferentiableAt 𝕜 f x) :
∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (fderiv 𝕜 f x) r ε := by |
let δ := (ε / 2) / 2
obtain ⟨R, R_pos, hR⟩ :
∃ R > 0, ∀ y ∈ ball x R, ‖f y - f x - fderiv 𝕜 f x (y - x)‖ ≤ δ * ‖y - x‖ :=
eventually_nhds_iff_ball.1 <| hx.hasFDerivAt.isLittleO.bound <| by positivity
refine ⟨R, R_pos, fun r hr => ?_⟩
have : r ∈ Ioc (r / 2) r := right_mem_Ioc.2 <| half_lt_self hr.1
... | 0 |
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
import Mathlib.Init.Classical
#align_import algebra.group_with_zero.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists DenselyOrdered
variable {α M₀ G₀ M₀' G₀' F F' :... | Mathlib/Algebra/GroupWithZero/Semiconj.lean | 45 | 54 | theorem inv_right₀ (h : SemiconjBy a x y) : SemiconjBy a x⁻¹ y⁻¹ := by |
by_cases ha : a = 0
· simp only [ha, zero_left]
by_cases hx : x = 0
· subst x
simp only [SemiconjBy, mul_zero, @eq_comm _ _ (y * a), mul_eq_zero] at h
simp [h.resolve_right ha]
· have := mul_ne_zero ha hx
rw [h.eq, mul_ne_zero_iff] at this
exact @units_inv_right _ _ _ (Units.mk0 x hx) (Units.... | 0 |
import Mathlib.GroupTheory.Perm.Cycle.Basic
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {ι α β : Type*}
namespace Equiv.Perm
section Generation
variable [Finite β]
open Subgroup
| Mathlib/GroupTheory/Perm/Closure.lean | 37 | 41 | theorem closure_isCycle : closure { σ : Perm β | IsCycle σ } = ⊤ := by |
classical
cases nonempty_fintype β
exact
top_le_iff.mp (le_trans (ge_of_eq closure_isSwap) (closure_mono fun _ => IsSwap.isCycle))
| 0 |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 136 | 151 | theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by |
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a formula for the Beta function.
have conv_int := integral_posConvolution
(GammaIntegral_convergent hs) (GammaIntegral_convergent ht) (ContinuousLinearMap.mul ℝ ℂ)
simp_rw [ContinuousLinearMap.mul_a... | 0 |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 75 | 83 | theorem continuous_circleTransform {R : ℝ} (hR : 0 < R) {f : ℂ → E} {z w : ℂ}
(hf : ContinuousOn f <| sphere z R) (hw : w ∈ ball z R) :
Continuous (circleTransform R z w f) := by |
apply_rules [Continuous.smul, continuous_const]
· simp_rw [deriv_circleMap]
apply_rules [Continuous.mul, continuous_circleMap 0 R, continuous_const]
· exact continuous_circleMap_inv hw
· apply ContinuousOn.comp_continuous hf (continuous_circleMap z R)
exact fun _ => (circleMap_mem_sphere _ hR.le) _
| 0 |
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 | 78 | 192 | theorem IsOpen.exists_smooth_support_eq {s : Set E} (hs : IsOpen s) :
∃ f : E → ℝ, f.support = s ∧ ContDiff ℝ ⊤ f ∧ Set.range f ⊆ Set.Icc 0 1 := by |
/- For any given point `x` in `s`, one can construct a smooth function with support in `s` and
nonzero at `x`. By second-countability, it follows that we may cover `s` with the supports of
countably many such functions, say `g i`.
Then `∑ i, r i • g i` will be the desired function if `r i` is a sequence ... | 0 |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 145 | 147 | theorem norm_zero [Nontrivial S] [Module.Free R S] [Module.Finite R S] : norm R (0 : S) = 0 := by |
nontriviality
rw [norm_apply, coe_lmul_eq_mul, map_zero, LinearMap.det_zero' (Module.Free.chooseBasis R S)]
| 0 |
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
variable {T : Type u} [Category.{v} T]
... | Mathlib/CategoryTheory/Comma/Arrow.lean | 138 | 143 | theorem isIso_of_isIso_left_of_isIso_right {f g : Arrow T} (ff : f ⟶ g) [IsIso ff.left]
[IsIso ff.right] : IsIso ff where
out := by |
let inverse : g ⟶ f := ⟨inv ff.left, inv ff.right, (by simp)⟩
apply Exists.intro inverse
aesop_cat
| 0 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 143 | 149 | theorem map_empty_eq_zero {β} [AddCancelMonoid β] {T : Set α → β} (hT : FinMeasAdditive μ T) :
T ∅ = 0 := by |
have h_empty : μ ∅ ≠ ∞ := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne
specialize hT ∅ ∅ MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty ∅)
rw [Set.union_empty] at hT
nth_rw 1 [← add_zero (T ∅)] at hT
exact (add_left_cancel hT).symm
| 0 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 228 | 241 | theorem gcd_eq_gcd_filter_ne_zero [DecidablePred fun x : β ↦ f x = 0] :
s.gcd f = (s.filter fun x ↦ f x ≠ 0).gcd f := by |
classical
trans ((s.filter fun x ↦ f x = 0) ∪ s.filter fun x ↦ (f x ≠ 0)).gcd f
· rw [filter_union_filter_neg_eq]
rw [gcd_union]
refine Eq.trans (?_ : _ = GCDMonoid.gcd (0 : α) ?_) (?_ : GCDMonoid.gcd (0 : α) _ = _)
· exact (gcd (filter (fun x => (f x ≠ 0)) s) f)
· refine congr (congr rfl <| ... | 0 |
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 | 73 | 78 | theorem direction_perpBisector (p₁ p₂ : P) :
(perpBisector p₁ p₂).direction = (ℝ ∙ (p₂ -ᵥ p₁))ᗮ := by |
erw [perpBisector, comap_symm, map_direction, Submodule.map_id,
Submodule.toAffineSubspace_direction]
ext x
exact Submodule.mem_orthogonal_singleton_iff_inner_right.symm
| 0 |
