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.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : ℕ) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 101 | 103 | theorem coeff_add_of_eq {n : ℕ} {a b : R} {f g : R[X]}
(h_add_left : f.coeff n = a) (h_add_right : g.coeff n = b) :
(f + g).coeff n = a + b := by | subst ‹_› ‹_›; apply coeff_add
| 1 | 2.718282 | 0 | 1.4 | 5 | 1,474 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 109 | 111 | theorem dist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
dist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖ * dist p₁ p₂ := by |
simpa only [lineMap_apply_one, dist_eq_norm'] using dist_lineMap_lineMap p₁ p₂ c 1
| 1 | 2.718282 | 0 | 0.5 | 6 | 418 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 239 | 241 | theorem PreservesPushout.inl_iso_inv :
G.map pushout.inl ≫ (PreservesPushout.iso G f g).inv = pushout.inl := by |
simp [PreservesPushout.iso, Iso.comp_inv_eq]
| 1 | 2.718282 | 0 | 0.25 | 8 | 287 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 303 | 305 | theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0 := by |
erw [constr_basis b, constr_basis b]
simp only [eq_comm]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,196 |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 57 | 57 | theorem gal_X_pow_isSolvable (n : ℕ) : IsSolvable (X ^ n : F[X]).Gal := by | infer_instance
| 1 | 2.718282 | 0 | 0.909091 | 11 | 788 |
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import topology.c... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 127 | 128 | theorem sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i := by | simp [sigmaIsoSigma]
| 1 | 2.718282 | 0 | 0.5 | 4 | 450 |
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 161 | 162 | theorem cylinder_empty (s : Finset ι) : cylinder s (∅ : Set (∀ i : s, α i)) = ∅ := by |
rw [cylinder, preimage_empty]
| 1 | 2.718282 | 0 | 0.6875 | 16 | 636 |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
| Mathlib/Data/Vector/MapLemmas.lean | 60 | 68 | theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂ xs s₂).snd ys s₁)
= let m := (mapAccumr₂ (fun x y s =>
let r₂ := f₂ x s.snd
let r₁ := f₁ r₂.snd y s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs ys (s₁, s₂))
(m.f... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
| 1 | 2.718282 | 0 | 0.333333 | 24 | 337 |
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 68 | 68 | theorem congr_app {α β : F ⟶ G} (h : α = β) (X : C) : α.app X = β.app X := by | rw [h]
| 1 | 2.718282 | 0 | 0 | 4 | 163 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 52 | 53 | theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by |
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
| 1 | 2.718282 | 0 | 1.111111 | 9 | 1,194 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 126 | 128 | theorem PreservesPullback.iso_hom_snd :
(PreservesPullback.iso G f g).hom ≫ pullback.snd = G.map pullback.snd := by |
simp [PreservesPullback.iso]
| 1 | 2.718282 | 0 | 0.25 | 8 | 287 |
import Mathlib.Data.ENNReal.Real
#align_import data.real.conjugate_exponents from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open scoped ENNReal
namespace Real
@[mk_iff]
structure IsConjExponent (p q : ℝ) : Prop where
one_lt : 1 < p
inv_add_inv_conj : p⁻... | Mathlib/Data/Real/ConjExponents.lean | 115 | 118 | theorem inv_add_inv_conj_ennreal : (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = 1 := by |
rw [← ENNReal.ofReal_one, ← ENNReal.ofReal_inv_of_pos h.pos,
← ENNReal.ofReal_inv_of_pos h.symm.pos, ← ENNReal.ofReal_add h.inv_nonneg h.symm.inv_nonneg,
h.inv_add_inv_conj]
