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.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 163 | 180 | theorem IsSymmetric.inner_map_polarization {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (x y : E) :
⟪T x, y⟫ =
(⟪T (x + y), x + y⟫ - ⟪T (x - y), x - y⟫ - I * ⟪T (x + (I : 𝕜) • y), x + (I : 𝕜) • y⟫ +
I * ⟪T (x - (I : 𝕜) • y), x - (I : 𝕜) • y⟫) /
4 := by |
rcases@I_mul_I_ax 𝕜 _ with (h | h)
· simp_rw [h, zero_mul, sub_zero, add_zero, map_add, map_sub, inner_add_left,
inner_add_right, inner_sub_left, inner_sub_right, hT x, ← inner_conj_symm x (T y)]
suffices (re ⟪T y, x⟫ : 𝕜) = ⟪T y, x⟫ by
rw [conj_eq_iff_re.mpr this]
ring
rw [← re_add_im ... | 13 | 442,413.392009 | 2 | 1.5 | 6 | 1,614 |
import Mathlib.Data.List.Basic
import Mathlib.Data.Sigma.Basic
#align_import data.list.prod_sigma from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
variable {α β : Type*}
namespace List
@[simp]
theorem nil_product (l : List β) : (@nil α) ×ˢ l = [] :=
rfl
#align list.nil_product... | Mathlib/Data/List/ProdSigma.lean | 45 | 48 | theorem mem_product {l₁ : List α} {l₂ : List β} {a : α} {b : β} :
(a, b) ∈ l₁ ×ˢ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂ := by |
simp_all [SProd.sprod, product, mem_bind, mem_map, Prod.ext_iff, exists_prop, and_left_comm,
exists_and_left, exists_eq_left, exists_eq_right]
| 2 | 7.389056 | 1 | 1.25 | 4 | 1,323 |
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 64 | 66 | theorem toNat_strictMonoOn : StrictMonoOn toNat (Iio ℵ₀) := by |
simp only [← range_natCast, StrictMonoOn, forall_mem_range, toNat_natCast, Nat.cast_lt]
exact fun _ _ ↦ id
| 2 | 7.389056 | 1 | 0.4 | 5 | 391 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.Real.Sqrt
import Mathlib.Tactic.Polyrith
#align_import algebra.star.chsh from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
universe u
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure Is... | Mathlib/Algebra/Star/CHSH.lean | 121 | 138 | theorem CHSH_inequality_of_comm [OrderedCommRing R] [StarRing R] [StarOrderedRing R] [Algebra ℝ R]
[OrderedSMul ℝ R] (A₀ A₁ B₀ B₁ : R) (T : IsCHSHTuple A₀ A₁ B₀ B₁) :
A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁ ≤ 2 := by |
let P := 2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁
have i₁ : 0 ≤ P := by
have idem : P * P = 4 * P := CHSH_id T.A₀_inv T.A₁_inv T.B₀_inv T.B₁_inv
have idem' : P = (1 / 4 : ℝ) • (P * P) := by
have h : 4 * P = (4 : ℝ) • P := by simp [Algebra.smul_def]
rw [idem, h, ← mul_smul]
norm_num
h... | 15 | 3,269,017.372472 | 2 | 1.75 | 4 | 1,872 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 106 | 112 | theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by |
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
| 5 | 148.413159 | 2 | 1.6 | 10 | 1,740 |
import Batteries.Data.Nat.Gcd
import Batteries.Data.Int.DivMod
import Batteries.Lean.Float
-- `Rat` is not tagged with the `ext` attribute, since this is more often than not undesirable
structure Rat where
mk' ::
num : Int
den : Nat := 1
den_nz : den ≠ 0 := by decide
reduced : num.natAbs.C... | .lake/packages/batteries/Batteries/Data/Rat/Basic.lean | 60 | 66 | theorem Rat.normalize.reduced {num : Int} {den g : Nat} (den_nz : den ≠ 0)
(e : g = num.natAbs.gcd den) : (num.div g).natAbs.Coprime (den / g) :=
have : Int.natAbs (num.div ↑g) = num.natAbs / g := by |
match num, num.eq_nat_or_neg with
| _, ⟨_, .inl rfl⟩ => rfl
| _, ⟨_, .inr rfl⟩ => rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg]; rfl
this ▸ e ▸ Nat.coprime_div_gcd_div_gcd (Nat.gcd_pos_of_pos_right _ (Nat.pos_of_ne_zero den_nz))
| 4 | 54.59815 | 2 | 2 | 1 | 2,079 |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section TopologicalGroup
variable [CommGroup α] [TopologicalSpace α] [TopologicalGroup α]
variable {f g : β → α} {a a₁... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 50 | 53 | theorem HasProd.div (hf : HasProd f a₁) (hg : HasProd g a₂) :
HasProd (fun b ↦ f b / g b) (a₁ / a₂) := by |
simp only [div_eq_mul_inv]
exact hf.mul hg.inv
| 2 | 7.389056 | 1 | 0.833333 | 6 | 738 |
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 | 85 | 90 | theorem norm_approxOn_zero_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
(h₀ : (0 : E) ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
‖approxOn f hf s 0 h₀ n x‖ ≤ ‖f x‖ + ‖f x‖ := by |
have := edist_approxOn_y0_le hf h₀ x n
simp [edist_comm (0 : E), edist_eq_coe_nnnorm] at this
exact mod_cast this
| 3 | 20.085537 | 1 | 1.666667 | 6 | 1,781 |
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 | 67 | 77 | theorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :
↑u ≤ 𝓝 x ↔ x = joinM u := by |
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro h a ⟨xi, a, rfl⟩
exact h ... | 9 | 8,103.083928 | 2 | 2 | 5 | 2,361 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 97 | 99 | theorem int_valuation_ne_zero (x : R) (hx : x ≠ 0) : v.intValuationDef x ≠ 0 := by |
rw [intValuationDef, if_neg hx]
exact WithZero.coe_ne_zero
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,836 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 114 | 116 | theorem gauge_empty : gauge (∅ : Set E) = 0 := by |
ext
simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
| 2 | 7.389056 | 1 | 1.090909 | 11 | 1,185 |
