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.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
assert_not_exists Finset
assert_not_exists Module
assert_not_exists Submonoid
assert_not_exists FloorRing
structure Real w... | Mathlib/Data/Real/Basic.lean | 325 | 325 | theorem mk_mul {f g : CauSeq ℚ abs} : mk (f * g) = mk f * mk g := by | simp [mk, ← ofCauchy_mul]
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Real.Basic
import Mathlib.Data.Set.Image
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set Function
structure Complex : Type where... | Mathlib/Data/Complex/Basic.lean | 326 | 326 | theorem I_mul_re (z : ℂ) : (I * z).re = -z.im := by | simp
|
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
open Nat Function
namespace List
theorem rel_of_pairwise_cons (p : (a :: l).Pairwise R) : ∀ {a'}, a' ∈ l → R a a' :=
(pairwise_cons.1 p).1 _
theorem Pairwise.of_cons (p : (a :: l).Pairwise R) : Pairwise R l :=
(pairwise_cons.1 p).2
theorem... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 173 | 189 | theorem pairwise_iff_forall_sublist : l.Pairwise R ↔ (∀ {a b}, [a,b] <+ l → R a b) := by |
induction l with
| nil => simp
| cons hd tl IH =>
rw [List.pairwise_cons]
constructor <;> intro h
· intro
| a, b, .cons _ hab => exact IH.mp h.2 hab
| _, b, .cons₂ _ hab => refine h.1 _ (hab.subset ?_); simp
· constructor
· intro x hx
apply h
rw [List.cons_sublis... |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 244 | 245 | theorem Gamma_cong_eq_self (N : ℕ) (g : ConjAct SL(2, ℤ)) : g • Gamma N = Gamma N := by |
apply Subgroup.Normal.conjAct (Gamma_normal N)
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousFunction.Polynomial
import Mathlib.Topology.ContinuousFunction.Compact
#align_import analysis.special_functions.bernstein from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba14... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 61 | 64 | theorem bernstein_apply (n ν : ℕ) (x : I) :
bernstein n ν x = (n.choose ν : ℝ) * (x : ℝ) ^ ν * (1 - (x : ℝ)) ^ (n - ν) := by |
dsimp [bernstein, Polynomial.toContinuousMapOn, Polynomial.toContinuousMap, bernsteinPolynomial]
simp
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 184 | 187 | theorem sum_totient' (n : ℕ) : (∑ m ∈ (range n.succ).filter (· ∣ n), φ m) = n := by |
convert sum_totient _ using 1
simp only [Nat.divisors, sum_filter, range_eq_Ico]
rw [sum_eq_sum_Ico_succ_bot] <;> simp
|
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
import Mathlib.CategoryTheory.MorphismProperty.Composition
universe v u
namespace CategoryTheory
open Limits
namespace MorphismProperty
variable {C : Type u} [Category.{v} C]
def StableUnderBaseChange (P : ... | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | 256 | 268 | theorem universally_respectsIso (P : MorphismProperty C) : P.universally.RespectsIso := by |
constructor
· intro X Y Z e f hf X' Z' i₁ i₂ f' H
have : IsPullback (𝟙 _) (i₁ ≫ e.hom) i₁ e.inv :=
IsPullback.of_horiz_isIso
⟨by rw [Category.id_comp, Category.assoc, e.hom_inv_id, Category.comp_id]⟩
exact hf _ _ _
(by simpa only [Iso.inv_hom_id_assoc, Category.id_comp] using this.past... |
import Mathlib.Topology.MetricSpace.PseudoMetric
open Filter
open scoped Uniformity Topology
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
theorem Metric.complete_of_convergent_controlled_sequences (B : ℕ → Real) (hB : ∀ n, 0 < B n)
(H : ∀ u : ℕ → α, (∀ N n m... | Mathlib/Topology/MetricSpace/Cauchy.lean | 113 | 123 | theorem cauchySeq_bdd {u : ℕ → α} (hu : CauchySeq u) : ∃ R > 0, ∀ m n, dist (u m) (u n) < R := by |
rcases Metric.cauchySeq_iff'.1 hu 1 zero_lt_one with ⟨N, hN⟩
rsuffices ⟨R, R0, H⟩ : ∃ R > 0, ∀ n, dist (u n) (u N) < R
· exact ⟨_, add_pos R0 R0, fun m n =>
lt_of_le_of_lt (dist_triangle_right _ _ _) (add_lt_add (H m) (H n))⟩
let R := Finset.sup (Finset.range N) fun n => nndist (u n) (u N)
refine ⟨↑R +... |
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.polynomials from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset Asymptotic... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 64 | 70 | theorem tendsto_atTop_iff_leadingCoeff_nonneg :
Tendsto (fun x => eval x P) atTop atTop ↔ 0 < P.degree ∧ 0 ≤ P.leadingCoeff := by |
refine ⟨fun h => ?_, fun h => tendsto_atTop_of_leadingCoeff_nonneg P h.1 h.2⟩
have : Tendsto (fun x => P.leadingCoeff * x ^ P.natDegree) atTop atTop :=
(isEquivalent_atTop_lead P).tendsto_atTop h
rw [tendsto_const_mul_pow_atTop_iff, ← pos_iff_ne_zero, natDegree_pos_iff_degree_pos] at this
exact ⟨this.1, th... |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 122 | 127 | theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
(star s).topologicalClosure = star s.topologicalClosure := by |
