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 | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
|---|---|---|---|---|---|---|---|---|---|---|---|
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 82 | 98 | theorem Sphere.eq_or_eq_secondInter_of_mem_mk'_span_singleton_iff_mem {s : Sphere P} {p : P}
(hp : p ∈ s) {v : V} {p' : P} (hp' : p' ∈ AffineSubspace.mk' p (ℝ ∙ v)) :
p' = p ∨ p' = s.secondInter p v ↔ p' ∈ s := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· rcases h with (h | h)
· rwa [h]
· rwa [h, Sphere.secondInter_mem]
· rw [AffineSubspace.mem_mk'_iff_vsub_mem, Submodule.mem_span_singleton] at hp'
rcases hp' with ⟨r, hr⟩
rw [eq_comm, ← eq_vadd_iff_vsub_eq] at hr
subst hr
by_cases hv : v = 0
· simp [... | 14 | 1,202,604.284165 | 2 | 1.25 | 8 | 1,314 |
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c9754d62860110295cde0d7181"
open Function (Injective Surjective)
noncomputable section
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) (R : Type*)
local notation "𝕎" =>... | Mathlib/RingTheory/WittVector/Truncated.lean | 118 | 122 | theorem out_injective : Injective (@out p n R _) := by |
intro x y h
ext i
rw [WittVector.ext_iff] at h
simpa only [coeff_out] using h ↑i
| 4 | 54.59815 | 2 | 0.666667 | 3 | 612 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 186 | 218 | theorem exists_countable_union_perfect_of_isClosed [SecondCountableTopology α]
(hclosed : IsClosed C) : ∃ V D : Set α, V.Countable ∧ Perfect D ∧ C = V ∪ D := by |
obtain ⟨b, bct, _, bbasis⟩ := TopologicalSpace.exists_countable_basis α
let v := { U ∈ b | (U ∩ C).Countable }
let V := ⋃ U ∈ v, U
let D := C \ V
have Vct : (V ∩ C).Countable := by
simp only [V, iUnion_inter, mem_sep_iff]
apply Countable.biUnion
· exact Countable.mono inter_subset_left bct
· ... | 31 | 29,048,849,665,247.426 | 2 | 1.666667 | 9 | 1,822 |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 105 | 116 | theorem weightedTotalDegree_coe (w : σ → M) (p : MvPolynomial σ R) (hp : p ≠ 0) :
weightedTotalDegree' w p = ↑(weightedTotalDegree w p) := by |
rw [Ne, ← weightedTotalDegree'_eq_bot_iff w p, ← Ne, WithBot.ne_bot_iff_exists] at hp
obtain ⟨m, hm⟩ := hp
apply le_antisymm
· simp only [weightedTotalDegree, weightedTotalDegree', Finset.sup_le_iff, WithBot.coe_le_coe]
intro b
exact Finset.le_sup
· simp only [weightedTotalDegree]
have hm' : weig... | 10 | 22,026.465795 | 2 | 1.125 | 8 | 1,208 |
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Topology
namespa... | Mathlib/Probability/Process/Adapted.lean | 188 | 198 | theorem progMeasurable_of_tendsto' {γ} [MeasurableSpace ι] [PseudoMetrizableSpace β]
(fltr : Filter γ) [fltr.NeBot] [fltr.IsCountablyGenerated] {U : γ → ι → Ω → β}
(h : ∀ l, ProgMeasurable f (U l)) (h_tendsto : Tendsto U fltr (𝓝 u)) : ProgMeasurable f u := by |
intro i
apply @stronglyMeasurable_of_tendsto (Set.Iic i × Ω) β γ
(MeasurableSpace.prod _ (f i)) _ _ fltr _ _ _ _ fun l => h l i
rw [tendsto_pi_nhds] at h_tendsto ⊢
intro x
specialize h_tendsto x.fst
rw [tendsto_nhds] at h_tendsto ⊢
exact fun s hs h_mem => h_tendsto {g | g x.snd ∈ s} (hs.preimage (con... | 8 | 2,980.957987 | 2 | 2 | 2 | 2,288 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 114 | 121 | theorem Matrix.Represents.mul {A A' : Matrix ι ι R} {f f' : Module.End R M} (h : A.Represents b f)
(h' : Matrix.Represents b A' f') : (A * A').Represents b (f * f') := by |
delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_mul]
ext
dsimp [PiToModule.fromEnd]
rw [← h'.congr_fun, ← h.congr_fun]
rfl
| 6 | 403.428793 | 2 | 1.444444 | 9 | 1,527 |
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.Nat.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Util.AssertExists
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSub
namespace List
universe u v
variable {α : Type u} {β : Type v} (l :... | Mathlib/Data/List/GetD.lean | 38 | 44 | theorem getD_eq_get {n : ℕ} (hn : n < l.length) : l.getD n d = l.get ⟨n, hn⟩ := by |
induction l generalizing n with
| nil => simp at hn
| cons head tail ih =>
cases n
· exact getD_cons_zero
· exact ih _
| 6 | 403.428793 | 2 | 1.375 | 8 | 1,471 |
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathlib"@"6c351a8fb9b06e5a542fdf427bfb9f46724f9453"
suppress_compilation
universe u v w
open TensorProduct
open TensorProduct
open Algebra.TensorProduct
open Matri... | Mathlib/RingTheory/MatrixAlgebra.lean | 124 | 130 | theorem left_inv (M : A ⊗[R] Matrix n n R) : invFun R A n (toFunAlgHom R A n M) = M := by |
induction M using TensorProduct.induction_on with
| zero => simp
| tmul a m => simp
| add x y hx hy =>
rw [map_add]
conv_rhs => rw [← hx, ← hy, ← invFun_add]
| 6 | 403.428793 | 2 | 1 | 6 | 859 |
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 | 54 | 58 | theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by |
rcases eq_or_ne x 0 with (rfl | hx)
· simp
· have : abs x ≠ 0 := abs.ne_zero hx
apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]
| 4 | 54.59815 | 2 | 1.571429 | 7 | 1,706 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
suppress_compilation
open Bornology Metric Set Real
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special cases about linear (as o... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 70 | 86 | theorem tendsto_of_tendsto_pointwise_of_cauchySeq {f : ℕ → E' →SL[σ₁₂] F} {g : E' →SL[σ₁₂] F}
(hg : Tendsto (fun n x => f n x) atTop (𝓝 g)) (hf : CauchySeq f) : Tendsto f atTop (𝓝 g) := by |
/- Since `f` is a Cauchy sequence, there exists `b → 0` such that `‖f n - f m‖ ≤ b N` for any
`m, n ≥ N`. -/
rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, hb₀, hfb, hb_lim⟩
-- Since `b → 0`, it suffices to show that `‖f n x - g x‖ ≤ b n * ‖x‖` for all `n` and `x`.
