Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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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 | 542 | 542 | theorem degree_neg (p : R[X]) : degree (-p) = degree p := by | unfold degree; rw [support_neg]
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Set.Lattice
#align_import order.upper_lower.basic from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function OrderDual Set
variable {α β... | Mathlib/Order/UpperLower/Basic.lean | 1,633 | 1,634 | theorem bddBelow_upperClosure : BddBelow (upperClosure s : Set α) ↔ BddBelow s := by |
simp_rw [BddBelow, lowerBounds_upperClosure]
|
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 97 | 113 | theorem tendsto_natCast_div_add_atTop {𝕜 : Type*} [DivisionRing 𝕜] [TopologicalSpace 𝕜]
[CharZero 𝕜] [Algebra ℝ 𝕜] [ContinuousSMul ℝ 𝕜] [TopologicalDivisionRing 𝕜] (x : 𝕜) :
Tendsto (fun n : ℕ ↦ (n : 𝕜) / (n + x)) atTop (𝓝 1) := by |
convert Tendsto.congr' ((eventually_ne_atTop 0).mp (eventually_of_forall fun n hn ↦ _)) _
· exact fun n : ℕ ↦ 1 / (1 + x / n)
· field_simp [Nat.cast_ne_zero.mpr hn]
· have : 𝓝 (1 : 𝕜) = 𝓝 (1 / (1 + x * (0 : 𝕜))) := by
rw [mul_zero, add_zero, div_one]
rw [this]
refine tendsto_const_nhds.div (t... |
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 | 499 | 501 | theorem toExterior_ι (m : M) :
TensorAlgebra.toExterior (TensorAlgebra.ι R m) = ExteriorAlgebra.ι R m := by |
simp [toExterior]
|
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.NonemptyFinLinOrd
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import algebraic_topology.simplex_category from "leanprover-community/mathlib"@"e8ac6315bcfcbaf2d19a046719c3b5... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 197 | 199 | theorem hom_zero_zero (f : ([0] : SimplexCategory) ⟶ [0]) : f = 𝟙 _ := by |
ext : 3
apply @Subsingleton.elim (Fin 1)
|
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 2,064 | 2,066 | theorem factorization_mul {a b : α} (ha : a ≠ 0) (hb : b ≠ 0) :
factorization (a * b) = factorization a + factorization b := by |
simp [factorization, normalizedFactors_mul ha hb]
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 422 | 428 | theorem Monic.sub_of_right {p q : R[X]} (hq : q.leadingCoeff = -1) (hpq : degree p < degree q) :
Monic (p - q) := by |
have : (-q).coeff (-q).natDegree = 1 := by
rw [natDegree_neg, coeff_neg, show q.coeff q.natDegree = -1 from hq, neg_neg]
rw [sub_eq_add_neg]
apply Monic.add_of_right this
rwa [degree_neg]
|
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.cardinal.cofinality from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputable section
open Function Cardinal Set Order
open scoped Classical
open Cardinal Ordinal
un... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,125 | 1,132 | theorem card_iUnion_lt_iff_forall_of_isRegular {ι : Type u} {α : Type u} {t : ι → Set α}
{c : Cardinal} (hc : c.IsRegular) (hι : #ι < c) : #(⋃ i, t i) < c ↔ ∀ i, #(t i) < c := by |
refine ⟨card_lt_of_card_iUnion_lt, fun h ↦ ?_⟩
apply lt_of_le_of_lt (Cardinal.mk_sUnion_le _)
apply Cardinal.mul_lt_of_lt hc.aleph0_le
(lt_of_le_of_lt Cardinal.mk_range_le hι)
apply Cardinal.iSup_lt_of_isRegular hc (lt_of_le_of_lt Cardinal.mk_range_le hι)
simpa
|
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 739 | 740 | theorem subtype_eq_empty {p : α → Prop} [DecidablePred p] {s : Finset α} :
s.subtype p = ∅ ↔ ∀ x, p x → x ∉ s := by | simp [ext_iff, Subtype.forall, Subtype.coe_mk]
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 2,591 | 2,601 | theorem getLast_pmap (p : α → Prop) (f : ∀ a, p a → β) (l : List α)
(hl₁ : ∀ a ∈ l, p a) (hl₂ : l ≠ []) :
(l.pmap f hl₁).getLast (mt List.pmap_eq_nil.1 hl₂) =
f (l.getLast hl₂) (hl₁ _ (List.getLast_mem hl₂)) := by |
induction' l with l_hd l_tl l_ih
· apply (hl₂ rfl).elim
· by_cases hl_tl : l_tl = []
· simp [hl_tl]
· simp only [pmap]
rw [getLast_cons, l_ih _ hl_tl]
simp only [getLast_cons hl_tl]
|
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 | 211 | 220 | theorem Equiv.Perm.fin_5_not_solvable : ¬IsSolvable (Equiv.Perm (Fin 5)) := by |
let x : Equiv.Perm (Fin 5) := ⟨![1, 2, 0, 3, 4], ![2, 0, 1, 3, 4], by decide, by decide⟩
let y : Equiv.Perm (Fin 5) := ⟨![3, 4, 2, 0, 1], ![3, 4, 2, 0, 1], by decide, by decide⟩
let z : Equiv.Perm (Fin 5) := ⟨![0, 3, 2, 1, 4], ![0, 3, 2, 1, 4], by decide, by decide⟩
have key : x = z * ⁅x, y * x * y⁻¹⁆ * z⁻¹ :=... |
import Mathlib.Order.SuccPred.LinearLocallyFinite
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_sampling from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open scoped MeasureTheory ENNReal
open TopologicalSpace
namespace MeasureTheory
nam... | Mathlib/Probability/Martingale/OptionalSampling.lean | 77 | 85 | theorem stoppedValue_ae_eq_restrict_eq (h : Martingale f ℱ μ) (hτ : IsStoppingTime ℱ τ)
(hτ_le : ∀ x, τ x ≤ n) [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_le hτ_le))] (i : ι) :
stoppedValue f τ =ᵐ[μ.restrict {x | τ x = i}] μ[f n|hτ.measurableSpace] := by |
refine Filter.EventuallyEq.trans ?_
(condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const h hτ hτ_le i).symm
rw [Filter.EventuallyEq, ae_restrict_iff' (ℱ.le _ _ (hτ.measurableSet_eq i))]
refine Filter.eventually_of_forall fun x hx => ?_
rw [Set.mem_setOf_eq] at hx
simp_rw [stoppedValue, hx]
|
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import category_theory.limits.shapes.diagonal from "leanprover-community/mathlib"@"f6bab67886fb92c3e2f539cc90a83815f69a189d"
open CategoryTheory
