Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
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import Mathlib.CategoryTheory.FinCategory.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.P... | Mathlib/CategoryTheory/Filtered/Basic.lean | 372 | 388 | theorem of_cocone_nonempty (h : ∀ {J : Type w} [SmallCategory J] [FinCategory J] (F : J ⥤ C),
Nonempty (Cocone F)) : IsFiltered C := by |
have : Nonempty C := by
obtain ⟨c⟩ := h (Functor.empty _)
exact ⟨c.pt⟩
have : IsFilteredOrEmpty C := by
refine ⟨?_, ?_⟩
· intros X Y
obtain ⟨c⟩ := h (ULiftHom.down ⋙ ULift.downFunctor ⋙ pair X Y)
exact ⟨c.pt, c.ι.app ⟨⟨WalkingPair.left⟩⟩, c.ι.app ⟨⟨WalkingPair.right⟩⟩, trivial⟩
· in... | false |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 95 | 99 | theorem const_smul [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (c • f) μ := by |
rcases hf with ⟨f', h_f'_meas, hff'⟩
refine ⟨c • f', h_f'_meas.const_smul c, ?_⟩
exact EventuallyEq.fun_comp hff' fun x => c • x
| false |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 99 | 105 | theorem zpow_log_le_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) : (b : R) ^ log b r ≤ r := by |
rcases le_total 1 r with hr1 | hr1
· rw [log_of_one_le_right _ hr1]
rw [zpow_natCast, ← Nat.cast_pow, ← Nat.le_floor_iff hr.le]
exact Nat.pow_log_le_self b (Nat.floor_pos.mpr hr1).ne'
· rw [log_of_right_le_one _ hr1, zpow_neg, zpow_natCast, ← Nat.cast_pow]
exact inv_le_of_inv_le hr (Nat.ceil_le.1 <| ... | false |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 103 | 106 | theorem support_opRingEquiv (p : R[X]ᵐᵒᵖ) : (opRingEquiv R p).support = (unop p).support := by |
induction' p using MulOpposite.rec' with p
cases p
exact Finsupp.support_mapRange_of_injective (map_zero _) _ op_injective
| false |
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 33 | 38 | theorem forall_sum {γ : α ⊕ β → Sort _} (p : (∀ ab, γ ab) → Prop) :
(∀ fab, p fab) ↔ (∀ fa fb, p (Sum.rec fa fb)) := by |
refine ⟨fun h fa fb => h _, fun h fab => ?_⟩
have h1 : fab = Sum.rec (fun a => fab (Sum.inl a)) (fun b => fab (Sum.inr b)) := by
ext ab; cases ab <;> rfl
rw [h1]; exact h _ _
| false |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
open Set Function
open scoped MeasureTheory
namespace MeasurableSpace
variable {α : Type*}
def invariants [m : MeasurableSpace α] (f : α → α) : MeasurableSpace α :=
{ m ⊓ ⟨fun s ↦ f ⁻¹' s = s, by simp, by simp, fun f hf ↦ by simp [hf]⟩ with
MeasurableSet' :... | Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean | 50 | 54 | theorem le_invariants_iterate (f : α → α) (n : ℕ) :
invariants f ≤ invariants (f^[n]) := by |
induction n with
| zero => simp [invariants_le]
| succ n ihn => exact le_trans (le_inf ihn le_rfl) (inf_le_invariants_comp _ _)
| false |
import Mathlib.LinearAlgebra.Dimension.Constructions
import Mathlib.LinearAlgebra.Dimension.Finite
universe u v
open Function Set Cardinal
variable {R} {M M₁ M₂ M₃ : Type u} {M' : Type v} [Ring R]
variable [AddCommGroup M] [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [AddCommGroup M']
variable [Module R M... | Mathlib/LinearAlgebra/Dimension/RankNullity.lean | 81 | 84 | theorem lift_rank_eq_of_surjective {f : M →ₗ[R] M'} (h : Surjective f) :
lift.{v} (Module.rank R M) =
lift.{u} (Module.rank R M') + lift.{v} (Module.rank R (LinearMap.ker f)) := by |
rw [← lift_rank_range_add_rank_ker f, ← rank_range_of_surjective f h]
| false |
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 | 490 | 490 | theorem div_one (a : G) : a / 1 = a := by | simp [div_eq_mul_inv]
| false |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 60 | 61 | theorem factorization_def (n : ℕ) {p : ℕ} (pp : p.Prime) : n.factorization p = padicValNat p n := by |