| 3 | 20.085537 | 1 | 0.75 | 4 | 653 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 88 | 90 | theorem reduceOption_concat_of_some (l : List (Option α)) (x : α) :
(l.concat (some x)).reduceOption = l.reduceOption.concat x := by |
simp only [reduceOption_nil, concat_eq_append, reduceOption_append, reduceOption_cons_of_some]
| 1 | 2.718282 | 0 | 0.615385 | 13 | 540 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 126 | 128 | theorem finSuccEquiv'_above {i : Fin (n + 1)} {m : Fin n} (h : i ≤ Fin.castSucc m) :
(finSuccEquiv' i) m.succ = some m := by |
rw [← Fin.succAbove_of_le_castSucc _ _ h, finSuccEquiv'_succAbove]
| 1 | 2.718282 | 0 | 0.25 | 4 | 298 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 251 | 252 | theorem finrank_matrix (m n : Type*) [Fintype m] [Fintype n] :
finrank R (Matrix m n R) = card m * card n := by | simp [finrank]
| 1 | 2.718282 | 0 | 0.75 | 24 | 667 |
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.hom.units from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u v w
namespace Units
variable {α : Ty... | Mathlib/Algebra/Group/Units/Hom.lean | 150 | 152 | theorem mul_liftRight_inv {f : M →* N} {g : M → Nˣ} (h : ∀ x, ↑(g x) = f x) (x) :
f x * ↑(liftRight f g h x)⁻¹ = 1 := by |
rw [Units.mul_inv_eq_iff_eq_mul, one_mul, coe_liftRight]
| 1 | 2.718282 | 0 | 0 | 4 | 43 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 153 | 154 | theorem U_add_one (n : ℤ) : U R (n + 1) = 2 * X * U R n - U R (n - 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
| 1 | 2.718282 | 0 | 0.166667 | 12 | 265 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 113 | 114 | theorem univ_eq_empty_iff : (univ : Finset α) = ∅ ↔ IsEmpty α := by |
rw [← not_nonempty_iff, ← univ_nonempty_iff, not_nonempty_iff_eq_empty]
| 1 | 2.718282 | 0 | 0.111111 | 9 | 248 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 171 | 171 | theorem nonpos_iff {a : SignType} : a ≤ 0 ↔ a = -1 ∨ a = 0 := by | cases a <;> decide
| 1 | 2.718282 | 0 | 0 | 6 | 104 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 58 | 59 | theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by |
simp only [chain_cons, Chain.nil, and_true_iff]
| 1 | 2.718282 | 0 | 0.428571 | 7 | 405 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 369 | 370 | theorem setAverage_congr_fun (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
⨍ x in s, f x ∂μ = ⨍ x in s, g x ∂μ := by | simp only [average_eq, setIntegral_congr_ae hs h]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 28 | 28 | theorem join_singleton (l : List α) : [l].join = l := by | rw [join, join, append_nil]
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
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 | 79 | 80 | theorem fold_congr {g : α → β} (H : ∀ x ∈ s, f x = g x) : s.fold op b f = s.fold op b g := by |
rw [fold, fold, map_congr rfl H]
| 1 | 2.718282 | 0 | 0.909091 | 11 | 789 |
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 102 | 102 | theorem or_inr {a b : Bool} (H : b) : a || b := by | cases a <;> simp [H]
| 1 | 2.718282 | 0 | 0 | 6 | 186 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatorics.hales_jewett from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open scoped Classical
universe u v
namespace ... | Mathlib/Combinatorics/HalesJewett.lean | 211 | 212 | theorem diagonal_apply {α ι} [Nonempty ι] (x : α) : Line.diagonal α ι x = fun _ => x := by |
simp_rw [Line.diagonal, Option.getD_none]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 522 |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 133 | 137 | theorem ball_subset_univBall_target (c : P) (r : ℝ) : ball c r ⊆ (univBall c r).target := by |
by_cases hr : 0 < r
· rw [univBall_target c hr]
· rw [univBall, dif_neg hr]