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 | 101 | 104 | theorem sign_mul_pos_of_ne_zero (r : ℝ) (hr : r ≠ 0) : 0 < sign r * r := by |
refine lt_of_le_of_ne (sign_mul_nonneg r) fun h => hr ?_
have hs0 := (zero_eq_mul.mp h).resolve_right hr
exact sign_eq_zero_iff.mp hs0
| 3 | 20.085537 | 1 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 126 | 136 | theorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' := by |
dsimp [destruct]
induction' f0 : s.1 0 with a' <;> intro h
· injection h with h'
rw [← h']
cases' s with f al
apply Subtype.eq
dsimp [think, tail]
rw [← f0]
exact (Stream'.eta f).symm
· contradiction
| 10 | 22,026.465795 | 2 | 1.333333 | 3 | 1,401 |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 77 | 80 | theorem comap_comp (f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R)
(g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R) : comap (g.comp f) = comap f ∘ comap g := by |
funext x
exact comap_comp_apply _ _ _
| 2 | 7.389056 | 1 | 1.166667 | 6 | 1,243 |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 107 | 109 | theorem exists_irreducible : ∃ ϖ : R, Irreducible ϖ := by |
simp_rw [irreducible_iff_uniformizer]
exact (IsPrincipalIdealRing.principal <| maximalIdeal R).principal
| 2 | 7.389056 | 1 | 1.666667 | 6 | 1,772 |
import Mathlib.GroupTheory.GroupAction.Prod
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Cast.Basic
assert_not_exists DenselyOrdered
variable {M : Type*}
class NatPowAssoc (M : Type*) [MulOneClass M] [Pow M ℕ] : Prop where
protected npow_add : ∀ (k n: ℕ) (x : M), x ^ (k + n) = x ^ k * x ^ n
... | Mathlib/Algebra/Group/NatPowAssoc.lean | 72 | 75 | theorem npow_mul (x : M) (m n : ℕ) : x ^ (m * n) = (x ^ m) ^ n := by |
induction n with
| zero => rw [npow_zero, Nat.mul_zero, npow_zero]
| succ n ih => rw [mul_add, npow_add, ih, mul_one, npow_add, npow_one]
| 3 | 20.085537 | 1 | 0.5 | 4 | 502 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 249 | 251 | theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by |
ext x
exact exists_congr fun a ↦ and_congr_right fun ha ↦ by rw [h a ha]
| 2 | 7.389056 | 1 | 0.666667 | 15 | 590 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 102 | 105 | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y := by |
let ⟨a, b, H⟩ := H1
rw [← mul_one y, ← H, mul_add, ← mul_assoc, mul_left_comm]
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
| 3 | 20.085537 | 1 | 1.142857 | 7 | 1,219 |
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Filter
variable {ι α β : Type*}
class Bornology (α : Type*) where
cobounded' : Filter α
le_cofinite' : cobounded' ≤ cofinite
#align borno... | Mathlib/Topology/Bornology/Basic.lean | 161 | 163 | theorem isBounded_empty : IsBounded (∅ : Set α) := by |
rw [isBounded_def, compl_empty]
exact univ_mem
| 2 | 7.389056 | 1 | 0.75 | 4 | 665 |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 100 | 102 | theorem eqToIso_refl {n : ℕ} (h : n = n) : X.eqToIso h = Iso.refl _ := by |
ext
simp [eqToIso]
| 2 | 7.389056 | 1 | 1 | 8 | 1,050 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 106 | 116 | theorem cramer_transpose_row_self (i : n) : Aᵀ.cramer (A i) = Pi.single i A.det := by |
ext j
rw [cramer_apply, Pi.single_apply]
split_ifs with h
· -- i = j: this entry should be `A.det`
subst h
simp only [updateColumn_transpose, det_transpose, updateRow_eq_self]
· -- i ≠ j: this entry should be 0
rw [updateColumn_transpose, det_transpose]
apply det_zero_of_row_eq h
rw [upda... | 10 | 22,026.465795 | 2 | 1.222222 | 9 | 1,297 |
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 | 118 | 120 | theorem degrees_zero : degrees (0 : MvPolynomial σ R) = 0 := by |
rw [← C_0]
exact degrees_C 0
| 2 | 7.389056 | 1 | 0.846154 | 13 | 743 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 69 | 73 | theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by |
by_cases hx : gauge t x = 0
· simp [gaugeRescale, hx, gauge_nonneg]
· exact (gauge_gaugeRescale' s hx).le
| 3 | 20.085537 | 1 | 1.285714 | 7 | 1,358 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 161 | 164 | theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x := by |
have H := f.sum_eq_one hx
contrapose! H
simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
| 3 | 20.085537 | 1 | 1.3 | 10 | 1,365 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 62 | 81 | theorem lieIdeal_oper_eq_linear_span :
(↑⁅I, N⁆ : Submodule R M) =
Submodule.span R { m | ∃ (x : I) (n : N), ⁅(x : L), (n : M)⁆ = m } := by |
apply le_antisymm
· let s := { m : M | ∃ (x : ↥I) (n : ↥N), ⁅(x : L), (n : M)⁆ = m }
have aux : ∀ (y : L), ∀ m' ∈ Submodule.span R s, ⁅y, m'⁆ ∈ Submodule.span R s := by
intro y m' hm'
refine Submodule.span_induction (R := R) (M := M) (s := s)
(p := fun m' ↦ ⁅y, m'⁆ ∈ Submodule.span R s) hm'... | 17 | 24,154,952.753575 | 2 | 1.285714 | 7 | 1,363 |
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 130 | 136 | theorem splitUpper_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, update I.lower i x j < I.upper j :=
(forall_update_iff I.lower fun j y => y < I.upper j).2
⟨h.2, fun j _ => I.lower_lt_upper _⟩) :
I.splitUpper i x = (⟨update I.lower i x, I.upper, h'⟩ : Box ι) := by |
simp (config := { unfoldPartialApp := true }) only [splitUpper, mk'_eq_coe, max_eq_left h.1.le,
