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.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 85 | 89 | theorem bottom_row_coprime {R : Type*} [CommRing R] (g : SL(2, R)) :
IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) R) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) R) 1 1) := by |
use -(↑g : Matrix (Fin 2) (Fin 2) R) 0 1, (↑g : Matrix (Fin 2) (Fin 2) R) 0 0
rw [add_comm, neg_mul, ← sub_eq_add_neg, ← det_fin_two]
exact g.det_coe
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,812 |
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 | 137 | 139 | theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ := by |
rw [← image_inv]
exact ((OrderIso.inv α).map_csSup' hs₀ hs₁).symm
| 2 | 7.389056 | 1 | 1.25 | 16 | 1,340 |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Localization.FractionRing
#align_import algebra.char_p.algebra from "leanprover-community/mathlib"@"96782a2d6dcded92116d8ac9ae48efb41d46a27c"
theorem charP_of_injective_ringHom {R A : Type*} [NonAssocSemiring R] [NonAs... | Mathlib/Algebra/CharP/Algebra.lean | 64 | 67 | theorem RingHom.charP {R A : Type*} [NonAssocSemiring R] [NonAssocSemiring A] (f : R →+* A)
(H : Function.Injective f) (p : ℕ) [CharP A p] : CharP R p := by |
obtain ⟨q, h⟩ := CharP.exists R
exact CharP.eq _ (charP_of_injective_ringHom H q) ‹CharP A p› ▸ h
| 2 | 7.389056 | 1 | 0.666667 | 3 | 598 |
import Mathlib.Data.List.Range
import Mathlib.Algebra.Order.Ring.Nat
variable {α : Type*}
namespace List
@[simp]
theorem length_iterate (f : α → α) (a : α) (n : ℕ) : length (iterate f a n) = n := by
induction n generalizing a <;> simp [*]
@[simp]
theorem iterate_eq_nil {f : α → α} {a : α} {n : ℕ} : iterate f ... | Mathlib/Data/List/Iterate.lean | 48 | 52 | theorem iterate_add (f : α → α) (a : α) (m n : ℕ) :
iterate f a (m + n) = iterate f a m ++ iterate f (f^[m] a) n := by |
induction m generalizing a with
| zero => simp
| succ n ih => rw [iterate, add_right_comm, iterate, ih, Nat.iterate, cons_append]
| 3 | 20.085537 | 1 | 0.166667 | 6 | 264 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {𝕜 E ι : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 114 | 118 | theorem Balanced.balancedHull_subset_of_subset (ht : Balanced 𝕜 t) (h : s ⊆ t) :
balancedHull 𝕜 s ⊆ t := by |
intros x hx
obtain ⟨r, hr, y, hy, rfl⟩ := mem_balancedHull_iff.1 hx
exact ht.smul_mem hr (h hy)
| 3 | 20.085537 | 1 | 0.75 | 4 | 666 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.Probability.Kernel.Disintegration.CdfToKernel
#align_import probability.kernel.cond_cdf from "leanprover-community/mathlib"@"3b88f4005dc2e28d42f974cc1ce838f0dafb39b8"
open MeasureTheory Set Filter TopologicalSpace
open scoped NNReal ENNReal Me... | Mathlib/Probability/Kernel/Disintegration/CondCdf.lean | 102 | 124 | theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0) := by |
simp_rw [ρ.IicSnd_apply _ hs]
have h_empty : ρ (s ×ˢ ∅) = 0 := by simp only [prod_empty, measure_empty]
rw [← h_empty, ← Real.iInter_Iic_rat, prod_iInter]
suffices h_neg :
Tendsto (fun r : ℚ ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r : ℚ, s ×ˢ Iic ↑(-r)))) by
have h_inter_eq : ⋂ r : ℚ, s ×ˢ Iic ↑(-r) = ... | 21 | 1,318,815,734.483215 | 2 | 1.333333 | 6 | 1,412 |
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import Mathlib.Analysis.Complex.Polynomial
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 91 | 96 | theorem IsNontrivial.isPrimitive {F : Type u} [Field F] {ψ : AddChar F R'} (hψ : IsNontrivial ψ) :
IsPrimitive ψ := by |
intro a ha
cases' hψ with x h
use a⁻¹ * x
rwa [mulShift_apply, mul_inv_cancel_left₀ ha]
| 4 | 54.59815 | 2 | 1.5 | 6 | 1,658 |
import Mathlib.Order.BoundedOrder
#align_import data.prod.lex from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α β γ : Type*}
namespace Prod.Lex
@[inherit_doc] notation:35 α " ×ₗ " β:34 => Lex (Prod α β)
instance decidableEq (α β : Type*) [DecidableEq α] [DecidableEq β] ... | Mathlib/Data/Prod/Lex.lean | 105 | 109 | theorem monotone_fst [Preorder α] [LE β] (t c : α ×ₗ β) (h : t ≤ c) :
(ofLex t).1 ≤ (ofLex c).1 := by |
cases (Prod.Lex.le_iff t c).mp h with
| inl h' => exact h'.le
| inr h' => exact h'.1.le
| 3 | 20.085537 | 1 | 1.666667 | 3 | 1,784 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 430 | 433 | theorem codRestrict_apply_cod {f : M →ₗ[S] M} (h : IsProj m f) (x : m) : h.codRestrict x = x := by |
ext
rw [codRestrict_apply]
exact h.map_id x x.2
| 3 | 20.085537 | 1 | 1.142857 | 7 | 1,220 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 162 | 181 | theorem mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : E} (hx : DifferentiableAt 𝕜 f x) :
∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (fderiv 𝕜 f x) r ε := by |
let δ := (ε / 2) / 2
obtain ⟨R, R_pos, hR⟩ :
∃ R > 0, ∀ y ∈ ball x R, ‖f y - f x - fderiv 𝕜 f x (y - x)‖ ≤ δ * ‖y - x‖ :=
eventually_nhds_iff_ball.1 <| hx.hasFDerivAt.isLittleO.bound <| by positivity
refine ⟨R, R_pos, fun r hr => ?_⟩
have : r ∈ Ioc (r / 2) r := right_mem_Ioc.2 <| half_lt_self hr.1
... | 18 | 65,659,969.137331 | 2 | 1.4 | 10 | 1,489 |
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 | 39 | 42 | theorem not_nonempty_eq (s : Set γ) : (¬ s.Nonempty) = (s = ∅) := by |
have A : ∀ (x : γ), ¬(x ∈ (∅ : Set γ)) := fun x ↦ id
simp only [Set.Nonempty, not_exists, eq_iff_iff]
exact ⟨fun h ↦ Set.ext (fun x ↦ by simp only [h x, false_iff, A]), fun h ↦ by rwa [h]⟩
| 3 | 20.085537 | 1 | 0.333333 | 3 | 366 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Tactic.LiftLets
#align_import linear_algebra.clifford_algebra.even_equiv from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36d... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 69 | 71 | theorem ι_eq_v_add_smul_e0 (m : M) (r : R) : ι (Q' Q) (m, r) = v Q m + r • e0 Q := by |
rw [e0, v, LinearMap.comp_apply, LinearMap.inl_apply, ← LinearMap.map_smul, Prod.smul_mk,