exact subset_univ _
| 4 | 54.59815 | 2 | 0.375 | 8 | 380 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 164 | 170 | theorem isPiSystem_Ixx_mem {Ixx : α → α → Set α} {p : α → α → Prop}
(Hne : ∀ {a b}, (Ixx a b).Nonempty → p a b)
(Hi : ∀ {a₁ b₁ a₂ b₂}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)) (s t : Set α) :
IsPiSystem { S | ∃ᵉ (l ∈ s) (u ∈ t), p l u ∧ Ixx l u = S } := by |
rintro _ ⟨l₁, hls₁, u₁, hut₁, _, rfl⟩ _ ⟨l₂, hls₂, u₂, hut₂, _, rfl⟩
simp only [Hi]
exact fun H => ⟨l₁ ⊔ l₂, sup_ind l₁ l₂ hls₁ hls₂, u₁ ⊓ u₂, inf_ind u₁ u₂ hut₁ hut₂, Hne H, rfl⟩
| 3 | 20.085537 | 1 | 1.333333 | 9 | 1,451 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 70 | 71 | theorem convexJoin_singleton_right (s : Set E) (y : E) :
convexJoin 𝕜 s {y} = ⋃ x ∈ s, segment 𝕜 x y := by | simp [convexJoin]
| 1 | 2.718282 | 0 | 0.1 | 10 | 244 |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 154 | 156 | theorem Heap.size_deleteMin_lt {s : Heap α} (eq : s.deleteMin le = some (a, s')) :
s'.size < s.size := by |
cases s with cases eq | node a c => simp_arith [size_combine, size]
| 1 | 2.718282 | 0 | 1 | 13 | 899 |
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 | 151 | 151 | theorem rotation_pi_apply (x : V) : o.rotation π x = -x := by | simp
| 1 | 2.718282 | 0 | 0.857143 | 7 | 744 |
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 | 351 | 351 | theorem mk_mem_col_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.col j ↔ (i, j) ∈ μ := by | simp [col]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 683 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 74 | 75 | theorem length_ndinsert_of_mem {a : α} {s : Multiset α} (h : a ∈ s) :
card (ndinsert a s) = card s := by | simp [h]
| 1 | 2.718282 | 0 | 0.4 | 5 | 392 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 113 | 114 | theorem image_const_add_Ici : (fun x => a + x) '' Ici b = Ici (a + b) := by |
simp only [add_comm a, image_add_const_Ici]
| 1 | 2.718282 | 0 | 1.090909 | 11 | 1,187 |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Hom
open Set Pointwise
theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) := by
lift m to Mˣ using hm
rw [Functio... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 58 | 60 | theorem image_smul_setₛₗ :
h '' (c • s) = σ c • h '' s := by |
simp only [← image_smul, image_image, map_smulₛₗ h]
| 1 | 2.718282 | 0 | 1.2 | 5 | 1,282 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 120 | 121 | theorem interior_Ioc [NoMaxOrder α] {a b : α} : interior (Ioc a b) = Ioo a b := by |
rw [← Ioi_inter_Iic, interior_inter, interior_Ioi, interior_Iic, Ioi_inter_Iio]
| 1 | 2.718282 | 0 | 0.769231 | 13 | 685 |
import Mathlib.Topology.Category.LightProfinite.Basic
import Mathlib.Topology.Category.Profinite.Limits
namespace LightProfinite
universe u w
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
section Pullbacks
variable {X Y B : LightProfinite.{u}} (f : X ⟶ B) (g ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 202 | 204 | theorem Sigma.ι_comp_toFiniteCoproduct (a : α) :
(Limits.Sigma.ι X a) ≫ (coproductIsoCoproduct X).inv = finiteCoproduct.ι X a := by |
simp [coproductIsoCoproduct]
| 1 | 2.718282 | 0 | 0.666667 | 3 | 593 |
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 | 140 | 140 | theorem rotation_zero : o.rotation 0 = LinearIsometryEquiv.refl ℝ V := by | ext; simp [rotation]
| 1 | 2.718282 | 0 | 0.857143 | 7 | 744 |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 184 | 185 | theorem get_natCast' (x : ℕ) (h : (x : PartENat).Dom) : get (x : PartENat) h = x := by |
rw [← natCast_inj, natCast_get]