update, and_self]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,269 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 78 | 89 | theorem eventually_forall_le_dist_cast_div (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n) := by |
have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,076 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 39 | 44 | theorem Coprime.gcd_mul_left_cancel (m : Nat) (H : Coprime k n) : gcd (k * m) n = gcd m n :=
have H1 : Coprime (gcd (k * m) n) k := by |
rw [Coprime, Nat.gcd_assoc, H.symm.gcd_eq_one, gcd_one_right]
Nat.dvd_antisymm
(dvd_gcd (H1.dvd_of_dvd_mul_left (gcd_dvd_left _ _)) (gcd_dvd_right _ _))
(gcd_dvd_gcd_mul_left _ _ _)
| 4 | 54.59815 | 2 | 1 | 9 | 1,124 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 131 | 133 | theorem rdropWhile_prefix : l.rdropWhile p <+: l := by |
rw [← reverse_suffix, rdropWhile, reverse_reverse]
exact dropWhile_suffix _
| 2 | 7.389056 | 1 | 0.631579 | 19 | 550 |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace CochainComplex
@[simp... | Mathlib/Algebra/Homology/Augment.lean | 325 | 328 | theorem cochainComplex_d_succ_succ_zero (C : CochainComplex V ℕ) (i : ℕ) : C.d 0 (i + 2) = 0 := by |
rw [C.shape]
simp only [ComplexShape.up_Rel, zero_add]
exact (Nat.one_lt_succ_succ _).ne
| 3 | 20.085537 | 1 | 0.666667 | 3 | 587 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 111 | 116 | theorem degree_hermite (n : ℕ) : (hermite n).degree = n := by |
rw [degree_eq_of_le_of_coeff_ne_zero]
· simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
rintro m hnm
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n]
| 5 | 148.413159 | 2 | 1.1 | 10 | 1,191 |
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 118 | 135 | theorem lmarginal_union (f : (∀ i, π i) → ℝ≥0∞) (hf : Measurable f)
(hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_s, ∫⋯∫⁻_t, f ∂μ ∂μ := by |
ext1 x
let e := MeasurableEquiv.piFinsetUnion π hst
calc (∫⋯∫⁻_s ∪ t, f ∂μ) x
= ∫⁻ (y : (i : ↥(s ∪ t)) → π i), f (updateFinset x (s ∪ t) y)
∂.pi fun i' : ↥(s ∪ t) ↦ μ i' := rfl
_ = ∫⁻ (y : ((i : s) → π i) × ((j : t) → π j)), f (updateFinset x (s ∪ t) _)
∂(Measure.pi fun i : s ↦ μ i)... | 16 | 8,886,110.520508 | 2 | 1.333333 | 6 | 1,378 |
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
#align_import analysis.normed_space.M_structure from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
variable (X : Type*) [NormedAddCommGroup X]
... | Mathlib/Analysis/NormedSpace/MStructure.lean | 147 | 162 | theorem mul [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
IsLprojection X (P * Q) := by |
refine ⟨IsIdempotentElem.mul_of_commute (h₁.commute h₂) h₁.proj h₂.proj, ?_⟩
intro x
refine le_antisymm ?_ ?_
· calc
‖x‖ = ‖(P * Q) • x + (x - (P * Q) • x)‖ := by rw [add_sub_cancel ((P * Q) • x) x]
_ ≤ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖ := by apply norm_add_le
_ = ‖(P * Q) • x‖ + ‖(1 - P * Q)... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 2,170 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 140 | 142 | theorem count_le_card (hp : (setOf p).Finite) (n : ℕ) : count p n ≤ hp.toFinset.card := by |
rw [count_eq_card_filter_range]
exact Finset.card_mono fun x hx ↦ hp.mem_toFinset.2 (mem_filter.1 hx).2
| 2 | 7.389056 | 1 | 0.642857 | 14 | 554 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 170 | 174 | theorem bind_hcongr {β' : Type v} {m : Multiset α} {f : α → Multiset β} {f' : α → Multiset β'}
(h : β = β') (hf : ∀ a ∈ m, HEq (f a) (f' a)) : HEq (bind m f) (bind m f') := by |
subst h
simp only [heq_eq_eq] at hf
simp [bind_congr hf]
| 3 | 20.085537 | 1 | 0.384615 | 13 | 382 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 376 | 380 | theorem HasStrictFDerivAt.mul (hc : HasStrictFDerivAt c c' x) (hd : HasStrictFDerivAt d d' x) :
HasStrictFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x := by |
convert hc.mul' hd
ext z
apply mul_comm
| 3 | 20.085537 | 1 | 0.375 | 8 | 376 |
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 87 | 89 | theorem add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by |
rw [SModEq.def] at hxy₁ hxy₂ ⊢
simp_rw [Quotient.mk_add, hxy₁, hxy₂]
| 2 | 7.389056 | 1 | 0.714286 | 7 | 645 |
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 | 120 | 125 | theorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
(hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by |
apply IsCyclic.ext rfl (fun ζ ↦ ?_)
specialize hc ζ
suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this
rw [← toFun_spec g ζ, hc]
| 4 | 54.59815 | 2 | 1 | 4 | 1,078 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 109 | 114 | theorem dependent_pair_iff_eq (u v : ℙ K V) : Dependent ![u, v] ↔ u = v := by |
rw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2,
Function.comp_apply, Matrix.cons_val_one, Matrix.head_cons, Ne]
simp only [Matrix.cons_val_zero, not_and, not_forall, Classical.not_not, Function.comp_apply,
← mk_eq_mk_iff' K _ _ (rep_nonzero u) (rep_nonzero v), mk_rep, Classical.... | 5 | 148.413159 | 2 | 1.142857 | 7 | 1,223 |
import Mathlib.Algebra.Group.Fin
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Fintype.Card
#align_import data.zmod.defs from "leanprover-community/mathlib"@"3a2b5524a138b5d0b818b858b516d4ac8a484b03"