suffices ∀ n x, ‖f n x - g x‖ ≤ b n * ‖x... | 15 | 3,269,017.372472 | 2 | 2 | 2 | 2,021 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed.group.controlled_closure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset
open Topology
variable {G : Type*} [NormedAddCommGroup G] [CompleteSpace... | Mathlib/Analysis/Normed/Group/ControlledClosure.lean | 116 | 125 | theorem controlled_closure_range_of_complete {f : NormedAddGroupHom G H} {K : Type*}
[SeminormedAddCommGroup K] {j : NormedAddGroupHom K H} (hj : ∀ x, ‖j x‖ = ‖x‖) {C ε : ℝ}
(hC : 0 < C) (hε : 0 < ε) (hyp : ∀ k, ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖) :
f.SurjectiveOnWith j.range.topologicalClosure (C + ε) := by |
replace hyp : ∀ h ∈ j.range, ∃ g, f g = h ∧ ‖g‖ ≤ C * ‖h‖ := by
intro h h_in
rcases (j.mem_range _).mp h_in with ⟨k, rfl⟩
rw [hj]
exact hyp k
exact controlled_closure_of_complete hC hε hyp
| 6 | 403.428793 | 2 | 2 | 2 | 2,329 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 91 | 99 | theorem hasDerivAt_tsum_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
(h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable fun n => g n y₀)
(hy : y ∈ t) : HasDerivAt (fun z => ∑' n, g n z) (∑' n, g' n y) y := by |
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg ⊢
convert hasFDerivAt_tsum_of_isPreconnected hu ht h't hg ?_ hy₀ hg0 hy
· exact (ContinuousLinearMap.smulRightL 𝕜 𝕜 F 1).map_tsum <|
.of_norm_bounded u hu fun n ↦ hg' n y hy
· simpa? says simpa only [ContinuousLinearMap.norm_smulRight_apply, norm_one, one_mul]... | 5 | 148.413159 | 2 | 1.6 | 5 | 1,721 |
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 | 116 | 126 | theorem hyperoperation_ge_four_zero (n k : ℕ) :
hyperoperation (n + 4) 0 k = if Even k then 1 else 0 := by |
induction' k with kk kih
· rw [hyperoperation_ge_three_eq_one]
simp only [Nat.zero_eq, even_zero, if_true]
· rw [hyperoperation_recursion]
rw [kih]
simp_rw [Nat.even_add_one]
split_ifs
· exact hyperoperation_ge_two_eq_self (n + 1) 0
· exact hyperoperation_ge_three_eq_one n 0
| 9 | 8,103.083928 | 2 | 1.444444 | 9 | 1,532 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
#align_import linear_algebra.vandermonde from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Vandermonde.lean | 77 | 139 | theorem det_vandermonde {n : ℕ} (v : Fin n → R) :
det (vandermonde v) = ∏ i : Fin n, ∏ j ∈ Ioi i, (v j - v i) := by |
unfold vandermonde
induction' n with n ih
· exact det_eq_one_of_card_eq_zero (Fintype.card_fin 0)
calc
det (of fun i j : Fin n.succ => v i ^ (j : ℕ)) =
det
(of fun i j : Fin n.succ =>
Matrix.vecCons (v 0 ^ (j : ℕ)) (fun i => v (Fin.succ i) ^ (j : ℕ) - v 0 ^ (j : ℕ)) i) :=
... | 60 | 114,200,738,981,568,440,000,000,000 | 2 | 1 | 5 | 1,157 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 113 | 124 | theorem sinZeta_two_mul_nat_add_one' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) / (2 * k + 1) / Gammaℂ (2 * k + 1) *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [sinZeta_two_mul_nat_add_one hk hx]
congr 1
have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by
rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)),
Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul,
← Nat.cast_add_one, ← Nat.cast_m... | 9 | 8,103.083928 | 2 | 2 | 8 | 2,019 |
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
-- Porting note: due to `PresheafedSpace`, `SheafedSpace` and `Locally... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 133 | 141 | theorem isoRestrict_hom_ofRestrict : H.isoRestrict.hom ≫ Y.ofRestrict _ = f := by |
-- Porting note: `ext` did not pick up `NatTrans.ext`
refine PresheafedSpace.Hom.ext _ _ rfl <| NatTrans.ext _ _ <| funext fun x => ?_
simp only [isoRestrict_hom_c_app, NatTrans.comp_app, eqToHom_refl,
ofRestrict_c_app, Category.assoc, whiskerRight_id']
erw [Category.comp_id, comp_c_app, f.c.naturality_ass... | 8 | 2,980.957987 | 2 | 1 | 2 | 889 |
import Mathlib.CategoryTheory.Sites.IsSheafFor
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v u
namespace CategoryTheory
open Opposite ... | Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean | 246 | 261 | theorem sheaf_condition : R.IsSheafFor P ↔ Nonempty (IsLimit (Fork.ofι _ (w P R))) := by |
rw [Types.type_equalizer_iff_unique]
erw [← Equiv.forall_congr_left (firstObjEqFamily P R).toEquiv.symm]
simp_rw [← compatible_iff, ← Iso.toEquiv_fun, Equiv.apply_symm_apply]
apply forall₂_congr
intro x _
apply exists_unique_congr
intro t
rw [Equiv.eq_symm_apply]
constructor
· intro q
funext Y ... | 15 | 3,269,017.372472 | 2 | 1.833333 | 6 | 1,913 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 57 | 63 | theorem bernoulli_def (n : ℕ) : bernoulli n =
∑ i ∈ range (n + 1), Polynomial.monomial i (_root_.bernoulli (n - i) * choose n i) := by |
rw [← sum_range_reflect, add_succ_sub_one, add_zero, bernoulli]
apply sum_congr rfl
rintro x hx
rw [mem_range_succ_iff] at hx
rw [choose_symm hx, tsub_tsub_cancel_of_le hx]
| 5 | 148.413159 | 2 | 1.6 | 5 | 1,715 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 76 | 82 | theorem borel_eq_generateFrom_Ici_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Ici (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Iio_rat, iUnion_singleton_eq_range, iUnion_singleton_eq_range]
refine le_antisymm (generateFrom_le ?_) (generateFrom_le ?_) <;>
rintro _ ⟨q, rfl⟩ <;>
dsimp only <;>
[rw [← compl_Ici]; rw [← compl_Iio]] <;>
exact MeasurableSet.compl (GenerateMeasurable.basic _ (mem_range_self q))
| 6 | 403.428793 | 2 | 1.444444 | 9 | 1,528 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 188 | 194 | theorem convexBodyLT'_neg_mem (x : E K) (hx : x ∈ convexBodyLT' K f w₀) :
-x ∈ convexBodyLT' K f w₀ := by |
simp [Set.mem_prod, Prod.fst_neg, Set.mem_pi, Set.mem_univ, Pi.neg_apply,
mem_ball_zero_iff, norm_neg, Real.norm_eq_abs, forall_true_left, Subtype.forall,
Prod.snd_neg, Complex.norm_eq_abs] at hx ⊢
convert hx using 3
split_ifs <;> simp
| 5 | 148.413159 | 2 | 1.666667 | 15 | 1,803 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 80 | 90 | theorem integerNormalization_spec (p : S[X]) :
∃ b : M, ∀ i, algebraMap R S ((integerNormalization M p).coeff i) = (b : R) • p.coeff i := by |
use Classical.choose (exist_integer_multiples_of_finset M (p.support.image p.coeff))
intro i
rw [integerNormalization_coeff, coeffIntegerNormalization]
split_ifs with hi
· exact
Classical.choose_spec
(Classical.choose_spec (exist_integer_multiples_of_finset M (p.support.image p.coeff))
... | 9 | 8,103.083928 | 2 | 1.333333 | 6 | 1,454 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
| Mathlib/GroupTheory/Archimedean.lean | 40 | 54 | theorem AddSubgroup.cyclic_of_min {H : AddSubgroup G} {a : G}
(ha : IsLeast { g : G | g ∈ H ∧ 0 < g } a) : H = AddSubgroup.closure {a} := by |
obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
refine le_antisymm ?_ (H.closure_le.mpr <| by simp [a_in])
intro g g_in
obtain ⟨k, ⟨nonneg, lt⟩, _⟩ := existsUnique_zsmul_near_of_pos' a_pos g
have h_zero : g - k • a = 0 := by
by_contra h
have h : a ≤ g - k • a := by
refine a_min ⟨?_, ?_⟩
· exact AddSu... | 13 | 442,413.392009 | 2 | 1.666667 | 3 | 1,776 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 57 | 73 | theorem coeff_derivative (p : R[X]) (n : ℕ) :
coeff (derivative p) n = coeff p (n + 1) * (n + 1) := by |
rw [derivative_apply]
simp only [coeff_X_pow, coeff_sum, coeff_C_mul]
rw [sum, Finset.sum_eq_single (n + 1)]
· simp only [Nat.add_succ_sub_one, add_zero, mul_one, if_true, eq_self_iff_true]; norm_cast
· intro b
cases b
· intros
rw [Nat.cast_zero, mul_zero, zero_mul]
· intro _ H
rw [Na... | 15 | 3,269,017.372472 | 2 | 0.3 | 10 | 317 |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 177 | 191 | theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
(i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by |
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of choice) as a linear combination over `w`.