noncomput... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 100 | 109 | theorem pullback_diagonal_map_snd_snd_fst :
(pullback.snd :
pullback (diagonal f)
(map (i₁ ≫ snd) (i₂ ≫ snd) f f (i₁ ≫ fst) (i₂ ≫ fst) i (by simp [condition])
(by simp [condition])) ⟶
_) ≫
snd ≫ i₂ ≫ fst =
pullback.fst := by |
conv_rhs => rw [← Category.comp_id pullback.fst]
rw [← diagonal_snd f, pullback.condition_assoc, pullback.lift_snd]
|
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.fin from "leanprover-community/mathlib"@"7e1c1263b6a25eb90bf16e80d8f47a657e403c4c"
open Equiv
def Equiv.Perm.decomposeFin {n : ℕ} : ... | Mathlib/GroupTheory/Perm/Fin.lean | 29 | 31 | theorem Equiv.Perm.decomposeFin_symm_of_refl {n : ℕ} (p : Fin (n + 1)) :
Equiv.Perm.decomposeFin.symm (p, Equiv.refl _) = swap 0 p := by |
simp [Equiv.Perm.decomposeFin, Equiv.permCongr_def]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,910 | 1,915 | theorem reverse_transfer (hp) :
(p.transfer H hp).reverse =
p.reverse.transfer H (by simp only [edges_reverse, List.mem_reverse]; exact hp) := by |
induction p with
| nil => simp
| cons _ _ ih => simp only [transfer_append, Walk.transfer, reverse_nil, reverse_cons, ih]
|
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
#align_import category_theory.limits.shapes.binary_products from "leanprover-community/mathlib"@"fec1d95fc61c750c1ddbb5b1f7f48b8e811a80d7"
... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,300 | 1,301 | theorem inv_prodComparison_map_snd [IsIso (prodComparison F A B)] :
inv (prodComparison F A B) ≫ F.map prod.snd = prod.snd := by | simp [IsIso.inv_comp_eq]
|
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFuncDense
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import measure_theory.function.strongly_measurable.basic from "leanprover-community/mathlib"@"3b522... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,799 | 1,802 | theorem _root_.aestronglyMeasurable_add_measure_iff [PseudoMetrizableSpace β] {ν : Measure α} :
AEStronglyMeasurable f (μ + ν) ↔ AEStronglyMeasurable f μ ∧ AEStronglyMeasurable f ν := by |
rw [← sum_cond, aestronglyMeasurable_sum_measure_iff, Bool.forall_bool, and_comm]
rfl
|
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.FieldTheory.IsAlgClosed.Basic
#align_import field_theory.is_alg_closed.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
namespace spectrum
open Set Polynomial
open scoped Pointwise Polynomial
universe u v
section Scal... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 135 | 138 | theorem map_pow_of_pos [IsAlgClosed 𝕜] (a : A) {n : ℕ} (hn : 0 < n) :
σ (a ^ n) = (· ^ n) '' σ a := by |
simpa only [aeval_X_pow, eval_pow, eval_X]
using map_polynomial_aeval_of_degree_pos a (X ^ n : 𝕜[X]) (by rwa [degree_X_pow, Nat.cast_pos])
|
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.equipartition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset Fintype
namespace Finpartition
variable {α : Type*} [DecidableEq α] ... | Mathlib/Order/Partition/Equipartition.lean | 68 | 71 | theorem IsEquipartition.average_le_card_part (hP : P.IsEquipartition) (ht : t ∈ P.parts) :
s.card / P.parts.card ≤ t.card := by |
rw [← P.sum_card_parts]
exact Finset.EquitableOn.le hP ht
|
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 157 | 169 | theorem sInf_add {n : ℕ} {p : ℕ → Prop} (hn : n ≤ sInf { m | p m }) :
sInf { m | p (m + n) } + n = sInf { m | p m } := by |
obtain h | ⟨m, hm⟩ := { m | p (m + n) }.eq_empty_or_nonempty
· rw [h, Nat.sInf_empty, zero_add]
obtain hnp | hnp := hn.eq_or_lt
· exact hnp
suffices hp : p (sInf { m | p m } - n + n) from (h.subset hp).elim
rw [Nat.sub_add_cancel hn]
exact csInf_mem (nonempty_of_pos_sInf <| n.zero_le.trans_lt h... |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
universe u v
namespace SimpleGraph
@[ext]
structure Subgraph {V : Type u} (G : SimpleGra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 565 | 566 | theorem edgeSet_iSup (f : ι → G.Subgraph) :
(⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet := by | simp [iSup]
|
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 | 268 | 279 | theorem Monic.irreducible_iff_natDegree' (hp : p.Monic) : Irreducible p ↔ p ≠ 1 ∧
∀ f g : R[X], f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (p.natDegree / 2) := by |
simp_rw [hp.irreducible_iff_natDegree, mem_Ioc, Nat.le_div_iff_mul_le zero_lt_two, mul_two]
apply and_congr_right'
constructor <;> intro h f g hf hg he <;> subst he
· rw [hf.natDegree_mul hg, add_le_add_iff_right]
exact fun ha => (h f g hf hg rfl).elim (ha.1.trans_le ha.2).ne' ha.1.ne'
· simp_rw [hf.natD... |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 171 | 181 | theorem Submodule.rank_le_one_iff_isPrincipal (W : Submodule K V) [Module.Free K W] :
Module.rank K W ≤ 1 ↔ W.IsPrincipal := by |
simp only [rank_le_one_iff, Submodule.isPrincipal_iff, le_antisymm_iff, le_span_singleton_iff,
span_singleton_le_iff_mem]
constructor
· rintro ⟨⟨m, hm⟩, hm'⟩
choose f hf using hm'
exact ⟨m, ⟨fun v hv => ⟨f ⟨v, hv⟩, congr_arg ((↑) : W → V) (hf ⟨v, hv⟩)⟩, hm⟩⟩
· rintro ⟨a, ⟨h, ha⟩⟩
choose f hf us... |
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 | 332 | 343 | theorem dist_sq_add_dist_sq_eq_two_mul_dist_midpoint_sq_add_half_dist_sq (a b c : P) :
dist a b ^ 2 + dist a c ^ 2 = 2 * (dist a (midpoint ℝ b c) ^ 2 + (dist b c / 2) ^ 2) := by |
by_cases hbc : b = c
· simp [hbc, midpoint_self, dist_self, two_mul]
· let m := midpoint ℝ b c
have : dist b c ≠ 0 := (dist_pos.mpr hbc).ne'
have hm := dist_sq_mul_dist_add_dist_sq_mul_dist a b c m (angle_midpoint_eq_pi b c hbc)