suffices ∀ t : Subalgebra R A, (star t).topologicalClosure ≤ star t.topologicalClosure from
le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))
exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)
(isClosed_closure.preimage continu... |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 104 | 106 | theorem lcm_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).lcm = GCDMonoid.lcm s₁.lcm s₂.lcm := by |
rw [← lcm_dedup, dedup_ext.2, lcm_dedup, lcm_add]
simp
|
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
open scoped Classical
open Polynomial Set Function minpoly
namespace... | Mathlib/FieldTheory/Minpoly/Field.lean | 226 | 227 | theorem one : minpoly A (1 : B) = X - 1 := by |
simpa only [RingHom.map_one, C_1, sub_eq_add_neg] using eq_X_sub_C B (1 : A)
|
import Mathlib.Algebra.CharP.Basic
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.RingTheory.Coprime.Lemmas
#align_import algebra.char_p.char_and_card from "leanprover-community/mathlib"@"2fae5fd7f90711febdadf19c44dc60fae8834d1b"
theorem isUnit_iff_not_dvd_char_of_ringChar_ne_zero (R : Type*) [CommRin... | Mathlib/Algebra/CharP/CharAndCard.lean | 59 | 75 | theorem prime_dvd_char_iff_dvd_card {R : Type*} [CommRing R] [Fintype R] (p : ℕ) [Fact p.Prime] :
p ∣ ringChar R ↔ p ∣ Fintype.card R := by |
refine
⟨fun h =>
h.trans <|
Int.natCast_dvd_natCast.mp <|
(CharP.intCast_eq_zero_iff R (ringChar R) (Fintype.card R)).mp <|
mod_cast Nat.cast_card_eq_zero R,
fun h => ?_⟩
by_contra h₀
rcases exists_prime_addOrderOf_dvd_card p h with ⟨r, hr⟩
have hr₁ := addOrderOf_n... |
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.NumberTheory.Padics.PadicNumbers
| Mathlib/NumberTheory/Harmonic/Int.lean | 21 | 33 | theorem padicValRat_two_harmonic (n : ℕ) : padicValRat 2 (harmonic n) = -Nat.log 2 n := by |
induction' n with n ih
· simp
· rcases eq_or_ne n 0 with rfl | hn
· simp
rw [harmonic_succ]
have key : padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹ := by
rw [ih, padicValRat.inv, padicValRat.of_nat, Ne, neg_inj, Nat.cast_inj]
exact Nat.log_ne_padicValNat_succ hn
rw [padicVa... |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 152 | 153 | theorem encard_le_card (h : s ⊆ t) : s.encard ≤ t.encard := by |
rw [← union_diff_cancel h, encard_union_eq disjoint_sdiff_right]; exact le_self_add
|
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 | 93 | 95 | theorem intervalIntegrable_iff_integrableOn_Ioc_of_le (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ := by |
rw [intervalIntegrable_iff, uIoc_of_le hab]
|
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 261 | 263 | theorem mk_preimage_prod_right_fn_eq_if [DecidablePred (· ∈ s)] (g : δ → β) :
(fun b => (a, g b)) ⁻¹' s ×ˢ t = if a ∈ s then g ⁻¹' t else ∅ := by |
rw [← mk_preimage_prod_right_eq_if, prod_preimage_right, preimage_preimage]
|
import Mathlib.Init.Algebra.Classes
import Mathlib.Data.FunLike.Basic
import Mathlib.Logic.Embedding.Basic
import Mathlib.Order.RelClasses
#align_import order.rel_iso.basic from "leanprover-community/mathlib"@"f29120f82f6e24a6f6579896dfa2de6769fec962"
set_option autoImplicit true
open Function
universe u v w
v... | Mathlib/Order/RelIso/Basic.lean | 168 | 180 | theorem injective_of_increasing (r : α → α → Prop) (s : β → β → Prop) [IsTrichotomous α r]
[IsIrrefl β s] (f : α → β) (hf : ∀ {x y}, r x y → s (f x) (f y)) : Injective f := by |
intro x y hxy
rcases trichotomous_of r x y with (h | h | h)
· have := hf h
rw [hxy] at this
exfalso
exact irrefl_of s (f y) this
· exact h
· have := hf h
rw [hxy] at this
exfalso
exact irrefl_of s (f y) this
|
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.normed_space.hahn_banach.separation from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 198 | 203 | theorem geometric_hahn_banach_point_point [T1Space E] (hxy : x ≠ y) :
∃ f : E →L[ℝ] ℝ, f x < f y := by |
obtain ⟨f, s, t, hs, st, ht⟩ :=
geometric_hahn_banach_compact_closed (convex_singleton x) isCompact_singleton
(convex_singleton y) isClosed_singleton (disjoint_singleton.2 hxy)
exact ⟨f, by linarith [hs x rfl, ht y rfl]⟩
|
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 | 299 | 317 | theorem mem_cardinaleGenerate_iff {s : Set α} {hreg : c.IsRegular} :
s ∈ cardinalGenerate g (IsRegular.nat_lt hreg 2) ↔
∃ S : Set (Set α), S ⊆ g ∧ (#S < c) ∧ ⋂₀ S ⊆ s := by |
constructor <;> intro h
· induction' h with s hs s t _ st ih S Sct _ ih
· refine ⟨{s}, singleton_subset_iff.mpr hs, ?_⟩
norm_num; exact ⟨IsRegular.nat_lt hreg 1, subset_rfl⟩
· exact ⟨∅, ⟨empty_subset g, mk_eq_zero (∅ : Set <| Set α) ▸ IsRegular.nat_lt hreg 0, by simp⟩⟩
· exact Exists.imp (by taut... |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 137 | 143 | theorem div2_bit (b n) : div2 (bit b n) = n := by |
rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