-- We can do this linearly by constructing the map on a basis.
fapply card_le_o... | 12 | 162,754.791419 | 2 | 1.727273 | 11 | 1,843 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 116 | 123 | theorem polar_add_left_iff {f : M → R} {x x' y : M} :
polar f (x + x') y = polar f x y + polar f x' y ↔
f (x + x' + y) + (f x + f x' + f y) = f (x + x') + f (x' + y) + f (y + x) := by |
simp only [← add_assoc]
simp only [polar, sub_eq_iff_eq_add, eq_sub_iff_add_eq, sub_add_eq_add_sub, add_sub]
simp only [add_right_comm _ (f y) _, add_right_comm _ (f x') (f x)]
rw [add_comm y x, add_right_comm _ _ (f (x + y)), add_comm _ (f (x + y)),
add_right_comm (f (x + y)), add_left_inj]
| 5 | 148.413159 | 2 | 0.5 | 6 | 482 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.Matrix
import Mathlib.Analysis.RCLike.Basic
import Mathlib.LinearAlgebra.UnitaryGroup
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.star.matrix from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1... | Mathlib/Analysis/NormedSpace/Star/Matrix.lean | 83 | 90 | theorem entrywise_sup_norm_bound_of_unitary {U : Matrix n n 𝕜} (hU : U ∈ Matrix.unitaryGroup n 𝕜) :
‖U‖ ≤ 1 := by |
conv => -- Porting note: was `simp_rw [pi_norm_le_iff_of_nonneg zero_le_one]`
rw [pi_norm_le_iff_of_nonneg zero_le_one]
intro
rw [pi_norm_le_iff_of_nonneg zero_le_one]
intros
exact entry_norm_bound_of_unitary hU _ _
| 6 | 403.428793 | 2 | 2 | 2 | 2,259 |
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Preadditive.LeftExact
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.Algebra.Homology.Exact
import Mathli... | Mathlib/CategoryTheory/Abelian/Exact.lean | 115 | 120 | theorem exact_epi_comp_iff {W : C} (h : W ⟶ X) [Epi h] : Exact (h ≫ f) g ↔ Exact f g := by |
refine ⟨fun hfg => ?_, fun h => exact_epi_comp h⟩
let hc := isCokernelOfComp _ _ (colimit.isColimit (parallelPair (h ≫ f) 0))
(by rw [← cancel_epi h, ← Category.assoc, CokernelCofork.condition, comp_zero]) rfl
refine (exact_iff' _ _ (limit.isLimit _) hc).2 ⟨?_, ((exact_iff _ _).1 hfg).2⟩
exact zero_of_epi_... | 5 | 148.413159 | 2 | 1.8 | 5 | 1,886 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theory.constructions.pi from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Function Set MeasureTheory... | Mathlib/MeasureTheory/Constructions/Pi.lean | 87 | 95 | theorem IsCountablySpanning.pi {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsCountablySpanning (C i)) :
IsCountablySpanning (pi univ '' pi univ C) := by |
choose s h1s h2s using hC
cases nonempty_encodable (ι → ℕ)
let e : ℕ → ι → ℕ := fun n => (@decode (ι → ℕ) _ n).iget
refine ⟨fun n => Set.pi univ fun i => s i (e n i), fun n =>
mem_image_of_mem _ fun i _ => h1s i _, ?_⟩
simp_rw [(surjective_decode_iget (ι → ℕ)).iUnion_comp fun x => Set.pi univ fun i => s ... | 7 | 1,096.633158 | 2 | 1.2 | 10 | 1,268 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 114 | 118 | theorem card_insert_le (a : α) (s : Finset α) : card (insert a s) ≤ s.card + 1 := by |
by_cases h : a ∈ s
· rw [insert_eq_of_mem h]
exact Nat.le_succ _
· rw [card_insert_of_not_mem h]
| 4 | 54.59815 | 2 | 0.6 | 10 | 527 |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.nat.factorial.big_operators from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open Finset Nat
namespace Nat
lemma monotone_factorial : Monotone factorial := fun _ _ => fa... | Mathlib/Data/Nat/Factorial/BigOperators.lean | 34 | 38 | theorem prod_factorial_dvd_factorial_sum : (∏ i ∈ s, (f i)!) ∣ (∑ i ∈ s, f i)! := by |
induction' s using Finset.cons_induction_on with a s has ih
· simp
· rw [prod_cons, Finset.sum_cons]
exact (mul_dvd_mul_left _ ih).trans (Nat.factorial_mul_factorial_dvd_factorial_add _ _)
| 4 | 54.59815 | 2 | 1 | 2 | 884 |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
| Mathlib/Topology/Perfect.lean | 62 | 68 | theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by |
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_principal_iff]
exact mem_nhdsWithin_of_mem_nhds hU
rw [AccPt, ← inf_principal, ← inf_assoc, inf_of_le_left this]
exact h_acc
| 5 | 148.413159 | 2 | 1.666667 | 9 | 1,822 |
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 | 157 | 169 | theorem smul_single_apply (i : ℕ) (f : R[X]) (m : M) (n : ℕ) :
(f • single R i m) n = ite (i ≤ n) (f.coeff (n - i) • m) 0 := by |
induction' f using Polynomial.induction_on' with p q hp hq
· rw [add_smul, Finsupp.add_apply, hp, hq, coeff_add, add_smul]
split_ifs
exacts [rfl, zero_add 0]
· rw [monomial_smul_single, single_apply, coeff_monomial, ite_smul, zero_smul]
by_cases h : i ≤ n
· simp_rw [eq_tsub_iff_add_eq_of_le h, if... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,085 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 101 | 120 | theorem mirror_eval_one : p.mirror.eval 1 = p.eval 1 := by |
simp_rw [eval_eq_sum_range, one_pow, mul_one, mirror_natDegree]
refine Finset.sum_bij_ne_zero ?_ ?_ ?_ ?_ ?_
· exact fun n _ _ => revAt (p.natDegree + p.natTrailingDegree) n
· intro n hn hp
rw [Finset.mem_range_succ_iff] at *
rw [revAt_le (hn.trans (Nat.le_add_right _ _))]
rw [tsub_le_iff_tsub_le, ... | 19 | 178,482,300.963187 | 2 | 1.571429 | 7 | 1,704 |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 140 | 148 | theorem isEquivalent_const_iff_tendsto {c : β} (h : c ≠ 0) :
u ~[l] const _ c ↔ Tendsto u l (𝓝 c) := by |
simp (config := { unfoldPartialApp := true }) only [IsEquivalent, const, isLittleO_const_iff h]
constructor <;> intro h
· have := h.sub (tendsto_const_nhds (x := -c))
simp only [Pi.sub_apply, sub_neg_eq_add, sub_add_cancel, zero_add] at this
exact this
· have := h.sub (tendsto_const_nhds (x := c))
... | 7 | 1,096.633158 | 2 | 1.285714 | 7 | 1,354 |
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 138 | 150 | theorem trace_eq_contract_of_basis [Finite ι] (b : Basis ι R M) :
LinearMap.trace R M ∘ₗ dualTensorHom R M M = contractLeft R M := by |
classical
cases nonempty_fintype ι
apply Basis.ext (Basis.tensorProduct (Basis.dualBasis b) b)
rintro ⟨i, j⟩
simp only [Function.comp_apply, Basis.tensorProduct_apply, Basis.coe_dualBasis, coe_comp]
rw [trace_eq_matrix_trace R b, toMatrix_dualTensorHom]
by_cases hij : i = j
· rw [hij]
... | 11 | 59,874.141715 | 2 | 1.333333 | 6 | 1,414 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 98 | 128 | theorem Asymptotics.IsLittleO.sum_range {α : Type*} [NormedAddCommGroup α] {f : ℕ → α} {g : ℕ → ℝ}
(h : f =o[atTop] g) (hg : 0 ≤ g) (h'g : Tendsto (fun n => ∑ i ∈ range n, g i) atTop atTop) :
(fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => ∑ i ∈ range n, g i := by |
have A : ∀ i, ‖g i‖ = g i := fun i => Real.norm_of_nonneg (hg i)
have B : ∀ n, ‖∑ i ∈ range n, g i‖ = ∑ i ∈ range n, g i := fun n => by
rwa [Real.norm_eq_abs, abs_sum_of_nonneg']
apply isLittleO_iff.2 fun ε εpos => _
intro ε εpos
obtain ⟨N, hN⟩ : ∃ N : ℕ, ∀ b : ℕ, N ≤ b → ‖f b‖ ≤ ε / 2 * g b := by
si... | 28 | 1,446,257,064,291.475 | 2 | 1.25 | 8 | 1,299 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.valuation.quotient from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
namespace Valuation
variable {R Γ₀ : Type*} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀]
va... | Mathlib/RingTheory/Valuation/Quotient.lean | 66 | 74 | theorem supp_quot {J : Ideal R} (hJ : J ≤ supp v) :
supp (v.onQuot hJ) = (supp v).map (Ideal.Quotient.mk J) := by |
apply le_antisymm
· rintro ⟨x⟩ hx
apply Ideal.subset_span
exact ⟨x, hx, rfl⟩
· rw [Ideal.map_le_iff_le_comap]
intro x hx
exact hx