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Sub... | Mathlib/RingTheory/Localization/Ideal.lean | 78 | 89 | theorem comap_map_of_isPrime_disjoint (I : Ideal R) (hI : I.IsPrime) (hM : Disjoint (M : Set R) I) :
Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) = I := by |
refine le_antisymm ?_ Ideal.le_comap_map
refine (fun a ha => ?_)
obtain ⟨⟨b, s⟩, h⟩ := (mem_map_algebraMap_iff M S).1 (Ideal.mem_comap.1 ha)
replace h : algebraMap R S (s * a) = algebraMap R S b := by
simpa only [← map_mul, mul_comm] using h
obtain ⟨c, hc⟩ := (eq_iff_exists M S).1 h
have : ↑c * ↑s * a ... | 0 |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 134 | 137 | theorem content_X_pow {k : ℕ} : content ((X : R[X]) ^ k) = 1 := by |
induction' k with k hi
· simp
rw [pow_succ', content_X_mul, hi]
| 0 |
import Batteries.Classes.SatisfiesM
namespace Array
theorem SatisfiesM_foldlM [Monad m] [LawfulMonad m]
{as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → m β}
(hf : ∀ i : Fin as.size, ∀ b, motive i.1 b → SatisfiesM (motive (i.1 + 1)) (f b as[i])) :
SatisfiesM (motive... | .lake/packages/batteries/Batteries/Data/Array/Monadic.lean | 32 | 48 | theorem SatisfiesM_mapM [Monad m] [LawfulMonad m] (as : Array α) (f : α → m β)
(motive : Nat → Prop) (h0 : motive 0)
(p : Fin as.size → β → Prop)
(hs : ∀ i, motive i.1 → SatisfiesM (p i · ∧ motive (i + 1)) (f as[i])) :
SatisfiesM
(fun arr => motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i,... |
rw [mapM_eq_foldlM]
refine SatisfiesM_foldlM (m := m) (β := Array β)
(motive := fun i arr => motive i ∧ arr.size = i ∧ ∀ i h2, p i (arr[i.1]'h2)) ?z ?s
|>.imp fun ⟨h₁, eq, h₂⟩ => ⟨h₁, eq, fun _ _ => h₂ ..⟩
· case z => exact ⟨h0, rfl, nofun⟩
· case s =>
intro ⟨i, hi⟩ arr ⟨ih₁, eq, ih₂⟩
refine (h... | 0 |
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 131 | 141 | theorem X_pow_sub_one_eq_prod {ζ : R} {n : ℕ} (hpos : 0 < n) (h : IsPrimitiveRoot ζ n) :
X ^ n - 1 = ∏ ζ ∈ nthRootsFinset n R, (X - C ζ) := by |
classical
rw [nthRootsFinset, ← Multiset.toFinset_eq (IsPrimitiveRoot.nthRoots_one_nodup h)]
simp only [Finset.prod_mk, RingHom.map_one]
rw [nthRoots]
have hmonic : (X ^ n - C (1 : R)).Monic := monic_X_pow_sub_C (1 : R) (ne_of_lt hpos).symm
symm
apply prod_multiset_X_sub_C_of_monic_of_roots_card_eq hmoni... | 0 |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace List
variable [D... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 73 | 86 | theorem isCycle_formPerm (hl : Nodup l) (hn : 2 ≤ l.length) : IsCycle (formPerm l) := by |
cases' l with x l
· set_option tactic.skipAssignedInstances false in norm_num at hn
induction' l with y l generalizing x
· set_option tactic.skipAssignedInstances false in norm_num at hn
· use x
constructor
· rwa [formPerm_apply_mem_ne_self_iff _ hl _ (mem_cons_self _ _)]
· intro w hw
have ... | 0 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 222 | 228 | theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
(hε : ε ≠ 0) : ∃ K, K ⊆ U ∧ p K ∧ μ U < μ K + ε := by |
rcases eq_or_ne (μ U) 0 with h₀ | h₀
· refine ⟨∅, empty_subset _, h0, ?_⟩
rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
· rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
| 0 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
def hyperoperation : ℕ → ℕ → ℕ → ℕ
| 0, _, k => k + 1
| 1, m, 0 => m
| 2, _, 0 => 0
| _ + 3, _, 0 => 1
| n + 1, m, k + 1 ... | Mathlib/Data/Nat/Hyperoperation.lean | 91 | 95 | theorem hyperoperation_ge_two_eq_self (n m : ℕ) : hyperoperation (n + 2) m 1 = m := by |
induction' n with nn nih
· rw [hyperoperation_two]
ring
· rw [hyperoperation_recursion, hyperoperation_ge_three_eq_one, nih]
| 0 |
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 | 592 | 597 | theorem oangle_left_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₃ p₁ p₂ = Real.arccos (dist p₁ p₂ / dist p₁ p₃) := 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,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h),
dist_comm p₁ p₃]
| 0 |
import Mathlib.Algebra.Order.Kleene
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.List.Join
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.DeriveFintype
#align_import computability.language from "leanprover-community/mathlib"@"a239cd3e7ac2c7cde36c913808f9d40c411344f6"
open List Set Computability... | Mathlib/Computability/Language.lean | 175 | 176 | theorem map_map (g : β → γ) (f : α → β) (l : Language α) : map g (map f l) = map (g ∘ f) l := by |
simp [map, image_image]
| 0 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 99 | 106 | theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
(vanishingIdeal t : Set A) =
{ f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, ← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf,
Submodule.mem_iInf]
refine forall_congr' fun x => ?_
rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
| 0 |
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 | 262 | 269 | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by |
induction' n with n hn