smul_zero, smul_eq_mul, mul_one, ← LinearMap.map_add, Prod.mk_add_mk, zero_add, add_zero]
| 2 | 7.389056 | 1 | 1 | 4 | 1,132 |
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine Matrix
open Set
universe u₁ u₂ u₃ u₄
variable {ι : Type u₁} {k : Type... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 124 | 127 | theorem toMatrix_mul_toMatrix : b.toMatrix b₂ * b₂.toMatrix b = 1 := by |
ext l m
change (b.coords (b₂ l) ᵥ* b₂.toMatrix b) m = _
rw [toMatrix_vecMul_coords, coords_apply, ← toMatrix_apply, toMatrix_self]
| 3 | 20.085537 | 1 | 1.428571 | 7 | 1,512 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 32 | 35 | theorem comm {a b c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c := by |
delta Fermat42
rw [add_comm]
tauto
| 3 | 20.085537 | 1 | 1.666667 | 9 | 1,779 |
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 | 126 | 129 | theorem relindex_mul_relindex (hHK : H ≤ K) (hKL : K ≤ L) :
H.relindex K * K.relindex L = H.relindex L := by |
rw [← relindex_subgroupOf hKL]
exact relindex_mul_index fun x hx => hHK hx
| 2 | 7.389056 | 1 | 0.666667 | 6 | 611 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 65 | 69 | theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h
| 3 | 20.085537 | 1 | 1.642857 | 14 | 1,752 |
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.Topology.ContinuousFunction.Algebra
#align_import topology.continuous_function.units from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {X M R 𝕜 : Type*} [TopologicalSpace X]
nam... | Mathlib/Topology/ContinuousFunction/Units.lean | 70 | 79 | theorem continuous_isUnit_unit {f : C(X, R)} (h : ∀ x, IsUnit (f x)) :
Continuous fun x => (h x).unit := by |
refine
continuous_induced_rng.2
(Continuous.prod_mk f.continuous
(MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))
have := NormedRing.inverse_continuousAt (h x).unit
simp only
simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢
exact this.comp (f.contin... | 8 | 2,980.957987 | 2 | 2 | 1 | 1,980 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 109 | 112 | theorem dvd_of_mem_divisors {m : ℕ} (h : n ∈ divisors m) : n ∣ m := by |
cases m
· apply dvd_zero
· simp [mem_divisors.1 h]
| 3 | 20.085537 | 1 | 1 | 11 | 1,070 |
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathlib"@"6c351a8fb9b06e5a542fdf427bfb9f46724f9453"
suppress_compilation
universe u v w
open TensorProduct
open TensorProduct
open Algebra.TensorProduct
open Matri... | Mathlib/RingTheory/MatrixAlgebra.lean | 105 | 110 | theorem invFun_algebraMap (M : Matrix n n R) : invFun R A n (M.map (algebraMap R A)) = 1 ⊗ₜ M := by |
dsimp [invFun]
simp only [Algebra.algebraMap_eq_smul_one, smul_tmul, ← tmul_sum, mul_boole]
congr
conv_rhs => rw [matrix_eq_sum_std_basis M]
convert Finset.sum_product (β := Matrix n n R); simp
| 5 | 148.413159 | 2 | 1 | 6 | 859 |
import Mathlib.Data.List.Forall2
#align_import data.list.sections from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat Function
namespace List
variable {α β : Type*}
| Mathlib/Data/List/Sections.lean | 23 | 34 | theorem mem_sections {L : List (List α)} {f} : f ∈ sections L ↔ Forall₂ (· ∈ ·) f L := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· induction L generalizing f
· cases mem_singleton.1 h
exact Forall₂.nil
simp only [sections, bind_eq_bind, mem_bind, mem_map] at h
rcases h with ⟨_, _, _, _, rfl⟩
simp only [*, forall₂_cons, true_and_iff]
· induction' h with a l f L al fL fs
· simp onl... | 11 | 59,874.141715 | 2 | 2 | 1 | 2,167 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 93 | 100 | theorem rotate'_length_mul (l : List α) : ∀ n : ℕ, l.rotate' (l.length * n) = l
| 0 => by simp
| n + 1 =>
calc
l.rotate' (l.length * (n + 1)) =
(l.rotate' (l.length * n)).rotate' (l.rotate' (l.length * n)).length := by |
simp [-rotate'_length, Nat.mul_succ, rotate'_rotate']
_ = l := by rw [rotate'_length, rotate'_length_mul l n]
| 2 | 7.389056 | 1 | 0.153846 | 13 | 257 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 124 | 127 | theorem Integrable.comp_div_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
(g : G) : Integrable (fun t => f (t / g)) μ := by |
simp_rw [div_eq_mul_inv]
exact hf.comp_mul_right g⁻¹
| 2 | 7.389056 | 1 | 0.75 | 8 | 660 |
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Closeds
open Function Set Filter TopologicalSpace
open scoped Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace Y]
theorem TopologicalSpace.Clopens.exists_prod_subset (W : Clopens (X × Y)) {a : X × Y} (h : a ∈ W... | Mathlib/Topology/ClopenBox.lean | 50 | 61 | theorem TopologicalSpace.Clopens.exists_finset_eq_sup_prod (W : Clopens (X × Y)) :
∃ (I : Finset (Clopens X × Clopens Y)), W = I.sup fun i ↦ i.1 ×ˢ i.2 := by |
choose! U hxU V hxV hUV using fun x ↦ W.exists_prod_subset (a := x)
rcases W.2.1.isCompact.elim_nhds_subcover (fun x ↦ U x ×ˢ V x) (fun x hx ↦
(U x ×ˢ V x).2.isOpen.mem_nhds ⟨hxU x hx, hxV x hx⟩) with ⟨I, hIW, hWI⟩
classical
use I.image fun x ↦ (U x, V x)
rw [Finset.sup_image]
refine le_antisymm (fun x... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,073 |
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open Set
open Computability
-- "ε_NFA"
set_option linter.uppercaseLean3 false
universe u v
structure εNFA (α : Type u) (σ : Type v) where
step : σ → Opt... | Mathlib/Computability/EpsilonNFA.lean | 116 | 119 | theorem evalFrom_empty (x : List α) : M.evalFrom ∅ x = ∅ := by |
induction' x using List.reverseRecOn with x a ih
· rw [evalFrom_nil, εClosure_empty]
· rw [evalFrom_append_singleton, ih, stepSet_empty]
| 3 | 20.085537 | 1 | 0.25 | 4 | 291 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 223 | 231 | theorem norm_eq_pow_val {f : PadicSeq p} (hf : ¬f ≈ 0) : f.norm = (p : ℚ) ^ (-f.valuation : ℤ) := by |
rw [norm, valuation, dif_neg hf, dif_neg hf, padicNorm, if_neg]
intro H
apply CauSeq.not_limZero_of_not_congr_zero hf
intro ε hε
use stationaryPoint hf
intro n hn
rw [stationaryPoint_spec hf le_rfl hn]