| 7 | 1,096.633158 | 2 | 1.333333 | 3 | 1,431 |
import Mathlib.Order.Interval.Set.OrdConnectedComponent
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.t5 from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter Set Function OrderDual Topology Interval
variable {X : Type*} [LinearOrder X] [Topological... | Mathlib/Topology/Order/T5.lean | 74 | 79 | theorem compl_section_ordSeparatingSet_mem_nhds (hd : Disjoint s (closure t)) (ha : a ∈ s) :
(ordConnectedSection <| ordSeparatingSet s t)ᶜ ∈ 𝓝 a := by |
rw [← nhds_left_sup_nhds_right, mem_sup]
exact
⟨compl_section_ordSeparatingSet_mem_nhdsWithin_Iic hd ha,
compl_section_ordSeparatingSet_mem_nhdsWithin_Ici hd ha⟩
| 4 | 54.59815 | 2 | 1.75 | 4 | 1,855 |
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 | 40 | 50 | theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by |
rw [arg]
split_ifs with h₁ h₂
· rw [Real.cos_arcsin]
field_simp [Real.sqrt_sq, (abs.pos hx).le, *]
· rw [Real.cos_add_pi, Real.cos_arcsin]
field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs,
_root_.abs_of_neg (not_le.1 h₁), *]
· rw [Real.cos_sub_pi, Real.cos_arcsin]
field_simp [Re... | 10 | 22,026.465795 | 2 | 1.571429 | 7 | 1,706 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 84 | 89 | theorem mk_one_mul_one_sub_eq_one : (mk 1 : S⟦X⟧) * (1 - X) = 1 := by |
rw [mul_comm, ext_iff]
intro n
cases n with
| zero => simp
| succ n => simp [sub_mul]
| 5 | 148.413159 | 2 | 0.857143 | 14 | 748 |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 326 | 330 | theorem rowLen_anti (μ : YoungDiagram) (i1 i2 : ℕ) (hi : i1 ≤ i2) : μ.rowLen i2 ≤ μ.rowLen i1 := by |
by_contra! h_lt
rw [← lt_self_iff_false (μ.rowLen i1)]
rw [← mem_iff_lt_rowLen] at h_lt ⊢
exact μ.up_left_mem hi (by rfl) h_lt
| 4 | 54.59815 | 2 | 0.769231 | 13 | 683 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
open CategoryTheory
namespace ModuleCat
variable {ι ι' R : Type*} [Ring R] {S : ShortComplex (ModuleCat R)}
(hS : S.Exact) (hS' : S.ShortExact) {v : ι → S.X₁}
open CategoryTheory Submodule Set
section Span
the... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 129 | 138 | theorem span_rightExact {w : ι' → S.X₃} (hv : ⊤ ≤ span R (range v))
(hw : ⊤ ≤ span R (range w)) (hE : Epi S.g) :
⊤ ≤ span R (range (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w))) := by |
refine span_exact hS ?_ hv ?_
· simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inl]
· convert hw
simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inr]
rw [ModuleCat.epi_iff_surjective] at hE
rw [← Function.comp.assoc, Function.RightInverse.comp_eq_id (Funct... | 7 | 1,096.633158 | 2 | 2 | 5 | 2,042 |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 160 | 167 | theorem neg {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (-f) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp
| 7 | 1,096.633158 | 2 | 1.625 | 8 | 1,750 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 137 | 142 | theorem zero_of_padicNorm_eq_zero {q : ℚ} (h : padicNorm p q = 0) : q = 0 := by |
apply by_contradiction; intro hq
unfold padicNorm at h; rw [if_neg hq] at h
apply absurd h
apply zpow_ne_zero
exact mod_cast hp.1.ne_zero
| 5 | 148.413159 | 2 | 1 | 4 | 1,075 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDomain
#align_import data.polynomial.field_division from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Polynomial
namespace Polynomial
u... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 40 | 57 | theorem derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors
{p : R[X]} {t : R} (hpt : Polynomial.IsRoot p t)
(hnzd : (p.rootMultiplicity t : R) ∈ nonZeroDivisors R) :
(derivative p).rootMultiplicity t = p.rootMultiplicity t - 1 := by |
by_cases h : p = 0
· simp only [h, map_zero, rootMultiplicity_zero]
obtain ⟨g, hp, hndvd⟩ := p.exists_eq_pow_rootMultiplicity_mul_and_not_dvd h t
set m := p.rootMultiplicity t
have hm : m - 1 + 1 = m := Nat.sub_add_cancel <| (rootMultiplicity_pos h).2 hpt
have hndvd : ¬(X - C t) ^ m ∣ derivative p := by
... | 14 | 1,202,604.284165 | 2 | 2 | 4 | 2,297 |
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 312 | 317 | theorem _root_.IsAntichain.finite_of_partiallyWellOrderedOn (ha : IsAntichain r s)
(hp : s.PartiallyWellOrderedOn r) : s.Finite := by |
refine not_infinite.1 fun hi => ?_
obtain ⟨m, n, hmn, h⟩ := hp (fun n => hi.natEmbedding _ n) fun n => (hi.natEmbedding _ n).2
exact hmn.ne ((hi.natEmbedding _).injective <| Subtype.val_injective <|
ha.eq (hi.natEmbedding _ m).2 (hi.natEmbedding _ n).2 h)
| 4 | 54.59815 | 2 | 1.5 | 10 | 1,649 |
import Mathlib.Order.Monotone.Odd
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
#align_import analysis.special_functions.trigonometric.deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open s... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | 35 | 41 | theorem hasStrictDerivAt_sin (x : ℂ) : HasStrictDerivAt sin (cos x) x := by |
simp only [cos, div_eq_mul_inv]
convert ((((hasStrictDerivAt_id x).neg.mul_const I).cexp.sub
((hasStrictDerivAt_id x).mul_const I).cexp).mul_const I).mul_const (2 : ℂ)⁻¹ using 1
simp only [Function.comp, id]
rw [sub_mul, mul_assoc, mul_assoc, I_mul_I, neg_one_mul, neg_neg, mul_one, one_mul, mul_assoc,
... | 6 | 403.428793 | 2 | 2 | 4 | 2,295 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 185 | 190 | theorem gcd_map_mul (a : α) (s : Multiset α) : (s.map (a * ·)).gcd = normalize a * s.gcd := by |
refine s.induction_on ?_ fun b s ih ↦ ?_
· simp_rw [map_zero, gcd_zero, mul_zero]
· simp_rw [map_cons, gcd_cons, ← gcd_mul_left]
rw [ih]
apply ((normalize_associated a).mul_right _).gcd_eq_right
| 5 | 148.413159 | 2 | 1.333333 | 9 | 1,390 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 51 | 55 | theorem sign_apply_eq (r : ℝ) : sign r = -1 ∨ sign r = 0 ∨ sign r = 1 := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· exact Or.inl <| sign_of_neg hn
· exact Or.inr <| Or.inl <| sign_zero
· exact Or.inr <| Or.inr <| sign_of_pos hp
| 4 | 54.59815 | 2 | 1.363636 | 11 | 1,467 |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 161 | 168 | theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idealOfSet R) := by |
refine fun I s => ⟨fun h f hf => ?_, fun h x hx => ?_⟩
· by_contra h'
rcases not_mem_idealOfSet.mp h' with ⟨x, hx, hfx⟩
exact hfx (not_mem_setOfIdeal.mp (mt (@h x) hx) hf)
· obtain ⟨f, hf, hfx⟩ := mem_setOfIdeal.mp hx
by_contra hx'
exact not_mem_idealOfSet.mpr ⟨x, hx', hfx⟩ (h hf)
| 7 | 1,096.633158 | 2 | 0.625 | 8 | 544 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 117 | 128 | theorem gramSchmidt_inv_triangular (v : ι → E) {i j : ι} (hij : i < j) :
⟪gramSchmidt 𝕜 v j, v i⟫ = 0 := by |
rw [gramSchmidt_def'' 𝕜 v]
simp only [inner_add_right, inner_sum, inner_smul_right]
set b : ι → E := gramSchmidt 𝕜 v
convert zero_add (0 : 𝕜)
· exact gramSchmidt_orthogonal 𝕜 v hij.ne'
apply Finset.sum_eq_zero
rintro k hki'
have hki : k < i := by simpa using hki'
have : ⟪b j, b k⟫ = 0 := gramSchm... | 10 | 22,026.465795 | 2 | 1.125 | 8 | 1,201 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 177 | 181 | theorem set_lintegral_pdf_le_map {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
(μ : Measure E := by | volume_tac) (s : Set E) :
∫⁻ x in s, pdf X ℙ μ x ∂μ ≤ map X ℙ s := by
apply (withDensity_apply_le _ s).trans
exact withDensity_pdf_le_map _ _ _ s
| 4 | 54.59815 | 2 | 1.428571 | 7 | 1,513 |
import Mathlib.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