simpa [factorization] using absurd pp
| false |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 94 | 94 | theorem sInf_div : sInf (s / t) = sInf s / sSup t := by | simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 81 | 81 | theorem logb_inv (x : ℝ) : logb b x⁻¹ = -logb b x := by | simp [logb, neg_div]
| false |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 115 | 120 | theorem IsSymmetric.coe_reApplyInnerSelf_apply {T : E →L[𝕜] E} (hT : IsSymmetric (T : E →ₗ[𝕜] E))
(x : E) : (T.reApplyInnerSelf x : 𝕜) = ⟪T x, x⟫ := by |
rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ⟪T x, x⟫ = r
· simp [hr, T.reApplyInnerSelf_apply]
rw [← conj_eq_iff_real]
exact hT.conj_inner_sym x x
| false |
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.MonotoneConvergence
#align_import analysis.box_integral.box.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Fu... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 157 | 168 | theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by |
tfae_have 1 ↔ 2
· exact Iff.rfl
tfae_have 2 → 3
· intro h
simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
tfae_have 3 ↔ 4
· exact Icc_subset_Icc_iff I.lower_le_upper
tfae_have 4 → 2
· exact fun h x hx i ↦ Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i)
tfae_finish
| false |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 32 | 38 | theorem tan_add {x y : ℝ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2) ∨
(∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) := by |
simpa only [← Complex.ofReal_inj, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_div,
Complex.ofReal_mul, Complex.ofReal_tan] using
@Complex.tan_add (x : ℂ) (y : ℂ) (by convert h <;> norm_cast)
| false |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 131 | 134 | theorem pullback_fst_eq :
CompHaus.pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
| false |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 72 | 77 | theorem tendsto_zero_smul_of_tendsto_zero_of_bounded {ι 𝕜 𝔸 : Type*} [NormedDivisionRing 𝕜]
[NormedAddCommGroup 𝔸] [Module 𝕜 𝔸] [BoundedSMul 𝕜 𝔸] {l : Filter ι} {ε : ι → 𝕜} {f : ι → 𝔸}
(hε : Tendsto ε l (𝓝 0)) (hf : Filter.IsBoundedUnder (· ≤ ·) l (norm ∘ f)) :
Tendsto (ε • f) l (𝓝 0) := by |
rw [← isLittleO_one_iff 𝕜] at hε ⊢
simpa using IsLittleO.smul_isBigO hε (hf.isBigO_const (one_ne_zero : (1 : 𝕜) ≠ 0))
| false |
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 | 146 | 147 | theorem type_cardinal : @type Cardinal (· < ·) _ = Ordinal.univ.{u, u + 1} := by |
rw [Ordinal.univ_id]; exact Quotient.sound ⟨alephIdx.relIso⟩
| false |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions.Fin... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 83 | 87 | theorem NatTrans.equifibered_of_discrete {ι : Type*} {F G : Discrete ι ⥤ C}
(α : F ⟶ G) : NatTrans.Equifibered α := by |
rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
simp only [Discrete.functor_map_id]
exact IsPullback.of_horiz_isIso ⟨by rw [Category.id_comp, Category.comp_id]⟩
| false |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 56 | 59 | theorem fold_insert [DecidableEq α] (h : a ∉ s) :
(insert a s).fold op b f = f a * s.fold op b f := by |
unfold fold
rw [insert_val, ndinsert_of_not_mem h, Multiset.map_cons, fold_cons_left]
| false |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 150 | 150 | theorem fourier_one {x : AddCircle T} : fourier 1 x = toCircle x := by | rw [fourier_apply, one_zsmul]
| false |
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 130 | 136 | theorem liftOn_condition_of_liftOn'_condition {P : Sort v} {f : K[X] → K[X] → P}
