Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 207 | 228 | theorem natDegree_ne_zero_induction_on {M : R[X] → Prop} {f : R[X]} (f0 : f.natDegree ≠ 0)
(h_C_add : ∀ {a p}, M p → M (C a + p)) (h_add : ∀ {p q}, M p → M q → M (p + q))
(h_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (monomial n a)) : M f := by |
suffices f.natDegree = 0 ∨ M f from Or.recOn this (fun h => (f0 h).elim) id
refine Polynomial.induction_on f ?_ ?_ ?_
· exact fun a => Or.inl (natDegree_C _)
· rintro p q (hp | hp) (hq | hq)
· refine Or.inl ?_
rw [eq_C_of_natDegree_eq_zero hp, eq_C_of_natDegree_eq_zero hq, ← C_add, natDegree_C]
·... |
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : ℕ → ℕ → ℕ
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 378 | 378 | theorem multichoose_zero_right (n : ℕ) : multichoose n 0 = 1 := by | cases n <;> simp [multichoose]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 226 | 248 | theorem _root_.IsOpen.exists_iUnion_isClosed {U : Set α} (hU : IsOpen U) :
∃ F : ℕ → Set α, (∀ n, IsClosed (F n)) ∧ (∀ n, F n ⊆ U) ∧ ⋃ n, F n = U ∧ Monotone F := by |
obtain ⟨a, a_pos, a_lt_one⟩ : ∃ a : ℝ≥0∞, 0 < a ∧ a < 1 := exists_between zero_lt_one
let F := fun n : ℕ => (fun x => infEdist x Uᶜ) ⁻¹' Ici (a ^ n)
have F_subset : ∀ n, F n ⊆ U := fun n x hx ↦ by
by_contra h
have : infEdist x Uᶜ ≠ 0 := ((ENNReal.pow_pos a_pos _).trans_le hx).ne'
exact this (infEdist... |
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 | 507 | 510 | theorem index_ne_zero_of_finite [hH : Finite (G ⧸ H)] : H.index ≠ 0 := by |
cases nonempty_fintype (G ⧸ H)
rw [index_eq_card]
exact Fintype.card_ne_zero
|
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 78 | 80 | theorem mem_graph_iff {c : α × M} {f : α →₀ M} : c ∈ f.graph ↔ f c.1 = c.2 ∧ c.2 ≠ 0 := by |
cases c
exact mk_mem_graph_iff
|
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 83 | 87 | theorem comap_eq_id_of_eq_id (f : MvPolynomial σ R →ₐ[R] MvPolynomial σ R) (hf : ∀ φ, f φ = φ)
(x : σ → R) : comap f x = x := by |
convert comap_id_apply x
ext1 φ
simp [hf, AlgHom.id_apply]
|
import Mathlib.MeasureTheory.Covering.DensityTheorem
#align_import measure_theory.covering.liminf_limsup from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
open Set Filter Metric MeasureTheory TopologicalSpace
open scoped NNReal ENNReal Topology
variable {α : Type*} [MetricSpace α] [... | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | 41 | 150 | theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
(hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
{M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop... |
/- Sketch of proof:
Assume that `p` is identically true for simplicity. Let `Y₁ i = cthickening (r₁ i) (s i)`, define
`Y₂` similarly except using `r₂`, and let `(Z i) = ⋃_{j ≥ i} (Y₂ j)`. Our goal is equivalent to
showing that `μ ((limsup Y₁) \ (Z i)) = 0` for all `i`.
Assume for contradiction that `μ ((li... |
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,977 | 1,983 | theorem isNormal_iff_lt_succ_and_blsub_eq {f : Ordinal.{u} → Ordinal.{max u v}} :
IsNormal f ↔ (∀ a, f a < f (succ a)) ∧
∀ o, IsLimit o → (blsub.{_, v} o fun x _ => f x) = f o := by |
rw [isNormal_iff_lt_succ_and_bsup_eq.{u, v}, and_congr_right_iff]
intro h
constructor <;> intro H o ho <;> have := H o ho <;>
rwa [← bsup_eq_blsub_of_lt_succ_limit ho fun a _ => h a] at *
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 544 | 545 | theorem eq_ones_iff_le_length {c : Composition n} : c = ones n ↔ n ≤ c.length := by |
simp [eq_ones_iff_length, le_antisymm_iff, c.length_le]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 376 | 379 | theorem coeff_succ_X_mul (n : ℕ) (φ : R⟦X⟧) : coeff R (n + 1) (X * φ) = coeff R n φ := by |
simp only [coeff, Finsupp.single_add, add_comm n 1]
convert φ.coeff_add_monomial_mul (single () 1) (single () n) _
rw [one_mul]; rfl
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory.measure.content from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
universe u v w
noncomputable section
open Set Topologic... | Mathlib/MeasureTheory/Measure/Content.lean | 219 | 223 | theorem is_mul_left_invariant_innerContent [Group G] [TopologicalGroup G]
(h : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (g : G)
(U : Opens G) :
μ.innerContent (Opens.comap (Homeomorph.mulLeft g).toContinuousMap U) = μ.innerContent U := by |
convert μ.innerContent_comap (Homeomorph.mulLeft g) (fun K => h g) U
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.fixed_point from "leanprover-community/mathlib"@"0dd4319a17376eda5763cd0a7e0d35bbaaa50e83"
noncomputable section
universe u v
open Function Order
namespace Ordinal
section
variable {ι ... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 435 | 437 | theorem iterate_le_nfp (f a n) : f^[n] a ≤ nfp f a := by |
rw [← sup_iterate_eq_nfp]
exact le_sup _ n
|
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
import Mathlib.Data.Sigma.Basic
import Mathlib.Logic.Equiv.Basic
import Mathlib.Tactic.Common