#align_import algebraic_topology.dold_kan.faces from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Limits CategoryTheory.Category
CategoryTheory.Preadditive CategoryTheor... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 53 | 58 | theorem comp_δ_eq_zero {Y : C} {n : ℕ} {q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(j : Fin (n + 2)) (hj₁ : j ≠ 0) (hj₂ : n + 2 ≤ (j : ℕ) + q) : φ ≫ X.δ j = 0 := by |
obtain ⟨i, rfl⟩ := Fin.eq_succ_of_ne_zero hj₁
apply v i
simp only [Fin.val_succ] at hj₂
omega
| 4 | 54.59815 | 2 | 2 | 2 | 2,494 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 83 | 91 | theorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1 := by |
induction a using limitRecOn with
| H₁ => simp only [opow_zero]
| H₂ _ ih =>
simp only [opow_succ, ih, mul_one]
| H₃ b l IH =>
refine eq_of_forall_ge_iff fun c => ?_
rw [opow_le_of_limit Ordinal.one_ne_zero l]
exact ⟨fun H => by simpa only [opow_zero] using H 0 l.pos, fun H b' h => by rwa [IH _... | 8 | 2,980.957987 | 2 | 0.555556 | 9 | 514 |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 100 | 104 | theorem commutator_eq_bot_iff_le_centralizer : ⁅H₁, H₂⁆ = ⊥ ↔ H₁ ≤ centralizer H₂ := by |
rw [eq_bot_iff, commutator_le]
refine forall_congr' fun p =>
forall_congr' fun _hp => forall_congr' fun q => forall_congr' fun hq => ?_
rw [mem_bot, commutatorElement_eq_one_iff_mul_comm, eq_comm]
| 4 | 54.59815 | 2 | 0.8 | 5 | 707 |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 66 | 72 | theorem reverse_inj [h : HasInvolutiveReverse V] {a b : V}
(f g : a ⟶ b) : reverse f = reverse g ↔ f = g := by |
constructor
· rintro h
simpa using congr_arg Quiver.reverse h
· rintro h
congr
| 5 | 148.413159 | 2 | 1.375 | 8 | 1,472 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 33 | 40 | theorem Int.isCoprime_iff_gcd_eq_one {m n : ℤ} : IsCoprime m n ↔ Int.gcd m n = 1 := by |
constructor
· rintro ⟨a, b, h⟩
have : 1 = m * a + n * b := by rwa [mul_comm m, mul_comm n, eq_comm]
exact Nat.dvd_one.mp (Int.gcd_dvd_iff.mpr ⟨a, b, this⟩)
· rw [← Int.ofNat_inj, IsCoprime, Int.gcd_eq_gcd_ab, mul_comm m, mul_comm n, Nat.cast_one]
intro h
exact ⟨_, _, h⟩
| 7 | 1,096.633158 | 2 | 1.111111 | 18 | 1,195 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 106 | 112 | theorem summable_inner (f g : lp G 2) : Summable fun i => ⟪f i, g i⟫ := by |
-- Apply the Direct Comparison Test, comparing with ∑' i, ‖f i‖ * ‖g i‖ (summable by Hölder)
refine .of_norm_bounded (fun i => ‖f i‖ * ‖g i‖) (lp.summable_mul ?_ f g) ?_
· rw [Real.isConjExponent_iff]; norm_num
intro i
-- Then apply Cauchy-Schwarz pointwise
exact norm_inner_le_norm (𝕜 := 𝕜) _ _
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,398 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 418 | 433 | theorem HasFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
(hg : ContinuousAt g a) (hf : HasFDerivAt f (f' : E →L[𝕜] F) (g a))
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) : HasFDerivAt g (f'.symm : F →L[𝕜] E) a := by |
have : (fun x : F => g x - g a - f'.symm (x - a)) =O[𝓝 a]
fun x : F => f' (g x - g a) - (x - a) := by
refine ((f'.symm : F →L[𝕜] E).isBigO_comp _ _).congr (fun x => ?_) fun _ => rfl
simp
refine HasFDerivAtFilter.of_isLittleO <| this.trans_isLittleO ?_
clear this
refine ((hf.isLittleO.comp_tends... | 13 | 442,413.392009 | 2 | 1.555556 | 9 | 1,700 |
import Mathlib.Analysis.BoxIntegral.DivergenceTheorem
import Mathlib.Analysis.BoxIntegral.Integrability
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_impo... | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | 143 | 245 | theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (B... |
/- Choose a monotone sequence `J k` of subboxes that cover the interior of `I` and prove that
these boxes satisfy the assumptions of the previous lemma. -/
rcases I.exists_seq_mono_tendsto with ⟨J, hJ_sub, hJl, hJu⟩
have hJ_sub' : ∀ k, Box.Icc (J k) ⊆ Box.Icc I := fun k => (hJ_sub k).trans I.Ioo_subset_Icc
... | 93 | 24,512,455,429,200,860,000,000,000,000,000,000,000,000 | 2 | 2 | 2 | 2,475 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
open Set Function Filter MeasureTheory MeasureTheory.Measure
open ENNReal
variable {α : Type*} {m : MeasurableSpace α} (f : α → α) {s : Set α}
... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 109 | 115 | theorem ergodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
Ergodic (e ∘ f ∘ e.symm) μ' ↔ Ergodic f μ := by |
have : MeasurePreserving (e ∘ f ∘ e.symm) μ' μ' ↔ MeasurePreserving f μ μ := by
rw [h.comp_left_iff, (MeasurePreserving.symm e h).comp_right_iff]
replace h : PreErgodic (e ∘ f ∘ e.symm) μ' ↔ PreErgodic f μ := h.preErgodic_conjugate_iff
exact ⟨fun hf => { this.mp hf.toMeasurePreserving, h.mp hf.toPreErgodic w... | 5 | 148.413159 | 2 | 1.166667 | 6 | 1,244 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 118 | 134 | theorem comp_P_eq_self {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ) :
φ ≫ (P q).f (n + 1) = φ := by |
induction' q with q hq
· simp only [P_zero]
apply comp_id
· simp only [P_succ, comp_add, HomologicalComplex.comp_f, HomologicalComplex.add_f_apply,
comp_id, ← assoc, hq v.of_succ, add_right_eq_self]
by_cases hqn : n < q
· exact v.of_succ.comp_Hσ_eq_zero hqn
· obtain ⟨a, ha⟩ := Nat.le.dest (... | 15 | 3,269,017.372472 | 2 | 1.2 | 5 | 1,271 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Matrix.CharP
#align_import linear_algebra.matrix.charpoly.finite_field from "leanprover-community/mathlib"@"b95b8c7a484a298228805c72c142f6b062eb0d70"
noncomputable section
open Polynomial Matrix
open s... | Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean | 53 | 58 | theorem FiniteField.trace_pow_card {K : Type*} [Field K] [Fintype K] (M : Matrix n n K) :
trace (M ^ Fintype.card K) = trace M ^ Fintype.card K := by |
cases isEmpty_or_nonempty n
· simp [Matrix.trace]
rw [Matrix.trace_eq_neg_charpoly_coeff, Matrix.trace_eq_neg_charpoly_coeff,
FiniteField.Matrix.charpoly_pow_card, FiniteField.pow_card]
| 4 | 54.59815 | 2 | 1.25 | 4 | 1,317 |
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 | 159 | 168 | theorem exists_mem_multiset_le_of_prime {s : Multiset (Associates α)} {p : Associates α}
(hp : Prime p) : p ≤ s.prod → ∃ a ∈ s, p ≤ a :=
Multiset.induction_on s (fun ⟨d, Eq⟩ => (hp.ne_one (mul_eq_one_iff.1 Eq.symm).1).elim)
fun a s ih h =>
have : p ≤ a * s.prod := by | simpa using h
match Prime.le_or_le hp this with
| Or.inl h => ⟨a, Multiset.mem_cons_self a s, h⟩
| Or.inr h =>
let ⟨a, has, h⟩ := ih h
⟨a, Multiset.mem_cons_of_mem has, h⟩
| 6 | 403.428793 | 2 | 2 | 6 | 2,348 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.lagrange from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Polynomial
section PolynomialDetermination
namespace Poly... | Mathlib/LinearAlgebra/Lagrange.lean | 74 | 83 | theorem eq_of_degree_le_of_eval_finset_eq
(h_deg_le : f.degree ≤ s.card)
(h_deg_eq : f.degree = g.degree)
(hlc : f.leadingCoeff = g.leadingCoeff)
(h_eval : ∀ x ∈ s, f.eval x = g.eval x) :
f = g := by |
rcases eq_or_ne f 0 with rfl | hf
· rwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq
· exact eq_of_degree_sub_lt_of_eval_finset_eq s
(lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_le) h_eval
| 4 | 54.59815 | 2 | 1.75 | 4 | 1,852 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 106 | 137 | theorem norm_max_aux₁ [CompleteSpace F] {f : ℂ → F} {z w : ℂ}
(hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ := by |
-- Consider a circle of radius `r = dist w z`.