(H : ∀ {p q a} (hq : q ≠ 0) (_ha : a ≠ 0), f (a * p) (a * q) = f p q) ⦃p q p' q' : K[X]⦄
(hq : q ≠ 0) (hq' : q' ≠ 0) (h : q' * p = q * p') : f p q = f p' q' :=
calc
f p q = f (q' * p) (q' * q) := (H hq hq').symm
_ = f (q ... | rw [h, mul_comm q']
_ = f p' q' := H hq' hq
| false |
import Mathlib.Data.Vector.Basic
set_option autoImplicit true
namespace Vector
def snoc : Vector α n → α → Vector α (n+1) :=
fun xs x => append xs (x ::ᵥ Vector.nil)
section Simp
variable (xs : Vector α n)
@[simp]
theorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=
rfl
@[simp]
theorem snoc_nil... | Mathlib/Data/Vector/Snoc.lean | 48 | 52 | theorem reverse_snoc : reverse (xs.snoc x) = x ::ᵥ (reverse xs) := by |
cases xs
simp only [reverse, snoc, cons, toList_mk]
congr
simp [toList, Vector.append, Append.append]
| false |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 295 | 297 | theorem pow_left (H : IsRelPrime x y) : IsRelPrime (x ^ m) y := by |
rw [← Finset.card_range m, ← Finset.prod_const]
exact IsRelPrime.prod_left fun _ _ ↦ H
| false |
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 65 | 66 | theorem isCaratheodory_compl : IsCaratheodory m s₁ → IsCaratheodory m s₁ᶜ := by |
simp [IsCaratheodory, diff_eq, add_comm]
| false |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 126 | 135 | theorem gradedComm_tmul_of_zero (a : ⨁ i, 𝒜 i) (b : ℬ 0) :
gradedComm R 𝒜 ℬ (a ⊗ₜ lof R _ ℬ 0 b) = lof R _ ℬ _ b ⊗ₜ a := by |
suffices
(gradedComm R 𝒜 ℬ).toLinearMap ∘ₗ
(TensorProduct.mk R (⨁ i, 𝒜 i) (⨁ i, ℬ i)).flip (lof R _ ℬ 0 b) =
TensorProduct.mk R _ _ (lof R _ ℬ 0 b) from
DFunLike.congr_fun this a
ext i a
dsimp
rw [gradedComm_of_tmul_of, zero_mul, uzpow_zero, one_smul]
| false |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 225 | 236 | theorem smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioc]
constructor
· rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
constructor
· exact (mul_lt_mul_left hr).mpr a_h_left_left
· exact (mul_le_mul_left hr).mpr a_h_left_right
· rintro ⟨a_left, a_right⟩
use x / r
refine ⟨⟨(lt_div_iff' hr).mpr... | false |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
#align_import linear_algebra.clifford_algebra.star from "leanprover-community/mathlib"@"4d66277cfec381260ba05c68f9ae6ce2a118031d"
variable {R : Type*} [CommRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {Q : QuadraticForm R M}
namespac... | Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean | 50 | 50 | theorem star_ι (m : M) : star (ι Q m) = -ι Q m := by | rw [star_def, involute_ι, map_neg, reverse_ι]
| false |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 117 | 123 | theorem fourier_coe_apply {n : ℤ} {x : ℝ} :
fourier n (x : AddCircle T) = Complex.exp (2 * π * Complex.I * n * x / T) := by |
rw [fourier_apply, ← QuotientAddGroup.mk_zsmul, toCircle, Function.Periodic.lift_coe,
expMapCircle_apply, Complex.ofReal_mul, Complex.ofReal_div, Complex.ofReal_mul, zsmul_eq_mul,
Complex.ofReal_mul, Complex.ofReal_intCast]
norm_num
congr 1; ring
| false |
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
#align_import analysis.special_functions.trigonometric.complex from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable secti... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 60 | 61 | theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π := by |
rw [← not_exists, not_iff_not, sin_eq_zero_iff]
| false |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_import ring_theory.chain_of_divisors from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {M : Type*} [CancelCommMonoidWithZero... | Mathlib/RingTheory/ChainOfDivisors.lean | 99 | 108 | theorem second_of_chain_is_irreducible {q : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c) (h₂ : ∀ {r}, r ≤ q ↔ ∃ i, r = c i)