#align_import combinatorics.quiver.covering from "leanprover-community/mathlib"@"188a411e916e1119e502dbe35b8b475716362401"
open Funct... | Mathlib/Combinatorics/Quiver/Covering.lean | 307 | 309 | theorem Prefunctor.costar_conj_star (u : U) :
φ.costar u = Quiver.starEquivCostar (φ.obj u) ∘ φ.star u ∘ (Quiver.starEquivCostar u).symm := by |
ext ⟨v, f⟩ <;> simp
|
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 := _ ∈ _)]
|
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finsupp.Fin
import Mathlib.Data.Finsupp.Indicator
#align_import algebra.bi... | Mathlib/Algebra/BigOperators/Finsupp.lean | 124 | 127 | theorem sum_ite_self_eq_aux [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) :
(if a ∈ f.support then f a else 0) = f a := by |
simp only [mem_support_iff, ne_eq, ite_eq_left_iff, not_not]
exact fun h ↦ h.symm
|
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
|
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 675 | 678 | theorem Coplanar.finiteDimensional_vectorSpan {s : Set P} (h : Coplanar k s) :
FiniteDimensional k (vectorSpan k s) := by |
refine IsNoetherian.iff_fg.1 (IsNoetherian.iff_rank_lt_aleph0.2 (lt_of_le_of_lt h ?_))
exact Cardinal.lt_aleph0.2 ⟨2, rfl⟩
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 147 | 148 | theorem quotient_norm_sub_rev {S : AddSubgroup M} (x y : M ⧸ S) : ‖x - y‖ = ‖y - x‖ := by |
rw [← neg_sub, quotient_norm_neg]
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.rat.defs from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- TODO: If `Inv` was defined earlier than `Algebra.Group.De... | Mathlib/Data/Rat/Defs.lean | 537 | 540 | theorem coe_int_num_of_den_eq_one {q : ℚ} (hq : q.den = 1) : (q.num : ℚ) = q := by |
conv_rhs => rw [← num_divInt_den q, hq]
rw [intCast_eq_divInt]
rfl
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 99 | 101 | theorem coeff_of_lt_order (n : ℕ) (h : ↑n < order φ) : coeff R n φ = 0 := by |
contrapose! h
exact order_le _ h
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 108 | 116 | theorem measurable_measure_mul_right (hs : MeasurableSet s) :
Measurable fun x => μ ((fun y => y * x) ⁻¹' s) := by |
suffices
Measurable fun y =>
μ ((fun x => (x, y)) ⁻¹' ((fun z : G × G => ((1 : G), z.1 * z.2)) ⁻¹' univ ×ˢ s))
by convert this using 1; ext1 x; congr 1 with y : 1; simp
apply measurable_measure_prod_mk_right
apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs)
infer_instance... |
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.exterior_algebra.basic from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
universe u1 u2 u3 u4 u5
variable (R : Type u1) [CommRing R]
variable (M : Type u2) [... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 274 | 286 | theorem ι_mul_prod_list {n : ℕ} (f : Fin n → M) (i : Fin n) :
(ι R <| f i) * (List.ofFn fun i => ι R <| f i).prod = 0 := by |
induction' n with n hn
· exact i.elim0
· rw [List.ofFn_succ, List.prod_cons, ← mul_assoc]
by_cases h : i = 0
· rw [h, ι_sq_zero, zero_mul]
· replace hn :=
congr_arg (ι R (f 0) * ·) <| hn (fun i => f <| Fin.succ i) (i.pred h)
simp only at hn
rw [Fin.succ_pred, ← mul_assoc, mul_zero... |
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsupp from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
variable {ι : Type*} {R : Type*} {S : Type*} {M : ι → Type*} {N : Type*}
n... | Mathlib/LinearAlgebra/DFinsupp.lean | 206 | 209 | theorem mapRange.linearMap_id :
(mapRange.linearMap fun i => (LinearMap.id : β₂ i →ₗ[R] _)) = LinearMap.id := by |
ext
simp [linearMap]
|
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 | 53 | 56 | theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) := by |
rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact,
CompleteLattice.isSupFiniteCompact_iff_all_elements_compact]
exact forall_congr' Opens.isCompactElement_iff
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.Sylow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.TFAE
#align_import group_theory.nilpotent from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144... | Mathlib/GroupTheory/Nilpotent.lean | 384 | 392 | theorem least_ascending_central_series_length_eq_nilpotencyClass :
Nat.find ((nilpotent_iff_finite_ascending_central_series G).mp hG) =
Group.nilpotencyClass G := by |
refine le_antisymm (Nat.find_mono ?_) (Nat.find_mono ?_)
· intro n hn
exact ⟨upperCentralSeries G, upperCentralSeries_isAscendingCentralSeries G, hn⟩
· rintro n ⟨H, ⟨hH, hn⟩⟩
rw [← top_le_iff, ← hn]
exact ascending_central_series_le_upper H hH n
|
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 98 | 100 | theorem kernelSubobject_arrow :
(kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow := by |
simp [kernelSubobjectIso]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 439 | 440 | theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by |
by_cases h : arg x = π <;> simp [arg_inv, h]
|
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 107 | 110 | theorem Monotone.isChain_range [LinearOrder α] [Preorder β] {f : α → β} (hf : Monotone f) :