def ZMod : ℕ → Type
| 0 => ℤ
| n + 1 => Fin (n + 1)
#align zmod ZMod
insta... | Mathlib/Data/ZMod/Defs.lean | 124 | 127 | theorem card (n : ℕ) [Fintype (ZMod n)] : Fintype.card (ZMod n) = n := by |
cases n with
| zero => exact (not_finite (ZMod 0)).elim
| succ n => convert Fintype.card_fin (n + 1) using 2
| 3 | 20.085537 | 1 | 1 | 1 | 823 |
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 96 | 100 | theorem toDualRight_symm_toDualLeft (x : M) :
p.toDualRight.symm.dualMap (p.toDualLeft x) = Dual.eval R M x := by |
ext f
simp only [LinearEquiv.dualMap_apply, Dual.eval_apply]
exact toDualLeft_of_toDualRight_symm p x f
| 3 | 20.085537 | 1 | 1 | 6 | 845 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 103 | 114 | theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
(ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
Continuous (gaugeRescale s t) := by |
have hta : Absorbent ℝ t := absorbent_nhds_zero ht₀
refine continuous_iff_continuousAt.2 fun x ↦ ?_
rcases eq_or_ne x 0 with rfl | hx
· rw [ContinuousAt, gaugeRescale_zero]
nth_rewrite 2 [← comap_gauge_nhds_zero htb ht₀]
simp only [tendsto_comap_iff, (· ∘ ·), gauge_gaugeRescale _ hta htb]
exact ten... | 9 | 8,103.083928 | 2 | 1.285714 | 7 | 1,358 |
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]
| 3 | 20.085537 | 1 | 1 | 5 | 1,110 |
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 | 82 | 97 | theorem coeff_mirror (n : ℕ) :
p.mirror.coeff n = p.coeff (revAt (p.natDegree + p.natTrailingDegree) n) := by |
by_cases h2 : p.natDegree < n
· rw [coeff_eq_zero_of_natDegree_lt (by rwa [mirror_natDegree])]
by_cases h1 : n ≤ p.natDegree + p.natTrailingDegree
· rw [revAt_le h1, coeff_eq_zero_of_lt_natTrailingDegree]
exact (tsub_lt_iff_left h1).mpr (Nat.add_lt_add_right h2 _)
· rw [← revAtFun_eq, revAtFun, i... | 14 | 1,202,604.284165 | 2 | 1.571429 | 7 | 1,704 |
import Mathlib.Data.Matroid.Restrict
variable {α : Type*} {M : Matroid α} {E B I X R J : Set α}
namespace Matroid
open Set
section EmptyOn
def emptyOn (α : Type*) : Matroid α where
E := ∅
Base := (· = ∅)
Indep := (· = ∅)
indep_iff' := by simp [subset_empty_iff]
exists_base := ⟨∅, rfl⟩
base_exchange... | Mathlib/Data/Matroid/Constructions.lean | 67 | 69 | theorem eq_emptyOn_or_nonempty (M : Matroid α) : M = emptyOn α ∨ Matroid.Nonempty M := by |
rw [← ground_eq_empty_iff]
exact M.E.eq_empty_or_nonempty.elim Or.inl (fun h ↦ Or.inr ⟨h⟩)
| 2 | 7.389056 | 1 | 1 | 3 | 1,145 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.RingTheory.Derivation.Basic
noncomputable section
namespace Polynomial
section CommSemiring
variable {R A : Type*} [CommSemiring R]
@[simps]
def derivative' : D... | Mathlib/Algebra/Polynomial/Derivation.lean | 43 | 46 | theorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :
C a • D f = a • D f := by |
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul]
| 2 | 7.389056 | 1 | 0.5 | 2 | 479 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 76 | 79 | theorem one_div_lt_one_div {n m : ℕ} (h : n < m) : 1 / ((m : α) + 1) < 1 / ((n : α) + 1) := by |
refine one_div_lt_one_div_of_lt ?_ ?_
· exact Nat.cast_add_one_pos _
· simpa
| 3 | 20.085537 | 1 | 1.166667 | 6 | 1,240 |
set_option autoImplicit true
namespace Array
@[simp]
theorem extract_eq_nil_of_start_eq_end {a : Array α} :
a.extract i i = #[] := by
refine extract_empty_of_stop_le_start a ?h
exact Nat.le_refl i
theorem extract_append_left {a b : Array α} {i j : Nat} (h : j ≤ a.size) :
(a ++ b).extract i j = a.extrac... | Mathlib/Data/Array/ExtractLemmas.lean | 29 | 38 | theorem extract_append_right {a b : Array α} {i j : Nat} (h : a.size ≤ i) :
(a ++ b).extract i j = b.extract (i - a.size) (j - a.size) := by |
apply ext
· rw [size_extract, size_extract, size_append]
omega
· intro k hi h2
rw [get_extract, get_extract,
get_append_right (show size a ≤ i + k by omega)]
congr
omega
| 8 | 2,980.957987 | 2 | 1.4 | 5 | 1,495 |
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.Topology.Algebra.ConstMulAction
#align_import dynamics.minimal from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Pointwise
class AddAction.IsMinimal (M α : Type*) [AddMonoid M] [TopologicalSpace α] [AddAction M α] :
... | Mathlib/Dynamics/Minimal.lean | 119 | 126 | theorem isMinimal_iff_closed_smul_invariant [ContinuousConstSMul M α] :
IsMinimal M α ↔ ∀ s : Set α, IsClosed s → (∀ c : M, c • s ⊆ s) → s = ∅ ∨ s = univ := by |
constructor
· intro _ _
exact eq_empty_or_univ_of_smul_invariant_closed M
refine fun H ↦ ⟨fun _ ↦ dense_iff_closure_eq.2 <| (H _ ?_ ?_).resolve_left ?_⟩
exacts [isClosed_closure, fun _ ↦ smul_closure_orbit_subset _ _,
(orbit_nonempty _).closure.ne_empty]
| 6 | 403.428793 | 2 | 2 | 1 | 2,355 |
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 | 64 | 68 | theorem map_invUnitsSub (f : R →+* S) (u : Rˣ) :
map f (invUnitsSub u) = invUnitsSub (Units.map (f : R →* S) u) := by |
ext
simp only [← map_pow, coeff_map, coeff_invUnitsSub, one_divp]
rfl
| 3 | 20.085537 | 1 | 0.857143 | 14 | 748 |
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 | 100 | 103 | theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
ENNReal.ofReal (variance X μ) = evariance X μ := by |
rw [variance, ENNReal.ofReal_toReal]