| 1 | 2.718282 | 0 | 0 | 4 | 215 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 | theorem lcm_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.lcm f = s₂.lcm g := by |
subst hs
exact Finset.fold_congr hfg
| 2 | 7.389056 | 1 | 1 | 13 | 925 |
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
#align_import data.nat.size from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
namespace Nat
section
set_option linter.deprecated false
theorem shiftLeft_eq_mul_pow (m) : ∀ n, m <<< n = m * 2 ^ n := shiftLeft_eq _
#align nat.... | Mathlib/Data/Nat/Size.lean | 55 | 61 | theorem size_bit {b n} (h : bit b n ≠ 0) : size (bit b n) = succ (size n) := by |
rw [size]
conv =>
lhs
rw [binaryRec]
simp [h]
rw [div2_bit]
| 6 | 403.428793 | 2 | 0.666667 | 9 | 569 |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
#align_import ring_theory.roots_of_unity.minpoly from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 53 | 58 | theorem minpoly_dvd_x_pow_sub_one : minpoly ℤ μ ∣ X ^ n - 1 := by |
rcases n.eq_zero_or_pos with (rfl | h0)
· simp
apply minpoly.isIntegrallyClosed_dvd (isIntegral h h0)
simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, aeval_X_pow, eq_intCast, Int.cast_one,
aeval_one, AlgHom.map_sub, sub_self]
| 5 | 148.413159 | 2 | 2 | 6 | 2,240 |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 106 | 106 | theorem singleton_mem_nhds_of_ne_zero (h : γ ≠ 0) : ({γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by | simp [h]
| 1 | 2.718282 | 0 | 0.454545 | 11 | 414 |
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v... | Mathlib/Data/Set/NAry.lean | 68 | 69 | theorem image2_subset_iff_left : image2 f s t ⊆ u ↔ ∀ a ∈ s, (fun b => f a b) '' t ⊆ u := by |
simp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 336 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
CategoryTheory.Subobject CategoryTheory.Idempotents DoldKan
non... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 91 | 92 | theorem PInfty_comp_PInftyToNormalizedMooreComplex (X : SimplicialObject A) :
PInfty ≫ PInftyToNormalizedMooreComplex X = PInftyToNormalizedMooreComplex X := by | aesop_cat
| 1 | 2.718282 | 0 | 0.8 | 5 | 699 |
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72"
universe uR uS uT uA u₄
variable {R : Type uR} [Semiring R]
variable {S : Type uS} [CommSemiring S]
variable {T : Type uT}
vari... | Mathlib/Algebra/RingQuot.lean | 79 | 80 | theorem Rel.smul {r : A → A → Prop} (k : S) ⦃a b : A⦄ (h : Rel r a b) : Rel r (k • a) (k • b) := by |
simp only [Algebra.smul_def, Rel.mul_right h]
| 1 | 2.718282 | 0 | 0.5 | 6 | 490 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 167 | 169 | theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
sInf (s / t) = sInf s / sSup t := by |
rw [div_eq_mul_inv, csInf_mul hs₀ hs₁ ht₀.inv ht₁.inv, csInf_inv ht₀ ht₁, div_eq_mul_inv]
| 1 | 2.718282 | 0 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 144 | 145 | theorem div_mem {x y : M} (hx : x ∈ H) (hy : y ∈ H) : x / y ∈ H := by |
rw [div_eq_mul_inv]; exact mul_mem hx (inv_mem hy)
| 1 | 2.718282 | 0 | 0.5 | 2 | 466 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 147 | 147 | theorem mem_of_mem_inter_right (h : a ∈ l₁ ∩ l₂) : a ∈ l₂ := by | simpa using of_mem_filter h
| 1 | 2.718282 | 0 | 0.3 | 10 | 318 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 92 | 93 | theorem gronwallBound_ε0_δ0 (K x : ℝ) : gronwallBound 0 K 0 x = 0 := by |
simp only [gronwallBound_ε0, zero_mul]
| 1 | 2.718282 | 0 | 1.428571 | 7 | 1,514 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 91 | 92 | theorem rel_iff {x y z w : α} : Rel α (x, y) (z, w) ↔ x = z ∧ y = w ∨ x = w ∧ y = z := by |
simp
| 1 | 2.718282 | 0 | 0 | 4 | 173 |
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
open CategoryTheory