cases b
· decide
· show ofNat _ = _
rw [Nat.div_eq_of_lt] <;> simp
· decide
|
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 337 | 340 | theorem isRightDescent_iff_not_isRightDescent_mul {w : W} {i : B} :
cs.IsRightDescent w i ↔ ¬cs.IsRightDescent (w * s i) i := by |
rw [isRightDescent_iff, not_isRightDescent_iff, simple_mul_simple_cancel_right]
tauto
|
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 96 | 101 | theorem dualTensorHom_prodMap_zero (f : Module.Dual R M) (p : P) :
((dualTensorHom R M P) (f ⊗ₜ[R] p)).prodMap (0 : N →ₗ[R] Q) =
dualTensorHom R (M × N) (P × Q) ((f ∘ₗ fst R M N) ⊗ₜ inl R P Q p) := by |
ext <;>
simp only [coe_comp, coe_inl, Function.comp_apply, prodMap_apply, dualTensorHom_apply,
fst_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 854 | 856 | theorem union_get_eq [Union α] (a b : Part α) (hab : Dom (a ∪ b)) :
(a ∪ b).get hab = a.get (left_dom_of_union_dom hab) ∪ b.get (right_dom_of_union_dom hab) := by |
simp [union_def]; aesop
|
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 294 | 295 | theorem hom_f_injective {C₁ C₂ : HomologicalComplex V c} :
Function.Injective fun f : Hom C₁ C₂ => f.f := by | aesop_cat
|
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 | 336 | 340 | theorem abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq {p₁ p₂ p₃ : P}
(h : dist p₁ p₂ = dist p₁ p₃) : |(∡ p₁ p₂ p₃).toReal| < π / 2 := by |
simp_rw [dist_eq_norm_vsub V] at h
rw [oangle, ← vsub_sub_vsub_cancel_left p₃ p₂ p₁]
exact o.abs_oangle_sub_right_toReal_lt_pi_div_two h
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 448 | 449 | theorem aroots_C [CommRing S] [IsDomain S] [Algebra T S] (a : T) : (C a).aroots S = 0 := by |
rw [aroots_def, map_C, roots_C]
|
import Mathlib.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift
import Mathlib.Data.Option.Basic
#align_import order.with_bot from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
variabl... | Mathlib/Order/WithBot.lean | 213 | 214 | theorem coe_le_coe : (a : WithBot α) ≤ b ↔ a ≤ b := by |
simp [LE.le]
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 618 | 621 | theorem tail_init_eq_init_tail {β : Type*} (q : Fin (n + 2) → β) :
tail (init q) = init (tail q) := by |
ext i
simp [tail, init, castSucc_fin_succ]
|
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 58 | 59 | theorem get_ofFn {n} (f : Fin n → α) (i) : get (ofFn f) i = f (Fin.cast (by simp) i) := by |
cases i; simp [ofFn, get_ofFn_go]
|
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 | 228 | 240 | theorem innerContent_pos_of_is_mul_left_invariant [Group G] [TopologicalGroup G]
(h3 : ∀ (g : G) {K : Compacts G}, μ (K.map _ <| continuous_mul_left g) = μ K) (K : Compacts G)
(hK : μ K ≠ 0) (U : Opens G) (hU : (U : Set G).Nonempty) : 0 < μ.innerContent U := by |
have : (interior (U : Set G)).Nonempty := by rwa [U.isOpen.interior_eq]
rcases compact_covered_by_mul_left_translates K.2 this with ⟨s, hs⟩
suffices μ K ≤ s.card * μ.innerContent U by
exact (ENNReal.mul_pos_iff.mp <| hK.bot_lt.trans_le this).2
have : (K : Set G) ⊆ ↑(⨆ g ∈ s, Opens.comap (Homeomorph.mulLeft... |
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 | 63 | 71 | theorem le_rank_iff_exists_linearIndependent_finset
[Module.Free K V] {n : ℕ} : ↑n ≤ Module.rank K V ↔
∃ s : Finset V, s.card = n ∧ LinearIndependent K ((↑) : ↥(s : Set V) → V) := by |
simp only [le_rank_iff_exists_linearIndependent, mk_set_eq_nat_iff_finset]
constructor
· rintro ⟨s, ⟨t, rfl, rfl⟩, si⟩
exact ⟨t, rfl, si⟩
· rintro ⟨s, rfl, si⟩
exact ⟨s, ⟨s, rfl, rfl⟩, si⟩
|
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 318 | 323 | theorem linfty_opNNNorm_diagonal [DecidableEq m] (v : m → α) : ‖diagonal v‖₊ = ‖v‖₊ := by |
rw [linfty_opNNNorm_def, Pi.nnnorm_def]
congr 1 with i : 1
refine (Finset.sum_eq_single_of_mem _ (Finset.mem_univ i) fun j _hj hij => ?_).trans ?_
· rw [diagonal_apply_ne' _ hij, nnnorm_zero]
· rw [diagonal_apply_eq]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 106 | 109 | theorem IsQuasiSeparated.of_subset {s t : Set α} (ht : IsQuasiSeparated t) (h : s ⊆ t) :
IsQuasiSeparated s := by |
intro U V hU hU' hU'' hV hV' hV''
exact ht U V (hU.trans h) hU' hU'' (hV.trans h) hV' hV''
|
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 166 | 171 | theorem gcd_insert [DecidableEq β] {b : β} :
(insert b s : Finset β).gcd f = GCDMonoid.gcd (f b) (s.gcd f) := by |
by_cases h : b ∈ s
· rw [insert_eq_of_mem h,
(gcd_eq_right_iff (f b) (s.gcd f) (Multiset.normalize_gcd (s.1.map f))).2 (gcd_dvd h)]
apply fold_insert h
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 84 | 87 | theorem eq_or_edist_eq_top_of_fixedPoints (hf : ContractingWith K f) {x y} (hx : IsFixedPt f x)