exact hX.evariance_lt_top.ne
| 2 | 7.389056 | 1 | 1.857143 | 7 | 1,927 |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 117 | 122 | theorem realize_functions_apply₂ {f : L.Functions 2} {t₁ t₂ : L.Term α} {v : α → M} :
(f.apply₂ t₁ t₂).realize v = funMap f ![t₁.realize v, t₂.realize v] := by |
rw [Functions.apply₂, Term.realize]
refine congr rfl (funext (Fin.cases ?_ ?_))
· simp only [Matrix.cons_val_zero]
· simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
| 4 | 54.59815 | 2 | 1.571429 | 7 | 1,709 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 123 | 125 | theorem isAlgebraic_int [Nontrivial R] (n : ℤ) : IsAlgebraic R (n : A) := by |
rw [← _root_.map_intCast (algebraMap R A)]
exact isAlgebraic_algebraMap (Int.cast n)
| 2 | 7.389056 | 1 | 1.125 | 8 | 1,205 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 156 | 158 | theorem inv_bot : (⊥ : Rel α β).inv = (⊥ : Rel β α) := by |
#adaptation_note /-- nightly-2024-03-16: simp was `simp [Bot.bot, inv, flip]` -/
simp [Bot.bot, inv, Function.flip_def]
| 2 | 7.389056 | 1 | 1 | 15 | 904 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 322 | 325 | theorem differentiableOn_euclidean :
DifferentiableOn 𝕜 f t ↔ ∀ i, DifferentiableOn 𝕜 (fun x => f x i) t := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableOn_iff, differentiableOn_pi]
rfl
| 2 | 7.389056 | 1 | 0.833333 | 12 | 734 |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 139 | 141 | theorem reflect_C_mul (f : R[X]) (r : R) (N : ℕ) : reflect N (C r * f) = C r * reflect N f := by |
ext
simp only [coeff_reflect, coeff_C_mul]
| 2 | 7.389056 | 1 | 1 | 12 | 945 |
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 | 82 | 86 | theorem hasPDF_iff_of_aemeasurable {X : Ω → E} {ℙ : Measure Ω}
{μ : Measure E} (hX : AEMeasurable X ℙ) :
HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by |
rw [hasPDF_iff]
simp only [hX, true_and]
| 2 | 7.389056 | 1 | 1.428571 | 7 | 1,513 |
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 | 108 | 110 | theorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n := by |
simp only [padicValNat, dite_eq_right_iff, PartENat.get_eq_iff_eq_coe, Nat.cast_zero,
multiplicity_eq_zero, and_imp, pos_iff_ne_zero, Ne, ← or_iff_not_imp_left]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,377 |
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 63 | 66 | theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a • b) := by |
rcases hb with ⟨b', hb⟩
use a * b'
rw [← hb, (algebraMap R S).map_mul, Algebra.smul_def]
| 3 | 20.085537 | 1 | 1.4 | 5 | 1,499 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 85 | 92 | theorem transpose_dualTensorHom (f : Module.Dual R M) (m : M) :
Dual.transpose (R := R) (dualTensorHom R M M (f ⊗ₜ m)) =
dualTensorHom R _ _ (Dual.eval R M m ⊗ₜ f) := by |
ext f' m'
simp only [Dual.transpose_apply, coe_comp, Function.comp_apply, dualTensorHom_apply,
LinearMap.map_smulₛₗ, RingHom.id_apply, Algebra.id.smul_eq_mul, Dual.eval_apply,
LinearMap.smul_apply]
exact mul_comm _ _
| 5 | 148.413159 | 2 | 1.5 | 6 | 1,604 |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 209 | 213 | theorem set_lintegral_condDistrib_of_measurableSet (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) {t : Set α} (ht : MeasurableSet[mβ.comap X] t) :
∫⁻ a in t, condDistrib Y X μ (X a) s ∂μ = μ (t ∩ Y ⁻¹' s) := by |
obtain ⟨t', ht', rfl⟩ := ht
rw [set_lintegral_preimage_condDistrib hX hY hs ht']
| 2 | 7.389056 | 1 | 0.777778 | 9 | 692 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 51 | 54 | theorem dNext_eq (f : ∀ i j, C.X i ⟶ D.X j) {i i' : ι} (w : c.Rel i i') :
dNext i f = C.d i i' ≫ f i' i := by |
obtain rfl := c.next_eq' w
rfl
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,477 |
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
set_option autoImplicit true
variable {C : Levenshtein.Cost α β δ} [CanonicallyLinearOrderedAddCommMonoid δ]
theorem suffixLevenshtein_minimum_le_levenshtein... | Mathlib/Data/List/EditDistance/Bounds.lean | 94 | 97 | theorem le_levenshtein_append (xs : List α) (ys₁ ys₂) :
∃ xs', xs' <:+ xs ∧ levenshtein C xs' ys₂ ≤ levenshtein C xs (ys₁ ++ ys₂) := by |
simpa [suffixLevenshtein_eq_tails_map, List.minimum_le_coe_iff] using
suffixLevenshtein_minimum_le_levenshtein_append (δ := δ) xs ys₁ ys₂
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,674 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Prod
import Mathlib.Tactic.Common
variable {ι G₁ G₂ : Type*} {G : ι → Type*} [Semigroup G₁] [Semigroup G₂] [∀ i, Semigroup (G i)]
| Mathlib/Algebra/Divisibility/Prod.lean | 16 | 20 | theorem prod_dvd_iff {x y : G₁ × G₂} :
x ∣ y ↔ x.1 ∣ y.1 ∧ x.2 ∣ y.2 := by |
cases x; cases y
simp only [dvd_def, Prod.exists, Prod.mk_mul_mk, Prod.mk.injEq,
exists_and_left, exists_and_right, and_self, true_and]
| 3 | 20.085537 | 1 | 0.5 | 2 | 447 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 122 | 128 | theorem cotangentIdeal_square (I : Ideal R) : I.cotangentIdeal ^ 2 = ⊥ := by |
rw [eq_bot_iff, pow_two I.cotangentIdeal, ← smul_eq_mul]
intro x hx
refine Submodule.smul_induction_on hx ?_ ?_
· rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩; apply (Submodule.Quotient.eq _).mpr _
rw [sub_zero, pow_two]; exact Ideal.mul_mem_mul hx hy