open CategoryTheory.Category
open CategoryTheory.Limits
open CategoryTheory.Preadditive
open Opposite
namespace Catego... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 99 | 101 | theorem idem_of_id_sub_idem [Preadditive C] {X : C} (p : X ⟶ X) (hp : p ≫ p = p) :
(𝟙 _ - p) ≫ (𝟙 _ - p) = 𝟙 _ - p := by |
simp only [comp_sub, sub_comp, id_comp, comp_id, hp, sub_self, sub_zero]
| 1 | 2.718282 | 0 | 1.6 | 5 | 1,735 |
import Lean.Elab.Tactic.Location
import Mathlib.Logic.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Tactic.Conv
import Mathlib.Init.Set
import Lean.Elab.Tactic.Location
set_option autoImplicit true
namespace Mathlib.Tactic.PushNeg
open Lean Meta Elab.Tactic Parser.Tactic
variable (p q : Prop) (s : α → Prop)... | Mathlib/Tactic/PushNeg.lean | 44 | 45 | theorem ne_empty_eq_nonempty (s : Set γ) : (s ≠ ∅) = s.Nonempty := by |
rw [ne_eq, ← not_nonempty_eq s, not_not]
| 1 | 2.718282 | 0 | 0.333333 | 3 | 366 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {α : Type*}
def CutExpand (r : α → α → Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 101 | 104 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by |
classical
rw [cutExpand_iff]
rintro ⟨_, _, _, ⟨⟩, _⟩
| 3 | 20.085537 | 1 | 1.833333 | 6 | 1,915 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 188 | 188 | theorem half_sub (a : R) : a / 2 - a = -(a / 2) := by | rw [← neg_sub, sub_half]
| 1 | 2.718282 | 0 | 0.5 | 12 | 426 |
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 | 31 | 31 | theorem prod_factorial_pos : 0 < ∏ i ∈ s, (f i)! := by | positivity
| 1 | 2.718282 | 0 | 1 | 2 | 884 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 87 | 88 | theorem fib_add_two {n : ℕ} : fib (n + 2) = fib n + fib (n + 1) := by |
simp [fib, Function.iterate_succ_apply']
| 1 | 2.718282 | 0 | 1.181818 | 11 | 1,246 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 113 | 114 | theorem angle_sub_const (v₁ v₂ v₃ : V) (v : V) : ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃ := by |
simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v
| 1 | 2.718282 | 0 | 0.4 | 5 | 403 |
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation
open scoped Topology
variable {X Y A} [TopologicalSpace X] [TopologicalSpace A]
theorem embedding_toPullbackDiag (f : X → Y) : Embedding (toPullbackDiag f) :=
Embedding.mk' _ (injective_toPullbackDiag f) fun x ↦ by
rw [toPullbackDiag,... | Mathlib/Topology/SeparatedMap.lean | 101 | 106 | theorem IsSeparatedMap.pullback {f : X → Y} (sep : IsSeparatedMap f) (g : A → Y) :
IsSeparatedMap (@snd X Y A f g) := by |
rw [isSeparatedMap_iff_isClosed_diagonal] at sep ⊢
rw [← preimage_map_fst_pullbackDiagonal]
refine sep.preimage (Continuous.mapPullback ?_ ?_) <;>
apply_rules [continuous_fst, continuous_subtype_val, Continuous.comp]
| 4 | 54.59815 | 2 | 1.5 | 4 | 1,617 |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 48 | 48 | theorem xgcd_zero_left {s t r' s' t'} : xgcdAux 0 s t r' s' t' = (r', s', t') := by | simp [xgcdAux]
| 1 | 2.718282 | 0 | 1.090909 | 11 | 1,188 |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 32 | 32 | theorem lift_top (g : Set α → Filter β) : (⊤ : Filter α).lift g = g univ := by | simp [Filter.lift]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 581 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 200 | 200 | theorem not_lt_top {x : ℝ≥0∞} : ¬x < ∞ ↔ x = ∞ := by | rw [lt_top_iff_ne_top, Classical.not_not]
| 1 | 2.718282 | 0 | 0.666667 | 12 | 570 |
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Pointwise
universe u v
namespace MonoidHom
o... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 35 | 36 | theorem ker_eq_bot_of_cancel {f : A →* B} (h : ∀ u v : f.ker →* A, f.comp u = f.comp v → u = v) :