· simp
· have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1
have : natDegree (X - (1 : R[X])) ≠ 0 := ne_zero_of_eq_one <| natDegree_X_sub_C (1 : R)
rw [descPochhammer_succ_left, Monic.def, leadingCoeff_mul, leadingCoeff_comp this, hn, monic_X,
one_mul, one_m... | 0 |
import Mathlib.Order.RelClasses
#align_import data.sigma.lex from "leanprover-community/mathlib"@"41cf0cc2f528dd40a8f2db167ea4fb37b8fde7f3"
namespace Sigma
variable {ι : Type*} {α : ι → Type*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
{a b : Σ i, α i}
inductive Lex (r : ι → ι → Prop) (s : ∀ ... | Mathlib/Data/Sigma/Lex.lean | 45 | 55 | theorem lex_iff : Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by |
constructor
· rintro (⟨a, b, hij⟩ | ⟨a, b, hab⟩)
· exact Or.inl hij
· exact Or.inr ⟨rfl, hab⟩
· obtain ⟨i, a⟩ := a
obtain ⟨j, b⟩ := b
dsimp only
rintro (h | ⟨rfl, h⟩)
· exact Lex.left _ _ h
· exact Lex.right _ _ h
| 0 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (𝕜 A : Type*) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A]
variable [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 89 | 101 | theorem splitMul_injective_of_clm_mul_injective
(h : Function.Injective (mul 𝕜 A)) :
Function.Injective (splitMul 𝕜 A) := by |
rw [injective_iff_map_eq_zero]
intro x hx
induction x
rw [map_add] at hx
simp only [splitMul_apply, fst_inl, snd_inl, map_zero, add_zero, fst_inr, snd_inr,
zero_add, Prod.mk_add_mk, Prod.mk_eq_zero] at hx
obtain ⟨rfl, hx⟩ := hx
simp only [map_zero, zero_add, inl_zero] at hx ⊢
rw [← map_zero (mul 𝕜... | 0 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 444 | 446 | theorem det_one_add_col_mul_row (u v : m → α) : det (1 + col u * row v) = 1 + v ⬝ᵥ u := by |
rw [det_one_add_mul_comm, det_unique, Pi.add_apply, Pi.add_apply, Matrix.one_apply_eq,
Matrix.row_mul_col_apply]
| 0 |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Unitization
#align_import analysis.normed_space.star.mul from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open ContinuousLinearMap
local postfix:max "⋆" => star
variable (𝕜 : Type*) {E : Type*}
varia... | Mathlib/Analysis/NormedSpace/Star/Unitization.lean | 87 | 124 | theorem Unitization.norm_splitMul_snd_sq (x : Unitization 𝕜 E) :
‖(Unitization.splitMul 𝕜 E x).snd‖ ^ 2 ≤ ‖(Unitization.splitMul 𝕜 E (star x * x)).snd‖ := by |
/- The key idea is that we can use `sSup_closed_unit_ball_eq_norm` to make this about
applying this linear map to elements of norm at most one. There is a bit of `sqrt` and `sq`
shuffling that needs to occur, which is primarily just an annoyance. -/
refine (Real.le_sqrt (norm_nonneg _) (norm_nonneg _)).mp ?_
... | 0 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 61 | 67 | theorem ExpChar.eq {p q : ℕ} (hp : ExpChar R p) (hq : ExpChar R q) : p = q := by |
cases' hp with hp _ hp' hp
· cases' hq with hq _ hq' hq
exacts [rfl, False.elim (Nat.not_prime_zero (CharP.eq R hq (CharP.ofCharZero R) ▸ hq'))]
· cases' hq with hq _ hq' hq
exacts [False.elim (Nat.not_prime_zero (CharP.eq R hp (CharP.ofCharZero R) ▸ hp')),
CharP.eq R hp hq]
| 0 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.RingTheory.Nilpotent.Defs
#align_import algebra.char_p.basic from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
open Finset
section
variable (R : Type*) [CommRing R] [IsReduced R] (p n : ℕ) [ExpChar R p]
theorem iterateFrobenius_in... | Mathlib/Algebra/CharP/Reduced.lean | 35 | 40 | theorem isSquare_of_charTwo' {R : Type*} [Finite R] [CommRing R] [IsReduced R] [CharP R 2]
(a : R) : IsSquare a := by |
cases nonempty_fintype R
exact
Exists.imp (fun b h => pow_two b ▸ Eq.symm h)
(((Fintype.bijective_iff_injective_and_card _).mpr ⟨frobenius_inj R 2, rfl⟩).surjective a)
| 0 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
th... | Mathlib/Analysis/Convex/StoneSeparation.lean | 81 | 109 | theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by |
let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
obtain ⟨C, hC, hsC, hCmax⟩ :=
zorn_subset_nonempty S
(fun c hcS hc ⟨_, _⟩ =>
⟨⋃₀ c,
⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
fun s => subset_sUnion... | 0 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 138 | 143 | theorem induced_topology_pure :
TopologicalSpace.induced (pure : α → Ultrafilter α) Ultrafilter.topologicalSpace = ⊥ := by |
apply eq_bot_of_singletons_open
intro x
use { u : Ultrafilter α | {x} ∈ u }, ultrafilter_isOpen_basic _
simp
| 0 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 74 | 80 | theorem toOuterMeasure_pure_apply : (pure a).toOuterMeasure s = if a ∈ s then 1 else 0 := by |
refine (toOuterMeasure_apply (pure a) s).trans ?_
split_ifs with ha
· refine (tsum_congr fun b => ?_).trans (tsum_ite_eq a 1)
exact ite_eq_left_iff.2 fun hb => symm (ite_eq_right_iff.2 fun h => (hb <| h.symm ▸ ha).elim)
· refine (tsum_congr fun b => ?_).trans tsum_zero
exact ite_eq_right_iff.2 fun hb =... | 0 |
import Mathlib.Algebra.Polynomial.UnitTrinomial