IsChain (· ≤ ·) (range f) := by |
rw [← image_univ]
exact (isChain_of_trichotomous _).image (· ≤ ·) _ _ hf
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 186 | 191 | theorem trace_one : trace R M 1 = (finrank R M : R) := by |
cases subsingleton_or_nontrivial R
· simp [eq_iff_true_of_subsingleton]
have b := Module.Free.chooseBasis R M
rw [trace_eq_matrix_trace R b, toMatrix_one, finrank_eq_card_chooseBasisIndex]
simp
|
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.deriv.zpow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter
open Filter Asymptotics Set
variable {𝕜 : Typ... | Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 138 | 140 | theorem iter_deriv_inv (k : ℕ) (x : 𝕜) :
deriv^[k] Inv.inv x = (∏ i ∈ Finset.range k, (-1 - i : 𝕜)) * x ^ (-1 - k : ℤ) := by |
simpa only [zpow_neg_one, Int.cast_neg, Int.cast_one] using iter_deriv_zpow (-1) x k
|
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Filter
variable {ι α β : Type*}
class Bornology (α : Type*) where
cobounded' : Filter α
le_cofinite' : cobounded' ≤ cofinite
#align borno... | Mathlib/Topology/Bornology/Basic.lean | 294 | 295 | theorem isBounded_sUnion {S : Set (Set α)} (hs : S.Finite) :
IsBounded (⋃₀ S) ↔ ∀ s ∈ S, IsBounded s := by | rw [sUnion_eq_biUnion, isBounded_biUnion hs]
|
import Mathlib.AlgebraicTopology.DoldKan.Normalized
#align_import algebraic_topology.dold_kan.homotopy_equivalence from "leanprover-community/mathlib"@"f951e201d416fb50cc7826171d80aa510ec20747"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
CategoryTheory.Preadditive Simplicial DoldKan
nonco... | Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean | 52 | 58 | theorem homotopyPToId_eventually_constant {q n : ℕ} (hqn : n < q) :
((homotopyPToId X (q + 1)).hom n (n + 1) : X _[n] ⟶ X _[n + 1]) =
(homotopyPToId X q).hom n (n + 1) := by |
simp only [homotopyHσToZero, AlternatingFaceMapComplex.obj_X, Nat.add_eq, Homotopy.trans_hom,
Homotopy.ofEq_hom, Pi.zero_apply, Homotopy.add_hom, Homotopy.compLeft_hom, add_zero,
Homotopy.nullHomotopy'_hom, ComplexShape.down_Rel, hσ'_eq_zero hqn (c_mk (n + 1) n rfl),
dite_eq_ite, ite_self, comp_zero, zer... |
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
#align_import analysis.complex.phragmen_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Asymptotics Metric Complex Bornology
open scoped Topology Filter R... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 529 | 550 | theorem quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
(hB : ∃ c < (2 : ℝ), ∃ B,
f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
(hre : ∀ x : ℝ, x ≤ 0 → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C) (hz_re : z.re ≤ 0)
(hz_im : z.im ≤ 0) : ‖f z‖ ≤ C := by |
obtain ⟨z, rfl⟩ : ∃ z', -z' = z := ⟨-z, neg_neg z⟩
simp only [neg_re, neg_im, neg_nonpos] at hz_re hz_im
change ‖(f ∘ Neg.neg) z‖ ≤ C
have H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0) := by
intro w hw
simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, mem_Iio] using hw
refine
quadrant... |
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 | 223 | 225 | theorem chain'_append_cons_cons {b c : α} {l₁ l₂ : List α} :
Chain' R (l₁ ++ b :: c :: l₂) ↔ Chain' R (l₁ ++ [b]) ∧ R b c ∧ Chain' R (c :: l₂) := by |
rw [chain'_split, chain'_cons]
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 191 | 194 | theorem Convex.sum_mem (hs : Convex R s) (h₀ : ∀ i ∈ t, 0 ≤ w i) (h₁ : ∑ i ∈ t, w i = 1)
(hz : ∀ i ∈ t, z i ∈ s) : (∑ i ∈ t, w i • z i) ∈ s := by |
simpa only [h₁, centerMass, inv_one, one_smul] using
hs.centerMass_mem h₀ (h₁.symm ▸ zero_lt_one) hz
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ι R M σ : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 145 | 147 | theorem degree_eq_of_mem_mem {x : M} {i j : ι} (hxi : x ∈ ℳ i) (hxj : x ∈ ℳ j) (hx : x ≠ 0) :
i = j := by |
contrapose! hx; rw [← decompose_of_mem_same ℳ hxj, decompose_of_mem_ne ℳ hxi hx]
|
import Mathlib.Topology.Sheaves.PUnit
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Topology.Sheaves.Functors
#align_import topology.sheaves.skyscraper from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open TopologicalSpace TopCat CategoryTheory CategoryT... | Mathlib/Topology/Sheaves/Skyscraper.lean | 68 | 74 | theorem skyscraperPresheaf_eq_pushforward
[hd : ∀ U : Opens (TopCat.of PUnit.{u + 1}), Decidable (PUnit.unit ∈ U)] :
skyscraperPresheaf p₀ A =
ContinuousMap.const (TopCat.of PUnit) p₀ _*
skyscraperPresheaf (X := TopCat.of PUnit) PUnit.unit A := by |
convert_to @skyscraperPresheaf X p₀ (fun U => hd <| (Opens.map <| ContinuousMap.const _ p₀).obj U)
C _ _ A = _ <;> congr
|
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 284 | 289 | theorem summable_norm_apply (p : FormalMultilinearSeries 𝕜 E F) {x : E}
(hx : x ∈ EMetric.ball (0 : E) p.radius) : Summable fun n : ℕ => ‖p n fun _ => x‖ := by |
rw [mem_emetric_ball_zero_iff] at hx
refine .of_nonneg_of_le