(hq : q ≠ 0) : Irreducible (c 1) := by |
cases' n with n; · contradiction
refine (Associates.isAtom_iff (ne_zero_of_dvd_ne_zero hq (h₂.2 ⟨1, rfl⟩))).mp ⟨?_, fun b hb => ?_⟩
· exact ne_bot_of_gt (h₁ (show (0 : Fin (n + 2)) < 1 from Fin.one_pos))
obtain ⟨⟨i, hi⟩, rfl⟩ := h₂.1 (hb.le.trans (h₂.2 ⟨1, rfl⟩))
cases i
· exact (Associates.isUnit_iff_eq_o... | false |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 131 | 132 | theorem ofFractionRing_one : (ofFractionRing 1 : RatFunc K) = 1 := by |
simp only [One.one, OfNat.ofNat, RatFunc.one]
| false |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 65 | 66 | theorem map_commutatorElement : (f ⁅g₁, g₂⁆ : G') = ⁅f g₁, f g₂⁆ := by |
simp_rw [commutatorElement_def, map_mul f, map_inv f]
| false |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.lym from "leanprover-co... | Mathlib/Combinatorics/SetFamily/LYM.lean | 65 | 87 | theorem card_mul_le_card_shadow_mul (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
𝒜.card * r ≤ (∂ 𝒜).card * (Fintype.card α - r + 1) := by |
let i : DecidableRel ((· ⊆ ·) : Finset α → Finset α → Prop) := fun _ _ => Classical.dec _
refine card_mul_le_card_mul' (· ⊆ ·) (fun s hs => ?_) (fun s hs => ?_)
· rw [← h𝒜 hs, ← card_image_of_injOn s.erase_injOn]
refine card_le_card ?_
simp_rw [image_subset_iff, mem_bipartiteBelow]
exact fun a ha =>... | false |
import Mathlib.Data.Multiset.Sum
import Mathlib.Data.Finset.Card
#align_import data.finset.sum from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Multiset Sum
namespace Finset
variable {α β : Type*} (s : Finset α) (t : Finset β)
def disjSum : Finset (Sum α β) :=
⟨s.... | Mathlib/Data/Finset/Sum.lean | 54 | 56 | theorem disjoint_map_inl_map_inr : Disjoint (s.map Embedding.inl) (t.map Embedding.inr) := by |
simp_rw [disjoint_left, mem_map]
rintro x ⟨a, _, rfl⟩ ⟨b, _, ⟨⟩⟩
| false |
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 | 219 | 221 | theorem gcd_ndinsert (a : α) (s : Multiset α) : (ndinsert a s).gcd = GCDMonoid.gcd a s.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_cons]
simp
| false |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Measure.Haar.Quotient
import Mathlib.MeasureTheory.Constructions.Polish
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Topology.Algebra.Order.Floor
#align_import measure_theory.integral.periodic from "leanprover-c... | Mathlib/MeasureTheory/Integral/Periodic.lean | 256 | 262 | theorem intervalIntegral_add_eq_of_pos (hf : Periodic f T) (hT : 0 < T) (t s : ℝ) :
∫ x in t..t + T, f x = ∫ x in s..s + T, f x := by |
simp only [integral_of_le, hT.le, le_add_iff_nonneg_right]
haveI : VAddInvariantMeasure (AddSubgroup.zmultiples T) ℝ volume :=
⟨fun c s _ => measure_preimage_add _ _ _⟩
apply IsAddFundamentalDomain.setIntegral_eq (G := AddSubgroup.zmultiples T)
exacts [isAddFundamentalDomain_Ioc hT t, isAddFundamentalDomai... | false |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 53 | 62 | theorem cpow_eq_nhds' {p : ℂ × ℂ} (hp_fst : p.fst ≠ 0) :
(fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) := by |
suffices ∀ᶠ x : ℂ × ℂ in 𝓝 p, x.1 ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [cpow_def_of_ne_zero hx]
refine IsOpen.eventually_mem ?_ hp_fst
change IsOpen { x : ℂ × ℂ | x.1 = 0 }ᶜ
rw [isOpen_compl_iff]
exact isClosed_eq continuous_fst continuous_const
| false |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.List.Zip
#align_import data.vector.zip from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
namespace Vector
section ZipWith