(hy : IsFixedPt f y) : x = y ∨ edist x y = ∞ := by |
refine or_iff_not_imp_right.2 fun h ↦ edist_le_zero.1 ?_
simpa only [hx.eq, edist_self, add_zero, ENNReal.zero_div] using hf.edist_le_of_fixedPoint h hy
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 62 | 65 | theorem IsConformalMap.smul (hf : IsConformalMap f) {c : R} (hc : c ≠ 0) :
IsConformalMap (c • f) := by |
rcases hf with ⟨c', hc', li, rfl⟩
exact ⟨c * c', mul_ne_zero hc hc', li, smul_smul _ _ _⟩
|
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 89 | 95 | theorem condCount_singleton (ω : Ω) (t : Set Ω) [Decidable (ω ∈ t)] :
condCount {ω} t = if ω ∈ t then 1 else 0 := by |
rw [condCount, cond_apply _ (measurableSet_singleton ω), Measure.count_singleton, inv_one,
one_mul]
split_ifs
· rw [(by simpa : ({ω} : Set Ω) ∩ t = {ω}), Measure.count_singleton]
· rw [(by simpa : ({ω} : Set Ω) ∩ t = ∅), Measure.count_empty]
|
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 176 | 196 | theorem hasDerivWithinAt_taylorWithinEval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : Set ℝ}
(hs'_unique : UniqueDiffWithinAt ℝ s' y) (hs_unique : UniqueDiffOn ℝ s) (hs' : s' ∈ 𝓝[s] y)
(hy : y ∈ s') (h : s' ⊆ s) (hf : ContDiffOn ℝ n f s)
(hf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y) :
HasD... |
induction' n with k hk
· simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one (hs_unique _ (h hy))]
exact hf'.hasDerivWithinAt.mono h
simp_rw [Nat.add_succ, t... |
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,098 | 1,102 | theorem liftRel_cons (R : α → β → Prop) (a b s t) :
LiftRel R (cons a s) (cons b t) ↔ R a b ∧ LiftRel R s t := by |
rw [liftRel_destruct_iff]
-- Porting note: These 2 theorems should be excluded.
simp [-liftRel_pure_left, -liftRel_pure_right]
|
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.cont_mdiff_mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners Bundle
open sc... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 541 | 546 | theorem ContMDiff.continuous_tangentMap (hf : ContMDiff I I' n f) (hmn : 1 ≤ n) :
Continuous (tangentMap I I' f) := by |
rw [← contMDiffOn_univ] at hf
rw [continuous_iff_continuousOn_univ]
convert hf.continuousOn_tangentMapWithin hmn uniqueMDiffOn_univ
rw [tangentMapWithin_univ]
|
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
namespace PNat
open Nat
def gcd (n m : ℕ+) : ℕ+ :=
⟨Nat.gcd (n : ℕ) (m : ℕ), Nat.gcd_pos_of_pos_left (m : ℕ) n.pos⟩
#align pnat.gcd PNat.gc... | Mathlib/Data/PNat/Prime.lean | 204 | 207 | theorem gcd_comm {m n : ℕ+} : m.gcd n = n.gcd m := by |
apply eq
simp only [gcd_coe]
apply Nat.gcd_comm
|
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 144 | 146 | theorem count_ne_zero [MeasurableSingletonClass α] (hs' : s.Nonempty) : count s ≠ 0 := by |
rw [Ne, count_eq_zero_iff]
exact hs'.ne_empty
|
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 | 333 | 335 | theorem div_le_div (hc : 0 ≤ c) (hac : a ≤ c) (hd : 0 < d) (hbd : d ≤ b) : a / b ≤ c / d := by |
rw [div_le_div_iff (hd.trans_le hbd) hd]
exact mul_le_mul hac hbd hd.le hc
|
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 190 | 190 | theorem nim_one_moveRight (x) : (nim 1).moveRight x = nim 0 := by | simp
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 305 | 308 | theorem deriv.comp_of_eq (hh₂ : DifferentiableAt 𝕜' h₂ y) (hh : DifferentiableAt 𝕜 h x)
(hy : y = h x) :
deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x := by |
rw [hy] at hh₂; exact deriv.comp x hh₂ hh
|
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 | 800 | 802 | theorem norm_deriv_le_of_lipschitzOn {f : 𝕜 → F} {x₀ : 𝕜} {s : Set 𝕜} (hs : s ∈ 𝓝 x₀)
{C : ℝ≥0} (hlip : LipschitzOnWith C f s) : ‖deriv f x₀‖ ≤ C := by |
simpa [norm_deriv_eq_norm_fderiv] using norm_fderiv_le_of_lipschitzOn 𝕜 hs hlip
|
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Subsingleton
import Mathlib.Topology.Category.TopCat.Limits.Konig
import Mathlib.Tactic.AdaptationNote
#align_import category_theory.cofiltered_system from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e... | Mathlib/CategoryTheory/CofilteredSystem.lean | 166 | 171 | theorem eventualRange_eq_range_precomp (f : i ⟶ j) (g : j ⟶ k)
(h : F.eventualRange k = range (F.map g)) : F.eventualRange k = range (F.map <| f ≫ g) := by |
apply subset_antisymm
· apply iInter₂_subset
· rw [h, F.map_comp]
apply range_comp_subset_range
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 120 | 129 | theorem mul_isClassified (k : ℤ) (hc : h.IsClassified) : (h.mul k).IsClassified := by |
obtain ⟨l, m, n, ⟨⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, co⟩⟩ := hc
· use k * l, m, n
apply And.intro _ co
left
constructor <;> ring
· use k * l, m, n
apply And.intro _ co
right
constructor <;> ring
|