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.selmer from "leanprover-community/mathlib"@"3e00d81bdcbf77c8188bbd18f5524ddc3ed8cac6"
namespace Polynomial
open scoped Polynomial
variable ... | Mathlib/RingTheory/Polynomial/Selmer.lean | 71 | 82 | theorem X_pow_sub_X_sub_one_irreducible_rat (hn1 : n ≠ 1) : Irreducible (X ^ n - X - 1 : ℚ[X]) := by |
by_cases hn0 : n = 0
· rw [hn0, pow_zero, sub_sub, add_comm, ← sub_sub, sub_self, zero_sub]
exact Associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := by
simp only [trinomial, C_neg, C_1]; ring
have hn : 1 < n := Nat.one_lt_iff_ne_ze... | 0 |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Path
#align_import category_theory.path_category from "leanprover-community/mathlib"@"c6dd521ebdce53bb372c527569dd7c25de53a08b"
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
section
def Paths (V : ... | Mathlib/CategoryTheory/PathCategory.lean | 124 | 135 | theorem ext_functor {C} [Category C] {F G : Paths V ⥤ C} (h_obj : F.obj = G.obj)
(h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath =
eqToHom (congr_fun h_obj a) ≫ G.map e.toPath ≫ eqToHom (congr_fun h_obj.symm b)) :
F = G := by |
fapply Functor.ext
· intro X
rw [h_obj]
· intro X Y f
induction' f with Y' Z' g e ih
· erw [F.map_id, G.map_id, Category.id_comp, eqToHom_trans, eqToHom_refl]
· erw [F.map_comp g (Quiver.Hom.toPath e), G.map_comp g (Quiver.Hom.toPath e), ih, h]
simp only [Category.id_comp, eqToHom_refl, eqT... | 0 |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 106 | 113 | theorem mem_join : ∀ {L : Array (Array α)}, a ∈ L.join ↔ ∃ l, l ∈ L ∧ a ∈ l := by |
simp only [mem_def, join_data, List.mem_join, List.mem_map]
intro l
constructor
· rintro ⟨_, ⟨s, m, rfl⟩, h⟩
exact ⟨s, m, h⟩
· rintro ⟨s, h₁, h₂⟩
refine ⟨s.data, ⟨⟨s, h₁, rfl⟩, h₂⟩⟩
| 0 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 102 | 131 | theorem derivative_succ_aux (n ν : ℕ) :
Polynomial.derivative (bernsteinPolynomial R (n + 1) (ν + 1)) =
(n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1)) := by |
rw [bernsteinPolynomial]
suffices ((n + 1).choose (ν + 1) : R[X]) * ((↑(ν + 1 : ℕ) : R[X]) * X ^ ν) * (1 - X) ^ (n - ν) -
((n + 1).choose (ν + 1) : R[X]) * X ^ (ν + 1) * ((↑(n - ν) : R[X]) * (1 - X) ^ (n - ν - 1)) =
(↑(n + 1) : R[X]) * ((n.choose ν : R[X]) * X ^ ν * (1 - X) ^ (n - ν) -
(n.choos... | 0 |
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 142 | 164 | theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] (hg : G ≠ ⊥) :
∑ x : G, (x.val : K) = 0 := by |
rw [Subgroup.ne_bot_iff_exists_ne_one] at hg
rcases hg with ⟨a, ha⟩
-- The action of a on G as an embedding
let a_mul_emb : G ↪ G := mulLeftEmbedding a
-- ... and leaves G unchanged
have h_unchanged : Finset.univ.map a_mul_emb = Finset.univ := by simp
-- Therefore the sum of x over a G is the sum of a x ... | 0 |
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 145 | 153 | theorem isBigO_cocompact_zpow_neg_nat (k : ℕ) :
f =O[cocompact E] fun x => ‖x‖ ^ (-k : ℤ) := by |
obtain ⟨d, _, hd'⟩ := f.decay k 0
simp only [norm_iteratedFDeriv_zero] at hd'
simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith]
refine ⟨d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite ?_⟩
refine (Filter.eventually_cofinite_ne 0).mono fun x hx => ?_
rw [Real.norm_of_nonneg (zpow_nonneg (nor... | 0 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 123 | 164 | theorem increasing_cantorFunction (h1 : 0 < c) (h2 : c < 1 / 2) {n : ℕ} {f g : ℕ → Bool}
(hn : ∀ k < n, f k = g k) (fn : f n = false) (gn : g n = true) :
cantorFunction c f < cantorFunction c g := by |
have h3 : c < 1 := by
apply h2.trans
norm_num
induction' n with n ih generalizing f g
· let f_max : ℕ → Bool := fun n => Nat.rec false (fun _ _ => true) n
have hf_max : ∀ n, f n → f_max n := by
intro n hn
cases n
· rw [fn] at hn
contradiction
apply rfl
let g_min : ... | 0 |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 115 | 118 | theorem chain_range_succ (r : ℕ → ℕ → Prop) (n a : ℕ) :
Chain r a (range n.succ) ↔ r a 0 ∧ ∀ m < n, r m m.succ := by |
rw [range_succ_eq_map, chain_cons, and_congr_right_iff, ← chain'_range_succ, range_succ_eq_map]
exact fun _ => Iff.rfl
| 0 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Prime
#align_import data.nat.choose.dvd from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
open Nat
namespace Prime
variable {p a b k : ℕ}
| Mathlib/Data/Nat/Choose/Dvd.lean | 24 | 29 | theorem dvd_choose_add (hp : Prime p) (hap : a < p) (hbp : b < p) (h : p ≤ a + b) :
p ∣ choose (a + b) a := by |
have h₁ : p ∣ (a + b)! := hp.dvd_factorial.2 h
rw [← add_choose_mul_factorial_mul_factorial, ← choose_symm_add, hp.dvd_mul, hp.dvd_mul,
hp.dvd_factorial, hp.dvd_factorial] at h₁
exact (h₁.resolve_right hbp.not_le).resolve_right hap.not_le
| 0 |
import Mathlib.NumberTheory.SmoothNumbers
import Mathlib.Analysis.PSeries
open Set Nat
open scoped Topology
-- This needs `Mathlib.Analysis.RCLike.Basic`, so we put it here
-- instead of in `Mathlib.NumberTheory.SmoothNumbers`.