· intro x y hx hy; exact add_mem hx hy
| 6 | 403.428793 | 2 | 1.333333 | 6 | 1,400 |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Function.Iterate
import Mathlib.GroupTheory.Perm.Basic
#align_import dynamics.fixed_points.basic from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Equiv
universe u v
variable {α : Type u} {β : Type v} {f fa g : α → α} {x y :... | Mathlib/Dynamics/FixedPoints/Basic.lean | 97 | 100 | theorem preimage_iterate {s : Set α} (h : IsFixedPt (Set.preimage f) s) (n : ℕ) :
IsFixedPt (Set.preimage f^[n]) s := by |
rw [Set.preimage_iterate_eq]
exact h.iterate n
| 2 | 7.389056 | 1 | 1 | 1 | 875 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 189 | 191 | theorem orderOf_r [NeZero n] (i : ZMod n) : orderOf (r i) = n / Nat.gcd n i.val := by |
conv_lhs => rw [← ZMod.natCast_zmod_val i]
rw [← r_one_pow, orderOf_pow, orderOf_r_one]
| 2 | 7.389056 | 1 | 1.125 | 8 | 1,206 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 64 | 66 | theorem goldConj_mul_gold : ψ * φ = -1 := by |
rw [mul_comm]
exact gold_mul_goldConj
| 2 | 7.389056 | 1 | 0.894737 | 19 | 776 |
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 | 176 | 180 | theorem cast_zero : (cast (0 : ZMod n) : R) = 0 := by |
delta ZMod.cast
cases n
· exact Int.cast_zero
· simp
| 4 | 54.59815 | 2 | 1 | 11 | 900 |
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.CategoryTheory.Linear.Yoneda
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Symmetric
#align_import algebra.category.Module.monoidal.closed from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe ... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean | 88 | 91 | theorem ihom_ev_app (M N : ModuleCat.{u} R) :
(ihom.ev M).app N = TensorProduct.uncurry _ _ _ _ LinearMap.id.flip := by |
apply TensorProduct.ext'
apply ModuleCat.monoidalClosed_uncurry
| 2 | 7.389056 | 1 | 1 | 1 | 929 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 80 | 91 | theorem χ₄_eq_neg_one_pow {n : ℕ} (hn : n % 2 = 1) : χ₄ n = (-1) ^ (n / 2) := by |
rw [χ₄_nat_eq_if_mod_four]
simp only [hn, Nat.one_ne_zero, if_false]
conv_rhs => -- Porting note: was `nth_rw`
arg 2; rw [← Nat.div_add_mod n 4]
enter [1, 1, 1]; rw [(by norm_num : 4 = 2 * 2)]
rw [mul_assoc, add_comm, Nat.add_mul_div_left _ _ (by norm_num : 0 < 2), pow_add, pow_mul,
neg_one_sq, one... | 11 | 59,874.141715 | 2 | 1.25 | 12 | 1,332 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 85 | 91 | theorem toFunLinear_mul_tmul_mul_aux_2 (k : ℕ) (a₁ a₂ : A) (p₁ p₂ : R[X]) :
a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) =
(Finset.antidiagonal k).sum fun x =>
a₁ * (algebraMap R A) (coeff p₁ x.1) * (a₂ * (algebraMap R A) (coeff p₂ x.2)) := by |
simp_rw [mul_assoc, Algebra.commutes, ← Finset.mul_sum, mul_assoc, ← Finset.mul_sum]
congr
simp_rw [Algebra.commutes (coeff p₂ _), coeff_mul, map_sum, RingHom.map_mul]
| 3 | 20.085537 | 1 | 0.833333 | 6 | 728 |
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 | 78 | 90 | theorem nhds_generateFrom {g : Set (Set α)} {a : α} :
@nhds α (generateFrom g) a = ⨅ s ∈ { s | a ∈ s ∧ s ∈ g }, 𝓟 s := by |
letI := generateFrom g
rw [nhds_def]
refine le_antisymm (biInf_mono fun s ⟨as, sg⟩ => ⟨as, .basic _ sg⟩) <| le_iInf₂ ?_
rintro s ⟨ha, hs⟩
induction hs with
| basic _ hs => exact iInf₂_le _ ⟨ha, hs⟩
| univ => exact le_top.trans_eq principal_univ.symm
| inter _ _ _ _ hs ht => exact (le_inf (hs ha.1) (ht ... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,445 |
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 | 83 | 86 | theorem getD_append (l l' : List α) (d : α) (n : ℕ) (h : n < l.length) :
(l ++ l').getD n d = l.getD n d := by |
rw [getD_eq_get _ _ (Nat.lt_of_lt_of_le h (length_append _ _ ▸ Nat.le_add_right _ _)),
get_append _ h, getD_eq_get]
| 2 | 7.389056 | 1 | 1.375 | 8 | 1,471 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 129 | 138 | theorem isCodetecting_op_iff (𝒢 : Set C) : IsCodetecting 𝒢.op ↔ IsDetecting 𝒢 := by |
refine ⟨fun h𝒢 X Y f hf => ?_, fun h𝒢 X Y f hf => ?_⟩
· refine (isIso_op_iff _).1 (h𝒢 _ fun G hG h => ?_)
obtain ⟨t, ht, ht'⟩ := hf (unop G) (Set.mem_op.1 hG) h.unop
exact
⟨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))⟩
· refine (isIso_unop_iff _).1... | 9 | 8,103.083928 | 2 | 1.333333 | 6 | 1,424 |
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.doubling from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filter Metric MeasureTheory TopologicalSpace ENNReal NN... | Mathlib/MeasureTheory/Measure/Doubling.lean | 132 | 136 | theorem eventually_measure_le_scaling_constant_mul (K : ℝ) :
∀ᶠ r in 𝓝[>] 0, ∀ x, μ (closedBall x (K * r)) ≤ scalingConstantOf μ K * μ (closedBall x r) := by |
filter_upwards [Classical.choose_spec
(exists_eventually_forall_measure_closedBall_le_mul μ K)] with r hr x
exact (hr x K le_rfl).trans (mul_le_mul_right' (ENNReal.coe_le_coe.2 (le_max_left _ _)) _)
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,819 |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w u₀ u₁ v₀ v₁
structure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Typ... | Mathlib/Control/Monad/Cont.lean | 101 | 105 | theorem monadLift_bind [Monad m] [LawfulMonad m] {α β} (x : m α) (f : α → m β) :
(monadLift (x >>= f) : ContT r m β) = monadLift x >>= monadLift ∘ f := by |