variable {α β γ : Type*} {n : ℕ} (f : α → β → γ)
def zipWith : Vector α n → Vector β n → Vector γ n := fun... | Mathlib/Data/Vector/Zip.lean | 33 | 36 | theorem zipWith_get (x : Vector α n) (y : Vector β n) (i) :
(Vector.zipWith f x y).get i = f (x.get i) (y.get i) := by |
dsimp only [Vector.zipWith, Vector.get]
simp only [List.get_zipWith, Fin.cast]
| false |
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Sum
import Mathlib.Logic.Embedding.Set
#align_import data.fintype.sum from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {α β : Type*}
open Finset
instance (α : Type u) (β : Type v) [Fintype α] [Fintyp... | Mathlib/Data/Fintype/Sum.lean | 105 | 115 | theorem Set.MapsTo.exists_equiv_extend_of_card_eq [Fintype α] {t : Finset β}
(hαt : Fintype.card α = t.card) {s : Set α} {f : α → β} (hfst : s.MapsTo f t)
(hfs : Set.InjOn f s) : ∃ g : α ≃ t, ∀ i ∈ s, (g i : β) = f i := by |
classical
let s' : Finset α := s.toFinset
have hfst' : s'.image f ⊆ t := by simpa [s', ← Finset.coe_subset] using hfst
have hfs' : Set.InjOn f s' := by simpa [s'] using hfs
obtain ⟨g, hg⟩ := Finset.exists_equiv_extend_of_card_eq hαt hfst' hfs'
refine ⟨g, fun i hi => ?_⟩
apply hg
simpa [s'... | false |
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {𝕜 E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 72 | 74 | theorem dslope_sub_smul_of_ne (f : 𝕜 → E) (h : b ≠ a) :
dslope (fun x => (x - a) • f x) a b = f b := by |
rw [dslope_of_ne _ h, slope_sub_smul _ h.symm]
| false |
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Complex Matrix
open scoped MatrixGroups
namespace EisensteinSeries
variable (N : ℕ) (a : Fin 2 → ZMod N)
variable {N a}
section eisSum... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean | 92 | 100 | theorem eisSummand_SL2_apply (k : ℤ) (i : (Fin 2 → ℤ)) (A : SL(2, ℤ)) (z : ℍ) :
eisSummand k i (A • z) = (z.denom A) ^ k * eisSummand k (i ᵥ* A) z := by |
simp only [eisSummand, specialLinearGroup_apply, algebraMap_int_eq, eq_intCast, ofReal_intCast,
one_div, vecMul, vec2_dotProduct, Int.cast_add, Int.cast_mul]
have h (a b c d u v : ℂ) (hc : c * z + d ≠ 0) : ((u * ((a * z + b) / (c * z + d)) + v) ^ k)⁻¹ =
(c * z + d) ^ k * (((u * a + v * c) * z + (u * b + ... | false |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 158 | 168 | theorem content_eq_zero_iff {p : R[X]} : content p = 0 ↔ p = 0 := by |
rw [content, Finset.gcd_eq_zero_iff]
constructor <;> intro h
· ext n
by_cases h0 : n ∈ p.support
· rw [h n h0, coeff_zero]
· rw [mem_support_iff] at h0
push_neg at h0
simp [h0]
· intro x
simp [h]
| false |
import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
import Mathlib.RingTheory.Norm
#align_import linear_algebra.free_module.norm from "leanprover-community/mathlib"@"90b0d53ee6ffa910e5c2a977ce7e2fc704647974"
open Ideal Polynomial
open scoped Polynomial
variable {R S ι : Type*} [CommRing R] [IsDomain R] [IsPri... | Mathlib/LinearAlgebra/FreeModule/Norm.lean | 30 | 50 | theorem associated_norm_prod_smith [Fintype ι] (b : Basis ι R S) {f : S} (hf : f ≠ 0) :
Associated (Algebra.norm R f) (∏ i, smithCoeffs b _ (span_singleton_eq_bot.not.2 hf) i) := by |
have hI := span_singleton_eq_bot.not.2 hf
let b' := ringBasis b (span {f}) hI
classical
rw [← Matrix.det_diagonal, ← LinearMap.det_toLin b']
let e :=
(b'.equiv ((span {f}).selfBasis b hI) <| Equiv.refl _).trans
((LinearEquiv.coord S S f hf).restrictScalars R)
refine (LinearMap.associated_det_of_e... | false |
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 | 394 | 394 | theorem lt_inv' : a < b⁻¹ ↔ b < a⁻¹ := by | rw [← inv_lt_inv_iff, inv_inv]
| false |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set MeasureTheory.MeasureSpace