set r : ℝ := dist w z
have hw : w ∈ closedBall z r := mem_closedBall.2 le_rfl
-- Assume the converse. Since `‖f w‖ ≤ ‖f z‖`, we have `‖f w‖ < ‖f z‖`.
refine (isMaxOn_iff.1 hz _ hw).antisymm (not_lt.1 ?_)
rintro hw_lt : ‖f w‖ < ‖f z‖
have hr : 0 < r := dist_p... | 29 | 3,931,334,297,144.042 | 2 | 2 | 4 | 2,117 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 122 | 151 | theorem card_sigma : #(Σn, L.Term (Sum α (Fin n))) = max ℵ₀ #(Sum α (Σi, L.Functions i)) := by |
refine le_antisymm ?_ ?_
· rw [mk_sigma]
refine (sum_le_iSup_lift _).trans ?_
rw [mk_nat, lift_aleph0, mul_eq_max_of_aleph0_le_left le_rfl, max_le_iff,
ciSup_le_iff' (bddAbove_range _)]
· refine ⟨le_max_left _ _, fun i => card_le.trans ?_⟩
refine max_le (le_max_left _ _) ?_
rw [← add_... | 29 | 3,931,334,297,144.042 | 2 | 2 | 3 | 2,074 |
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 138 | 146 | theorem of_exists_root (H : ∀ p : k[X], p.Monic → Irreducible p → ∃ x, p.eval x = 0) :
IsAlgClosed k := by |
refine ⟨fun p ↦ Or.inr ?_⟩
intro q hq _
have : Irreducible (q * C (leadingCoeff q)⁻¹) := by
rw [← coe_normUnit_of_ne_zero hq.ne_zero]
exact (associated_normalize _).irreducible hq
obtain ⟨x, hx⟩ := H (q * C (leadingCoeff q)⁻¹) (monic_mul_leadingCoeff_inv hq.ne_zero) this
exact degree_mul_leadingCoeff... | 7 | 1,096.633158 | 2 | 1.5 | 6 | 1,683 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 169 | 176 | theorem condexp_zero : μ[(0 : α → F')|m] = 0 := by |
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
exact
condexp_of_stronglyMeasurable hm (@stronglyMeasurable_zero _ _ m _ _) (integrable_zero _ _ _)
| 7 | 1,096.633158 | 2 | 1.222222 | 9 | 1,296 |
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 | 116 | 124 | theorem gradedComm_of_tmul_of (i j : ι) (a : 𝒜 i) (b : ℬ j) :
gradedComm R 𝒜 ℬ (lof R _ 𝒜 i a ⊗ₜ lof R _ ℬ j b) =
(-1 : ℤˣ)^(j * i) • (lof R _ ℬ _ b ⊗ₜ lof R _ 𝒜 _ a) := by |
rw [gradedComm]
dsimp only [LinearEquiv.trans_apply, LinearEquiv.ofLinear_apply]
rw [TensorProduct.directSum_lof_tmul_lof, gradedCommAux_lof_tmul, Units.smul_def,
-- Note: #8386 specialized `map_smul` to `LinearEquiv.map_smul` to avoid timeouts.
zsmul_eq_smul_cast R, LinearEquiv.map_smul, TensorProduct.d... | 6 | 403.428793 | 2 | 1.666667 | 6 | 1,805 |
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type*}
namespace Set
section LT
varia... | Mathlib/Order/Height.lean | 127 | 131 | theorem chainHeight_eq_top_iff : s.chainHeight = ⊤ ↔ ∀ n, ∃ l ∈ s.subchain, length l = n := by |
refine ⟨fun h n ↦ le_chainHeight_iff.1 (le_top.trans_eq h.symm), fun h ↦ ?_⟩
contrapose! h; obtain ⟨n, hn⟩ := WithTop.ne_top_iff_exists.1 h
exact ⟨n + 1, fun l hs ↦ (Nat.lt_succ_iff.2 <| Nat.cast_le.1 <|
(length_le_chainHeight_of_mem_subchain hs).trans_eq hn.symm).ne⟩
| 4 | 54.59815 | 2 | 1.285714 | 7 | 1,356 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 114 | 119 | theorem isPerfectMatching_iff : M.IsPerfectMatching ↔ ∀ v, ∃! w, M.Adj v w := by |
refine ⟨?_, fun hm => ⟨fun v _ => hm v, fun v => ?_⟩⟩
· rintro ⟨hm, hs⟩ v
exact hm (hs v)
· obtain ⟨w, hw, -⟩ := hm v
exact M.edge_vert hw
| 5 | 148.413159 | 2 | 0.888889 | 9 | 772 |
import Mathlib.Order.Filter.CountableInter
set_option autoImplicit true
open Function Set Filter
class HasCountableSeparatingOn (α : Type*) (p : Set α → Prop) (t : Set α) : Prop where
exists_countable_separating : ∃ S : Set (Set α), S.Countable ∧ (∀ s ∈ S, p s) ∧
∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ S, x ∈ s ↔ y ∈ s) ... | Mathlib/Order/Filter/CountableSeparatingOn.lean | 103 | 109 | theorem exists_seq_separating (α : Type*) {p : Set α → Prop} {s₀} (hp : p s₀) (t : Set α)
[HasCountableSeparatingOn α p t] :
∃ S : ℕ → Set α, (∀ n, p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ n, x ∈ S n ↔ y ∈ S n) → x = y := by |
rcases exists_nonempty_countable_separating α hp t with ⟨S, hSne, hSc, hS⟩
rcases hSc.exists_eq_range hSne with ⟨S, rfl⟩
use S
simpa only [forall_mem_range] using hS
| 4 | 54.59815 | 2 | 2 | 4 | 2,311 |
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Preadditive.LeftExact
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.Algebra.Homology.Exact
import Mathli... | Mathlib/CategoryTheory/Abelian/Exact.lean | 66 | 81 | theorem exact_iff : Exact f g ↔ f ≫ g = 0 ∧ kernel.ι g ≫ cokernel.π f = 0 := by |
constructor
· exact fun h ↦ ⟨h.1, kernel_comp_cokernel f g h⟩
· refine fun h ↦ ⟨h.1, ?_⟩
suffices hl : IsLimit
(KernelFork.ofι (imageSubobject f).arrow (imageSubobject_arrow_comp_eq_zero h.1)) by
have : imageToKernel f g h.1 = (hl.conePointUniqueUpToIso (limit.isLimit _)).hom ≫
(kerne... | 15 | 3,269,017.372472 | 2 | 1.8 | 5 | 1,886 |
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 | 98 | 106 | theorem continuousOn_prod_circle_transform_function {R r : ℝ} (hr : r < R) {z : ℂ} :
ContinuousOn (fun w : ℂ × ℝ => (circleMap z R w.snd - w.fst)⁻¹ ^ 2)
(closedBall z r ×ˢ univ) := by |
simp_rw [← one_div]
apply_rules [ContinuousOn.pow, ContinuousOn.div, continuousOn_const]
· exact ((continuous_circleMap z R).comp_continuousOn continuousOn_snd).sub continuousOn_fst
· rintro ⟨a, b⟩ ⟨ha, -⟩
have ha2 : a ∈ ball z R := closedBall_subset_ball hr ha
exact sub_ne_zero.2 (circleMap_ne_mem_bal... | 6 | 403.428793 | 2 | 1.777778 | 9 | 1,880 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 71 | 77 | theorem reflects_epi_of_reflectsColimit {X Y : C} (f : X ⟶ Y) [ReflectsColimit (span f f) F]
[Epi (F.map f)] : Epi f := by |
have := PushoutCocone.isColimitMkIdId (F.map f)
simp_rw [← F.map_id] at this
apply
PushoutCocone.epi_of_isColimitMkIdId _
(isColimitOfIsColimitPushoutCoconeMap F _ this)
| 5 | 148.413159 | 2 | 1.25 | 4 | 1,309 |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 296 | 322 | theorem measure_preimage_lt_top_of_memℒp (hp_pos : p ≠ 0) (hp_ne_top : p ≠ ∞) (f : α →ₛ E)
(hf : Memℒp f p μ) (y : E) (hy_ne : y ≠ 0) : μ (f ⁻¹' {y}) < ∞ := by |
have hp_pos_real : 0 < p.toReal := ENNReal.toReal_pos hp_pos hp_ne_top
have hf_snorm := Memℒp.snorm_lt_top hf
rw [snorm_eq_snorm' hp_pos hp_ne_top, f.snorm'_eq, ←
@ENNReal.lt_rpow_one_div_iff _ _ (1 / p.toReal) (by simp [hp_pos_real]),