ext
simp only [monadLift, MonadLift.monadLift, (· ∘ ·), (· >>= ·), bind_assoc, id, run,
ContT.monadLift]
| 3 | 20.085537 | 1 | 0.333333 | 3 | 361 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 107 | 119 | theorem countP_mono_left (h : ∀ x ∈ l, p x → q x) : countP p l ≤ countP q l := by |
induction l with
| nil => apply Nat.le_refl
| cons a l ihl =>
rw [forall_mem_cons] at h
have ⟨ha, hl⟩ := h
simp [countP_cons]
cases h : p a
. simp
apply Nat.le_trans ?_ (Nat.le_add_right _ _)
apply ihl hl
. simp [ha h]
apply ihl hl
| 12 | 162,754.791419 | 2 | 0.8 | 10 | 703 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 109 | 119 | theorem _root_.Function.Injective.wOppSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WOppSide (f x) (f y) ↔ s.WOppSide x y := by |
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
| 8 | 2,980.957987 | 2 | 1.6 | 5 | 1,714 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 123 | 129 | theorem t'_comp_eq_pullbackSymmetry (i j k : D.J) :
D.t' j k i ≫ D.t' k i j =
(pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom := by |
trans inv (D.t' i j k)
· exact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)
· rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)]
simp [t_fac, t_fac_assoc]
| 4 | 54.59815 | 2 | 1.5 | 6 | 1,591 |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
noncomputable section
namespace Finsupp
variable {n : ℕ} (i : Fin n) {M : Type*} [Zero M] (y : M) (t : Fin (n + 1) →₀ M) (s : Fin n →₀ M)
def tail (s : Fin (n + 1) →₀ ... | Mathlib/Data/Finsupp/Fin.lean | 78 | 80 | theorem cons_ne_zero_of_left (h : y ≠ 0) : cons y s ≠ 0 := by |
contrapose! h with c
rw [← cons_zero y s, c, Finsupp.coe_zero, Pi.zero_apply]
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,500 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 80 | 83 | theorem Ioc_mul_Ico_subset' (a b c d : α) : Ioc a b * Ico c d ⊆ Ioo (a * c) (b * d) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_lt_mul_of_lt_of_le hya hzc, mul_lt_mul_of_le_of_lt hyb hzd⟩
| 3 | 20.085537 | 1 | 0.37931 | 29 | 381 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 95 | 106 | theorem StrictConvexSpace.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
StrictConvexSpace ℝ E := by |
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lin... | 9 | 8,103.083928 | 2 | 1.833333 | 6 | 1,919 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Sum
namespace Multiset
variable {α β : Type*} (s : Multiset α) (t : Multiset β)
def disjSum : Multiset (Sum α β) :=
s.map inl + t.map inr
#align multiset.dis... | Mathlib/Data/Multiset/Sum.lean | 55 | 60 | theorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by |
rw [mem_disjSum, or_iff_left]
-- Porting note: Previous code for L62 was: simp only [exists_eq_right]
· simp only [inl.injEq, exists_eq_right]
rintro ⟨b, _, hb⟩
exact inr_ne_inl hb
| 5 | 148.413159 | 2 | 1 | 4 | 1,085 |
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import linear_algebra.finite_dimensional from "leanprover-community/mathlib"@"e95e4f92c8f8da3c7f693c3ec948bcf9b6683f51"
universe u v v' w
open Cardina... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 123 | 126 | theorem finite_of_finite [Finite K] [FiniteDimensional K V] : Finite V := by |
cases nonempty_fintype K
haveI := fintypeOfFintype K V
infer_instance
| 3 | 20.085537 | 1 | 1 | 1 | 1,058 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 265 | 268 | theorem cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) {δ : ℝ} (hδ : 0 ≤ δ) :
cthickening δ ({x} : Set α) = closedBall x δ := by |
ext y
simp [cthickening, edist_dist, ENNReal.ofReal_le_ofReal_iff hδ]
| 2 | 7.389056 | 1 | 0.777778 | 9 | 686 |
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 | 144 | 151 | theorem norm_max_aux₂ {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 |
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
have he : ∀ x, ‖e x‖ = ‖x‖ := UniformSpace.Completion.norm_coe
replace hz : IsMaxOn (norm ∘ e ∘ f) (closedBall z (dist w z)) z := by
simpa only [IsMaxOn, (· ∘ ·), he] using hz
simpa only [he, (· ∘ ·)]
using norm_max_aux₁ (e.differentiable.comp_diff... | 6 | 403.428793 | 2 | 2 | 4 | 2,117 |
import Mathlib.Algebra.EuclideanDomain.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
universe u v
variable {R : Type u} {M : Type v... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 104 | 106 | theorem generator_mem (S : Submodule R M) [S.IsPrincipal] : generator S ∈ S := by |
conv_rhs => rw [← span_singleton_generator S]
exact subset_span (mem_singleton _)
| 2 | 7.389056 | 1 | 0.333333 | 3 | 325 |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 55 | 57 | theorem not_squarefree_zero [MonoidWithZero R] [Nontrivial R] : ¬Squarefree (0 : R) := by |
erw [not_forall]
exact ⟨0, by simp⟩
| 2 | 7.389056 | 1 | 1.714286 | 7 | 1,837 |
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 | 140 | 152 | theorem exists_finite_submodule_of_finite' (s : Set (M₁ ⊗[R] N₁)) (hs : s.Finite) :
∃ (M' : Submodule R M) (N' : Submodule R N) (hM : M' ≤ M₁) (hN : N' ≤ N₁),
Module.Finite R M' ∧ Module.Finite R N' ∧
s ⊆ LinearMap.range (TensorProduct.map (inclusion hM) (inclusion hN)) := by |
obtain ⟨M', N', _, _, h⟩ := exists_finite_submodule_of_finite s hs