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 110 | 112 | theorem completedRiemannZeta_one_sub (s : ℂ) :
completedRiemannZeta (1 - s) = completedRiemannZeta s := by |
rw [← completedHurwitzZetaEven_zero, ← completedCosZeta_zero, completedHurwitzZetaEven_one_sub]
|
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Topology.Algebra.Polynomial
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
namespace Polynomial
| Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean | 36 | 46 | theorem card_roots_toFinset_le_card_roots_derivative_diff_roots_succ (p : ℝ[X]) :
p.roots.toFinset.card ≤ (p.derivative.roots.toFinset \ p.roots.toFinset).card + 1 := by |
rcases eq_or_ne (derivative p) 0 with hp' | hp'
· rw [eq_C_of_derivative_eq_zero hp', roots_C, Multiset.toFinset_zero, Finset.card_empty]
exact zero_le _
have hp : p ≠ 0 := ne_of_apply_ne derivative (by rwa [derivative_zero])
refine Finset.card_le_diff_of_interleaved fun x hx y hy hxy hxy' => ?_
rw [Mult... |
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v₁ v₂ u₁ u₂
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 594 | 597 | theorem IsSeparatedFor.isSheafFor (t : IsSeparatedFor P R) :
(∀ x : FamilyOfElements P R, x.Compatible → ∃ t, x.IsAmalgamation t) → IsSheafFor P R := by |
rw [← isSeparatedFor_and_exists_isAmalgamation_iff_isSheafFor]
exact And.intro t
|
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 42 | 42 | theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a := by | simp only [opow_def, if_true]
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 352 | 361 | theorem condexp_mono {E} [NormedLatticeAddCommGroup E] [CompleteSpace E] [NormedSpace ℝ E]
[OrderedSMul ℝ E] {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) (hfg : f ≤ᵐ[μ] g) :
μ[f|m] ≤ᵐ[μ] μ[g|m] := by |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm _).trans_le
((condexpL1_mono hf hg hfg).trans_eq (condexp_ae_eq_conde... |
import Mathlib.Algebra.Regular.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.regular from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
variable {α : Type*}
| Mathlib/Algebra/Ring/Regular.lean | 20 | 23 | theorem isLeftRegular_of_non_zero_divisor [NonUnitalNonAssocRing α] (k : α)
(h : ∀ x : α, k * x = 0 → x = 0) : IsLeftRegular k := by |
refine fun x y (h' : k * x = k * y) => sub_eq_zero.mp (h _ ?_)
rw [mul_sub, sub_eq_zero, h']
|
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
universe u v w x
structure PEquiv (α : Type u) (β : Type v) where
toFun : α → Option β
invFun : β → Option α
inv : ∀ (a : α) (b :... | Mathlib/Data/PEquiv.lean | 161 | 165 | theorem trans_eq_none (f : α ≃. β) (g : β ≃. γ) (a : α) :
f.trans g a = none ↔ ∀ b c, b ∉ f a ∨ c ∉ g b := by |
simp only [eq_none_iff_forall_not_mem, mem_trans, imp_iff_not_or.symm]
push_neg
exact forall_swap
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 208 | 222 | theorem snorm'_le_snorm'_mul_snorm' {p q r : ℝ} (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) (b : E → F → G)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) (hp0_lt : 0 < p) (hpq : p < q)
(hpqr : 1 / p = 1 / q + 1 / r) :
snorm' (fun x => b (f x) (g x)) p μ ≤ snorm' f q μ * snorm' g ... |
rw [snorm']
calc
(∫⁻ a : α, ↑‖b (f a) (g a)‖₊ ^ p ∂μ) ^ (1 / p) ≤
(∫⁻ a : α, ↑(‖f a‖₊ * ‖g a‖₊) ^ p ∂μ) ^ (1 / p) :=
(ENNReal.rpow_le_rpow_iff <| one_div_pos.mpr hp0_lt).mpr <|
lintegral_mono_ae <|
h.mono fun a ha => (ENNReal.rpow_le_rpow_iff hp0_lt).mpr <| ENNReal.coe_le_coe.mp... |
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 198 | 202 | theorem Measurable.map_prod_mk_right {μ : Measure α} [SFinite μ] :
Measurable fun y : β => map (fun x : α => (x, y)) μ := by |
apply measurable_of_measurable_coe; intro s hs
simp_rw [map_apply measurable_prod_mk_right hs]
exact measurable_measure_prod_mk_right hs
|
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Function Set Metric TopologicalSpace intervalIntegral MeasureTheory
open MeasureTh... | Mathlib/Analysis/ODE/PicardLindelof.lean | 222 | 223 | theorem vComp_apply_coe (t : Icc v.tMin v.tMax) : f.vComp t = v t (f t) := by |
simp only [vComp, proj_coe]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 628 | 630 | theorem lintegral_zero_measure {m : MeasurableSpace α} (f : α → ℝ≥0∞) :
∫⁻ a, f a ∂(0 : Measure α) = 0 := by |
simp [lintegral]
|
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 | 949 | 950 | theorem mul_div_cancel (a) {b : Ordinal} (b0 : b ≠ 0) : b * a / b = a := by |
simpa only [add_zero, zero_div] using mul_add_div a b0 0
|
import Mathlib.Analysis.Analytic.Basic
variable {𝕜 E F G : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
[NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
open scoped Classical
open Topology NNReal Filter ENNReal
open Set Filter Asymptotics