simpa [H] using hε
| 8 | 2,980.957987 | 2 | 1.4 | 5 | 1,505 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Data.ZMod.Algebra
#align_import ring_theory.polynomial.cyclotomic.expand from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"
namespace Polynomial
@[simp]
theorem cyclotomic_expand_eq_cyclotomic_mul {p n : ℕ} (hp : Na... | Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 100 | 110 | theorem cyclotomic_irreducible_pow_of_irreducible_pow {p : ℕ} (hp : Nat.Prime p) {R} [CommRing R]
[IsDomain R] {n m : ℕ} (hmn : m ≤ n) (h : Irreducible (cyclotomic (p ^ n) R)) :
Irreducible (cyclotomic (p ^ m) R) := by |
rcases m.eq_zero_or_pos with (rfl | hm)
· simpa using irreducible_X_sub_C (1 : R)
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hmn
induction' k with k hk
· simpa using h
have : m + k ≠ 0 := (add_pos_of_pos_of_nonneg hm k.zero_le).ne'
rw [Nat.add_succ, pow_succ, ← cyclotomic_expand_eq_cyclotomic hp <| dvd_p... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,215 |
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.Topology.Algebra.ValuedField
#align_import number_theory.function_field from "leanprover-community/mathlib"@"70fd9563... | Mathlib/NumberTheory/FunctionField.lean | 62 | 80 | theorem functionField_iff (Fqt : Type*) [Field Fqt] [Algebra Fq[X] Fqt]
[IsFractionRing Fq[X] Fqt] [Algebra (RatFunc Fq) F] [Algebra Fqt F] [Algebra Fq[X] F]
[IsScalarTower Fq[X] Fqt F] [IsScalarTower Fq[X] (RatFunc Fq) F] :
FunctionField Fq F ↔ FiniteDimensional Fqt F := by |
let e := IsLocalization.algEquiv Fq[X]⁰ (RatFunc Fq) Fqt
have : ∀ (c) (x : F), e c • x = c • x := by
intro c x
rw [Algebra.smul_def, Algebra.smul_def]
congr
refine congr_fun (f := fun c => algebraMap Fqt F (e c)) ?_ c -- Porting note: Added `(f := _)`
refine IsLocalization.ext (nonZeroDivisors ... | 15 | 3,269,017.372472 | 2 | 1.428571 | 7 | 1,522 |
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.mul_p from "leanprover-community/mathlib"@"7abfbc92eec87190fba3ed3d5ec58e7c167e7144"
namespace WittVector
variable {p : ℕ} {R : Type*} [hp : Fact p.Prime] [CommRing R]
local notation "𝕎" => WittVector p -- type as `\bbW`
open Mv... | Mathlib/RingTheory/WittVector/MulP.lean | 50 | 60 | theorem mulN_coeff (n : ℕ) (x : 𝕎 R) (k : ℕ) :
(x * n).coeff k = aeval x.coeff (wittMulN p n k) := by |
induction' n with n ih generalizing k
· simp only [Nat.zero_eq, Nat.cast_zero, mul_zero, zero_coeff, wittMulN,
AlgHom.map_zero, Pi.zero_apply]
· rw [wittMulN, Nat.cast_add, Nat.cast_one, mul_add, mul_one, aeval_bind₁, add_coeff]
apply eval₂Hom_congr (RingHom.ext_int _ _) _ rfl
ext1 ⟨b, i⟩
fin_c... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,138 |
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 83 | 93 | theorem genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[hX : IrreducibleSpace X.carrier] [IrreducibleSpace Y.carrier] :
f.1.base (genericPoint X.carrier : _) = (genericPoint Y.carrier : _) := by |
apply ((genericPoint_spec Y).eq _).symm
convert (genericPoint_spec X.carrier).image (show Continuous f.1.base by continuity)
symm
rw [eq_top_iff, Set.top_eq_univ, Set.top_eq_univ]
convert subset_closure_inter_of_isPreirreducible_of_isOpen _ H.base_open.isOpen_range _
· rw [Set.univ_inter, Set.image_univ]
... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,098 |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {α : Type*} {β : α →... | Mathlib/Control/LawfulFix.lean | 57 | 60 | theorem approx_mono' {i : ℕ} : Fix.approx f i ≤ Fix.approx f (succ i) := by |
induction i with
| zero => dsimp [approx]; apply @bot_le _ _ _ (f ⊥)
| succ _ i_ih => intro; apply f.monotone; apply i_ih
| 3 | 20.085537 | 1 | 1.6 | 5 | 1,725 |
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.BilinearMap
#align_import topology.algebra.module.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Filter
open Topology
variable {α 𝕜 𝕝 R E F M : Type*}
section WeakTo... | Mathlib/Topology/Algebra/Module/WeakDual.lean | 133 | 137 | theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by |
rw [← tendsto_pi_nhds, Embedding.tendsto_nhds_iff (embedding hB)]
rfl
| 2 | 7.389056 | 1 | 1 | 1 | 932 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 71 | 75 | theorem angle_sub_eq_angle_sub_rev_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) :
angle x (x - y) = angle y (y - x) := by |
refine Real.injOn_cos ⟨angle_nonneg _ _, angle_le_pi _ _⟩ ⟨angle_nonneg _ _, angle_le_pi _ _⟩ ?_
rw [cos_angle, cos_angle, h, ← neg_sub, norm_neg, neg_sub, inner_sub_right, inner_sub_right,
real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm, h, real_inner_comm x y]
| 3 | 20.085537 | 1 | 1.75 | 4 | 1,862 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 78 | 80 | theorem card_le_one_iff_subsingleton [Finite α] : Nat.card α ≤ 1 ↔ Subsingleton α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_le_one_iff_subsingleton]
| 2 | 7.389056 | 1 | 1.2 | 10 | 1,279 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 66 | 70 | theorem localization_localization_map_units [IsLocalization N T]
(y : localizationLocalizationSubmodule M N) : IsUnit (algebraMap R T y) := by |
obtain ⟨y', z, eq⟩ := mem_localizationLocalizationSubmodule.mp y.prop
rw [IsScalarTower.algebraMap_apply R S T, eq, RingHom.map_mul, IsUnit.mul_iff]
exact ⟨IsLocalization.map_units T y', (IsLocalization.map_units _ z).map (algebraMap S T)⟩
| 3 | 20.085537 | 1 | 1.8 | 5 | 1,899 |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 86 | 92 | theorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)
(hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by |
by_cases h : S = ∅
· rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]
exact h𝒜.univ_mem
· rw [← ne_eq, ← Finset.nonempty_iff_ne_empty] at h
exact h𝒜.isSetRing.biInter_mem S h hs
| 5 | 148.413159 | 2 | 2 | 4 | 1,933 |
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 | 67 | 74 | theorem Subalgebra.isAlgebraic_iff (S : Subalgebra R A) :