@ENNReal.top_rpow_of_pos (1 / (1 / p.toReal)) (by simp [hp_pos_real])... | 25 | 72,004,899,337.38586 | 2 | 1.666667 | 6 | 1,781 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 119 | 124 | theorem maxPowDiv_eq_multiplicity {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv n = multiplicity p n := by |
apply multiplicity.unique <| pow_dvd p n
intro h
apply Nat.not_lt.mpr <| le_of_dvd hp hn h
simp
| 4 | 54.59815 | 2 | 1.333333 | 6 | 1,377 |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 106 | 113 | theorem Duplicate.mono_sublist {l' : List α} (hx : x ∈+ l) (h : l <+ l') : x ∈+ l' := by |
induction' h with l₁ l₂ y _ IH l₁ l₂ y h IH
· exact hx
· exact (IH hx).duplicate_cons _
· rw [duplicate_cons_iff] at hx ⊢
rcases hx with (⟨rfl, hx⟩ | hx)
· simp [h.subset hx]
· simp [IH hx]
| 7 | 1,096.633158 | 2 | 0.818182 | 11 | 719 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
#align_import ring_theory.localization.module from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open nonZ... | Mathlib/RingTheory/Localization/Module.lean | 56 | 71 | theorem LinearIndependent.of_isLocalizedModule {ι : Type*} {v : ι → M}
(hv : LinearIndependent R v) : LinearIndependent Rₛ (f ∘ v) := by |
rw [linearIndependent_iff'] at hv ⊢
intro t g hg i hi
choose! a g' hg' using IsLocalization.exist_integer_multiples S t g
have h0 : f (∑ i ∈ t, g' i • v i) = 0 := by
apply_fun ((a : R) • ·) at hg
rw [smul_zero, Finset.smul_sum] at hg
rw [map_sum, ← hg]
refine Finset.sum_congr rfl fun i hi => ?_... | 14 | 1,202,604.284165 | 2 | 1.5 | 2 | 1,599 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousFunction.Polynomial
import Mathlib.Topology.ContinuousFunction.Compact
#align_import analysis.special_functions.bernstein from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba14... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 117 | 136 | theorem variance {n : ℕ} (h : 0 < (n : ℝ)) (x : I) :
(∑ k : Fin (n + 1), (x - k/ₙ : ℝ) ^ 2 * bernstein n k x) = (x : ℝ) * (1 - x) / n := by |
have h' : (n : ℝ) ≠ 0 := ne_of_gt h
apply_fun fun x : ℝ => x * n using GroupWithZero.mul_right_injective h'
apply_fun fun x : ℝ => x * n using GroupWithZero.mul_right_injective h'
dsimp
conv_lhs => simp only [Finset.sum_mul, z]
conv_rhs => rw [div_mul_cancel₀ _ h']
have := bernsteinPolynomial.variance ℝ ... | 18 | 65,659,969.137331 | 2 | 1.75 | 4 | 1,857 |
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine Matrix
open Set
universe u₁ u₂ u₃ u₄
variable {ι : Type u₁} {k : Type... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 61 | 76 | theorem affineIndependent_of_toMatrix_right_inv [Fintype ι] [Finite ι'] [DecidableEq ι']
(p : ι' → P) {A : Matrix ι ι' k} (hA : b.toMatrix p * A = 1) : AffineIndependent k p := by |
cases nonempty_fintype ι'
rw [affineIndependent_iff_eq_of_fintype_affineCombination_eq]
intro w₁ w₂ hw₁ hw₂ hweq
have hweq' : w₁ ᵥ* b.toMatrix p = w₂ ᵥ* b.toMatrix p := by
ext j
change (∑ i, w₁ i • b.coord j (p i)) = ∑ i, w₂ i • b.coord j (p i)
-- Porting note: Added `u` because `∘` was causing tro... | 14 | 1,202,604.284165 | 2 | 1.428571 | 7 | 1,512 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f... | Mathlib/Algebra/Polynomial/GroupRingAction.lean | 31 | 39 | theorem smul_eq_map [MulSemiringAction M R] (m : M) :
HSMul.hSMul m = map (MulSemiringAction.toRingHom M R m) := by |
suffices DistribMulAction.toAddMonoidHom R[X] m =
(mapRingHom (MulSemiringAction.toRingHom M R m)).toAddMonoidHom by
ext1 r
exact DFunLike.congr_fun this r
ext n r : 2
change m • monomial n r = map (MulSemiringAction.toRingHom M R m) (monomial n r)
rw [Polynomial.map_monomial, Polynomial.smul_mon... | 7 | 1,096.633158 | 2 | 0.666667 | 3 | 603 |
import Mathlib.Algebra.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#align_import field_theory.is_alg_closed.algebraic_closure from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
univ... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 138 | 148 | theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by |
let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
rw [← hp]
induction p using MvPolynomial.induction_on generalizing z with
| h_C => exact isIntegral_algebraMap
| h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
| h_X p f ih =>
refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _)... | 10 | 22,026.465795 | 2 | 1.666667 | 3 | 1,763 |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 129 | 138 | theorem nhds_mkOfNhds_single [DecidableEq α] {a₀ : α} {l : Filter α} (h : pure a₀ ≤ l) (b : α) :
@nhds α (TopologicalSpace.mkOfNhds (update pure a₀ l)) b =
(update pure a₀ l : α → Filter α) b := by |
refine nhds_mkOfNhds _ _ (le_update_iff.mpr ⟨h, fun _ _ => le_rfl⟩) fun a s hs => ?_
rcases eq_or_ne a a₀ with (rfl | ha)
· filter_upwards [hs] with b hb
rcases eq_or_ne b a with (rfl | hb)
· exact hs
· rwa [update_noteq hb]
· simpa only [update_noteq ha, mem_pure, eventually_pure] using hs
| 7 | 1,096.633158 | 2 | 2 | 3 | 2,445 |
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open sco... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 71 | 103 | theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ := by |
let f : Module.Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ _)
let e : (index K) ≃ Module.Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
let M := (mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)
let N := Alg... | 30 | 10,686,474,581,524.463 | 2 | 1.6 | 5 | 1,738 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 76 | 82 | theorem bernoulli_eval_zero (n : ℕ) : (bernoulli n).eval 0 = _root_.bernoulli n := by |
rw [bernoulli, eval_finset_sum, sum_range_succ]
have : ∑ x ∈ range n, _root_.bernoulli x * n.choose x * 0 ^ (n - x) = 0 := by
apply sum_eq_zero fun x hx => _
intros x hx
simp [tsub_eq_zero_iff_le, mem_range.1 hx]
simp [this]
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,715 |
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 | 178 | 185 | theorem degrees_add_of_disjoint [DecidableEq σ] {p q : MvPolynomial σ R}
(h : Multiset.Disjoint p.degrees q.degrees) : (p + q).degrees = p.degrees ∪ q.degrees := by |
apply le_antisymm
· apply degrees_add
· apply Multiset.union_le
· apply le_degrees_add h
· rw [add_comm]
apply le_degrees_add h.symm
| 6 | 403.428793 | 2 | 0.846154 | 13 | 743 |