have hM := map_subtype_le M₁ M'
have hN := map_subtype_le N₁ N'
refine ⟨_, _, hM, hN, .map _ _, .map _ _, ?_⟩
rw [mapIncl,
show M'.subtype = inclusion hM ∘ₗ M₁.subtype.submoduleMap M' by ext; simp,
show N'.subtype = inclusion hN ∘ₗ N₁... | 9 | 8,103.083928 | 2 | 1.875 | 8 | 1,929 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 60 | 62 | theorem count_eq_card_fintype (n : ℕ) : count p n = Fintype.card { k : ℕ // k < n ∧ p k } := by |
rw [count_eq_card_filter_range, ← Fintype.card_ofFinset, ← CountSet.fintype]
rfl
| 2 | 7.389056 | 1 | 0.642857 | 14 | 554 |
import Mathlib.RingTheory.UniqueFactorizationDomain
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.away.basic from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
section CommSemiring
variable {R : Type*} [CommSemiring R] (M : Submonoid R) {S : Type*... | Mathlib/RingTheory/Localization/Away/Basic.lean | 58 | 61 | theorem mul_invSelf : algebraMap R S x * invSelf x = 1 := by |
convert IsLocalization.mk'_mul_mk'_eq_one (M := Submonoid.powers x) (S := S) _ 1
symm
apply IsLocalization.mk'_one
| 3 | 20.085537 | 1 | 1 | 1 | 995 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Option
#align_import algebra.big_operators.option from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Function
namespace Finset
variable {α M : Type*} [CommMonoid M]
@[to_additive (attr := simp)]
theorem ... | Mathlib/Algebra/BigOperators/Option.lean | 36 | 42 | theorem prod_eraseNone (f : α → M) (s : Finset (Option α)) :
∏ x ∈ eraseNone s, f x = ∏ x ∈ s, Option.elim' 1 f x := by |
classical calc
∏ x ∈ eraseNone s, f x = ∏ x ∈ (eraseNone s).map Embedding.some, Option.elim' 1 f x :=
(prod_map (eraseNone s) Embedding.some <| Option.elim' 1 f).symm
_ = ∏ x ∈ s.erase none, Option.elim' 1 f x := by rw [map_some_eraseNone]
_ = ∏ x ∈ s, Option.elim' 1 f x := prod_erase _ rfl... | 5 | 148.413159 | 2 | 1 | 2 | 894 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 91 | 99 | theorem tendsto_polynomial_inv_mul_zero (p : ℝ[X]) :
Tendsto (fun x ↦ p.eval x⁻¹ * expNegInvGlue x) (𝓝 0) (𝓝 0) := by |
simp only [expNegInvGlue, mul_ite, mul_zero]
refine tendsto_const_nhds.if ?_
simp only [not_le]
have : Tendsto (fun x ↦ p.eval x⁻¹ / exp x⁻¹) (𝓝[>] 0) (𝓝 0) :=
p.tendsto_div_exp_atTop.comp tendsto_inv_zero_atTop
refine this.congr' <| mem_of_superset self_mem_nhdsWithin fun x hx ↦ ?_
simp [expNegInvGl... | 7 | 1,096.633158 | 2 | 1.166667 | 6 | 1,237 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.LinearAlgebra.Projection
#align_import linear_algebra.basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
set_option ... | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 127 | 131 | theorem ofVectorSpaceIndex.linearIndependent :
LinearIndependent K ((↑) : ofVectorSpaceIndex K V → V) := by |
convert (ofVectorSpace K V).linearIndependent
ext x
rw [ofVectorSpace_apply_self]
| 3 | 20.085537 | 1 | 0.666667 | 3 | 559 |
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 | 289 | 293 | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 := by |
cases n
· simp
· simp [X_mul, Nat.succ_ne_zero, descPochhammer_succ_left]
| 3 | 20.085537 | 1 | 0.96 | 25 | 796 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 145 | 147 | theorem rotation_pi : o.rotation π = LinearIsometryEquiv.neg ℝ := by |
ext x
simp [rotation]
| 2 | 7.389056 | 1 | 0.857143 | 7 | 744 |
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Types
namespace CategoryTheory.FunctorToTypes
open CategoryTheory.Limits
universe w v₁ v₂ u₁ u₂
variable {J : Type u₁} [Category.{v₁} J] {K : Type u₂} [Category.{v₂} K]
variable (F : J ⥤ K ⥤ TypeMax.{u₁, w})
| Mathlib/CategoryTheory/Limits/FunctorToTypes.lean | 25 | 29 | theorem jointly_surjective (k : K) {t : Cocone F} (h : IsColimit t) (x : t.pt.obj k) :
∃ j y, x = (t.ι.app j).app k y := by |
let hev := isColimitOfPreserves ((evaluation _ _).obj k) h
obtain ⟨j, y, rfl⟩ := Types.jointly_surjective _ hev x
exact ⟨j, y, by simp⟩
| 3 | 20.085537 | 1 | 1 | 1 | 1,109 |
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import probability.martingale.borel_cantelli from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Filter
open scoped NNRea... | Mathlib/Probability/Martingale/BorelCantelli.lean | 120 | 132 | theorem norm_stoppedValue_leastGE_le (hr : 0 ≤ r) (hf0 : f 0 = 0)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
∀ᵐ ω ∂μ, stoppedValue f (leastGE f r i) ω ≤ r + R := by |
filter_upwards [hbdd] with ω hbddω
change f (leastGE f r i ω) ω ≤ r + R
by_cases heq : leastGE f r i ω = 0
· rw [heq, hf0, Pi.zero_apply]
exact add_nonneg hr R.coe_nonneg
· obtain ⟨k, hk⟩ := Nat.exists_eq_succ_of_ne_zero heq
rw [hk, add_comm, ← sub_le_iff_le_add]
have := not_mem_of_lt_hitting (hk... | 10 | 22,026.465795 | 2 | 1.75 | 4 | 1,870 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operators.pi from "leanprover-community/mathlib"@"fa2309577c7009ea243cffdf990cd6c84f0ad497"
@[to_additive (attr := simp)]
theorem Finset.prod_apply {α : Type*} {β : α... | Mathlib/Algebra/BigOperators/Pi.lean | 81 | 84 | theorem Finset.univ_prod_mulSingle [Fintype I] (f : ∀ i, Z i) :
(∏ i, Pi.mulSingle i (f i)) = f := by |
ext a
simp
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,391 |
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