simp only [m, dist_left_midpoint, dist_right_midpoint, Real.norm_two] at... |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Int.GCD
instance : GCDMonoid ℕ where
gcd := Nat.gcd
lcm := Nat.lcm
gcd_dvd_left := Nat.gcd_dvd_left
gcd_dvd_right := Nat.gcd_dvd_right
dvd_gcd := Nat.dvd_gcd
gcd_mul_lcm a b := by rw [Nat.gcd_mul_lcm]; rfl
... | Mathlib/Algebra/GCDMonoid/Nat.lean | 71 | 75 | theorem normalize_of_nonpos {z : ℤ} (h : z ≤ 0) : normalize z = -z := by |
obtain rfl | h := h.eq_or_lt
· simp
· rw [normalize_apply, normUnit_eq, if_neg (not_le_of_gt h), Units.val_neg, Units.val_one,
mul_neg_one]
|
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [Algebra F K]
section per... | Mathlib/FieldTheory/PurelyInseparable.lean | 318 | 323 | theorem le_perfectClosure (L : IntermediateField F E) [h : IsPurelyInseparable F L] :
L ≤ perfectClosure F E := by |
rw [isPurelyInseparable_iff_pow_mem F (ringExpChar F)] at h
intro x hx
obtain ⟨n, y, hy⟩ := h ⟨x, hx⟩
exact ⟨n, y, congr_arg (algebraMap L E) hy⟩
|
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 1,287 | 1,291 | theorem valMinAbs_natCast_of_le_half (ha : a ≤ n / 2) : (a : ZMod n).valMinAbs = a := by |
cases n
· simp
· simp [valMinAbs_def_pos, val_natCast, Nat.mod_eq_of_lt (ha.trans_lt <| Nat.div_lt_self' _ 0),
ha]
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 910 | 914 | theorem integral_congr {a b : ℝ} (h : EqOn f g [[a, b]]) :
∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ := by |
rcases le_total a b with hab | hab <;>
simpa [hab, integral_of_le, integral_of_ge] using
setIntegral_congr measurableSet_Ioc (h.mono Ioc_subset_Icc_self)
|
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 426 | 445 | theorem Submartingale.sum_mul_upcrossingStrat_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) :
μ[∑ k ∈ Finset.range n, upcrossingStrat a b f N k * (f (k + 1) - f k)] ≤ μ[f n] - μ[f 0] := by |
have h₁ : (0 : ℝ) ≤
μ[∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)] := by
have := (hf.sum_sub_upcrossingStrat_mul a b N).setIntegral_le (zero_le n) MeasurableSet.univ
rw [integral_univ, integral_univ] at this
refine le_trans ?_ this
simp only [Finset.range_zero, Fin... |
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 327 | 330 | theorem inclusion_inclusion {L : S} (hHK : H ≤ K) (hKL : K ≤ L) (x : H) :
inclusion hKL (inclusion hHK x) = inclusion (hHK.trans hKL) x := by |
cases x
rfl
|
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 1,177 | 1,177 | theorem inv_lt_self_iff : a⁻¹ < a ↔ 1 < a := by | simp [inv_lt_iff_one_lt_mul]
|
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 801 | 801 | theorem one_div_mul_eq_div : 1 / a * b = b / a := by | simp
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.GroupAction.Opposite
#align_import ring_theory.non_zero_divisors from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
variable (M₀ : Type*) [... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 129 | 130 | theorem mul_left_mem_nonZeroDivisors_eq_zero_iff {r x : M₁} (hr : r ∈ M₁⁰) : r * x = 0 ↔ x = 0 := by |
rw [mul_comm, mul_right_mem_nonZeroDivisors_eq_zero_iff hr]
|
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
#align_import algebraic_geometry.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
set_opt... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 71 | 74 | theorem t_fst_snd (i j : 𝒰.J) :
t 𝒰 f g i j ≫ pullback.fst ≫ pullback.snd = pullback.fst ≫ pullback.snd := by |
simp only [t, Category.assoc, pullbackSymmetry_hom_comp_fst_assoc, pullbackAssoc_hom_snd_snd,
pullback.lift_snd, Category.comp_id, pullbackAssoc_inv_snd, pullbackSymmetry_hom_comp_snd_assoc]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 507 | 510 | theorem periodicOrbit_eq_nil_iff_not_periodic_pt :
periodicOrbit f x = Cycle.nil ↔ x ∉ periodicPts f := by |
simp only [periodicOrbit.eq_1, Cycle.coe_eq_nil, List.map_eq_nil, List.range_eq_nil]
exact minimalPeriod_eq_zero_iff_nmem_periodicPts
|
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 105 | 117 | theorem boundary_le_boundary_sup_sup_boundary_inf_left : ∂ a ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) := by |
-- Porting note: the following simp generates the same term as mathlib3 if you remove
-- sup_inf_right from both. With sup_inf_right included, mathlib4 and mathlib3 generate
-- different terms
simp only [boundary, sup_inf_left, sup_inf_right, sup_right_idem, le_inf_iff, sup_assoc,
sup_comm _ a]
refine ⟨⟨... |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 1,395 | 1,400 | theorem quotDualCoannihilatorToDual_nondegenerate (W : Submodule R (Dual R M)) :
W.quotDualCoannihilatorToDual.Nondegenerate := by |
rw [Nondegenerate, separatingLeft_iff_ker_eq_bot, separatingRight_iff_flip_ker_eq_bot]
letI : AddCommGroup W := inferInstance
simp_rw [ker_eq_bot]
exact ⟨W.quotDualCoannihilatorToDual_injective, W.flip_quotDualCoannihilatorToDual_injective⟩
|
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Euler
section Legendre
open ZMod
variable (p : ℕ) [Fact p.Prime]
def legendreSym (a : ℤ) : ℤ :=
... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 195 | 199 | theorem eq_one_iff' {a : ℕ} (ha0 : (a : ZMod p) ≠ 0) :
legendreSym p a = 1 ↔ IsSquare (a : ZMod p) := by |
rw [eq_one_iff]
· norm_cast
· exact mod_cast ha0
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 169 | 213 | theorem tietze_extension_step (f : X →ᵇ ℝ) (e : C(X, Y)) (he : ClosedEmbedding e) :
∃ g : Y →ᵇ ℝ, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖ := by |