S.IsAlgebraic ↔ @Algebra.IsAlgebraic R S _ _ S.algebra := by |
delta Subalgebra.IsAlgebraic
rw [Subtype.forall', Algebra.isAlgebraic_def]
refine forall_congr' fun x => exists_congr fun p => and_congr Iff.rfl ?_
have h : Function.Injective S.val := Subtype.val_injective
conv_rhs => rw [← h.eq_iff, AlgHom.map_zero]
rw [← aeval_algHom_apply, S.val_apply]
| 6 | 403.428793 | 2 | 1.125 | 8 | 1,205 |
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*}
theorem powerset_insert (s : Set α) (a : α)... | Mathlib/Data/Set/Image.lean | 1,490 | 1,496 | theorem injective_iff {α β} {f : Option α → β} :
Injective f ↔ Injective (f ∘ some) ∧ f none ∉ range (f ∘ some) := by |
simp only [mem_range, not_exists, (· ∘ ·)]
refine
⟨fun hf => ⟨hf.comp (Option.some_injective _), fun x => hf.ne <| Option.some_ne_none _⟩, ?_⟩
rintro ⟨h_some, h_none⟩ (_ | a) (_ | b) hab
exacts [rfl, (h_none _ hab.symm).elim, (h_none _ hab).elim, congr_arg some (h_some hab)]
| 5 | 148.413159 | 2 | 0.666667 | 15 | 590 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 61 | 64 | theorem coeffIntegerNormalization_mem_support (p : S[X]) (i : ℕ)
(h : coeffIntegerNormalization M p i ≠ 0) : i ∈ p.support := by |
contrapose h
rw [Ne, Classical.not_not, coeffIntegerNormalization, dif_neg h]
| 2 | 7.389056 | 1 | 1.333333 | 6 | 1,454 |
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
attribute [local instance] Classical.propDecidable
open F... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 143 | 149 | theorem uniformContinuous_extend : UniformContinuous (pkg.extend f) := by |
by_cases hf : UniformContinuous f
· rw [pkg.extend_def hf]
exact uniformContinuous_uniformly_extend pkg.uniformInducing pkg.dense hf
· change UniformContinuous (ite _ _ _)
rw [if_neg hf]
exact uniformContinuous_of_const fun a b => by congr 1
| 6 | 403.428793 | 2 | 1.333333 | 3 | 1,403 |
import Mathlib.Topology.GDelta
import Mathlib.MeasureTheory.Group.Arithmetic
import Mathlib.Topology.Instances.EReal
import Mathlib.Analysis.Normed.Group.Basic
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
noncomputable ... | Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | 63 | 69 | theorem borel_eq_top_of_countable [TopologicalSpace α] [T1Space α] [Countable α] : borel α = ⊤ := by |
refine top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ ?_
apply MeasurableSet.biUnion s.to_countable
intro x _
apply MeasurableSet.of_compl
apply GenerateMeasurable.basic
exact isClosed_singleton.isOpen_compl
| 6 | 403.428793 | 2 | 2 | 1 | 2,484 |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 91 | 97 | theorem coev_expComparison (A B : C) :
F.map ((exp.coev A).app B) ≫ (expComparison F A).app (A ⨯ B) =
(exp.coev _).app (F.obj B) ≫ (exp (F.obj A)).map (inv (prodComparison F A B)) := by |
convert unit_transferNatTrans _ _ (prodComparisonNatIso F A).inv B using 3
apply IsIso.inv_eq_of_hom_inv_id -- Porting note: was `ext`
dsimp
simp
| 4 | 54.59815 | 2 | 1.142857 | 7 | 1,210 |
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 | 114 | 119 | theorem δ_comp_δ' {n} {i : Fin (n + 2)} {j : Fin (n + 3)} (H : Fin.castSucc i < j) :
X.δ j ≫ X.δ i =
X.δ (Fin.castSucc i) ≫
X.δ (j.pred fun (hj : j = 0) => by simp [hj, Fin.not_lt_zero] at H) := by |
dsimp [δ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ' H]
| 2 | 7.389056 | 1 | 1 | 8 | 1,050 |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 50 | 54 | theorem pow_sub' (A : M) {m n : ℕ} (ha : IsUnit A.det) (h : n ≤ m) :
A ^ (m - n) = A ^ m * (A ^ n)⁻¹ := by |
rw [← tsub_add_cancel_of_le h, pow_add, Matrix.mul_assoc, mul_nonsing_inv,
tsub_add_cancel_of_le h, Matrix.mul_one]
simpa using ha.pow n
| 3 | 20.085537 | 1 | 1.25 | 4 | 1,330 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 123 | 136 | theorem pdf_eq {X : Ω → E} {s : Set E} (hms : MeasurableSet s)
(hu : IsUniform X s ℙ μ) : pdf X ℙ μ =ᵐ[μ] s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) := by |
by_cases hnt : μ s = ∞
· simp [pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inr hnt), hnt]
by_cases hns : μ s = 0
· filter_upwards [measure_zero_iff_ae_nmem.mp hns,
pdf_eq_zero_of_measure_eq_zero_or_top hu (Or.inl hns)] with x hx h'x
simp [hx, h'x, hns]
have : HasPDF X ℙ μ := hasPDF hns hnt hu
ha... | 12 | 162,754.791419 | 2 | 1.428571 | 7 | 1,510 |
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 | 51 | 53 | theorem Iic_mul_Iic_subset' (a b : α) : Iic a * Iic b ⊆ Iic (a * b) := by |
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_le_mul' hya hzb
| 2 | 7.389056 | 1 | 0.37931 | 29 | 381 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import number_theory.von_mangoldt from "leanprover-community/mathlib"@"c946d6097a6925ad16d7ec55677bbc977f9846de"
namespace ArithmeticFunction
open Finset Nat
open scoped Arit... | Mathlib/NumberTheory/VonMangoldt.lean | 144 | 160 | theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d ∈ n.divisors, (μ d : ℝ) * log d) = -Λ n := by |
simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
have : (∑ i ∈ n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
∑ i ∈ n.divisors, ((μ i : ℝ) * Rea... | 16 | 8,886,110.520508 | 2 | 0.636364 | 11 | 552 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 140 | 145 | theorem natDegree_pow (p : R[X]) (n : ℕ) : natDegree (p ^ n) = n * natDegree p := by |
classical
obtain rfl | hp := eq_or_ne p 0
· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [← leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
| 5 | 148.413159 | 2 | 1.5 | 32 | 1,561 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 141 | 144 | theorem Matrix.Represents.smul {A : Matrix ι ι R} {f : Module.End R M} (h : A.Represents b f)
(r : R) : (r • A).Represents b (r • f) := by |
delta Matrix.Represents at h ⊢
rw [_root_.map_smul, _root_.map_smul, h]
| 2 | 7.389056 | 1 | 1.444444 | 9 | 1,527 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 68 | 71 | theorem Equiv.pointReflection_midpoint_left (x y : P) :