(fun _ ↦ norm_nonneg _) (fun n ↦ ((p n).le_opNorm _).trans_eq ?_) (p.summable_norm_mul_pow hx)
simp
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 348 | 352 | theorem sum_le' {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α}
(hF : ∀ i ∈ s, padicNorm p (F i) ≤ t) (ht : 0 ≤ t) : padicNorm p (∑ i ∈ s, F i) ≤ t := by |
obtain rfl | hs := Finset.eq_empty_or_nonempty s
· simp [ht]
· exact sum_le hs hF
|
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a fu... | Mathlib/MeasureTheory/Group/LIntegral.lean | 46 | 49 | theorem lintegral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (x * g) ∂μ) = ∫⁻ x, f x ∂μ := by |
convert (lintegral_map_equiv f <| MeasurableEquiv.mulRight g).symm using 1
simp [map_mul_right_eq_self μ g]
|
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 136 | 138 | theorem coe_le_one (p : PMF α) (a : α) : p a ≤ 1 := by |
refine hasSum_le (fun b => ?_) (hasSum_ite_eq a (p a)) (hasSum_coe_one p)
split_ifs with h <;> simp only [h, zero_le', le_rfl]
|
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 98 | 99 | theorem Duplicate.of_duplicate_cons {y : α} (h : x ∈+ y :: l) (hx : x ≠ y) : x ∈+ l := by |
simpa [duplicate_cons_iff, hx.symm] using h
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 199 | 205 | theorem left_bot_eq_left_quot (H : Subgroup G) :
Quotient (⊥ : Subgroup G).1 (H : Set G) = (G ⧸ H) := by |
unfold Quotient
congr
ext
simp_rw [← bot_rel_eq_leftRel H]
rfl
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876... | Mathlib/Analysis/Convex/Topology.lean | 228 | 230 | theorem Convex.add_smul_mem_interior' {s : Set E} (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ closure s)
(hy : x + y ∈ interior s) {t : 𝕜} (ht : t ∈ Ioc (0 : 𝕜) 1) : x + t • y ∈ interior s := by |
simpa only [add_sub_cancel_left] using hs.add_smul_sub_mem_interior' hx hy ht
|
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 94 | 97 | theorem Filter.Tendsto.atBot_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg
simpa [(· ∘ ·)] using tendsto_neg_atTop_atBot.comp this
|
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : ℕ → ℕ → ℕ
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 257 | 261 | theorem choose_eq_asc_factorial_div_factorial (n k : ℕ) :
(n + k).choose k = (n + 1).ascFactorial k / k ! := by |
apply Nat.mul_left_cancel k.factorial_pos
rw [← ascFactorial_eq_factorial_mul_choose]
exact (Nat.mul_div_cancel' <| factorial_dvd_ascFactorial _ _).symm
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Order.Monotone.Monovary
import Mathlib.Tactic.Abel
#align_impo... | Mathlib/Algebra/Order/Rearrangement.lean | 308 | 310 | theorem Antivary.sum_smul_lt_sum_smul_comp_perm_iff (hfg : Antivary f g) :
((∑ i, f i • g i) < ∑ i, f i • g (σ i)) ↔ ¬Antivary f (g ∘ σ) := by |
simp [(hfg.antivaryOn _).sum_smul_lt_sum_smul_comp_perm_iff fun _ _ ↦ mem_univ _]
|
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 | 268 | 270 | theorem insert_inj_on' (s : Finset α) : Set.InjOn (fun a => insert a s) (sᶜ : Finset α) := by |
rw [coe_compl]
exact s.insert_inj_on
|
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory
open CategoryTheory.Limits
open Opposite
universe w v u
variable {C : Type u} [Ca... | Mathlib/CategoryTheory/Sites/Plus.lean | 323 | 330 | theorem plus_hom_ext {P Q : Cᵒᵖ ⥤ D} (η γ : J.plusObj P ⟶ Q) (hQ : Presheaf.IsSheaf J Q)
(h : J.toPlus P ≫ η = J.toPlus P ≫ γ) : η = γ := by |
have : γ = J.plusLift (J.toPlus P ≫ γ) hQ := by
apply plusLift_unique
rfl
rw [this]
apply plusLift_unique
exact h
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 178 | 185 | theorem isCyclic_of_surjective {H G F : Type*} [Group H] [Group G] [hH : IsCyclic H]
[FunLike F H G] [MonoidHomClass F H G] (f : F) (hf : Function.Surjective f) :
IsCyclic G := by |
obtain ⟨x, hx⟩ := hH
refine ⟨f x, fun a ↦ ?_⟩
obtain ⟨a, rfl⟩ := hf a
obtain ⟨n, rfl⟩ := hx a
exact ⟨n, (map_zpow _ _ _).symm⟩
|
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.pfunctor.multivariate.M from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
set_option linter.uppercaseLean3 false
universe u
open MvFunctor
namespace MvPFunctor
open TypeVec... | Mathlib/Data/PFunctor/Multivariate/M.lean | 318 | 325 | theorem M.dest_map {α β : TypeVec n} (g : α ⟹ β) (x : P.M α) :
M.dest P (g <$$> x) = (appendFun g fun x => g <$$> x) <$$> M.dest P x := by |
cases' x with a f
rw [map_eq]
conv =>
rhs
rw [M.dest, M.dest', map_eq, appendFun_comp_splitFun]
rfl
|
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.GCD.BigOperators
namespace Nat
variable {ι : Type*}
lemma modEq_list_prod_iff {a b} {l : List ℕ} (co : l.Pairwise Coprime) :
a ≡ b [MOD l.prod] ↔ ∀ i, a ≡ b [MOD l.get i] := by
induction' l with m l ih
· si... | Mathlib/Data/Nat/ChineseRemainder.lean | 93 | 105 | theorem chineseRemainderOfList_modEq_unique (l : List ι)
(co : l.Pairwise (Coprime on s)) {z} (hz : ∀ i ∈ l, z ≡ a i [MOD s i]) :
z ≡ chineseRemainderOfList a s l co [MOD (l.map s).prod] := by |
induction' l with i l ih
· simp [modEq_one]
· simp only [List.map_cons, List.prod_cons, chineseRemainderOfList]