have h3 : (0 : ℝ) < 3 := by norm_num1
have h23 : 0 < (2 / 3 : ℝ) := by norm_num1
-- In the trivial case `f = 0`, we take `g = 0`
rcases eq_or_ne f 0 with (rfl | hf)
· use 0
simp
replace hf : 0 < ‖f‖ := norm_pos_iff.2 hf
/- Otherwise, the closed sets `e '' (f ⁻¹' (Iic (-‖f‖ / 3)))` and `e '' (f ⁻¹' (I... |
import Mathlib.Logic.Function.Conjugate
#align_import logic.function.iterate from "leanprover-community/mathlib"@"792a2a264169d64986541c6f8f7e3bbb6acb6295"
universe u v
variable {α : Type u} {β : Type v}
def Nat.iterate {α : Sort u} (op : α → α) : ℕ → α → α
| 0, a => a
| succ k, a => iterate op k (op a)
#a... | Mathlib/Logic/Function/Iterate.lean | 196 | 197 | theorem iterate_pred_comp_of_pos {n : ℕ} (hn : 0 < n) : f^[n.pred] ∘ f = f^[n] := by |
rw [← iterate_succ, Nat.succ_pred_eq_of_pos hn]
|
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Euler
section Legendre
open ZMod
variable (p : ℕ) [Fact p.Prime]
def legendreSym (a : ℤ) : ℤ :=
... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 243 | 249 | theorem eq_one_of_sq_sub_mul_sq_eq_zero' {p : ℕ} [Fact p.Prime] {a : ℤ} (ha : (a : ZMod p) ≠ 0)
{x y : ZMod p} (hx : x ≠ 0) (hxy : x ^ 2 - a * y ^ 2 = 0) : legendreSym p a = 1 := by |
haveI hy : y ≠ 0 := by
rintro rfl
rw [zero_pow two_ne_zero, mul_zero, sub_zero, sq_eq_zero_iff] at hxy
exact hx hxy
exact eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 461 | 468 | theorem mul_rightUnitor {M : Mon_ C} :
(tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by |
rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
simp only [tensorHom_id, id_tensorHom]
slice_lhs 3 4 => rw [rightUnitor_naturality]
slice_lhs 1 3 => rw [← rightUnitor_monoidal]
simp only [Category.assoc, Category.id_comp]
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,035 | 1,038 | theorem power_nat_le_max {c : Cardinal.{u}} {n : ℕ} : c ^ (n : Cardinal.{u}) ≤ max c ℵ₀ := by |
rcases le_or_lt ℵ₀ c with hc | hc
· exact le_max_of_le_left (power_nat_le hc)
· exact le_max_of_le_right (power_lt_aleph0 hc (nat_lt_aleph0 _)).le
|
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 75 | 83 | theorem mem_supported : p ∈ supported R s ↔ ↑p.vars ⊆ s := by |
classical
rw [supported_eq_range_rename, AlgHom.mem_range]
constructor
· rintro ⟨p, rfl⟩
refine _root_.trans (Finset.coe_subset.2 (vars_rename _ _)) ?_
simp
· intro hs
exact exists_rename_eq_of_vars_subset_range p ((↑) : s → σ) Subtype.val_injective (by simpa)
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 663 | 668 | theorem card_Ioo_eq_card_Ico_sub_one (a b : α) : (Ioo a b).card = (Ico a b).card - 1 := by |
classical
by_cases h : a < b
· rw [Ico_eq_cons_Ioo h, card_cons]
exact (Nat.add_sub_cancel _ _).symm
· rw [Ioo_eq_empty h, Ico_eq_empty h, card_empty, Nat.zero_sub]
|
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 766 | 769 | theorem property_of_mem_map_subtype {p : α → Prop} (s : Finset { x // p x }) {a : α}
(h : a ∈ s.map (Embedding.subtype _)) : p a := by |
rcases mem_map.1 h with ⟨x, _, rfl⟩
exact x.2
|
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalClosure
import Mathlib.RingTheory.Polynomial.SeparableDegree
open scoped Classical Polynomial
open FiniteDimensional Polynomial Interm... | Mathlib/FieldTheory/SeparableDegree.lean | 347 | 349 | theorem natSepDegree_map : (f.map (algebraMap F E)).natSepDegree = f.natSepDegree := by |
simp_rw [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure E), aroots_def, map_map,
← IsScalarTower.algebraMap_eq]
|
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.BoundedVariation
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff
import Mathlib.... | Mathlib/Analysis/Calculus/Rademacher.lean | 63 | 77 | theorem ae_lineDifferentiableAt (hf : LipschitzWith C f) (v : E) :
∀ᵐ p ∂μ, LineDifferentiableAt ℝ f p v := by |
let L : ℝ →L[ℝ] E := ContinuousLinearMap.smulRight (1 : ℝ →L[ℝ] ℝ) v
suffices A : ∀ p, ∀ᵐ (t : ℝ) ∂volume, LineDifferentiableAt ℝ f (p + t • v) v from
ae_mem_of_ae_add_linearMap_mem L.toLinearMap volume μ
(measurableSet_lineDifferentiableAt hf.continuous) A
intro p
have : ∀ᵐ (s : ℝ), DifferentiableAt... |
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.BilinearForm.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Algebra.Bilinear
open LinearMap (BilinForm)
universe u v w
variable {R : Type*} {M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
variable {R₁ : Type*} {M₁ :... | Mathlib/LinearAlgebra/BilinearForm/Hom.lean | 240 | 243 | theorem comp_id_left (B : BilinForm R M) (r : M →ₗ[R] M) :
B.comp LinearMap.id r = B.compRight r := by |
ext
rfl
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_character from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
namespace MulChar
section Properties
section quadratic_an... | Mathlib/NumberTheory/MulChar/Basic.lean | 472 | 477 | theorem IsNontrivial.comp {χ : MulChar R R'} (hχ : χ.IsNontrivial) {f : R' →+* R''}
(hf : Function.Injective f) : (χ.ringHomComp f).IsNontrivial := by |
obtain ⟨a, ha⟩ := hχ
use a
simp_rw [ringHomComp_apply, ← RingHom.map_one f]
exact fun h => ha (hf h)
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 476 | 477 | theorem oangle_eq_pi_iff_sbtw {p₁ p₂ p₃ : P} : ∡ p₁ p₂ p₃ = π ↔ Sbtw ℝ p₁ p₂ p₃ := by |
rw [oangle_eq_pi_iff_angle_eq_pi, angle_eq_pi_iff_sbtw]
|
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 160 | 167 | theorem neg {f : ∀ i, E i} (hf : Memℓp f p) : Memℓp (-f) p := by |
rcases p.trichotomy with (rfl | rfl | hp)
· apply memℓp_zero
simp [hf.finite_dsupport]
· apply memℓp_infty