variable {x₀ : ℝ} {a b : ℕ} {f : ℝ → ℝ}
theorem AntitoneOn.in... | Mathlib/Analysis/SumIntegralComparisons.lean | 98 | 123 | theorem AntitoneOn.sum_le_integral (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
(∑ i ∈ Finset.range a, f (x₀ + (i + 1 : ℕ))) ≤ ∫ x in x₀..x₀ + a, f x := by |
have hint : ∀ k : ℕ, k < a → IntervalIntegrable f volume (x₀ + k) (x₀ + (k + 1 : ℕ)) := by
intro k hk
refine (hf.mono ?_).intervalIntegrable
rw [uIcc_of_le]
· apply Icc_subset_Icc
· simp only [le_add_iff_nonneg_right, Nat.cast_nonneg]
· simp only [add_le_add_iff_left, Nat.cast_le, Nat.suc... | false |
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 | 141 | 144 | theorem map_cast_heq {α β} (h : α = β) (s : Finset α) :
HEq (s.map (Equiv.cast h).toEmbedding) s := by |
subst h
simp
| false |
import Mathlib.Algebra.Module.PID
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.finite_abelian from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915347dafd749ad6"
open scoped DirectSum
private def directSumNeZeroMulHom {ι : Type} [DecidableEq ι] (p : ι → ℕ) (n : ι → ℕ) :
(⨁ i : {i ... | Mathlib/GroupTheory/FiniteAbelian.lean | 114 | 126 | theorem equiv_free_prod_directSum_zmod [hG : AddGroup.FG G] :
∃ (n : ℕ) (ι : Type) (_ : Fintype ι) (p : ι → ℕ) (_ : ∀ i, Nat.Prime <| p i) (e : ι → ℕ),
Nonempty <| G ≃+ (Fin n →₀ ℤ) × ⨁ i : ι, ZMod (p i ^ e i) := by |
obtain ⟨n, ι, fι, p, hp, e, ⟨f⟩⟩ :=
@Module.equiv_free_prod_directSum _ _ _ _ _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG)
refine ⟨n, ι, fι, fun i => (p i).natAbs, fun i => ?_, e, ⟨?_⟩⟩
· rw [← Int.prime_iff_natAbs_prime, ← irreducible_iff_prime]; exact hp i
exact
f.toAddEquiv.trans
((AddEquiv.re... | false |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
theorem tan_add {x y : ℝ}
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 56 | 60 | theorem continuousOn_tan : ContinuousOn tan {x | cos x ≠ 0} := by |
suffices ContinuousOn (fun x => sin x / cos x) {x | cos x ≠ 0} by
have h_eq : (fun x => sin x / cos x) = tan := by ext1 x; rw [tan_eq_sin_div_cos]
rwa [h_eq] at this
exact continuousOn_sin.div continuousOn_cos fun x => id
| false |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 142 | 142 | theorem bind_add : (s.bind fun a => f a + g a) = s.bind f + s.bind g := by | simp [bind, join]
| false |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 174 | 179 | theorem fib_two_mul (n : ℕ) : fib (2 * n) = fib n * (2 * fib (n + 1) - fib n) := by |
cases n
· simp
· rw [two_mul, ← add_assoc, fib_add, fib_add_two, two_mul]
simp only [← add_assoc, add_tsub_cancel_right]
ring
| false |
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 121 | 126 | theorem exists_finite_submodule_left_of_finite (s : Set (M ⊗[R] N)) (hs : s.Finite) :
∃ M' : Submodule R M, Module.Finite R M' ∧ s ⊆ LinearMap.range (M'.subtype.rTensor N) := by |
obtain ⟨M', _, hfin, _, h⟩ := exists_finite_submodule_of_finite s hs
refine ⟨M', hfin, ?_⟩
rw [mapIncl, ← LinearMap.rTensor_comp_lTensor] at h
exact h.trans (LinearMap.range_comp_le_range _ _)
| false |
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 | 137 | 140 | theorem digits_of_lt (b x : ℕ) (hx : x ≠ 0) (hxb : x < b) : digits b x = [x] := by |
rcases exists_eq_succ_of_ne_zero hx with ⟨x, rfl⟩
rcases Nat.exists_eq_add_of_le' ((Nat.le_add_left 1 x).trans_lt hxb) with ⟨b, rfl⟩
rw [digits_add_two_add_one, div_eq_of_lt hxb, digits_zero, mod_eq_of_lt hxb]
| false |
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v... | Mathlib/Data/Set/NAry.lean | 68 | 69 | theorem image2_subset_iff_left : image2 f s t ⊆ u ↔ ∀ a ∈ s, (fun b => f a b) '' t ⊆ u := by |
simp_rw [image2_subset_iff, image_subset_iff, subset_def, mem_preimage]