lemma Nat.roughNumbersUpTo_card_le' (N k : ℕ) :
(roughNumbersUpTo N k).card ≤
... | Mathlib/NumberTheory/SumPrimeReciprocals.lean | 82 | 83 | theorem Nat.Primes.not_summable_one_div : ¬ Summable (fun p : Nat.Primes ↦ (1 / p : ℝ)) := by |
convert summable_subtype_iff_indicator.mp.mt not_summable_one_div_on_primes
| 0 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 86 | 89 | theorem gronwallBound_ε0 (δ K x : ℝ) : gronwallBound δ K 0 x = δ * exp (K * x) := by |
by_cases hK : K = 0
· simp only [gronwallBound_K0, hK, zero_mul, exp_zero, add_zero, mul_one]
· simp only [gronwallBound_of_K_ne_0 hK, zero_div, zero_mul, add_zero]
| 0 |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 156 | 159 | theorem frobeniusMorphism_iso_of_expComparison_iso (h : L ⊣ F) (A : C)
[i : IsIso (expComparison F A)] : IsIso (frobeniusMorphism F h A) := by |
rw [← frobeniusMorphism_mate F h] at i
exact @transferNatTransSelf_of_iso _ _ _ _ _ _ _ _ _ _ _ i
| 0 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 63 | 66 | theorem FiniteDimensional.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by |
simp_rw [finrank]
rw [← toNat_lift.{w} (Module.rank F K), ← toNat_lift.{v} (Module.rank K A), ← toNat_mul,
lift_rank_mul_lift_rank, toNat_lift]
| 0 |
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 112 | 121 | theorem MonoidHom.isOpenMap_of_sigmaCompact
{H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
(f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
IsOpenMap f := by |
let A : MulAction G H := MulAction.compHom _ f
have : ContinuousSMul G H := continuousSMul_compHom h'f
have : IsPretransitive G H := isPretransitive_compHom hf
have : f = (fun (g : G) ↦ g • (1 : H)) := by simp [MulAction.compHom_smul_def]
rw [this]
exact isOpenMap_smul_of_sigmaCompact _
| 0 |
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
open Real Set MeasureTheory MeasureTheory.Measure
section real
| Mathlib/MeasureTheory/Integral/Gamma.lean | 21 | 37 | theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
∫ x in Ioi (0:ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q + 1) / p) := by |
calc
_ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by
rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),
abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]
refine setIntegral_congr measurableSet_Ioi (fun _ hx => ?_)
rw [← rpow_mul (le_... | 0 |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 108 | 117 | theorem IsFilteredOrEmpty.of_exists_of_isFiltered_of_fullyFaithful [IsFilteredOrEmpty D] [F.Full]
[F.Faithful] (h : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) : IsFilteredOrEmpty C where
cocone_objs c c' := by |
obtain ⟨c₀, ⟨f⟩⟩ := h (IsFiltered.max (F.obj c) (F.obj c'))
exact ⟨c₀, F.preimage (IsFiltered.leftToMax _ _ ≫ f),
F.preimage (IsFiltered.rightToMax _ _ ≫ f), trivial⟩
cocone_maps {c c'} f g := by
obtain ⟨c₀, ⟨f₀⟩⟩ := h (IsFiltered.coeq (F.map f) (F.map g))
refine ⟨_, F.preimage (IsFiltered.coeq... | 0 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 65 | 72 | theorem _root_.MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
evariance X μ < ∞ := by |
have := ENNReal.pow_lt_top (hX.sub <| memℒp_const <| μ[X]).2 2
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
simp only [coe_two, Pi.sub_apply, ENNReal.one_toReal, one_div] at this
rw [← ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_o... | 0 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finite.Card
#align_import group_theory.subgroup.finite from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
variable {G : Type*} [Group G]
variable {A : Type*} [AddGroup A]
n... | Mathlib/Algebra/Group/Subgroup/Finite.lean | 259 | 270 | theorem mem_normalizer_fintype {S : Set G} [Finite S] {x : G} (h : ∀ n, n ∈ S → x * n * x⁻¹ ∈ S) :
x ∈ Subgroup.setNormalizer S := by |
haveI := Classical.propDecidable; cases nonempty_fintype S;
haveI := Set.fintypeImage S fun n => x * n * x⁻¹;
exact fun n =>
⟨h n, fun h₁ =>
have heq : (fun n => x * n * x⁻¹) '' S = S :=
Set.eq_of_subset_of_card_le (fun n ⟨y, hy⟩ => hy.2 ▸ h y hy.1)
(by rw [Set.card_imag... | 0 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 252 | 259 | theorem multiequalizer_ext {I : MulticospanIndex.{w} C} [HasMultiequalizer I]
[PreservesLimit I.multicospan (forget C)] (x y : ↑(multiequalizer I))
(h : ∀ t : I.L, Multiequalizer.ι I t x = Multiequalizer.ι I t y) : x = y := by |
apply Concrete.limit_ext
rintro (a | b)
· apply h
· rw [← limit.w I.multicospan (WalkingMulticospan.Hom.fst b), comp_apply, comp_apply]
simp [h]
| 0 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 72 | 81 | theorem measurableSet_lineDifferentiableAt_uncurry (hf : Continuous f) :
MeasurableSet {p : E × E | LineDifferentiableAt 𝕜 f p.1 p.2} := by |
borelize 𝕜
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
have M_meas : MeasurableSet {q : (E × E) × 𝕜 | DifferentiableAt 𝕜 (g q.1) q.2} :=
meas... | 0 |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 92 | 108 | theorem norm_dslope_le_div_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
‖dslope f c z‖ ≤ R₂ / R₁ := by |
have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
have hR₂ : 0 < R₂ := nonempty_ball.1 ⟨f z, h_maps hz⟩
rcases eq_or_ne (dslope f c z) 0 with hc | hc
· rw [hc, norm_zero]; exact div_nonneg hR₂.le hR₁.le
rcases exists_dual_vector ℂ _ hc with ⟨g, hg, hgf⟩
have hg' : ‖g‖₊ = 1 := NNReal.eq hg
have hg₀ : ‖g‖₊ ≠ 0 ... | 0 |
import Mathlib.Analysis.Normed.Field.Basic
#align_import topology.metric_space.cau_seq_filter from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
universe u v