simpa using hf.bddAbove
· apply memℓp_gen
simpa using hf.summable hp
|
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 98 | 99 | theorem inverse_unit (u : M₀ˣ) : inverse (u : M₀) = (u⁻¹ : M₀ˣ) := by |
rw [inverse, dif_pos u.isUnit, IsUnit.unit_of_val_units]
|
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Order.Filter.AtTopBot
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheory.... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 441 | 448 | theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.NeBot] [l.IsCountablyGenerated]
{φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
(htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
Integrable f μ := by |
refine hφ.integrable_of_lintegral_nnnorm_bounded (max 1 (I + 1)) hfm ?_
refine htendsto.eventually (ge_mem_nhds ?_)
refine (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 ?_
exact lt_max_of_lt_right (lt_add_one I)
|
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 53 | 56 | theorem bernoulliFun_endpoints_eq_of_ne_one {k : ℕ} (hk : k ≠ 1) :
bernoulliFun k 1 = bernoulliFun k 0 := by |
rw [bernoulliFun_eval_zero, bernoulliFun, Polynomial.eval_one_map, Polynomial.bernoulli_eval_one,
bernoulli_eq_bernoulli'_of_ne_one hk, eq_ratCast]
|
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
universe u v w
def IsSeq {α : Type u} (s : Stream' (Option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 217 | 228 | theorem destruct_eq_cons {s : Seq α} {a s'} : destruct s = some (a, s') → s = cons a s' := by |
dsimp [destruct]
induction' f0 : get? s 0 with a' <;> intro h
· contradiction
· cases' s with f al
injections _ h1 h2
rw [← h2]
apply Subtype.eq
dsimp [tail, cons]
rw [h1] at f0
rw [← f0]
exact (Stream'.eta f).symm
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 316 | 321 | theorem volume_preimage_mul_left {a : ℝ} (h : a ≠ 0) (s : Set ℝ) :
volume ((a * ·) ⁻¹' s) = ENNReal.ofReal (abs a⁻¹) * volume s :=
calc
volume ((a * ·) ⁻¹' s) = Measure.map (a * ·) volume s :=
((Homeomorph.mulLeft₀ a h).toMeasurableEquiv.map_apply s).symm
_ = ENNReal.ofReal (abs a⁻¹) * volume s := b... | rw [map_volume_mul_left h]; rfl
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,659 | 1,666 | theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by |
obtain hx | hx := hx.symm.lt_or_lt
· exact add_one_lt_exp_of_pos hx
obtain h' | h' := le_or_lt 1 (-x)
· linarith [x.exp_pos]
have hx' : 0 < x + 1 := by linarith
simpa [add_comm, exp_neg, inv_lt_inv (exp_pos _) hx']
using exp_bound_div_one_sub_of_interval' (neg_pos.2 hx) h'
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 124 | 125 | theorem closure_setOf_lt_re (a : ℝ) : closure { z : ℂ | a < z.re } = { z | a ≤ z.re } := by |
simpa only [closure_Ioi] using closure_preimage_re (Ioi a)
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 596 | 598 | theorem oangle_map (x y : V') (f : V ≃ₗᵢ[ℝ] V') :
(Orientation.map (Fin 2) f.toLinearEquiv o).oangle x y = o.oangle (f.symm x) (f.symm y) := by |
simp [oangle, o.kahler_map]
|
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 | 439 | 449 | theorem formPerm_pow_length_eq_one_of_nodup (hl : Nodup l) : formPerm l ^ length l = 1 := by |
ext x
by_cases hx : x ∈ l
· obtain ⟨⟨k, hk⟩, rfl⟩ := get_of_mem hx
simp [formPerm_pow_apply_get _ hl, Nat.mod_eq_of_lt hk]
· have : x ∉ { x | (l.formPerm ^ l.length) x ≠ x } := by
intro H
refine hx ?_
replace H := set_support_zpow_subset l.formPerm l.length H
simpa using support_for... |
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 | 244 | 253 | theorem center_nontrivial [Nontrivial G] [Finite G] : Nontrivial (Subgroup.center G) := by |
classical
cases nonempty_fintype G
have := (hG.of_equiv ConjAct.toConjAct).exists_fixed_point_of_prime_dvd_card_of_fixed_point G
rw [ConjAct.fixedPoints_eq_center] at this
have dvd : p ∣ card G := by
obtain ⟨n, hn0, hn⟩ := hG.nontrivial_iff_card.mp inferInstance
exact hn.symm ▸ dvd_pow_se... |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Sym.Card
open Finset Function
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V) {e : Sym2 V}
section EdgeFinset
variable {G₁ G₂ : SimpleGraph V} [Fintype G.edgeSet] [Fintype G₁.edgeSet] [Finty... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 99 | 100 | theorem edgeFinset_inf [DecidableEq V] : (G₁ ⊓ G₂).edgeFinset = G₁.edgeFinset ∩ G₂.edgeFinset := by |
simp [edgeFinset]
|
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 215 | 222 | theorem iff_isSplitMono_eq_zero {X Y : C} (f : X ⟶ Y) [IsSplitMono f] : IsZero X ↔ f = 0 := by |
rw [iff_id_eq_zero]
constructor
· intro h
rw [← Category.id_comp f, h, zero_comp]
· intro h
rw [← IsSplitMono.id f]
simp only [h, zero_comp]
|
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u → Type u) [Functor F] where
P : PFunctor.{u}
abs : ∀ {α}, P α → F α
repr : ∀ {α}, F α → P α
abs_repr : ∀ {α} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 339 | 345 | theorem Fix.dest_mk (x : F (Fix F)) : Fix.dest (Fix.mk x) = x := by |
unfold Fix.dest; rw [Fix.rec_eq, ← Fix.dest, ← comp_map]
conv =>
rhs
rw [← id_map x]
congr with x
apply Fix.mk_dest
|
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 77 | 80 | theorem eqToHom_naturality {f g : β → C} (z : ∀ b, f b ⟶ g b) {j j' : β} (w : j = j') :
z j ≫ eqToHom (by simp [w]) = eqToHom (by simp [w]) ≫ z j' := by |
cases w
simp
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,003 | 1,016 | theorem integrableOn_iUnion_of_summable_integral_norm {f : X → E} {s : ι → Set X}
(hs : ∀ i : ι, MeasurableSet (s i)) (hi : ∀ i : ι, IntegrableOn f (s i) μ)