f.ker = ⊥ := by | simpa using _root_.congr_arg range (h f.ker.subtype 1 (by aesop_cat))
| 1 | 2.718282 | 0 | 1 | 2 | 1,113 |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 153 | 154 | theorem comp.mk_get (x : comp P Q α) : comp.mk (comp.get x) = x := by |
rfl
| 1 | 2.718282 | 0 | 0.857143 | 7 | 749 |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 46 | 48 | theorem length (n m : ℕ) : length (Ico n m) = m - n := by |
dsimp [Ico]
simp [length_range', autoParam]
| 2 | 7.389056 | 1 | 0.9375 | 16 | 794 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 125 | 125 | theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by | simp [nhds_eq]
| 1 | 2.718282 | 0 | 0.5 | 6 | 489 |
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 89 | 91 | theorem Injective.beq_eq {α β : Type*} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {f : α → β}
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by |
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
| 1 | 2.718282 | 0 | 0 | 2 | 166 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int
#align_import data.int.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Int
open Nat
variable {α : Type*}
@[norm_cast]
| Mathlib/Data/Int/Cast/Field.lean | 34 | 34 | theorem cast_neg_natCast {R} [DivisionRing R] (n : ℕ) : ((-n : ℤ) : R) = -n := by | simp
| 1 | 2.718282 | 0 | 0.5 | 2 | 480 |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 62 | 63 | theorem ball_eq_Ioo (x : ℤ) (r : ℝ) : ball x r = Ioo ⌊↑x - r⌋ ⌈↑x + r⌉ := by |
rw [← preimage_ball, Real.ball_eq_Ioo, preimage_Ioo]
| 1 | 2.718282 | 0 | 0.333333 | 6 | 327 |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 89 | 90 | theorem hasCompactSupport_normed : HasCompactSupport (f.normed μ) := by |
simp only [HasCompactSupport, f.tsupport_normed_eq (μ := μ), isCompact_closedBall]
| 1 | 2.718282 | 0 | 0.818182 | 11 | 722 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 796 | 797 | theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by | simp [h]
| 1 | 2.718282 | 0 | 1.083333 | 12 | 1,183 |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 57 | 60 | theorem rpow_mul_exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) (s : ℝ) :
(fun x : ℝ => x ^ s * exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
simp_rw [← rpow_two]
exact rpow_mul_exp_neg_mul_rpow_isLittleO_exp_neg s one_lt_two hb
| 2 | 7.389056 | 1 | 1.5 | 6 | 1,554 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 145 | 146 | theorem laverage_congr {f g : α → ℝ≥0∞} (h : f =ᵐ[μ] g) : ⨍⁻ x, f x ∂μ = ⨍⁻ x, g x ∂μ := by |
simp only [laverage_eq, lintegral_congr_ae h]
| 1 | 2.718282 | 0 | 0.347826 | 23 | 374 |
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 112 | 112 | theorem intrinsicInterior_empty : intrinsicInterior 𝕜 (∅ : Set P) = ∅ := by | simp [intrinsicInterior]
| 1 | 2.718282 | 0 | 0.571429 | 7 | 518 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 353 | 355 | theorem mul_inv_le_inv_mul_iff : a * b⁻¹ ≤ d⁻¹ * c ↔ d * a ≤ c * b := by |
rw [← mul_le_mul_iff_left d, ← mul_le_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
| 2 | 7.389056 | 1 | 0.4 | 25 | 400 |
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 | 187 | 192 | theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) := by |
rw [fibRec]
intro n
simp only
rw [Nat.fib_add_two, add_comm]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ']
| 5 | 148.413159 | 2 | 0.894737 | 19 | 776 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 151 | 154 | theorem symmₗ_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y := by |
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).left_inv y