(Equiv.pointReflection (midpoint R x y)) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
| 2 | 7.389056 | 1 | 0.444444 | 9 | 412 |
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Polynomial
namespace Multiset
open Polynomial
section Ring
variable {R : Type*} [CommRing R]
... | Mathlib/RingTheory/Polynomial/Vieta.lean | 94 | 101 | theorem esymm_neg (s : Multiset R) (k : ℕ) : (map Neg.neg s).esymm k = (-1) ^ k * esymm s k := by |
rw [esymm, esymm, ← Multiset.sum_map_mul_left, Multiset.powersetCard_map, Multiset.map_map,
map_congr rfl]
intro x hx
rw [(mem_powersetCard.mp hx).right.symm, ← prod_replicate, ← Multiset.map_const]
nth_rw 3 [← map_id' x]
rw [← prod_map_mul, map_congr rfl, Function.comp_apply]
exact fun z _ => neg_one_... | 7 | 1,096.633158 | 2 | 1.4 | 5 | 1,488 |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 153 | 165 | theorem exists_smul_eq_zero_and_mk_eq {z : M} (hz : Module.IsTorsionBy R M (p ^ pOrder hM z))
{k : ℕ} (f : (R ⧸ R ∙ p ^ k) →ₗ[R] M ⧸ R ∙ z) :
∃ x : M, p ^ k • x = 0 ∧ Submodule.Quotient.mk (p := span R {z}) x = f 1 := by |
have f1 := mk_surjective (R ∙ z) (f 1)
have : p ^ k • f1.choose ∈ R ∙ z := by
rw [← Quotient.mk_eq_zero, mk_smul, f1.choose_spec, ← f.map_smul]
convert f.map_zero; change _ • Submodule.Quotient.mk _ = _
rw [← mk_smul, Quotient.mk_eq_zero, Algebra.id.smul_eq_mul, mul_one]
exact Submodule.mem_span_si... | 10 | 22,026.465795 | 2 | 2 | 5 | 2,030 |
import Mathlib.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adhesive
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.subsheaf from "leanprover-community/mathl... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 110 | 113 | theorem Subpresheaf.homOfLe_ι {G G' : Subpresheaf F} (h : G ≤ G') :
Subpresheaf.homOfLe h ≫ G'.ι = G.ι := by |
ext
rfl
| 2 | 7.389056 | 1 | 1.333333 | 3 | 1,405 |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Measure.MeasureSpace
namespace MeasureTheory
namespace Measure
variable {M : Type*} [Monoid M] [MeasurableSpace M]
@[to_additive conv "Additive convolution of measures."]
noncomputable def mconv (μ : Measure M) (ν : Measure M) :
... | Mathlib/MeasureTheory/Group/Convolution.lean | 41 | 46 | theorem dirac_one_mconv [MeasurableMul₂ M] (μ : Measure M) [SFinite μ] :
(Measure.dirac 1) ∗ μ = μ := by |
unfold mconv
rw [MeasureTheory.Measure.dirac_prod, map_map]
· simp only [Function.comp_def, one_mul, map_id']
all_goals { measurability }
| 4 | 54.59815 | 2 | 1.428571 | 7 | 1,517 |
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
#align_import category_theory.limits.mono_coprod from "leanprover-community/mathli... | Mathlib/CategoryTheory/Limits/MonoCoprod.lean | 63 | 69 | theorem binaryCofan_inr {A B : C} [MonoCoprod C] (c : BinaryCofan A B) (hc : IsColimit c) :
Mono c.inr := by |
haveI hc' : IsColimit (BinaryCofan.mk c.inr c.inl) :=
BinaryCofan.IsColimit.mk _ (fun f₁ f₂ => hc.desc (BinaryCofan.mk f₂ f₁))
(by aesop_cat) (by aesop_cat)
(fun f₁ f₂ m h₁ h₂ => BinaryCofan.IsColimit.hom_ext hc (by aesop_cat) (by aesop_cat))
exact binaryCofan_inl _ hc'
| 5 | 148.413159 | 2 | 2 | 2 | 2,343 |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 149 | 151 | theorem iff_comp {r : Prop → α → Prop} : (· ↔ ·) ∘r r = r := by |
have : (· ↔ ·) = (· = ·) := by funext a b; exact iff_eq_eq
rw [this, eq_comp]
| 2 | 7.389056 | 1 | 1.6 | 15 | 1,743 |
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 47 | 50 | theorem _root_.Asymptotics.IsBigO.integrable (hfm : AEStronglyMeasurable f μ)
(hf : f =O[⊤] g) (hg : Integrable g μ) : Integrable f μ := by |
rewrite [← integrableAtFilter_top] at *
exact hf.integrableAtFilter ⟨univ, univ_mem, hfm.restrict⟩ hg
| 2 | 7.389056 | 1 | 1.125 | 8 | 1,209 |
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 139 | 142 | theorem noetherianSpace_set_iff (s : Set α) :
NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t := by |
simp only [noetherianSpace_iff_isCompact, embedding_subtype_val.isCompact_iff,
Subtype.forall_set_subtype]
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,545 |
import Mathlib.Algebra.Module.Submodule.Ker
open Function Submodule
namespace LinearMap
variable {R N M : Type*} [Semiring R] [AddCommMonoid N] [Module R N]
[AddCommMonoid M] [Module R M] (f i : N →ₗ[R] M)
def iterateMapComap (n : ℕ) := (fun K : Submodule R N ↦ (K.map i).comap f)^[n]
| Mathlib/Algebra/Module/Submodule/IterateMapComap.lean | 45 | 56 | theorem iterateMapComap_le_succ (K : Submodule R N) (h : K.map f ≤ K.map i) (n : ℕ) :
f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) K := by |
nth_rw 2 [iterateMapComap]
rw [iterate_succ', Function.comp_apply, ← iterateMapComap, ← map_le_iff_le_comap]
induction n with
| zero => exact h
| succ n ih =>
simp_rw [iterateMapComap, iterate_succ', Function.comp_apply]
calc
_ ≤ (f.iterateMapComap i n K).map i := map_comap_le _ _
_ ≤ (((... | 10 | 22,026.465795 | 2 | 1.666667 | 3 | 1,828 |
import Mathlib.CategoryTheory.CommSq
#align_import category_theory.lifting_properties.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v
namespace CategoryTheory
open Category
variable {C : Type*} [Category C] {A B B' X Y Y' : C} (i : A ⟶ B) (i' : B ⟶ B') (p : X ⟶ Y... | Mathlib/CategoryTheory/LiftingProperties/Basic.lean | 134 | 138 | theorem iff_of_arrow_iso_left {A B A' B' X Y : C} {i : A ⟶ B} {i' : A' ⟶ B'}
(e : Arrow.mk i ≅ Arrow.mk i') (p : X ⟶ Y) :
HasLiftingProperty i p ↔ HasLiftingProperty i' p := by |
constructor <;> intro
exacts [of_arrow_iso_left e p, of_arrow_iso_left e.symm p]
| 2 | 7.389056 | 1 | 1 | 3 | 916 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 36 | 42 | theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
| 5 | 148.413159 | 2 | 1.642857 | 14 | 1,752 |
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 | 173 | 179 | theorem liftP_iff' {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (a : P.A) (f : P.B a ⟹ α) :