(h : Summable fun i : ι => ∫ x : X in s i, ‖f x‖ ∂μ) : IntegrableOn f (iUnion s) μ := by |
refine ⟨AEStronglyMeasurable.iUnion fun i => (hi i).1, (lintegral_iUnion_le _ _).trans_lt ?_⟩
have B := fun i => lintegral_coe_eq_integral (fun x : X => ‖f x‖₊) (hi i).norm
rw [tsum_congr B]
have S' :
Summable fun i : ι =>
(⟨∫ x : X in s i, ‖f x‖₊ ∂μ, setIntegral_nonneg (hs i) fun x _ => NNReal.coe_n... |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 144 | 156 | theorem IsSRGWith.card_commonNeighbors_eq_of_adj_compl (h : G.IsSRGWith n k ℓ μ) {v w : V}
(ha : Gᶜ.Adj v w) : Fintype.card (Gᶜ.commonNeighbors v w) = n - (2 * k - μ) - 2 := by |
simp only [← Set.toFinset_card, commonNeighbors, Set.toFinset_inter, neighborSet_compl,
Set.toFinset_diff, Set.toFinset_singleton, Set.toFinset_compl, ← neighborFinset_def]
simp_rw [compl_neighborFinset_sdiff_inter_eq]
have hne : v ≠ w := ne_of_adj _ ha
rw [compl_adj] at ha
rw [card_sdiff, ← insert_eq, c... |
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 | 748 | 756 | theorem LinearIndependent.total_ne_of_not_mem_support [Nontrivial R] (hv : LinearIndependent R v)
{x : ι} (f : ι →₀ R) (h : x ∉ f.support) : Finsupp.total ι M R v f ≠ v x := by |
replace h : x ∉ (f.support : Set ι) := h
have p := hv.not_mem_span_image h
intro w
rw [← w] at p
rw [Finsupp.span_image_eq_map_total] at p
simp only [not_exists, not_and, mem_map] at p -- Porting note: `mem_map` isn't currently triggered
exact p f (f.mem_supported_support R) rfl
|
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Tactic.Monotonicity
#align_import algebra.continued_fractions.computation.approximations from "leanprover-commu... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 167 | 172 | theorem of_part_num_eq_one {a : K} (nth_part_num_eq : (of v).partialNumerators.get? n = some a) :
a = 1 := by |
obtain ⟨gp, nth_s_eq, gp_a_eq_a_n⟩ : ∃ gp, (of v).s.get? n = some gp ∧ gp.a = a :=
exists_s_a_of_part_num nth_part_num_eq
have : gp.a = 1 := (of_part_num_eq_one_and_exists_int_part_denom_eq nth_s_eq).left
rwa [gp_a_eq_a_n] at this
|
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Bases
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.Copy
#align_import topology.sets.opens from "leanprover-community/mathlib"@"dc6c365e751e34d100e8... | Mathlib/Topology/Sets/Opens.lean | 334 | 341 | theorem IsBasis.isCompact_open_iff_eq_finite_iUnion {ι : Type*} (b : ι → Opens α)
(hb : IsBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i : Set α)) (U : Set α) :
IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by |
apply isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis fun i : ι => (b i).1
· convert (config := {transparency := .default}) hb
ext
simp
· exact hb'
|
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 | 207 | 219 | theorem differentiable_set_subset_D :
{ x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K } ⊆ D f K := by |
intro x hx
rw [D, mem_iInter]
intro e
have : (0 : ℝ) < (1 / 2) ^ e := by positivity
rcases mem_A_of_differentiable this hx.1 with ⟨R, R_pos, hR⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, (1 / 2) ^ n < R :=
exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ) / 2 < 1)
simp only [mem_iUnion, mem_iInter, B, mem_inter_i... |
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 474 | 476 | theorem tendsto_prod_iff {f : α × β → γ} {x : Filter α} {y : Filter β} {z : Filter γ} :
Tendsto f (x ×ˢ y) z ↔ ∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W := by |
simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self_iff]
|
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 632 | 645 | theorem induction_on_all_ex {P : ∀ {m}, L.BoundedFormula α m → Prop} (φ : L.BoundedFormula α n)
(hqf : ∀ {m} {ψ : L.BoundedFormula α m}, IsQF ψ → P ψ)
(hall : ∀ {m} {ψ : L.BoundedFormula α (m + 1)}, P ψ → P ψ.all)
(hex : ∀ {m} {φ : L.BoundedFormula α (m + 1)}, P φ → P φ.ex)
(hse : ∀ {m} {φ₁ φ₂ : L.Bound... |
suffices h' : ∀ {m} {φ : L.BoundedFormula α m}, φ.IsPrenex → P φ from
(hse φ.semanticallyEquivalent_toPrenex).2 (h' φ.toPrenex_isPrenex)
intro m φ hφ
induction' hφ with _ _ hφ _ _ _ hφ _ _ _ hφ
· exact hqf hφ
· exact hall hφ
· exact hex hφ
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_theory.measure.stieltjes from "leanprover-community/mathlib"@"20d5763051978e9bc6428578ed070445df6a18b3"
noncomputable section
open scoped Classical
open Set Filter Function ENNReal NNReal T... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 334 | 337 | theorem borel_le_measurable : borel ℝ ≤ f.outer.caratheodory := by |
rw [borel_eq_generateFrom_Ioi]
refine MeasurableSpace.generateFrom_le ?_
simp (config := { contextual := true }) [f.measurableSet_Ioi]
|
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Polynomial.Content
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.polynomial.basic from "leanprover-commun... | Mathlib/RingTheory/Polynomial/Basic.lean | 1,349 | 1,356 | theorem Polynomial.exists_monic_irreducible_factor {F : Type*} [Field F] (f : F[X])
(hu : ¬IsUnit f) : ∃ g : F[X], g.Monic ∧ Irreducible g ∧ g ∣ f := by |
by_cases hf : f = 0
· exact ⟨X, monic_X, irreducible_X, hf ▸ dvd_zero X⟩
obtain ⟨g, hi, hf⟩ := WfDvdMonoid.exists_irreducible_factor hu hf
have ha : Associated g (g * C g.leadingCoeff⁻¹) := associated_mul_unit_right _ _ <|
isUnit_C.2 (leadingCoeff_ne_zero.2 hi.ne_zero).isUnit.inv
exact ⟨_, monic_mul_lead... |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Polynomial
variable {R : Type*} [CommRing R] {n : ℕ}