open Set Filter
open scoped Classical
open Topology
variable {β : Type v}
theorem CauSeq.tendsto_limit [NormedRing β] [hn : ... | Mathlib/Topology/MetricSpace/CauSeqFilter.lean | 67 | 82 | theorem CauSeq.cauchySeq (f : CauSeq β norm) : CauchySeq f := by |
refine cauchy_iff.2 ⟨by infer_instance, fun s hs => ?_⟩
rcases mem_uniformity_dist.1 hs with ⟨ε, ⟨hε, hεs⟩⟩
cases' CauSeq.cauchy₂ f hε with N hN
exists { n | n ≥ N }.image f
simp only [exists_prop, mem_atTop_sets, mem_map, mem_image, ge_iff_le, mem_setOf_eq]
constructor
· exists N
intro b hb
exis... | 0 |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 120 | 129 | theorem abs_circleTransformBoundingFunction_le {R r : ℝ} (hr : r < R) (hr' : 0 ≤ r) (z : ℂ) :
∃ x : closedBall z r ×ˢ [[0, 2 * π]], ∀ y : closedBall z r ×ˢ [[0, 2 * π]],
abs (circleTransformBoundingFunction R z y) ≤ abs (circleTransformBoundingFunction R z x) := by |
have cts := continuousOn_abs_circleTransformBoundingFunction hr z
have comp : IsCompact (closedBall z r ×ˢ [[0, 2 * π]]) := by
apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]
have none : (closedBall z r ×ˢ [[0, 2 * π]]).Nonempty :=
(nonempty_closedBall.2 hr').prod nonem... | 0 |
import Mathlib.Algebra.Field.ULift
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.Data.Rat.Denumerable
import Mathlib.FieldTheory.Finite.GaloisField
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.RingTheory.Localization.Cardinality
import Mathlib.... | Mathlib/FieldTheory/Cardinality.lean | 53 | 57 | theorem Fintype.nonempty_field_iff {α} [Fintype α] : Nonempty (Field α) ↔ IsPrimePow ‖α‖ := by |
refine ⟨fun ⟨h⟩ => Fintype.isPrimePow_card_of_field, ?_⟩
rintro ⟨p, n, hp, hn, hα⟩
haveI := Fact.mk hp.nat_prime
exact ⟨(Fintype.equivOfCardEq ((GaloisField.card p n hn.ne').trans hα)).symm.field⟩
| 0 |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 66 | 79 | theorem lintegral_mul_le_one_of_lintegral_rpow_eq_one {p q : ℝ} (hpq : p.IsConjExponent q)
{f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf_norm : ∫⁻ a, f a ^ p ∂μ = 1)
(hg_norm : ∫⁻ a, g a ^ q ∂μ = 1) : (∫⁻ a, (f * g) a ∂μ) ≤ 1 := by |
calc
(∫⁻ a : α, (f * g) a ∂μ) ≤
∫⁻ a : α, f a ^ p / ENNReal.ofReal p + g a ^ q / ENNReal.ofReal q ∂μ :=
lintegral_mono fun a => young_inequality (f a) (g a) hpq
_ = 1 := by
simp only [div_eq_mul_inv]
rw [lintegral_add_left']
· rw [lintegral_mul_const'' _ (hf.pow_const p), lint... | 0 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
#align_import ring_theory.graded_algebra.radical from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a"
open GradedRing DirectSum SetLike Finset
variable {ι σ A : Type*}
variable [CommRing A]
variable [LinearOrderedCancelAddCommMono... | Mathlib/RingTheory/GradedAlgebra/Radical.lean | 47 | 136 | theorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)
(I_ne_top : I ≠ ⊤)
(homogeneous_mem_or_mem :
∀ {x y : A}, Homogeneous 𝒜 x → Homogeneous 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :
Ideal.IsPrime I :=
⟨I_ne_top, by
intro x y hxy
by_contra! rid
... |
intro x hx
rw [filter_nonempty_iff]
contrapose! hx
simp_rw [proj_apply] at hx
rw [← sum_support_decompose 𝒜 x]
exact Ideal.sum_mem _ hx
set max₁ := set₁.max' (nonempty x rid₁)
set max₂ := set₂.max' (nonempty y rid₂)
have mem_max₁ : max₁ ∈ set₁ := max'_... | 0 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 170 | 179 | theorem contDiff_stereoInvFunAux : ContDiff ℝ ⊤ (stereoInvFunAux v) := by |
have h₀ : ContDiff ℝ ⊤ fun w : E => ‖w‖ ^ 2 := contDiff_norm_sq ℝ
have h₁ : ContDiff ℝ ⊤ fun w : E => (‖w‖ ^ 2 + 4)⁻¹ := by
refine (h₀.add contDiff_const).inv ?_
intro x
nlinarith
have h₂ : ContDiff ℝ ⊤ fun w => (4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v := by
refine (contDiff_const.smul contDiff_id).add ?_... | 0 |
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 | 56 | 59 | theorem stalkMap_germ {X Y : PresheafedSpace.{_, _, v} C} (α : X ⟶ Y) (U : Opens Y)
(x : (Opens.map α.base).obj U) :
Y.presheaf.germ ⟨α.base x.1, x.2⟩ ≫ stalkMap α ↑x = α.c.app (op U) ≫ X.presheaf.germ x := by |
rw [stalkMap, stalkFunctor_map_germ_assoc, stalkPushforward_germ]
| 0 |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 142 | 151 | theorem compress_injOn : Set.InjOn (compress u v) ↑(s.filter (compress u v · ∉ s)) := by |
intro a ha b hb hab
rw [mem_coe, mem_filter] at ha hb
rw [compress] at ha hab
split_ifs at ha hab with has
· rw [compress] at hb hab
split_ifs at hb hab with hbs
· exact sup_sdiff_injOn u v has hbs hab
· exact (hb.2 hb.1).elim
· exact (ha.2 ha.1).elim
| 0 |
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 169 | 174 | theorem filter_snd_eq_antidiagonal (n m : A) [DecidablePred (· = m)] [Decidable (m ≤ n)] :
filter (fun x : A × A ↦ x.snd = m) (antidiagonal n) = if m ≤ n then {(n - m, m)} else ∅ := by |
have : (fun x : A × A ↦ (x.snd = m)) ∘ Prod.swap = fun x : A × A ↦ x.fst = m := by
ext; simp
rw [← map_swap_antidiagonal, filter_map]
simp [this, filter_fst_eq_antidiagonal, apply_ite (Finset.map _)]
| 0 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 144 | 153 | theorem Walk.isPath_of_length_eq_dist {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v) :
p.IsPath := by |
classical
have : p.bypass = p := by
apply Walk.bypass_eq_self_of_length_le
calc p.length
_ = G.dist u v := hp
_ ≤ p.bypass.length := dist_le p.bypass
rw [← this]
apply Walk.bypass_isPath
| 0 |
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 | 120 | 122 | theorem lift_add (x y : ℍ[R,c₁,c₂]) : q.lift (x + y) = q.lift x + q.lift y := by |
simp only [lift, add_re, map_add, add_imI, add_smul, add_imJ, add_imK]
abel
| 0 |