@LiftP.{u} _ P.Obj _ α p ⟨a, f⟩ ↔ ∀ i x, p (f i x) := by |
simp only [liftP_iff, Sigma.mk.inj_iff]; constructor
· rintro ⟨_, _, ⟨⟩, _⟩
assumption
· intro
repeat' first |constructor|assumption
| 5 | 148.413159 | 2 | 0.857143 | 7 | 749 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 131 | 133 | theorem mem_of_formPerm_apply_mem (h : l.formPerm x ∈ l) : x ∈ l := by |
contrapose h
rwa [formPerm_apply_of_not_mem h]
| 2 | 7.389056 | 1 | 0.7 | 10 | 641 |
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 58 | 65 | theorem minFacHelper_0 (n : ℕ)
(h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
MinFacHelper n (nat_lit 3) := by |
refine ⟨by norm_num, by norm_num, ?_⟩
refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
rcases hp.eq_or_lt with rfl|h
· simp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2
· exact h
| 5 | 148.413159 | 2 | 1.8 | 5 | 1,898 |
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
#align_import cat... | Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean | 51 | 54 | theorem pullbackFst_eq_pullback_snd (F : WalkingParallelPair ⥤ C) :
pullbackFst F = pullback.snd := by |
convert (eq_whisker pullback.condition Limits.prod.fst :
(_ : constructEqualizer F ⟶ F.obj WalkingParallelPair.zero) = _) <;> simp
| 2 | 7.389056 | 1 | 1 | 1 | 957 |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 119 | 122 | theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re := by |
apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
intro z
apply LinearIsometry.im_apply_eq_im h
| 3 | 20.085537 | 1 | 1.571429 | 7 | 1,705 |
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
#align_import algebraic_geometry.locally_ringed_space.has_colimits from "leanprov... | Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean | 214 | 223 | theorem imageBasicOpen_image_open :
IsOpen ((coequalizer.π f.1 g.1).base '' (imageBasicOpen f g U s).1) := by |
rw [← (TopCat.homeoOfIso (PreservesCoequalizer.iso (SheafedSpace.forget _) f.1
g.1)).isOpen_preimage, TopCat.coequalizer_isOpen_iff, ← Set.preimage_comp]
erw [← TopCat.coe_comp]
rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison]
dsimp only [SheafedSpace.forget]
-- Porting note (#11224): chan... | 8 | 2,980.957987 | 2 | 2 | 2 | 2,498 |
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
open scoped Classical
open Polynomial Set Function minpoly
namespace... | Mathlib/FieldTheory/Minpoly/Field.lean | 86 | 90 | theorem dvd_map_of_isScalarTower (A K : Type*) {R : Type*} [CommRing A] [Field K] [CommRing R]
[Algebra A K] [Algebra A R] [Algebra K R] [IsScalarTower A K R] (x : R) :
minpoly K x ∣ (minpoly A x).map (algebraMap A K) := by |
refine minpoly.dvd K x ?_
rw [aeval_map_algebraMap, minpoly.aeval]
| 2 | 7.389056 | 1 | 1.5 | 4 | 1,544 |
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 | 68 | 70 | theorem countP_le_length : countP p l ≤ l.length := by |
simp only [countP_eq_length_filter]
apply length_filter_le
| 2 | 7.389056 | 1 | 0.8 | 10 | 703 |
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ι : Type*} {α : ι → Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 61 | 64 | theorem lex_lt_of_lt [∀ i, PartialOrder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : Pi.Lex r (· < ·) x y := by |
simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder r hlt
| 2 | 7.389056 | 1 | 1.666667 | 3 | 1,829 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 76 | 84 | theorem exists_ord_compl_eq_one_iff_isPrimePow {n : ℕ} (hn : n ≠ 1) :
IsPrimePow n ↔ ∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by |
refine ⟨fun h => IsPrimePow.exists_ord_compl_eq_one h, fun h => ?_⟩
rcases h with ⟨p, pp, h⟩
rw [isPrimePow_nat_iff]
rw [← Nat.eq_of_dvd_of_div_eq_one (Nat.ord_proj_dvd n p) h] at hn ⊢
refine ⟨p, n.factorization p, pp, ?_, by simp⟩
contrapose! hn
simp [Nat.le_zero.1 hn]
| 7 | 1,096.633158 | 2 | 1.714286 | 7 | 1,840 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 133 | 139 | theorem mem_span_gramSchmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
f i ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j) := by |
rw [gramSchmidt_def' 𝕜 f i]
simp_rw [orthogonalProjection_singleton]
exact Submodule.add_mem _ (subset_span <| mem_image_of_mem _ hij)
(Submodule.sum_mem _ fun k hk => smul_mem (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j)) _ <|
subset_span <| mem_image_of_mem (gramSchmidt 𝕜 f) <| (Finset.mem_Iio.1 hk).le... | 5 | 148.413159 | 2 | 1.125 | 8 | 1,201 |
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 66 | 68 | theorem at_neg_two : legendreSym p (-2) = χ₈' p := by |
have : (-2 : ZMod p) = (-2 : ℤ) := by norm_cast
rw [legendreSym, ← this, quadraticChar_neg_two ((ringChar_zmod_n p).substr hp), card p]
| 2 | 7.389056 | 1 | 1.5 | 8 | 1,563 |
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"
open FirstOrder Set
namespace FirstOrder
namespace Language
open Structure
variable {L : Language} {M : Type*} [L.Structure M]
namespace Substru... | Mathlib/ModelTheory/FinitelyGenerated.lean | 116 | 135 | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by |
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(Scount.union (Set.countable_singleton h.some)).exists_eq_range
(singleton_nonempty h.some).inr
refine Or.intro_right _ ⟨f, ?_⟩
rw [← h... | 18 | 65,659,969.137331 | 2 | 1.75 | 4 | 1,879 |
set_option autoImplicit true
namespace Array
@[simp]
| Mathlib/Data/Array/ExtractLemmas.lean | 16 | 19 | 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
| 2 | 7.389056 | 1 | 1.4 | 5 | 1,495 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 56 | 58 | theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by |
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl,
subset_compl_iff_disjoint_left]
| 2 | 7.389056 | 1 | 0.333333 | 9 | 339 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 51 | 54 | theorem stdBasisMatrix_add (i : m) (j : n) (a b : α) :
stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b := by |
unfold stdBasisMatrix; ext
split_ifs with h <;> simp [h]
| 2 | 7.389056 | 1 | 1.111111 | 9 | 1,192 |
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Functor.EpiMono
#align_import category_theory.adjunction.evaluation from "leanprover-community/mathlib"@"937c692d73f5130c7fecd3fd32e81419f4e04eb7"
namespace CategoryTheory
open CategoryTheory.Limits
universe v₁ v₂ u₁ u₂
variable... | Mathlib/CategoryTheory/Adjunction/Evaluation.lean | 81 | 86 | theorem NatTrans.mono_iff_mono_app {F G : C ⥤ D} (η : F ⟶ G) : Mono η ↔ ∀ c, Mono (η.app c) := by |
constructor
· intro h c
exact (inferInstance : Mono (((evaluation _ _).obj c).map η))
· intro _
apply NatTrans.mono_of_mono_app
| 5 | 148.413159 | 2 | 2 | 2 | 2,483 |
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
import Mathlib.CategoryTheory.Sites.Equivalence
namespace CategoryTheory
variable {C : Type*} [Category C]
open GrothendieckTopology
namespace Equivalence
variable {D : Type*} [Category D]
variable (e : C ≌ D)
section Regular
variable [Preregular C]
... | Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean | 101 | 106 | theorem preregular_isSheaf_iff (F : Cᵒᵖ ⥤ A) : haveI := e.preregular
IsSheaf (regularTopology C) F ↔ IsSheaf (regularTopology D) (e.inverse.op ⋙ F) := by |
refine ⟨fun hF ↦ ((e.sheafCongrPreregular A).functor.obj ⟨F, hF⟩).cond, fun hF ↦ ?_⟩
rw [isSheaf_of_iso_iff (P' := e.functor.op ⋙ e.inverse.op ⋙ F)]
· exact (e.sheafCongrPreregular A).inverse.obj ⟨e.inverse.op ⋙ F, hF⟩ |>.cond
· exact isoWhiskerRight e.op.unitIso F
| 4 | 54.59815 | 2 | 2 | 2 | 2,253 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 110 | 113 | theorem comp_differentiableOn_iff {f : G → E} {s : Set G} :
DifferentiableOn 𝕜 (iso ∘ f) s ↔ DifferentiableOn 𝕜 f s := by |
rw [DifferentiableOn, DifferentiableOn]
simp only [iso.comp_differentiableWithinAt_iff]
| 2 | 7.389056 | 1 | 1.555556 | 9 | 1,700 |
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 145 | 153 | theorem isBigO_cocompact_zpow_neg_nat (k : ℕ) :
f =O[cocompact E] fun x => ‖x‖ ^ (-k : ℤ) := by |
obtain ⟨d, _, hd'⟩ := f.decay k 0
simp only [norm_iteratedFDeriv_zero] at hd'
simp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith]
refine ⟨d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite ?_⟩
refine (Filter.eventually_cofinite_ne 0).mono fun x hx => ?_
rw [Real.norm_of_nonneg (zpow_nonneg (nor... | 7 | 1,096.633158 | 2 | 1.25 | 8 | 1,308 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 150 | 154 | theorem mapRange_comp (f : ∀ i, β₁ i → β₂ i) (f₂ : ∀ i, β i → β₁ i) (hf : ∀ i, f i 0 = 0)
(hf₂ : ∀ i, f₂ i 0 = 0) (h : ∀ i, (f i ∘ f₂ i) 0 = 0) (g : Π₀ i : ι, β i) :
mapRange (fun i => f i ∘ f₂ i) h g = mapRange f hf (mapRange f₂ hf₂ g) := by |
ext
simp only [mapRange_apply]; rfl
| 2 | 7.389056 | 1 | 1 | 3 | 915 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 58 | 62 | theorem exists_not_acc_lt_of_not_acc {a : α} {r} (h : ¬Acc r a) : ∃ b, ¬Acc r b ∧ r b a := by |
contrapose! h
refine ⟨_, fun b hr => ?_⟩
by_contra hb
exact h b hb hr
| 4 | 54.59815 | 2 | 1.6 | 5 | 1,718 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_import analysis.calculus.deriv.inverse from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
va... | Mathlib/Analysis/Calculus/Deriv/Inverse.lean | 120 | 124 | theorem not_differentiableAt_of_local_left_inverse_hasDerivAt_zero {f g : 𝕜 → 𝕜} {a : 𝕜}
(hf : HasDerivAt f 0 (g a)) (hfg : f ∘ g =ᶠ[𝓝 a] id) : ¬DifferentiableAt 𝕜 g a := by |
intro hg
have := (hf.comp a hg.hasDerivAt).congr_of_eventuallyEq hfg.symm
simpa using this.unique (hasDerivAt_id a)
| 3 | 20.085537 | 1 | 1 | 2 | 1,005 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 76 | 94 | theorem parallelepiped_comp_equiv (v : ι → E) (e : ι' ≃ ι) :
parallelepiped (v ∘ e) = parallelepiped v := by |
simp only [parallelepiped]
let K : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a : ι' => ℝ) e
have : Icc (0 : ι → ℝ) 1 = K '' Icc (0 : ι' → ℝ) 1 := by
rw [← Equiv.preimage_eq_iff_eq_image]
ext x
simp only [K, mem_preimage, mem_Icc, Pi.le_def, Pi.zero_apply, Equiv.piCongrLeft'_apply,
Pi.one_a... | 17 | 24,154,952.753575 | 2 | 1.5 | 6 | 1,672 |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 165 | 167 | theorem vsub_vadd_eq_vsub_sub (p₁ p₂ : P) (g : G) : p₁ -ᵥ (g +ᵥ p₂) = p₁ -ᵥ p₂ - g := by |
rw [← add_right_inj (p₂ -ᵥ p₁ : G), vsub_add_vsub_cancel, ← neg_vsub_eq_vsub_rev, vadd_vsub, ←
add_sub_assoc, ← neg_vsub_eq_vsub_rev, neg_add_self, zero_sub]
| 2 | 7.389056 | 1 | 0.555556 | 9 | 513 |
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 | 174 | 176 | theorem constantCoeff_exp : constantCoeff A (exp A) = 1 := by |
rw [← coeff_zero_eq_constantCoeff_apply, coeff_exp]
simp
| 2 | 7.389056 | 1 | 0.857143 | 14 | 748 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 77 | 83 | theorem symmetry (X Y : C) :
(Limits.BinaryFan.braiding (ℬ X Y).isLimit (ℬ Y X).isLimit).hom ≫
(Limits.BinaryFan.braiding (ℬ Y X).isLimit (ℬ X Y).isLimit).hom =
𝟙 (tensorObj ℬ X Y) := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext;
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| 3 | 20.085537 | 1 | 1.5 | 4 | 1,677 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 116 | 119 | theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by |
cases x
rfl
| 2 | 7.389056 | 1 | 0.833333 | 6 | 733 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 131 | 151 | theorem linearIndependent_iff' :
LinearIndependent R v ↔
∀ s : Finset ι, ∀ g : ι → R, ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0 :=
linearIndependent_iff.trans
⟨fun hf s g hg i his =>
have h :=
hf (∑ i ∈ s, Finsupp.single i (g i)) <| by
simpa only [map_sum, Finsupp.total_single] u... |
{ rw [Finsupp.lapply_apply, Finsupp.single_eq_same] }
_ = ∑ j ∈ s, (Finsupp.lapply i : (ι →₀ R) →ₗ[R] R) (Finsupp.single j (g j)) :=
Eq.symm <|
Finset.sum_eq_single i