theorem isRoot_... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 99 | 104 | theorem isRoot_cyclotomic_iff [NeZero (n : R)] {μ : R} :
IsRoot (cyclotomic n R) μ ↔ IsPrimitiveRoot μ n := by |
have hf : Function.Injective _ := IsFractionRing.injective R (FractionRing R)
haveI : NeZero (n : FractionRing R) := NeZero.nat_of_injective hf
rw [← isRoot_map_iff hf, ← IsPrimitiveRoot.map_iff_of_injective hf, map_cyclotomic, ←
isRoot_cyclotomic_iff']
|
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.Data.Int.Associated
import Mathlib.LinearAlgebra.FreeModule.Determinant
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathli... | Mathlib/RingTheory/Ideal/Norm.lean | 596 | 607 | theorem spanNorm_mul_of_bot_or_top [IsDomain R] [IsDomain S] [Module.Free R S] [Module.Finite R S]
(eq_bot_or_top : ∀ I : Ideal R, I = ⊥ ∨ I = ⊤) (I J : Ideal S) :
spanNorm R (I * J) = spanNorm R I * spanNorm R J := by |
refine le_antisymm ?_ (spanNorm_mul_spanNorm_le R _ _)
cases' eq_bot_or_top (spanNorm R I) with hI hI
· rw [hI, spanNorm_eq_bot_iff.mp hI, bot_mul, spanNorm_bot]
exact bot_le
rw [hI, Ideal.top_mul]
cases' eq_bot_or_top (spanNorm R J) with hJ hJ
· rw [hJ, spanNorm_eq_bot_iff.mp hJ, mul_bot, spanNorm_bot... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Positivity.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
open Function
... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 295 | 301 | theorem mul_card_image_le_card_of_maps_to {f : α → β} {s : Finset α} {t : Finset β}
(Hf : ∀ a ∈ s, f a ∈ t) (n : ℕ) (hn : ∀ a ∈ t, n ≤ (s.filter fun x ↦ f x = a).card) :
n * t.card ≤ s.card :=
calc
n * t.card = ∑ _a ∈ t, n := by | simp [mul_comm]
_ ≤ ∑ a ∈ t, (s.filter fun x ↦ f x = a).card := sum_le_sum hn
_ = s.card := by rw [← card_eq_sum_card_fiberwise Hf]
|
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal
open Set Function Filter
variable {α : Type*} [... | Mathlib/Topology/Semicontinuous.lean | 238 | 245 | theorem IsClosed.lowerSemicontinuous_indicator (hs : IsClosed s) (hy : y ≤ 0) :
LowerSemicontinuous (indicator s fun _x => y) := by |
intro x z hz
by_cases h : x ∈ s <;> simp [h] at hz
· refine Filter.eventually_of_forall fun x' => ?_
by_cases h' : x' ∈ s <;> simp [h', hz, hz.trans_le hy]
· filter_upwards [hs.isOpen_compl.mem_nhds h]
simp (config := { contextual := true }) [hz]
|
import Mathlib.Topology.MetricSpace.Antilipschitz
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
import Mathlib.Data.FunLike.Basic
#align_import topology.metric_space.dilation from "leanprover-community/mathlib"@"93f880918cb51905fd51b76add8273cbc27718ab"
noncomputable s... | Mathlib/Topology/MetricSpace/Dilation.lean | 398 | 402 | theorem ratio_mul (f g : α →ᵈ α) : ratio (f * g) = ratio f * ratio g := by |
by_cases h : ∀ x y : α, edist x y = 0 ∨ edist x y = ∞
· simp [ratio_of_trivial, h]
push_neg at h
exact ratio_comp' h
|
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
assert_not_exists MonoidWithZero
open Function
variable {α β ι M N : Type*}
namespace Set
section Monoid
variable ... | Mathlib/Algebra/Group/Indicator.lean | 447 | 454 | theorem mulIndicator_mul_compl_eq_piecewise [DecidablePred (· ∈ s)] (f g : α → M) :
s.mulIndicator f * sᶜ.mulIndicator g = s.piecewise f g := by |
ext x
by_cases h : x ∈ s
· rw [piecewise_eq_of_mem _ _ _ h, Pi.mul_apply, Set.mulIndicator_of_mem h,
Set.mulIndicator_of_not_mem (Set.not_mem_compl_iff.2 h), mul_one]
· rw [piecewise_eq_of_not_mem _ _ _ h, Pi.mul_apply, Set.mulIndicator_of_not_mem h,
Set.mulIndicator_of_mem (Set.mem_compl h), one_m... |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Directed
import Mathlib.Logic.Equiv.Set
#align_import order.complete_boolean_algebra from "leanprover-community/mathlib"@"71b36b6f3bbe3b44e6538673819324d3ee9fcc96"
set_option autoImplicit true
open Function Set
universe u v w
variable {α : Type u} {β : ... | Mathlib/Order/CompleteBooleanAlgebra.lean | 457 | 460 | theorem iSup_symmDiff_iSup_le {g : ι → α} : (⨆ i, f i) ∆ (⨆ i, g i) ≤ ⨆ i, ((f i) ∆ (g i)) := by |
simp_rw [symmDiff_le_iff, ← iSup_sup_eq]
exact ⟨iSup_mono fun i ↦ sup_comm (g i) _ ▸ le_symmDiff_sup_right ..,
iSup_mono fun i ↦ sup_comm (f i) _ ▸ symmDiff_comm (f i) _ ▸ le_symmDiff_sup_right ..⟩
|
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-community/mathlib"@"cca40788df1b8755d5baf17ab2f27dacc2e17acb"
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 273 | 284 | theorem GammaAux_recurrence1 (s : ℂ) (n : ℕ) (h1 : -s.re < ↑n) :
GammaAux n s = GammaAux n (s + 1) / s := by |
induction' n with n hn generalizing s
· simp only [Nat.zero_eq, CharP.cast_eq_zero, Left.neg_neg_iff] at h1
dsimp only [GammaAux]; rw [GammaIntegral_add_one h1]
rw [mul_comm, mul_div_cancel_right₀]; contrapose! h1; rw [h1]
simp
· dsimp only [GammaAux]
have hh1 : -(s + 1).re < n := by
rw [Na... |
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 1,565 | 1,569 | theorem uniformity_prod_eq_comap_prod [UniformSpace α] [UniformSpace β] :
𝓤 (α × β) =
comap (fun p : (α × β) × α × β => ((p.1.1, p.2.1), (p.1.2, p.2.2))) (𝓤 α ×ˢ 𝓤 β) := by |
dsimp [SProd.sprod]
rw [uniformity_prod, Filter.prod, comap_inf, comap_comap, comap_comap]; rfl
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 1,245 | 1,247 | theorem tendsto_mul_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
Tendsto (fun x => f x * r) l atTop ↔ r < 0 := by |
simp only [mul_comm _ r, tendsto_const_mul_atTop_iff_neg h]
|
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 | 685 | 687 | theorem aequiv_trans {U : Type*} [CommRing U] [Algebra R U] (h : IsAdjoinRoot S f)