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter unitInterval Set Fun... | Mathlib/Topology/Connected/PathConnected.lean | 178 | 181 | theorem symm_symm (γ : Path x y) : γ.symm.symm = γ := by |
ext t
show γ (σ (σ t)) = γ t
rw [unitInterval.symm_symm]
| 0 |
import Mathlib.GroupTheory.Sylow
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schur_zassenhaus from "leanprover-community/mathlib"@"d57133e49cf06508700ef69030cd099917e0f0de"
namespace Subgroup
section SchurZassenhausAbelian
open MulOpposite MulAction Subgroup.leftTransversals MemLeftTransversa... | Mathlib/GroupTheory/SchurZassenhaus.lean | 48 | 62 | theorem smul_diff_smul' [hH : Normal H] (g : Gᵐᵒᵖ) :
diff (MonoidHom.id H) (g • α) (g • β) =
⟨g.unop⁻¹ * (diff (MonoidHom.id H) α β : H) * g.unop,
hH.mem_comm ((congr_arg (· ∈ H) (mul_inv_cancel_left _ _)).mpr (SetLike.coe_mem _))⟩ := by |
letI := H.fintypeQuotientOfFiniteIndex
let ϕ : H →* H :=
{ toFun := fun h =>
⟨g.unop⁻¹ * h * g.unop,
hH.mem_comm ((congr_arg (· ∈ H) (mul_inv_cancel_left _ _)).mpr (SetLike.coe_mem _))⟩
map_one' := by rw [Subtype.ext_iff, coe_mk, coe_one, mul_one, inv_mul_self]
map_mul' := fun h₁ ... | 0 |
import Mathlib.Data.Real.NNReal
import Mathlib.RingTheory.Valuation.Basic
noncomputable section
open Function Multiplicative
open scoped NNReal
variable {R : Type*} [Ring R] {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀]
namespace Valuation
class RankOne (v : Valuation R Γ₀) where
hom : Γ₀ →*₀ ℝ≥0
st... | Mathlib/RingTheory/Valuation/RankOne.lean | 67 | 69 | theorem unit_ne_one : unit v ≠ 1 := by |
rw [Ne, ← Units.eq_iff, Units.val_one]
exact ((nontrivial v).choose_spec ).2
| 0 |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Finsupp
#align_import algebra.big_operators.associated from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β γ δ : Type*}
-- the same local notation used in `Algebra.Associated`
local infixl:50 " ~ᵤ " => ... | Mathlib/Algebra/BigOperators/Associated.lean | 103 | 111 | theorem Finset.prod_primes_dvd [CancelCommMonoidWithZero α] [Unique αˣ] {s : Finset α} (n : α)
(h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) : (∏ p ∈ s, p) ∣ n := by |
classical
exact
Multiset.prod_primes_dvd n (by simpa only [Multiset.map_id', Finset.mem_def] using h)
(by simpa only [Multiset.map_id', Finset.mem_def] using div)
(by
simp only [Multiset.map_id', associated_eq_eq, Multiset.countP_eq_card_filter,
← s.val.count_eq_card_f... | 0 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 104 | 106 | theorem sum_card : Cardinal.sum (fun i => #(Sequence₂ a₀ a₁ a₂ i)) = #a₀ + #a₁ + #a₂ := by |
rw [sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ]
simp [add_assoc, Sequence₂]
| 0 |
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 | 146 | 153 | theorem tangentMap_chart {p q : TangentBundle I M} (h : q.1 ∈ (chartAt H p.1).source) :
tangentMap I I (chartAt H p.1) q =
(TotalSpace.toProd _ _).symm
((chartAt (ModelProd H E) p : TangentBundle I M → ModelProd H E) q) := by |
dsimp [tangentMap]
rw [MDifferentiableAt.mfderiv]
· rfl
· exact mdifferentiableAt_atlas _ (chart_mem_atlas _ _) h
| 0 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 59 | 66 | theorem isTopologicalBasis_Iic_principal :
IsTopologicalBasis (range (Iic ∘ 𝓟 : Set α → Set (Filter α))) :=
{ exists_subset_inter := by |
rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl
exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩
sUnion_eq := sUnion_eq_univ_iff.2 fun l => ⟨Iic ⊤, ⟨univ, congr_arg Iic principal_univ⟩,
mem_Iic.2 le_top⟩
eq_generateFrom := rfl }
| 0 |
import Mathlib.GroupTheory.Perm.Cycle.Basic
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {ι α β : Type*}
namespace Equiv.Perm
section Generation
variable [Finite β]
open Subgroup
theorem closure... | Mathlib/GroupTheory/Perm/Closure.lean | 46 | 93 | theorem closure_cycle_adjacent_swap {σ : Perm α} (h1 : IsCycle σ) (h2 : σ.support = ⊤) (x : α) :
closure ({σ, swap x (σ x)} : Set (Perm α)) = ⊤ := by |
let H := closure ({σ, swap x (σ x)} : Set (Perm α))
have h3 : σ ∈ H := subset_closure (Set.mem_insert σ _)
have h4 : swap x (σ x) ∈ H := subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))
have step1 : ∀ n : ℕ, swap ((σ ^ n) x) ((σ ^ (n + 1) : Perm α) x) ∈ H := by
intro n
induction' n with n... | 0 |
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 | 122 | 126 | theorem addOrderOf_coe (a : ℕ) {n : ℕ} (n0 : n ≠ 0) : addOrderOf (a : ZMod n) = n / n.gcd a := by |
cases' a with a
· simp only [Nat.zero_eq, Nat.cast_zero, addOrderOf_zero, Nat.gcd_zero_right,
Nat.pos_of_ne_zero n0, Nat.div_self]
rw [← Nat.smul_one_eq_cast, addOrderOf_nsmul' _ a.succ_ne_zero, ZMod.addOrderOf_one]
| 0 |
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
#align_import algebra.homology.opposite from "leanprover-community/mathlib"@"8c75ef3517d4106e89fe524e6281d0b0545f47fc"
... | Mathlib/Algebra/Homology/Opposite.lean | 53 | 63 | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... |
ext
dsimp only [imageUnopUnop]
simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelUnopUnop_inv, Category.assoc,
imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, cokernel.π_desc, Iso.unop_inv,
← unop_comp, factorThruImage_comp_imageUnopOp_inv, Quiver.Hom.unop_op, imageSubobject_arrow]... | 0 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.OfFn
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.list_of_fn from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Set
variable {F α β γ : Type*}
variable [Monoid α] {s t : Set α}... | Mathlib/Data/Set/Pointwise/ListOfFn.lean | 36 | 47 | theorem mem_list_prod {l : List (Set α)} {a : α} :