import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
import Mathlib.Topology.Homotopy.Contractible
import Mathlib.CategoryTheory.PUnit
import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit
#align_import algebraic_topology.fundamental_groupoid.simply_connected from "leanprover-community/mathlib"@"3834... | Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean | 42 | 48 | theorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :
SimplyConnectedSpace X ↔
Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) := by |
simp only [simply_connected_def, equiv_punit_iff_unique,
FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]
intros
exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩
| 4 | 54.59815 | 2 | 2 | 1 | 2,241 |
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]
| 4 | 54.59815 | 2 | 1.642857 | 14 | 1,752 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | Mathlib/Geometry/Euclidean/Basic.lean | 78 | 87 | theorem inner_weightedVSub {ι₁ : Type*} {s₁ : Finset ι₁} {w₁ : ι₁ → ℝ} (p₁ : ι₁ → P)
(h₁ : ∑ i ∈ s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : Finset ι₂} {w₂ : ι₂ → ℝ} (p₂ : ι₂ → P)
(h₂ : ∑ i ∈ s₂, w₂ i = 0) :
⟪s₁.weightedVSub p₁ w₁, s₂.weightedVSub p₂ w₂⟫ =
(-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (dist (p₁ i₁) (p... |
rw [Finset.weightedVSub_apply, Finset.weightedVSub_apply,
inner_sum_smul_sum_smul_of_sum_eq_zero _ h₁ _ h₂]
simp_rw [vsub_sub_vsub_cancel_right]
rcongr (i₁ i₂) <;> rw [dist_eq_norm_vsub V (p₁ i₁) (p₂ i₂)]
| 4 | 54.59815 | 2 | 1.4 | 5 | 1,493 |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 88 | 93 | theorem exists_finset (x : M ⊗[R] N) :
∃ S : Finset (M × N), x = S.sum fun i ↦ i.1 ⊗ₜ[R] i.2 := by |
obtain ⟨S, h⟩ := exists_finsupp_left x
use S.graph
rw [h, Finsupp.sum]
apply Finset.sum_nbij' (fun m ↦ ⟨m, S m⟩) Prod.fst <;> simp
| 4 | 54.59815 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 112 | 116 | theorem base_mul_mem_lifts {p : S[X]} (r : R) (hp : p ∈ lifts f) : C (f r) * p ∈ lifts f := by |
simp only [lifts, RingHom.mem_rangeS] at hp ⊢
obtain ⟨p₁, rfl⟩ := hp
use C r * p₁
simp only [coe_mapRingHom, map_C, map_mul]
| 4 | 54.59815 | 2 | 1 | 13 | 1,139 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 59 | 69 | theorem sq_of_gcd_eq_one {a b c : ℤ} (h : Int.gcd a b = 1) (heq : a * b = c ^ 2) :
∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 := by |
have h' : IsUnit (GCDMonoid.gcd a b) := by
rw [← coe_gcd, h, Int.ofNat_one]
exact isUnit_one
obtain ⟨d, ⟨u, hu⟩⟩ := exists_associated_pow_of_mul_eq_pow h' heq
use d
rw [← hu]
cases' Int.units_eq_one_or u with hu' hu' <;>
· rw [hu']
simp
| 9 | 8,103.083928 | 2 | 1.153846 | 13 | 1,227 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 125 | 135 | theorem _root_.IsIntegrallyClosed.minpoly.unique {s : S} {P : R[X]} (hmo : P.Monic)
(hP : Polynomial.aeval s P = 0)
(Pmin : ∀ Q : R[X], Q.Monic → Polynomial.aeval s Q = 0 → degree P ≤ degree Q) :
P = minpoly R s := by |
have hs : IsIntegral R s := ⟨P, hmo, hP⟩
symm; apply eq_of_sub_eq_zero
by_contra hnz
refine IsIntegrallyClosed.degree_le_of_ne_zero hs hnz (by simp [hP]) |>.not_lt ?_
refine degree_sub_lt ?_ (ne_zero hs) ?_
· exact le_antisymm (min R s hmo hP) (Pmin (minpoly R s) (monic hs) (aeval R s))
· rw [(monic hs).... | 7 | 1,096.633158 | 2 | 1.571429 | 7 | 1,708 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
#align_import analysis.special_functions.trigonometric.euler_sine_prod from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Real Topology
open Real Set Filter intervalIntegra... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 49 | 56 | theorem antideriv_sin_comp_const_mul (hz : z ≠ 0) (x : ℝ) :
HasDerivAt (fun y : ℝ => -Complex.cos (2 * z * y) / (2 * z)) (Complex.sin (2 * z * x)) x := by |
have a : HasDerivAt (fun y : ℂ => y * (2 * z)) _ x := hasDerivAt_mul_const _
have b : HasDerivAt (fun y : ℂ => Complex.cos (y * (2 * z))) _ x :=
HasDerivAt.comp (x : ℂ) (Complex.hasDerivAt_cos (x * (2 * z))) a
have c := (b.comp_ofReal.div_const (2 * z)).neg
field_simp at c; simp only [fun y => mul_comm y (... | 6 | 403.428793 | 2 | 2 | 5 | 2,257 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 124 | 129 | theorem fold_insert_idem [DecidableEq α] [hi : Std.IdempotentOp op] :
(insert a s).fold op b f = f a * s.fold op b f := by |
by_cases h : a ∈ s
· rw [← insert_erase h]
simp [← ha.assoc, hi.idempotent]
· apply fold_insert h
| 4 | 54.59815 | 2 | 0.909091 | 11 | 789 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 56 | 61 | theorem wittPolyProd_vars (n : ℕ) : (wittPolyProd p n).vars ⊆ univ ×ˢ range (n + 1) := by |
rw [wittPolyProd]
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_ <;>
· refine Subset.trans (vars_rename _ _) ?_
simp [wittPolynomial_vars, image_subset_iff]
| 5 | 148.413159 | 2 | 1.833333 | 6 | 1,912 |
import Mathlib.Topology.StoneCech
import Mathlib.Topology.Algebra.Semigroup
import Mathlib.Data.Stream.Init
#align_import combinatorics.hindman from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Filter
@[to_additive
"Addition of ultrafilters given by `∀ᶠ m in U+V, p m ↔ ∀ᶠ... | Mathlib/Combinatorics/Hindman.lean | 119 | 131 | theorem FP.mul {M} [Semigroup M] {a : Stream' M} {m : M} (hm : m ∈ FP a) :
∃ n, ∀ m' ∈ FP (a.drop n), m * m' ∈ FP a := by |
induction' hm with a a m hm ih a m hm ih
· exact ⟨1, fun m hm => FP.cons a m hm⟩
· cases' ih with n hn
use n + 1
intro m' hm'
exact FP.tail _ _ (hn _ hm')
· cases' ih with n hn
use n + 1
intro m' hm'
rw [mul_assoc]
exact FP.cons _ _ (hn _ hm')
| 11 | 59,874.141715 | 2 | 2 | 2 | 2,280 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 132 | 192 | theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈ p) :
p ⊓ ⨆ μ, ⨆ k, f.genEigenspace μ k = ⨆ μ, ⨆ k, p ⊓ f.genEigenspace μ k := by |
simp_rw [← (f.genEigenspace _).mono.directed_le.inf_iSup_eq]
refine le_antisymm (fun m hm ↦ ?_)
(le_inf_iff.mpr ⟨iSup_le fun μ ↦ inf_le_left, iSup_mono fun μ ↦ inf_le_right⟩)
classical
obtain ⟨hm₀ : m ∈ p, hm₁ : m ∈ ⨆ μ, ⨆ k, f.genEigenspace μ k⟩ := hm
obtain ⟨m, hm₂, rfl⟩ := (mem_iSup_iff_exists_finsupp... | 59 | 42,012,104,037,905,144,000,000,000 | 2 | 1.25 | 4 | 1,338 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.