have : Coprime (s i) (l.map s).prod := by
simp only [coprime_list_prod_right_iff, List.mem_map, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂]
intro j hj
exact (L... |
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 605 | 606 | theorem tendsto_inv_nhdsWithin_Iio_inv {a : H} : Tendsto Inv.inv (𝓝[<] a⁻¹) (𝓝[>] a) := by |
simpa only [inv_inv] using @tendsto_inv_nhdsWithin_Iio _ _ _ _ a⁻¹
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 418 | 425 | theorem sum_fin [AddCommMonoid S] (f : ℕ → R → S) (hf : ∀ i, f i 0 = 0) {n : ℕ} {p : R[X]}
(hn : p.degree < n) : (∑ i : Fin n, f i (p.coeff i)) = p.sum f := by |
by_cases hp : p = 0
· rw [hp, sum_zero_index, Finset.sum_eq_zero]
intro i _
exact hf i
rw [sum_over_range' _ hf n ((natDegree_lt_iff_degree_lt hp).mpr hn),
Fin.sum_univ_eq_sum_range fun i => f i (p.coeff i)]
|
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import group_theory.free_abelian_group_finsupp from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e600... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 54 | 59 | theorem FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup :
toFinsupp.comp toFreeAbelianGroup = AddMonoidHom.id (X →₀ ℤ) := by |
ext x y; simp only [AddMonoidHom.id_comp]
rw [AddMonoidHom.comp_assoc, Finsupp.toFreeAbelianGroup_comp_singleAddHom]
simp only [toFinsupp, AddMonoidHom.coe_comp, Finsupp.singleAddHom_apply, Function.comp_apply,
one_smul, lift.of, AddMonoidHom.flip_apply, smulAddHom_apply, AddMonoidHom.id_apply]
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special cases about linear (as opposed to semilinear) maps
variable {𝕜 𝕜₂ 𝕜₃ E Eₗ F Fₗ G Gₗ 𝓕 : Type*}
... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 210 | 220 | theorem sSup_closed_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
[SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
[NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
sSup ((fun x => ‖f x‖₊) '' closedBall 0 1... |
have hbdd : ∀ y ∈ (fun x => ‖f x‖₊) '' closedBall 0 1, y ≤ ‖f‖₊ := by
rintro - ⟨x, hx, rfl⟩
exact f.unit_le_opNorm x (mem_closedBall_zero_iff.1 hx)
refine le_antisymm (csSup_le ((nonempty_closedBall.mpr zero_le_one).image _) hbdd) ?_
rw [← sSup_unit_ball_eq_nnnorm]
exact csSup_le_csSup ⟨‖f‖₊, hbdd⟩ ((n... |
import Mathlib.MeasureTheory.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
#align_import measure_theory.measure.ae_measurable from "leanprover-community/mathlib"@"3310acfa9787aa171db6d4cba3945f6f275fe9f2"
open scoped Classical
open MeasureTheory MeasureTheory.Measure Filter Set Funct... | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 238 | 243 | theorem aemeasurable_const' (h : ∀ᵐ (x) (y) ∂μ, f x = f y) : AEMeasurable f μ := by |
rcases eq_or_ne μ 0 with (rfl | hμ)
· exact aemeasurable_zero_measure
· haveI := ae_neBot.2 hμ
rcases h.exists with ⟨x, hx⟩
exact ⟨const α (f x), measurable_const, EventuallyEq.symm hx⟩
|
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 | 50 | 52 | theorem fold_cons (h : a ∉ s) : (cons a s h).fold op b f = f a * s.fold op b f := by |
dsimp only [fold]
rw [cons_val, Multiset.map_cons, fold_cons_left]
|
import Mathlib.Order.Partition.Equipartition
#align_import combinatorics.simple_graph.regularity.equitabilise from "leanprover-community/mathlib"@"bf7ef0e83e5b7e6c1169e97f055e58a2e4e9d52d"
open Finset Nat
namespace Finpartition
variable {α : Type*} [DecidableEq α] {s t : Finset α} {m n a b : ℕ} {P : Finpartitio... | Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean | 205 | 215 | theorem exists_equipartition_card_eq (hn : n ≠ 0) (hs : n ≤ s.card) :
∃ P : Finpartition s, P.IsEquipartition ∧ P.parts.card = n := by |
rw [← pos_iff_ne_zero] at hn
have : (n - s.card % n) * (s.card / n) + s.card % n * (s.card / n + 1) = s.card := by
rw [tsub_mul, mul_add, ← add_assoc,
tsub_add_cancel_of_le (Nat.mul_le_mul_right _ (mod_lt _ hn).le), mul_one, add_comm,
mod_add_div]
refine
⟨(indiscrete (card_pos.1 <| hn.trans_l... |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 168 | 173 | theorem weightedHomogeneousSubmodule_eq_finsupp_supported (w : σ → M) (m : M) :
weightedHomogeneousSubmodule R w m = Finsupp.supported R R { d | weightedDegree w d = m } := by |
ext x
rw [mem_supported, Set.subset_def]
simp only [Finsupp.mem_support_iff, mem_coe]
rfl
|
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 178 | 243 | theorem exists_partition_polynomial_aux (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : Fq[X]} (hb : b ≠ 0)
(A : Fin n → Fq[X]) : ∃ t : Fin n → Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊),
∀ i₀ i₁ : Fin n, t i₀ = t i₁ ↔
(cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε := by |
have hbε : 0 < cardPowDegree b • ε := by
rw [Algebra.smul_def, eq_intCast]
exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hε
-- We go by induction on the size `A`.
induction' n with n ih
· refine ⟨finZeroElim, finZeroElim⟩
-- Show `anti_archimedean` also holds for real distances.