(fun j _hjs hji => by rw [Finsupp.lapply_apply, Finsupp.single_eq_of_ne hji])
fun hn... | 11 | 59,874.141715 | 2 | 1 | 7 | 908 |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.FractionalIdeal.Norm
import Mathlib.RingTheory.FractionalIdeal.Operations
variable (K : Type*) [Field K] [NumberField K]
namespace NumberField
open scoped nonZeroDivisors
section Basis
open Module
-- This is necessary to avoid several time... | Mathlib/NumberTheory/NumberField/FractionalIdeal.lean | 93 | 96 | theorem fractionalIdeal_rank (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
finrank ℤ I = finrank ℤ (𝓞 K) := by |
rw [finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank,
finrank_eq_card_basis (basisOfFractionalIdeal K I)]
| 2 | 7.389056 | 1 | 1 | 2 | 1,153 |
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
import Mathlib.Init.Classical
#align_import algebra.group_with_zero.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists DenselyOrdered
variable {α M₀ G₀ M₀' G₀' F F' :... | Mathlib/Algebra/GroupWithZero/Semiconj.lean | 62 | 65 | theorem div_right (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
SemiconjBy a (x / x') (y / y') := by |
rw [div_eq_mul_inv, div_eq_mul_inv]
exact h.mul_right h'.inv_right₀
| 2 | 7.389056 | 1 | 0.75 | 4 | 654 |
import Mathlib.Combinatorics.SimpleGraph.Clique
open Finset
namespace SimpleGraph
variable {V : Type*} [Fintype V] [DecidableEq V] (G H : SimpleGraph V) [DecidableRel G.Adj]
{n r : ℕ}
def IsTuranMaximal (r : ℕ) : Prop :=
G.CliqueFree (r + 1) ∧ ∀ (H : SimpleGraph V) [DecidableRel H.Adj],
H.CliqueFree (r +... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 84 | 92 | theorem not_cliqueFree_of_isTuranMaximal (hn : r ≤ Fintype.card V) (hG : G.IsTuranMaximal r) :
¬G.CliqueFree r := by |
rintro h
obtain ⟨K, _, rfl⟩ := exists_smaller_set (univ : Finset V) r hn
obtain ⟨a, -, b, -, hab, hGab⟩ : ∃ a ∈ K, ∃ b ∈ K, a ≠ b ∧ ¬ G.Adj a b := by
simpa only [isNClique_iff, IsClique, Set.Pairwise, mem_coe, ne_eq, and_true, not_forall,
exists_prop, exists_and_right] using h K
exact hGab <| le_sup_... | 7 | 1,096.633158 | 2 | 2 | 3 | 2,273 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 68 | 76 | theorem edist_inequality (hf : ContractingWith K f) {x y} (h : edist x y ≠ ∞) :
edist x y ≤ (edist x (f x) + edist y (f y)) / (1 - K) :=
suffices edist x y ≤ edist x (f x) + edist y (f y) + K * edist x y by
rwa [ENNReal.le_div_iff_mul_le (Or.inl hf.one_sub_K_ne_zero) (Or.inl one_sub_K_ne_top),
mul_comm,... | rw [edist_comm y, add_right_comm]
_ ≤ edist x (f x) + edist y (f y) + K * edist x y := add_le_add le_rfl (hf.2 _ _)
| 2 | 7.389056 | 1 | 0.6 | 5 | 531 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.basis.bilinear from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
namespace LinearMap
variable {ι₁ ι₂ : Type*}
variable {R R₂ S S₂ M N P Rₗ : Type*}
variable {Mₗ Nₗ Pₗ : Type*}
--... | Mathlib/LinearAlgebra/Basis/Bilinear.lean | 55 | 60 | theorem sum_repr_mul_repr_mul {B : Mₗ →ₗ[Rₗ] Nₗ →ₗ[Rₗ] Pₗ} (x y) :
((b₁'.repr x).sum fun i xi => (b₂'.repr y).sum fun j yj => xi • yj • B (b₁' i) (b₂' j)) =
B x y := by |
conv_rhs => rw [← b₁'.total_repr x, ← b₂'.total_repr y]
simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smul₂,
LinearMap.map_smul]
| 3 | 20.085537 | 1 | 1 | 2 | 911 |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 85 | 88 | theorem convergents_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.TerminatedAt n) :
g.convergents m = g.convergents n := by |
simp only [convergents, denominators_stable_of_terminated n_le_m terminated_at_n,
numerators_stable_of_terminated n_le_m terminated_at_n]
| 2 | 7.389056 | 1 | 1 | 9 | 1,021 |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' :... | Mathlib/Order/SupIndep.lean | 120 | 126 | theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by |
refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
· rw [← sup_map]
exact hs (map_subset_map.2 ht) ((mem_map' _).2 hi) (by rwa [mem_map'])
· classical
rw [map_eq_image]
exact hs.image
| 6 | 403.428793 | 2 | 1.6 | 5 | 1,742 |
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 | 116 | 119 | theorem const.mk_get (x : const n A α) : const.mk n (const.get x) = x := by |
cases x
dsimp [const.get, const.mk]
congr with (_⟨⟩)
| 3 | 20.085537 | 1 | 0.857143 | 7 | 749 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 76 | 79 | theorem IsDiag.map [Zero α] [Zero β] {A : Matrix n n α} (ha : A.IsDiag) {f : α → β} (hf : f 0 = 0) :
(A.map f).IsDiag := by |
intro i j h
simp [ha h, hf]
| 2 | 7.389056 | 1 | 1.25 | 8 | 1,302 |
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
#align_import analysis.convex.specific_functions.basic from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open Real Set NNReal
theorem strictConvexOn_exp : St... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 99 | 122 | theorem one_add_mul_self_lt_rpow_one_add {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠ 0) {p : ℝ} (hp : 1 < p) :
1 + p * s < (1 + s) ^ p := by |
have hp' : 0 < p := zero_lt_one.trans hp
rcases eq_or_lt_of_le hs with rfl | hs
· rwa [add_right_neg, zero_rpow hp'.ne', mul_neg_one, add_neg_lt_iff_lt_add, zero_add]
have hs1 : 0 < 1 + s := neg_lt_iff_pos_add'.mp hs
rcases le_or_lt (1 + p * s) 0 with hs2 | hs2
· exact hs2.trans_lt (rpow_pos_of_pos hs1 _)
... | 22 | 3,584,912,846.131591 | 2 | 2 | 5 | 2,149 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fin.tuple.nat_antidiagonal from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
namespace List.Nat
def antidiagona... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 142 | 147 | theorem antidiagonalTuple_two (n : ℕ) :
antidiagonalTuple 2 n = (antidiagonal n).map fun i => ![i.1, i.2] := by |
rw [antidiagonalTuple]
simp_rw [antidiagonalTuple_one, List.map_singleton]
rw [List.map_eq_bind]
rfl
| 4 | 54.59815 | 2 | 2 | 4 | 2,023 |
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