| 2 | 7.389056 | 1 | 0.6 | 5 | 539 |
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 | 69 | 71 | theorem mem_toCotangent_ker {x : I} : x ∈ LinearMap.ker I.toCotangent ↔ (x : R) ∈ I ^ 2 := by |
rw [← I.map_toCotangent_ker]
simp
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,400 |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 79 | 80 | theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by |
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
| 1 | 2.718282 | 0 | 0.125 | 16 | 251 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 155 | 155 | theorem yn_one : yn a1 1 = 1 := by | simp
| 1 | 2.718282 | 0 | 0 | 2 | 100 |
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 121 | 121 | theorem a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a := by | rw [← coeff_eq_a, h, coeff_eq_a]
| 1 | 2.718282 | 0 | 0.1 | 10 | 246 |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 73 | 75 | theorem equiv_eq_conj (a : A) :
(of (φ a : G) : HNNExtension G A B φ) = t * of (a : G) * t⁻¹ := by |
rw [t_mul_of]; simp
| 1 | 2.718282 | 0 | 0.444444 | 9 | 413 |
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α R : Type*}
open Filter Set Function
@[simp]
theorem Nat.comap_cast_atTop [S... | Mathlib/Order/Filter/Archimedean.lean | 69 | 71 | theorem tendsto_intCast_atTop_iff [StrictOrderedRing R] [Archimedean R] {f : α → ℤ}
{l : Filter α} : Tendsto (fun n => (f n : R)) l atTop ↔ Tendsto f l atTop := by |
rw [← @Int.comap_cast_atTop R, tendsto_comap_iff]; rfl
| 1 | 2.718282 | 0 | 0 | 4 | 221 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 248 | 248 | theorem bihimp_self : a ⇔ a = ⊤ := by | rw [bihimp, inf_idem, himp_self]
| 1 | 2.718282 | 0 | 0.181818 | 22 | 266 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
import Mathlib.CategoryTheory.Elementwise
#align_import analysis.normed.group.SemiNormedGroup from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat.lean | 121 | 129 | theorem iso_isometry_of_normNoninc {V W : SemiNormedGroupCat} (i : V ≅ W) (h1 : i.hom.NormNoninc)
(h2 : i.inv.NormNoninc) : Isometry i.hom := by |
apply AddMonoidHomClass.isometry_of_norm
intro v
apply le_antisymm (h1 v)
calc
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
‖v‖ = ‖i.inv (i.hom v)‖ := by erw [Iso.hom_inv_id_apply]
_ ≤ ‖i.hom v‖ := h2 _
| 7 | 1,096.633158 | 2 | 1.5 | 2 | 1,640 |
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 | 106 | 107 | theorem count_succ_eq_succ_count_iff {n : ℕ} : count p (n + 1) = count p n + 1 ↔ p n := by |
by_cases h : p n <;> simp [h, count_succ]
| 1 | 2.718282 | 0 | 0.642857 | 14 | 554 |
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryTheory.Triangulated.TriangleShift
#align_import category_theory.triangulated.pretriangulated from "leanprover-community/mathlib"@"6876fa15e3158ff3e4a4e2af1fb6e194... | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 156 | 159 | theorem comp_distTriang_mor_zero₃₁ (T : Triangle C) (H : T ∈ distTriang C) :
T.mor₃ ≫ T.mor₁⟦1⟧' = 0 := by |
have H₂ := rot_of_distTriang T.rotate (rot_of_distTriang T H)
simpa using comp_distTriang_mor_zero₁₂ T.rotate.rotate H₂
| 2 | 7.389056 | 1 | 1.5 | 2 | 1,615 |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Filter.CountableInter
import Mathlib.Order.Filter.CardinalInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.Order.Filter.Bases
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u}... | Mathlib/Order/Filter/Cocardinal.lean | 70 | 72 | theorem frequently_cocardinal {p : α → Prop} :
(∃ᶠ x in cocardinal α hreg, p x) ↔ c ≤ # { x | p x } := by |
simp only [Filter.Frequently, eventually_cocardinal, not_not,coe_setOf, not_lt]
| 1 | 2.718282 | 0 | 0.75 | 4 | 659 |