have an... |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 331 | 337 | theorem value_at (a : ℤ) {R : Type*} [CommSemiring R] (χ : R →* ℤ)
(hp : ∀ (p : ℕ) (pp : p.Prime), p ≠ 2 → @legendreSym p ⟨pp⟩ a = χ p) {b : ℕ} (hb : Odd b) :
J(a | b) = χ b := by |
conv_rhs => rw [← prod_factors hb.pos.ne', cast_list_prod, map_list_prod χ]
rw [jacobiSym, List.map_map, ← List.pmap_eq_map Nat.Prime _ _ fun _ => prime_of_mem_factors]
congr 1; apply List.pmap_congr
exact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <| dvd_of_mem_factors h)
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 385 | 386 | theorem nodup_sublists' {l : List α} : Nodup (sublists' l) ↔ Nodup l := by |
rw [sublists'_eq_sublists, nodup_map_iff reverse_injective, nodup_sublists, nodup_reverse]
|
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 | 538 | 541 | theorem subalgebra_top_rank_eq_submodule_top_rank :
Module.rank F (⊤ : Subalgebra F E) = Module.rank F (⊤ : Submodule F E) := by |
rw [← Algebra.top_toSubmodule]
rfl
|
import Mathlib.Geometry.Euclidean.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
noncomputable section
open scoped Classical
open scoped RealInnerProductSpace
namespace Affine
namespace Simplex
open Finset AffineSubspac... | Mathlib/Geometry/Euclidean/MongePoint.lean | 297 | 327 | theorem eq_mongePoint_of_forall_mem_mongePlane {n : ℕ} {s : Simplex ℝ P (n + 2)} {i₁ : Fin (n + 3)}
{p : P} (h : ∀ i₂, i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂) : p = s.mongePoint := by |
rw [← @vsub_eq_zero_iff_eq V]
have h' : ∀ i₂, i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈
(ℝ ∙ s.points i₁ -ᵥ s.points i₂)ᗮ ⊓ vectorSpan ℝ (Set.range s.points) := by
intro i₂ hne
rw [← s.direction_mongePlane, vsub_right_mem_direction_iff_mem s.mongePoint_mem_mongePlane]
exact h i₂ hne
have hi : p -ᵥ s.mongeP... |
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace Multiset
open List
instance functor : Functor Multiset... | Mathlib/Data/Multiset/Functor.lean | 137 | 143 | theorem naturality {G H : Type _ → Type _} [Applicative G] [Applicative H] [CommApplicative G]
[CommApplicative H] (eta : ApplicativeTransformation G H) {α β : Type _} (f : α → G β)
(x : Multiset α) : eta (traverse f x) = traverse (@eta _ ∘ f) x := by |
refine Quotient.inductionOn x ?_
intro
simp only [quot_mk_to_coe, traverse, lift_coe, Function.comp_apply,
ApplicativeTransformation.preserves_map, LawfulTraversable.naturality]
|
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 117 | 120 | theorem lintegral_rpow_nnnorm_eq_rpow_snorm' {f : α → F} (hq0_lt : 0 < q) :
(∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) = snorm' f q μ ^ q := by |
rw [snorm', ← ENNReal.rpow_mul, one_div, inv_mul_cancel, ENNReal.rpow_one]
exact (ne_of_lt hq0_lt).symm
|
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 424 | 441 | theorem pullback_fst_image_snd_preimage (f : X ⟶ Z) (g : Y ⟶ Z) (U : Set Y) :
(pullback.fst : pullback f g ⟶ _) '' ((pullback.snd : pullback f g ⟶ _) ⁻¹' U) =
f ⁻¹' (g '' U) := by |
ext x
constructor
· rintro ⟨(y : (forget TopCat).obj _), hy, rfl⟩
exact
⟨(pullback.snd : pullback f g ⟶ _) y, hy,
(ConcreteCategory.congr_hom pullback.condition y).symm⟩
· rintro ⟨y, hy, eq⟩
-- next 5 lines were
-- `exact ⟨(TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, eq.symm⟩, by ... |
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.box_integral.box.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Finset Function Filter Metric Classical Topology Filter ENNReal
noncomputable... | Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean | 53 | 62 | theorem mem_splitCenterBox {s : Set ι} {y : ι → ℝ} :
y ∈ I.splitCenterBox s ↔ y ∈ I ∧ ∀ i, (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s := by |
simp only [splitCenterBox, mem_def, ← forall_and]
refine forall_congr' fun i ↦ ?_
dsimp only [Set.piecewise]
split_ifs with hs <;> simp only [hs, iff_true_iff, iff_false_iff, not_lt]
exacts [⟨fun H ↦ ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩,
fun H ↦ ⟨H.2, H.1.2⟩⟩,
⟨fun H... |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 158 | 158 | theorem one_rpow (x : ℝ) : (1 : ℝ) ^ x = 1 := by | simp [rpow_def]
|
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 216 | 218 | theorem union_ae_eq_right_of_ae_eq_empty (h : s =ᵐ[μ] (∅ : Set α)) : (s ∪ t : Set α) =ᵐ[μ] t := by |
convert ae_eq_set_union h (ae_eq_refl t)
rw [empty_union]
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
#align_import ring_theory.adjoin.basic fr... | Mathlib/RingTheory/Adjoin/Basic.lean | 247 | 258 | theorem mem_adjoin_of_map_mul {s} {x : A} {f : A →ₗ[R] B} (hf : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
(h : x ∈ adjoin R s) : f x ∈ adjoin R (f '' (s ∪ {1})) := by |
refine
@adjoin_induction R A _ _ _ _ (fun a => f a ∈ adjoin R (f '' (s ∪ {1}))) x h
(fun a ha => subset_adjoin ⟨a, ⟨Set.subset_union_left ha, rfl⟩⟩) (fun r => ?_)
(fun y z hy hz => by simpa [hy, hz] using Subalgebra.add_mem _ hy hz) fun y z hy hz => by
simpa [hy, hz, hf y z] using Subalgebra.mu... |
import Mathlib.Data.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
#align_import number_theory.dioph from "leanprover-community/mathlib"@"a66d07e27d5b5b8ac1147cacfe353478e5c14002"
open Fin2 Function Nat Sum
local infixr:67 " ::ₒ " => Option.elim'