a ∈ l.prod ↔
∃ l' : List (Σs : Set α, ↥s),
List.prod (l'.map fun x ↦ (Sigma.snd x : α)) = a ∧ l'.map Sigma.fst = l := by |
induction' l using List.ofFnRec with n f
simp only [mem_prod_list_ofFn, List.exists_iff_exists_tuple, List.map_ofFn, Function.comp,
List.ofFn_inj', Sigma.mk.inj_iff, and_left_comm, exists_and_left, exists_eq_left, heq_eq_eq]
constructor
· rintro ⟨fi, rfl⟩
exact ⟨fun i ↦ ⟨_, fi i⟩, rfl, rfl⟩
· rintro ... | 0 |
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 253 | 259 | theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
nextCoeff (t.map fun x => X - C x).prod = -t.sum := by |
rw [nextCoeff_multiset_prod]
· simp only [nextCoeff_X_sub_C]
exact t.sum_hom (-AddMonoidHom.id R)
· intros
apply monic_X_sub_C
| 0 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 151 | 156 | theorem measurePreserving_mul_prod_inv [IsMulLeftInvariant ν] :
MeasurePreserving (fun z : G × G => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) (μ.prod ν) := by |
convert (measurePreserving_prod_inv_mul_swap ν μ).comp (measurePreserving_prod_mul_swap μ ν)
using 1
ext1 ⟨x, y⟩
simp_rw [Function.comp_apply, mul_inv_rev, inv_mul_cancel_right]
| 0 |
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 70 | 75 | theorem IsCompact.induction_on (hs : IsCompact s) {p : Set X → Prop} (he : p ∅)
(hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s) (hunion : ∀ ⦃s t⦄, p s → p t → p (s ∪ t))
(hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s := by |
let f : Filter X := comk p he (fun _t ht _s hsub ↦ hmono hsub ht) (fun _s hs _t ht ↦ hunion hs ht)
have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using hnhds)
rwa [← compl_compl s]
| 0 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
variable (𝕜 : Type*) {S R M : Type*}
loca... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 91 | 100 | theorem hasSum_snd_expSeries_of_smul_comm (x : tsze R M)
(hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
(h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) := by |
rw [← hasSum_nat_add_iff' 1]
simp_rw [snd_expSeries_of_smul_comm _ _ hx]
simp_rw [expSeries_apply_eq] at *
rw [Finset.range_one, Finset.sum_singleton, Nat.factorial_zero, Nat.cast_one, pow_zero,
inv_one, one_smul, snd_one, sub_zero]
exact h.smul_const _
| 0 |
import Mathlib.RingTheory.RootsOfUnity.Basic
universe u
variable {L : Type u} [CommRing L] [IsDomain L]
variable (n : ℕ+)
theorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :
∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by
obtain ⟨m, hm⟩ := MonoidHom.map_cyclic ((g : L ≃* L).re... | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 105 | 109 | theorem toFun_spec (g : L ≃+* L) {n : ℕ+} (t : rootsOfUnity n L) :
g (t : Lˣ) = (t ^ (χ₀ n g).val : Lˣ) := by |
rw [ModularCyclotomicCharacter_aux_spec g n t, ← zpow_natCast, ModularCyclotomicCharacter.toFun,
ZMod.val_intCast, ← Subgroup.coe_zpow]
exact Units.ext_iff.1 <| SetCoe.ext_iff.2 <| zpow_eq_zpow_emod _ pow_card_eq_one
| 0 |
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 82 | 84 | theorem measure_inter_union (h : s₁ ∩ s₂ ⊆ ∅) (h₁ : IsCaratheodory m s₁) {t : Set α} :
m (t ∩ (s₁ ∪ s₂)) = m (t ∩ s₁) + m (t ∩ s₂) := by |
rw [h₁, Set.inter_assoc, Set.union_inter_cancel_left, inter_diff_assoc, union_diff_cancel_left h]
| 0 |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 175 | 185 | theorem add_lt_ack : ∀ m n, m + n < ack m n
| 0, n => by simp
| m + 1, 0 => by simpa using add_lt_ack m 1
| m + 1, n + 1 =>
calc
m + 1 + n + 1 ≤ m + (m + n + 2) := by | omega
_ < ack m (m + n + 2) := add_lt_ack _ _
_ ≤ ack m (ack (m + 1) n) :=
ack_mono_right m <| le_of_eq_of_le (by rw [succ_eq_add_one]; ring_nf)
<| succ_le_of_lt <| add_lt_ack (m + 1) n
_ = ack (m + 1) (n + 1) := (ack_succ_succ m n).symm
| 0 |
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace Multiset
open List
instance functor : Functor Multiset... | Mathlib/Data/Multiset/Functor.lean | 108 | 116 | theorem comp_traverse {G H : Type _ → Type _} [Applicative G] [Applicative H] [CommApplicative G]
[CommApplicative H] {α β γ : Type _} (g : α → G β) (h : β → H γ) (x : Multiset α) :
traverse (Comp.mk ∘ Functor.map h ∘ g) x =
Comp.mk (Functor.map (traverse h) (traverse g x)) := by |
refine Quotient.inductionOn x ?_
intro
simp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map,
functor_norm]
simp only [Function.comp, lift_coe]
| 0 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.complex.liouville from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Metric Set Filter Asymptotics ... | Mathlib/Analysis/Complex/Liouville.lean | 88 | 101 | theorem liouville_theorem_aux {f : ℂ → F} (hf : Differentiable ℂ f) (hb : IsBounded (range f))
(z w : ℂ) : f z = f w := by |
suffices ∀ c, deriv f c = 0 from is_const_of_deriv_eq_zero hf this z w
clear z w; intro c
obtain ⟨C, C₀, hC⟩ : ∃ C > (0 : ℝ), ∀ z, ‖f z‖ ≤ C := by
rcases isBounded_iff_forall_norm_le.1 hb with ⟨C, hC⟩
exact
⟨max C 1, lt_max_iff.2 (Or.inr zero_lt_one), fun z =>
(hC (f z) (mem_range_self _)).... | 0 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 158 | 166 | theorem convexIndependent_set_iff_not_mem_convexHull_diff {s : Set E} :
ConvexIndependent 𝕜 ((↑) : s → E) ↔ ∀ x ∈ s, x ∉ convexHull 𝕜 (s \ {x}) := by |
rw [convexIndependent_set_iff_inter_convexHull_subset]
constructor
· rintro hs x hxs hx
exact (hs _ Set.diff_subset ⟨hxs, hx⟩).2 (Set.mem_singleton _)
· rintro hs t ht x ⟨hxs, hxt⟩
by_contra h
exact hs _ hxs (convexHull_mono (Set.subset_diff_singleton ht h) hxt)
| 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.