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function Set
variable {α : Type*} (r r₁ r₂ : α → α → Prop) (s... | Mathlib/Order/Minimal.lean | 113 | 115 | theorem mem_minimals_iff_forall_lt_not_mem' (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt y x → y ∉ s := by |
simp [minimals, right_iff_left_not_left_of r rlt, not_imp_not, imp.swap (a := _ ∈ _)]
| 1 | 2.718282 | 0 | 0.75 | 4 | 676 |
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 | 43 | 43 | theorem sign_zero : sign 0 = 0 := by | rw [sign, if_neg (lt_irrefl _), if_neg (lt_irrefl _)]
| 1 | 2.718282 | 0 | 1.363636 | 11 | 1,467 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 74 | 76 | theorem image_restrict (f : α → β) (s t : Set α) :
s.restrict f '' (Subtype.val ⁻¹' t) = f '' (t ∩ s) := by |
rw [restrict_eq, image_comp, image_preimage_eq_inter_range, Subtype.range_coe]
| 1 | 2.718282 | 0 | 0.8 | 10 | 704 |
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)]
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... | Mathlib/Algebra/Divisibility/Prod.lean | 35 | 36 | theorem pi_dvd_iff {x y : ∀ i, G i} : x ∣ y ↔ ∀ i, x i ∣ y i := by |
simp_rw [dvd_def, Function.funext_iff, Classical.skolem]; rfl
| 1 | 2.718282 | 0 | 0.5 | 2 | 447 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 102 | 103 | theorem iUnion_Ico_left (b : α) : ⋃ a, Ico a b = Iio b := by |
simp only [← Ici_inter_Iio, ← iUnion_inter, iUnion_Ici, univ_inter]
| 1 | 2.718282 | 0 | 0.333333 | 18 | 364 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 152 | 155 | theorem single_mul_single [Fintype n] [DecidableEq k] [DecidableEq m] [DecidableEq n] [Semiring α]
(a : m) (b : n) (c : k) :
((single a b).toMatrix : Matrix _ _ α) * (single b c).toMatrix = (single a c).toMatrix := by |
rw [← toMatrix_trans, single_trans_single]
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,267 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 73 | 73 | theorem symm : PythagoreanTriple y x z := by | rwa [pythagoreanTriple_comm]
| 1 | 2.718282 | 0 | 1.2 | 10 | 1,251 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 154 | 163 | theorem wittPolynomial_zmod_self (n : ℕ) :
W_ (ZMod (p ^ (n + 1))) (n + 1) = expand p (W_ (ZMod (p ^ (n + 1))) n) := by |
simp only [wittPolynomial_eq_sum_C_mul_X_pow]
rw [sum_range_succ, ← Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0,
zero_mul, add_zero, AlgHom.map_sum, sum_congr rfl]
intro k hk
rw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, ← pow_mul, ← pow_succ']
congr
rw [mem_range... | 8 | 2,980.957987 | 2 | 1.181818 | 11 | 1,247 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 178 | 178 | theorem not_mem_compl : a ∉ sᶜ ↔ a ∈ s := by | rw [mem_compl, not_not]
| 1 | 2.718282 | 0 | 0.111111 | 9 | 248 |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 64 | 65 | theorem char_tensor (V W : FdRep k G) : (V ⊗ W).character = V.character * W.character := by |
ext g; convert trace_tensorProduct' (V.ρ g) (W.ρ g)
| 1 | 2.718282 | 0 | 0 | 5 | 148 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 44 | 44 | theorem join_concat (L : List (List α)) (l : List α) : join (L.concat l) = join L ++ l := by | simp
| 1 | 2.718282 | 0 | 0.9 | 10 | 782 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 134 | 135 | theorem inf_relindex_right : (H ⊓ K).relindex K = H.relindex K := by |
rw [relindex, relindex, inf_subgroupOf_right]
| 1 | 2.718282 | 0 | 0.666667 | 6 | 611 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 47 | 48 | theorem verschiebungFun_coeff_zero (x : 𝕎 R) : (verschiebungFun x).coeff 0 = 0 := by |
rw [verschiebungFun_coeff, if_pos rfl]
| 1 | 2.718282 | 0 | 1 | 6 | 1,101 |
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