local ... | Mathlib/NumberTheory/Dioph.lean | 225 | 232 | theorem induction {C : Poly α → Prop} (H1 : ∀ i, C (proj i)) (H2 : ∀ n, C (const n))
(H3 : ∀ f g, C f → C g → C (f - g)) (H4 : ∀ f g, C f → C g → C (f * g)) (f : Poly α) : C f := by |
cases' f with f pf
induction' pf with i n f g pf pg ihf ihg f g pf pg ihf ihg
· apply H1
· apply H2
· apply H3 _ _ ihf ihg
· apply H4 _ _ ihf ihg
|
import Mathlib.CategoryTheory.Preadditive.ProjectiveResolution
import Mathlib.Algebra.Homology.HomotopyCategory
import Mathlib.Tactic.SuppressCompilation
suppress_compilation
noncomputable section
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
open Category Limits Projective
set_... | Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean | 99 | 102 | theorem lift_commutes {Y Z : C} (f : Y ⟶ Z) (P : ProjectiveResolution Y)
(Q : ProjectiveResolution Z) : lift f P Q ≫ Q.π = P.π ≫ (ChainComplex.single₀ C).map f := by |
ext
simp [lift, liftFZero, liftFOne]
|
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 | 401 | 407 | theorem nmem_nonZeroDivisors_iff {P : R[X]} : P ∉ R[X]⁰ ↔ ∃ a : R, a ≠ 0 ∧ a • P = 0 := by |
refine ⟨fun hP ↦ ?_, fun ⟨a, ha, h⟩ h1 ↦ ha <| C_eq_zero.1 <| (h1 _) <| smul_eq_C_mul a ▸ h⟩
by_contra! h
obtain ⟨Q, hQ⟩ := _root_.nmem_nonZeroDivisors_iff.1 hP
refine hQ.2 (eq_zero_of_mul_eq_zero_of_smul P (fun a ha ↦ ?_) Q (mul_comm P _ ▸ hQ.1))
contrapose! ha
exact h a ha
|
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 234 | 237 | theorem d_pos_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : 0 < d := by |
refine pos_of_mul_pos_left ?_ (sq_nonneg a.y)
rw [a.prop_y, sub_pos]
exact one_lt_pow ha two_ne_zero
|
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α β M E : Type*}
open Set Filter
@[to_additive]
| Mathlib/Order/Filter/IndicatorFunction.lean | 63 | 66 | theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
(hs : Monotone s) (f : α → β) (a : α) :
(fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a := by |
classical exact hs.piecewise_eventually_eq_iUnion f 1 a
|
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 784 | 791 | theorem nf_repr_split {o o' m} [NF o] (h : split o = (o', m)) : NF o' ∧ repr o = repr o' + m := by |
cases' e : split' o with a n
cases' nf_repr_split' e with s₁ s₂
rw [split_eq_scale_split' e] at h
injection h; substs o' n
simp only [repr_scale, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero,
opow_one, s₂.symm, and_true]
infer_instance
|
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 561 | 569 | theorem derivWithin_Ioi_eq_Ici {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : ℝ → E)
(x : ℝ) : derivWithin f (Ioi x) x = derivWithin f (Ici x) x := by |
by_cases H : DifferentiableWithinAt ℝ f (Ioi x) x
· have A := H.hasDerivWithinAt.Ici_of_Ioi
have B := (differentiableWithinAt_Ioi_iff_Ici.1 H).hasDerivWithinAt
simpa using (uniqueDiffOn_Ici x).eq left_mem_Ici A B
· rw [derivWithin_zero_of_not_differentiableWithinAt H,
derivWithin_zero_of_not_differ... |
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
variable [TopologicalSpace α]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 75 | 76 | theorem nhdsWithin_univ (a : α) : 𝓝[Set.univ] a = 𝓝 a := by |
rw [nhdsWithin, principal_univ, inf_top_eq]
|
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 102 | 105 | theorem eventually_cardinal_forall {p : α → ι → Prop} (hic : #ι < c) :
(∀ᶠ x in l, ∀ i, p x i) ↔ ∀ i, ∀ᶠ x in l, p x i := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_iInter_mem hic
|
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 52 | 54 | theorem lift_le_continuum {c : Cardinal.{u}} : lift.{v} c ≤ 𝔠 ↔ c ≤ 𝔠 := by |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_le]
|
import Mathlib.CategoryTheory.Adjunction.Whiskering
import Mathlib.CategoryTheory.Sites.PreservesSheafification
#align_import category_theory.sites.adjunction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open GrothendieckTopology CategoryTheory Limits Op... | Mathlib/CategoryTheory/Sites/Adjunction.lean | 148 | 160 | theorem adjunctionToTypes_counit_app_val {G : Type max v u ⥤ D} (adj : G ⊣ forget D)
(X : Sheaf J D) :
((adjunctionToTypes J adj).counit.app X).val =
sheafifyLift J ((Functor.associator _ _ _).hom ≫ (adj.whiskerRight _).counit.app _) X.2 := by |
apply sheafifyLift_unique
dsimp only [adjunctionToTypes, Adjunction.comp, NatTrans.comp_app,
instCategorySheaf_comp_val, instCategorySheaf_id_val]
rw [adjunction_counit_app_val]
erw [Category.id_comp, sheafifyMap_sheafifyLift, toSheafify_sheafifyLift]
ext
dsimp [sheafEquivSheafOfTypes, Equivalence.symm... |
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
#align_import data.nat.parity from "leanpr... | Mathlib/Algebra/Order/Ring/Abs.lean | 192 | 193 | theorem abs_dvd (a b : α) : |a| ∣ b ↔ a ∣ b := by |
cases' abs_choice a with h h <;> simp only [h, neg_dvd]
|
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 | 63 | 73 | theorem IsPrimePow.exists_ord_compl_eq_one {n : ℕ} (h : IsPrimePow n) :
∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by |
rcases eq_or_ne n 0 with (rfl | hn0); · cases not_isPrimePow_zero h
rcases isPrimePow_iff_factorization_eq_single.mp h with ⟨p, k, hk0, h1⟩
rcases em' p.Prime with (pp | pp)
· refine absurd ?_ hk0.ne'
simp [← Nat.factorization_eq_zero_of_non_prime n pp, h1]
refine ⟨p, pp, ?_⟩