| false |
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {α : Type*}
namespace List
inductive Duplicate (x : α) : List α → Prop
| cons_mem {l : List α} : x ∈ l → Duplicate x (x :: l)
| cons_duplicate {y : α} {l ... | Mathlib/Data/List/Duplicate.lean | 141 | 142 | theorem duplicate_iff_two_le_count [DecidableEq α] : x ∈+ l ↔ 2 ≤ count x l := by |
simp [duplicate_iff_sublist, le_count_iff_replicate_sublist]
| false |
import Mathlib.RingTheory.RingHomProperties
import Mathlib.RingTheory.IntegralClosure
#align_import ring_theory.ring_hom.integral from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
theorem isIntegra... | Mathlib/RingTheory/RingHom/Integral.lean | 28 | 32 | theorem isIntegral_respectsIso : RespectsIso fun f => f.IsIntegral := by |
apply isIntegral_stableUnderComposition.respectsIso
introv x
rw [← e.apply_symm_apply x]
apply RingHom.isIntegralElem_map
| false |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
| Mathlib/Data/Nat/GCD/BigOperators.lean | 20 | 22 | theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by |
induction l <;> simp [Nat.coprime_mul_iff_left, *]
| false |
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 | 145 | 150 | theorem volume_Ioi {a : ℝ} : volume (Ioi a) = ∞ :=
top_unique <|
le_of_tendsto' ENNReal.tendsto_nat_nhds_top fun n =>
calc
(n : ℝ≥0∞) = volume (Ioo a (a + n)) := by | simp
_ ≤ volume (Ioi a) := measure_mono Ioo_subset_Ioi_self
| false |
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped ENNReal
namespace ProbabilityTheory
namespace kernel
variable {α β ι : Type*} {mα : MeasurableSpace α}... | Mathlib/Probability/Kernel/Composition.lean | 146 | 148 | theorem compProdFun_tsum_left (κ : kernel α β) (η : kernel (α × β) γ) [IsSFiniteKernel κ] (a : α)
(s : Set (β × γ)) : compProdFun κ η a s = ∑' n, compProdFun (seq κ n) η a s := by |
simp_rw [compProdFun, (measure_sum_seq κ _).symm, lintegral_sum_measure]
| false |
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.EffectiveEpi.Coproduct
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CategoryTheory
open Limits
variable {C : Type*} [Category C] [FinitaryPreExtensive C]
| Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean | 24 | 29 | theorem effectiveEpi_desc_iff_effectiveEpiFamily {α : Type} [Finite α]
{B : C} (X : α → C) (π : (a : α) → X a ⟶ B) :
EffectiveEpi (Sigma.desc π) ↔ EffectiveEpiFamily X π := by |
exact ⟨fun h ↦ ⟨⟨@effectiveEpiFamilyStructOfEffectiveEpiDesc _ _ _ _ X π _ h _ _ (fun g ↦
(FinitaryPreExtensive.sigma_desc_iso (fun a ↦ Sigma.ι X a) g inferInstance).epi_of_iso)⟩⟩,
fun _ ↦ inferInstance⟩
| false |
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 84 | 85 | theorem toList_push (x : α) (l : DList α) : toList (push l x) = toList l ++ [x] := by |
cases' l with _ l_invariant; simp; rw [l_invariant]
| false |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Tactic.ComputeDegree
#align_import linear_algebra.matrix.polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 73 | 86 | theorem coeff_det_X_add_C_card (A B : Matrix n n α) :
coeff (det ((X : α[X]) • A.map C + B.map C)) (Fintype.card n) = det A := by |
rw [det_apply, det_apply, finset_sum_coeff]
refine Finset.sum_congr rfl ?_
simp only [Algebra.id.smul_eq_mul, Finset.mem_univ, RingHom.mapMatrix_apply, forall_true_left,
map_apply, Pi.smul_apply]
intro g
convert coeff_smul (R := α) (sign g) _ _
rw [← mul_one (Fintype.card n)]
convert (coeff_prod_of_n... | false |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 107 | 108 | theorem zero_eq_nndist {x y : γ} : 0 = nndist x y ↔ x = y := by |
simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, zero_eq_dist]