var... | Mathlib/Analysis/Analytic/CPolynomial.lean | 194 | 197 | theorem HasFiniteFPowerSeriesOnBall.sub (hf : HasFiniteFPowerSeriesOnBall f pf x n r)
(hg : HasFiniteFPowerSeriesOnBall g pg x m r) :
HasFiniteFPowerSeriesOnBall (f - g) (pf - pg) x (max n m) r := by |
simpa only [sub_eq_add_neg] using hf.add hg.neg
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 393 | 395 | theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } := by |
ext
exact mem_iUnion
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.PowerBasis
import Mathlib.RingTheory.PrincipalI... | Mathlib/RingTheory/AdjoinRoot.lean | 350 | 352 | theorem root_isInv (r : R) : of _ r * root (C r * X - 1) = 1 := by |
convert sub_eq_zero.1 ((eval₂_sub _).symm.trans <| eval₂_root <| C r * X - 1) <;>
simp only [eval₂_mul, eval₂_C, eval₂_X, eval₂_one]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 458 | 460 | theorem ofDigits_digits_append_digits {b m n : ℕ} :
ofDigits b (digits b n ++ digits b m) = n + b ^ (digits b n).length * m := by |
rw [ofDigits_append, ofDigits_digits, ofDigits_digits]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 117 | 127 | theorem add_pow_eq_zero_of_add_le_succ_of_pow_eq_zero {m n k : ℕ}
(hx : x ^ m = 0) (hy : y ^ n = 0) (h : m + n ≤ k + 1) :
(x + y) ^ k = 0 := by |
rw [h_comm.add_pow']
apply Finset.sum_eq_zero
rintro ⟨i, j⟩ hij
suffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]
by_cases hi : m ≤ i
· rw [pow_eq_zero_of_le hi hx, zero_mul]
rw [pow_eq_zero_of_le ?_ hy, mul_zero]
linarith [Finset.mem_antidiagonal.mp hij]
|
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 193 | 197 | theorem RingHom.LocalizationPreserves.away (H : RingHom.LocalizationPreserves @P) (r : R)
[IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : P f) :
P (IsLocalization.Away.map R' S' f r) := by |
have : IsLocalization ((Submonoid.powers r).map f) S' := by rw [Submonoid.map_powers]; assumption
exact H f (Submonoid.powers r) R' S' hf
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 658 | 661 | theorem sum_affineCombinationSingleWeights [DecidableEq ι] {i : ι} (h : i ∈ s) :
∑ j ∈ s, affineCombinationSingleWeights k i j = 1 := by |
rw [← affineCombinationSingleWeights_apply_self k i]
exact sum_eq_single_of_mem i h fun j _ hj => affineCombinationSingleWeights_apply_of_ne k hj
|
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 | 537 | 545 | theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
(h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) := by |
have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
-- Porting note: Originally `by decide!`
fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
<;> simp (config := {decide := true}) only at h₁₂ h₁₃ h₂₃ ⊢
rw [affineIndependent_iff_not_collinear, ← Set.image_univ, ← Finset.coe_univ, hu,
Finse... |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
#align_import dynamics.ergodic.conservative from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open scoped Classi... | Mathlib/Dynamics/Ergodic/Conservative.lean | 121 | 130 | theorem measure_mem_forall_ge_image_not_mem_eq_zero (hf : Conservative f μ) (hs : MeasurableSet s)
(n : ℕ) : μ ({ x ∈ s | ∀ m ≥ n, f^[m] x ∉ s }) = 0 := by |
by_contra H
have : MeasurableSet (s ∩ { x | ∀ m ≥ n, f^[m] x ∉ s }) := by
simp only [setOf_forall, ← compl_setOf]
exact
hs.inter (MeasurableSet.biInter (to_countable _) fun m _ => hf.measurable.iterate m hs.compl)
rcases (hf.exists_gt_measure_inter_ne_zero this H) n with ⟨m, hmn, hm⟩
rcases nonem... |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Range
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
#align_import linear_algebra.basic from "leanprover-c... | Mathlib/LinearAlgebra/Basic.lean | 223 | 226 | theorem eq_bot_of_equiv [Module R₂ M₂] (e : p ≃ₛₗ[σ₁₂] (⊥ : Submodule R₂ M₂)) : p = ⊥ := by |
refine bot_unique (SetLike.le_def.2 fun b hb => (Submodule.mem_bot R).2 ?_)
rw [← p.mk_eq_zero hb, ← e.map_eq_zero_iff]
apply Submodule.eq_zero_of_bot_submodule
|
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 271 | 272 | theorem disjoint_product : Disjoint (s ×ˢ t) (s' ×ˢ t') ↔ Disjoint s s' ∨ Disjoint t t' := by |
simp_rw [← disjoint_coe, coe_product, Set.disjoint_prod]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 426 | 427 | theorem Submonoid.isSubmonoid (S : Submonoid M) : IsSubmonoid (S : Set M) := by |
exact ⟨S.2, S.1.2⟩
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 1,352 | 1,361 | theorem Valid'.balanceL_aux {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
(H₁ : size l = 0 → size r ≤ 1) (H₂ : 1 ≤ size l → 1 ≤ size r → size r ≤ delta * size l)
(H₃ : 2 * @size α l ≤ 9 * size r + 5 ∨ size l ≤ 3) : Valid' o₁ (@balanceL α l x r) o₂ := by |