(h' : IsAdjoinRoot T f) (h'' : IsAdjoinRoot U f) :
(h.aequiv h').trans (h'.aequiv h'') = h.aequiv h'' := by | ext z; exact h.aequiv_aequiv h' h'' z
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 397 | 398 | theorem one_div_lt_one_div_of_lt (ha : 0 < a) (h : a < b) : 1 / b < 1 / a := by |
rwa [lt_div_iff' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)]
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Sort
import Mathlib.Data.List.FinRange
import Mathlib.LinearAlgebra.Pi
import Mathlib.Logic.Equiv.Fintype
#align_import linear_algebra.multilinear.basic from ... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 342 | 344 | theorem cons_add (f : MultilinearMap R M M₂) (m : ∀ i : Fin n, M i.succ) (x y : M 0) :
f (cons (x + y) m) = f (cons x m) + f (cons y m) := by |
simp_rw [← update_cons_zero x m (x + y), f.map_add, update_cons_zero]
|
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ι α : Type*} {m : MeasurableSpace α} (μ : Measure α)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 106 | 107 | theorem union_left_iff : AEDisjoint μ (s ∪ t) u ↔ AEDisjoint μ s u ∧ AEDisjoint μ t u := by |
simp [union_eq_iUnion, and_comm]
|
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.complex.circle from "leanprover-community/mathlib"@"f333194f5ecd1482191452c5ea60b37d4d6afa08"
open Complex Function Set
open Real
theorem leftInverse_expMapCircle_arg : LeftInvers... | Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 179 | 183 | theorem toCircle_add (x : AddCircle T) (y : AddCircle T) :
@toCircle T (x + y) = toCircle x * toCircle y := by |
induction x using QuotientAddGroup.induction_on'
induction y using QuotientAddGroup.induction_on'
simp_rw [← coe_add, toCircle_apply_mk, mul_add, expMapCircle_add]
|
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 611 | 624 | theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
(tensor_μ C (X₁, X₂) (Y₁, Y₂) ▷ (Z₁ ⊗ Z₂)) ≫
tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =
(α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫
((X₁ ⊗ X₂) ◁ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗... |
dsimp only [tensor_obj, prodMonoidal_tensorObj, tensor_μ]
simp only [whiskerRight_tensor, comp_whiskerRight, whisker_assoc, assoc, Iso.inv_hom_id_assoc,
tensor_whiskerLeft, braiding_tensor_left, MonoidalCategory.whiskerLeft_comp,
braiding_tensor_right]
calc
_ = 𝟙 _ ⊗≫
X₁ ◁ ((β_ X₂ Y₁).hom ▷ (Y... |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 278 | 279 | theorem oangle_add_oangle_rev_neg_left (x y : V) : o.oangle (-x) y + o.oangle (-y) x = 0 := by |
rw [oangle_neg_left_eq_neg_right, oangle_rev, add_left_neg]
|
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 117 | 128 | theorem tendstoUniformlyOn_deriv_of_cthickening_subset (hf : TendstoLocallyUniformlyOn F f φ U)
(hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) {δ : ℝ} (hδ : 0 < δ) (hK : IsCompact K)
(hU : IsOpen U) (hKU : cthickening δ K ⊆ U) :
TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K := by |
have h1 : ∀ᶠ n in φ, ContinuousOn (F n) (cthickening δ K) := by
filter_upwards [hF] with n h using h.continuousOn.mono hKU
have h2 : IsCompact (cthickening δ K) := hK.cthickening
have h3 : TendstoUniformlyOn F f φ (cthickening δ K) :=
(tendstoLocallyUniformlyOn_iff_forall_isCompact hU).mp hf (cthickening... |
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 | 684 | 687 | theorem snorm_one_add_measure (f : α → F) (μ ν : Measure α) :
snorm f 1 (μ + ν) = snorm f 1 μ + snorm f 1 ν := by |
simp_rw [snorm_one_eq_lintegral_nnnorm]
rw [lintegral_add_measure _ μ ν]
|
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Int.Order.Lemmas
#align_import group_theory.submonoid.membership fro... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 639 | 641 | theorem sup_eq_range (s t : Submonoid N) : s ⊔ t = mrange (s.subtype.coprod t.subtype) := by |
rw [mrange_eq_map, ← mrange_inl_sup_mrange_inr, map_sup, map_mrange, coprod_comp_inl, map_mrange,
coprod_comp_inr, range_subtype, range_subtype]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.Perm.List
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Fintype
import Mathlib.GroupTheory.Perm.Cycle.Basic
#align_import grou... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 464 | 470 | theorem cycleFactorsFinset_eq_finset {σ : Perm α} {s : Finset (Perm α)} :
σ.cycleFactorsFinset = s ↔
(∀ f : Perm α, f ∈ s → f.IsCycle) ∧
∃ h : (s : Set (Perm α)).Pairwise Disjoint,
s.noncommProd id (h.mono' fun _ _ => Disjoint.commute) = σ := by |
obtain ⟨l, hl, rfl⟩ := s.exists_list_nodup_eq
simp [cycleFactorsFinset_eq_list_toFinset, hl]
|
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 | 110 | 113 | theorem Ici_mul_Ioi_subset' (a b : α) : Ici a * Ioi b ⊆ Ioi (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_le_of_lt hya hzb
|
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import analysis.normed.group.pointwise from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Metric Set Pointwise Topology
variable {E :... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 143 | 143 | theorem singleton_mul_ball_one : {x} * ball 1 δ = ball x δ := by | simp
|
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 524 | 527 | theorem Integrable.of_measure_le_smul {μ' : Measure α} (c : ℝ≥0∞) (hc : c ≠ ∞) (hμ'_le : μ' ≤ c • μ)
{f : α → β} (hf : Integrable f μ) : Integrable f μ' := by |
rw [← memℒp_one_iff_integrable] at hf ⊢
exact hf.of_measure_le_smul c hc hμ'_le
|
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