refine Nat.eq_of_factoriza... |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 1,045 | 1,047 | theorem clusterPt_principal_iff_frequently :
ClusterPt x (𝓟 s) ↔ ∃ᶠ y in 𝓝 x, y ∈ s := by |
simp only [clusterPt_principal_iff, frequently_iff, Set.Nonempty, exists_prop, mem_inter_iff]
|
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
#align_import group_theory.noncomm_pi_coprod from "leanprover-community/mathlib"@"6f9f36364eae3f42368b04858fd66d6d9ae730d8"
... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 125 | 137 | theorem noncommPiCoprod_mulSingle (i : ι) (y : N i) :
noncommPiCoprod ϕ hcomm (Pi.mulSingle i y) = ϕ i y := by |
change Finset.univ.noncommProd (fun j => ϕ j (Pi.mulSingle i y j)) (fun _ _ _ _ h => hcomm h _ _)
= ϕ i y
rw [← Finset.insert_erase (Finset.mem_univ i)]
rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ (Finset.not_mem_erase i _)]
rw [Pi.mulSingle_eq_same]
rw [Finset.noncommProd_eq_pow_card]
· rw [one_p... |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 869 | 875 | theorem reaches_eval {σ} {f : σ → Option σ} {a b} (ab : Reaches f a b) : eval f a = eval f b := by |
refine Part.ext fun _ ↦ ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· have ⟨ac, c0⟩ := mem_eval.1 h
exact mem_eval.2 ⟨(or_iff_left_of_imp fun cb ↦ (eval_maximal h).1 cb ▸ ReflTransGen.refl).1
(reaches_total ab ac), c0⟩
· have ⟨bc, c0⟩ := mem_eval.1 h
exact mem_eval.2 ⟨ab.trans bc, c0⟩
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.Tactic.IntervalCases
#align_import group_theory.p_gr... | Mathlib/GroupTheory/PGroup.lean | 144 | 152 | theorem card_eq_or_dvd : Nat.card G = 1 ∨ p ∣ Nat.card G := by |
cases fintypeOrInfinite G
· obtain ⟨n, hn⟩ := iff_card.mp hG
rw [Nat.card_eq_fintype_card, hn]
cases' n with n n
· exact Or.inl rfl
· exact Or.inr ⟨p ^ n, by rw [pow_succ']⟩
· rw [Nat.card_eq_zero_of_infinite]
exact Or.inr ⟨0, rfl⟩
|
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 864 | 866 | theorem continuousAt_of_tendsto_nhds [TopologicalSpace Y] [T1Space Y] {f : X → Y} {x : X} {y : Y}
(h : Tendsto f (𝓝 x) (𝓝 y)) : ContinuousAt f x := by |
rwa [ContinuousAt, eq_of_tendsto_nhds h]
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 113 | 124 | theorem sinZeta_two_mul_nat_add_one' (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) / (2 * k + 1) / Gammaℂ (2 * k + 1) *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [sinZeta_two_mul_nat_add_one hk hx]
congr 1
have : (2 * k + 1)! = (2 * k + 1) * Complex.Gamma (2 * k + 1) := by
rw [(by simp : Complex.Gamma (2 * k + 1) = Complex.Gamma (↑(2 * k) + 1)),
Complex.Gamma_nat_eq_factorial, ← Nat.cast_ofNat (R := ℂ), ← Nat.cast_mul,
← Nat.cast_add_one, ← Nat.cast_m... |
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal M... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 180 | 183 | theorem toOuterMeasure_apply_singleton (a : α) : p.toOuterMeasure {a} = p a := by |
refine (p.toOuterMeasure_apply {a}).trans ((tsum_eq_single a fun b hb => ?_).trans ?_)
· exact ite_eq_right_iff.2 fun hb' => False.elim <| hb hb'
· exact ite_eq_left_iff.2 fun ha' => False.elim <| ha' rfl
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 262 | 264 | theorem cauchySeq_iff {u : ℕ → α} :
CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V := by |
simp only [cauchySeq_iff', Filter.eventually_atTop_prod_self', mem_preimage, Prod.map_apply]
|
import Mathlib.Data.Fin.VecNotation
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Perm.ViaEmbedding
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.SetTheory.Cardinal.Basic
#align_import group_theory.solvable from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd298... | Mathlib/GroupTheory/Solvable.lean | 56 | 59 | theorem derivedSeries_normal (n : ℕ) : (derivedSeries G n).Normal := by |
induction' n with n ih
· exact (⊤ : Subgroup G).normal_of_characteristic
· exact @Subgroup.commutator_normal G _ (derivedSeries G n) (derivedSeries G n) ih ih
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,039 | 2,041 | theorem exists_of_lt_mex {ι} {f : ι → Ordinal} {a} (ha : a < mex f) : ∃ i, f i = a := by |
by_contra! ha'
exact ha.not_le (mex_le_of_ne ha')
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 201 | 202 | theorem coprime_mul_left_add_right (m n k : ℕ) : Coprime m (m * k + n) ↔ Coprime m n := by |
rw [Coprime, Coprime, gcd_mul_left_add_right]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 248 | 249 | theorem lift_algebraMap (h : IsAdjoinRoot S f) (a : R) :
h.lift i x hx (algebraMap R S a) = i a := by | rw [h.algebraMap_apply, lift_map, eval₂_C]
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import data.nat.choose.bounds from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
open Nat
variable {α : Type*} [LinearOrderedSemif... | Mathlib/Data/Nat/Choose/Bounds.lean | 32 | 37 | theorem choose_le_pow (r n : ℕ) : (n.choose r : α) ≤ (n ^ r : α) / r ! := by |
rw [le_div_iff']
· norm_cast
rw [← Nat.descFactorial_eq_factorial_mul_choose]
exact n.descFactorial_le_pow r
exact mod_cast r.factorial_pos
|
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 275 | 277 | theorem finish_isReduced : u.finish.IsReduced := by |
dsimp [IsReduced]
rfl
|
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 | 49 | 49 | theorem gal_X_isSolvable : IsSolvable (X : F[X]).Gal := by | infer_instance
|
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