rw [balanceL_eq_balance hl.2 hr.2 H₁ H₂, balance_eq_balance' hl.3 hr.3 hl.2 hr.2]
refine hl.balance'_aux hr (Or.inl ?_) H₃
rcases Nat.eq_zero_or_pos (size r) with r0 | r0
· rw [r0]; exact Nat.zero_le _
rcases Nat.eq_zero_or_pos (size l) with l0 | l0
· rw [l0]; exact le_trans (Nat.mul_le_mul_left _ (H₁ l0))... |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 429 | 439 | theorem sqLe_cancel {c d x y z w : ℕ} (zw : SqLe y d x c) (h : SqLe (x + z) c (y + w) d) :
SqLe z c w d := by |
apply le_of_not_gt
intro l
refine not_le_of_gt ?_ h
simp only [SqLe, mul_add, mul_comm, mul_left_comm, add_assoc, gt_iff_lt]
have hm := sqLe_add_mixed zw (le_of_lt l)
simp only [SqLe, mul_assoc, gt_iff_lt] at l zw
exact
lt_of_le_of_lt (add_le_add_right zw _)
(add_lt_add_left (add_lt_add_of_le_o... |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 162 | 169 | theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.snd, Prod.ext h.symm rfl⟩
|
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 646 | 654 | theorem den_mul_self_eq_num' (I : FractionalIdeal S P) :
spanSingleton S (algebraMap R P I.den) * I = I.num := by |
apply coeToSubmodule_injective
dsimp only
rw [coe_mul, ← smul_eq_mul, coe_spanSingleton, smul_eq_mul, Submodule.span_singleton_mul]
convert I.den_mul_self_eq_num using 1
ext
erw [Set.mem_smul_set, Set.mem_smul_set]
simp [Algebra.smul_def]
|
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Sum
namespace Multiset
variable {α β : Type*} (s : Multiset α) (t : Multiset β)
def disjSum : Multiset (Sum α β) :=
s.map inl + t.map inr
#align multiset.dis... | Mathlib/Data/Multiset/Sum.lean | 64 | 69 | theorem inr_mem_disjSum : inr b ∈ s.disjSum t ↔ b ∈ t := by |
rw [mem_disjSum, or_iff_right]
-- Porting note: Previous code for L72 was: simp only [exists_eq_right]
· simp only [inr.injEq, exists_eq_right]
rintro ⟨a, _, ha⟩
exact inl_ne_inr ha
|
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 114 | 128 | theorem mul_rat (h : LiouvilleWith p x) (hr : r ≠ 0) : LiouvilleWith p (x * r) := by |
rcases h.exists_pos with ⟨C, _hC₀, hC⟩
refine ⟨r.den ^ p * (|r| * C), (tendsto_id.nsmul_atTop r.pos).frequently (hC.mono ?_)⟩
rintro n ⟨_hn, m, hne, hlt⟩
have A : (↑(r.num * m) : ℝ) / ↑(r.den • id n) = m / n * r := by
simp [← div_mul_div_comm, ← r.cast_def, mul_comm]
refine ⟨r.num * m, ?_, ?_⟩
· rw [A]... |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 139 | 141 | theorem nhds_iInf (f : ι → Filter α) : 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i) := by |
simp only [nhds_eq]
apply lift'_iInf_of_map_univ <;> simp
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 292 | 301 | theorem degree_map_eq_iff {f : R →+* S} {p : Polynomial R} :
degree (map f p) = degree p ↔ f (leadingCoeff p) ≠ 0 ∨ p = 0 := by |
rcases eq_or_ne p 0 with h|h
· simp [h]
simp only [h, or_false]
refine ⟨fun h2 ↦ ?_, degree_map_eq_of_leadingCoeff_ne_zero f⟩
have h3 : natDegree (map f p) = natDegree p := by simp_rw [natDegree, h2]
have h4 : map f p ≠ 0 := by
rwa [ne_eq, ← degree_eq_bot, h2, degree_eq_bot]
rwa [← coeff_natDegree, ←... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 815 | 816 | theorem tan_toReal (θ : Angle) : Real.tan θ.toReal = tan θ := by |
conv_rhs => rw [← coe_toReal θ, tan_coe]
|
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equiv from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Lean Meta Elab Tactic
def mfld_cfg : Simps.Config where
attrs :=... | Mathlib/Logic/Equiv/PartialEquiv.lean | 458 | 464 | theorem leftInvOn_piecewise {e' : PartialEquiv α β} [∀ i, Decidable (i ∈ s)]
[∀ i, Decidable (i ∈ t)] (h : e.IsImage s t) (h' : e'.IsImage s t) :
LeftInvOn (t.piecewise e.symm e'.symm) (s.piecewise e e') (s.ite e.source e'.source) := by |
rintro x (⟨he, hs⟩ | ⟨he, hs : x ∉ s⟩)
· rw [piecewise_eq_of_mem _ _ _ hs, piecewise_eq_of_mem _ _ _ ((h he).2 hs), e.left_inv he]
· rw [piecewise_eq_of_not_mem _ _ _ hs, piecewise_eq_of_not_mem _ _ _ ((h'.compl he).2 hs),
e'.left_inv he]
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 70 | 72 | theorem inv_inv : inv (inv r) = r := by |
ext x y
rfl
|
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 | 189 | 190 | theorem ae_eq_set_compl_compl {s t : Set α} : sᶜ =ᵐ[μ] tᶜ ↔ s =ᵐ[μ] t := by |
simp only [← measure_symmDiff_eq_zero_iff, compl_symmDiff_compl]
|
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 | 340 | 341 | theorem closure_singleton_one : closure ({1} : Set M) = ⊥ := by |
simp [eq_bot_iff_forall, mem_closure_singleton]
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 95 | 103 | theorem ramificationIdx_lt {n : ℕ} (hgt : ¬map f p ≤ P ^ n) : ramificationIdx f p P < n := by |
cases' n with n n
· simp at hgt
· rw [Nat.lt_succ_iff]
have : ∀ k, map f p ≤ P ^ k → k ≤ n := by
refine fun k hk => le_of_not_lt fun hnk => ?_
exact hgt (hk.trans (Ideal.pow_le_pow_right hnk))
rw [ramificationIdx_eq_find ⟨n, this⟩]
exact Nat.find_min' ⟨n, this⟩ this
|
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {α : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 33 | 34 | theorem lt_of_eq_of_lt {α} [OrderedSemiring α] {a b : α} (ha : a = 0) (hb : b < 0) : a + b < 0 := by |
simp [*]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,528 | 1,531 | theorem lintegral_count' {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂count = ∑' a, f a := by |
rw [count, lintegral_sum_measure]
congr
exact funext fun a => lintegral_dirac' a hf
|
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 410 | 412 | theorem ringHom_eq_ofReal_of_continuous {f : ℝ →+* ℂ} (h : Continuous f) : f = Complex.ofReal := by |
convert congr_arg AlgHom.toRingHom <| Subsingleton.elim (AlgHom.mk' f <| map_real_smul f h)
(Algebra.ofId ℝ ℂ)
|
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 371 | 381 | theorem lintegral_withDensity_eq_lintegral_mul (μ : Measure α) {f : α → ℝ≥0∞}
(h_mf : Measurable f) :
∀ {g : α → ℝ≥0∞}, Measurable g → ∫⁻ a, g a ∂μ.withDensity f = ∫⁻ a, (f * g) a ∂μ := by |
apply Measurable.ennreal_induction
· intro c s h_ms
simp [*, mul_comm _ c, ← indicator_mul_right]
· intro g h _ h_mea_g _ h_ind_g h_ind_h
simp [mul_add, *, Measurable.mul]
· intro g h_mea_g h_mono_g h_ind
have : Monotone fun n a => f a * g n a := fun m n hmn x => mul_le_mul_left' (h_mono_g hmn x) _... |
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 469 | 476 | theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
{tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
(h : by haveI := ta1; haveI := tb1; exact Continuous fun p : X × Y => f p.1 p.2) : by
haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f ... |
have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _))
have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _))
have h_continuous_id := @Continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_c... |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | Mathlib/Data/DFinsupp/NeLocus.lean | 41 | 43 | theorem mem_neLocus {f g : Π₀ a, N a} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a := by |
simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,
and_iff_right_iff_imp] using Ne.ne_or_ne _
|
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 120 | 122 | theorem succ_singleton {n : ℕ} : Ico n (n + 1) = [n] := by |
dsimp [Ico]
simp [range', Nat.add_sub_cancel_left]
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 82 | 87 | theorem sup_sup : s.sup (f ⊔ g) = s.sup f ⊔ s.sup g := by |
induction s using Finset.cons_induction with
| empty => rw [sup_empty, sup_empty, sup_empty, bot_sup_eq]
| cons _ _ _ ih =>
rw [sup_cons, sup_cons, sup_cons, ih]
exact sup_sup_sup_comm _ _ _ _
|
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.limits.opposites from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa09372... | Mathlib/CategoryTheory/Limits/Opposites.lean | 519 | 526 | theorem opProductIsoCoproduct'_inv_comp_lift {f : Fan Z} {c : Cofan (op <| Z ·)}
(hf : IsLimit f) (hc : IsColimit c) (f' : Fan Z) :
(opProductIsoCoproduct' hf hc).inv ≫ (hf.lift f').op = hc.desc f'.op := by |
refine (Iso.inv_comp_eq _).mpr (Quiver.Hom.unop_inj (hf.hom_ext (fun ⟨j⟩ ↦ Quiver.Hom.op_inj ?_)))
simp only [Discrete.functor_obj, unop_op, Quiver.Hom.unop_op, IsLimit.fac, Fan.op, unop_comp,
Category.assoc, op_comp, op_unop, Quiver.Hom.op_unop]
erw [← Category.assoc, proj_comp_opProductIsoCoproduct'_hom, I... |
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 188 | 188 | theorem trans_refl (α : X ≅ Y) : α ≪≫ Iso.refl Y = α := by | ext; apply Category.comp_id
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 79 | 81 | theorem weightedVSubOfPoint_apply_const (w : ι → k) (p : P) (b : P) :
s.weightedVSubOfPoint (fun _ => p) b w = (∑ i ∈ s, w i) • (p -ᵥ b) := by |
rw [weightedVSubOfPoint_apply, sum_smul]
|
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
open Function Set
universe u v w z
variable {α : Sort u} {β : Sort v} {γ : Sort w}
namespace Equiv
@[simp]
th... | Mathlib/Logic/Equiv/Set.lean | 386 | 390 | theorem sumDiffSubset_symm_apply_of_mem {α} {s t : Set α} (h : s ⊆ t) [DecidablePred (· ∈ s)]
{x : t} (hx : x.1 ∈ s) : (Equiv.Set.sumDiffSubset h).symm x = Sum.inl ⟨x, hx⟩ := by |
apply (Equiv.Set.sumDiffSubset h).injective
simp only [apply_symm_apply, sumDiffSubset_apply_inl]
exact Subtype.eq rfl
|
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 160 | 161 | theorem reflect_C (r : R) (N : ℕ) : reflect N (C r) = C r * X ^ N := by |
conv_lhs => rw [← mul_one (C r), ← pow_zero X, reflect_C_mul_X_pow, revAt_zero]
|
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