| false |
import Mathlib.Analysis.SpecialFunctions.Bernstein
import Mathlib.Topology.Algebra.Algebra
#align_import topology.continuous_function.weierstrass from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open ContinuousMap Filter
open scoped unitInterval
theorem polynomialFunctions_closure... | Mathlib/Topology/ContinuousFunction/Weierstrass.lean | 99 | 105 | theorem exists_polynomial_near_continuousMap (a b : ℝ) (f : C(Set.Icc a b, ℝ)) (ε : ℝ)
(pos : 0 < ε) : ∃ p : ℝ[X], ‖p.toContinuousMapOn _ - f‖ < ε := by |
have w := mem_closure_iff_frequently.mp (continuousMap_mem_polynomialFunctions_closure _ _ f)
rw [Metric.nhds_basis_ball.frequently_iff] at w
obtain ⟨-, H, ⟨m, ⟨-, rfl⟩⟩⟩ := w ε pos
rw [Metric.mem_ball, dist_eq_norm] at H
exact ⟨m, H⟩
| false |
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 141 | 144 | theorem antidiagonal.snd_le {n : A} {kl : A × A} (hlk : kl ∈ antidiagonal n) : kl.2 ≤ n := by |
rw [le_iff_exists_add]
use kl.1
rwa [mem_antidiagonal, eq_comm, add_comm] at hlk
| false |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 209 | 210 | theorem weightedSMul_null {s : Set α} (h_zero : μ s = 0) : (weightedSMul μ s : F →L[ℝ] F) = 0 := by |
ext1 x; rw [weightedSMul_apply, h_zero]; simp
| false |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 32 | 46 | theorem Complex.hasSum_cos' (z : ℂ) :
HasSum (fun n : ℕ => (z * Complex.I) ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) := by |
rw [Complex.cos, Complex.exp_eq_exp_ℂ]
have := ((expSeries_div_hasSum_exp ℂ (z * Complex.I)).add
(expSeries_div_hasSum_exp ℂ (-z * Complex.I))).div_const 2
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [← mul_comm 2 _] at this
refine this.prod_fiberwi... | false |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 41 | 53 | theorem isOpen_singleton_iff : IsOpen ({a} : Set Ordinal) ↔ ¬IsLimit a := by |
refine ⟨fun h ⟨h₀, hsucc⟩ => ?_, fun ha => ?_⟩
· obtain ⟨b, c, hbc, hbc'⟩ :=
(mem_nhds_iff_exists_Ioo_subset' ⟨0, Ordinal.pos_iff_ne_zero.2 h₀⟩ ⟨_, lt_succ a⟩).1
(h.mem_nhds rfl)
have hba := hsucc b hbc.1
exact hba.ne (hbc' ⟨lt_succ b, hba.trans hbc.2⟩)
· rcases zero_or_succ_or_limit a with... | false |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 92 | 92 | theorem coe_univ : ↑(univ : Finset α) = (Set.univ : Set α) := by | ext; simp
| false |
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
#align_import data.int.cast.basic from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u
open Nat
namespace Int
variable {R : Type u} [AddGroupWithOne R]
@[simp, norm_cas... | Mathlib/Data/Int/Cast/Basic.lean | 74 | 76 | theorem cast_ofNat (n : ℕ) [n.AtLeastTwo] :
((no_index (OfNat.ofNat n) : ℤ) : R) = OfNat.ofNat n := by |
simpa only [OfNat.ofNat] using AddGroupWithOne.intCast_ofNat (R := R) n
| false |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open scoped ENNReal
namespace MeasureTheory
variable {α E : Type*} {m0 : MeasurableSpace α} [NormedAddCommGroup E]
{p : ℝ≥0∞} (μ... | Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean | 23 | 28 | theorem pow_mul_meas_ge_le_snorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
(ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }) ^ (1 / p.toReal) ≤ snorm f p μ := by |
rw [snorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top]
gcongr
exact mul_meas_ge_le_lintegral₀ (hf.ennnorm.pow_const _) ε
| false |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 64 | 66 | theorem lift_lt_continuum {c : Cardinal.{u}} : lift.{v} c < 𝔠 ↔ c < 𝔠 := by |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_lt]
| false |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 326 | 348 | theorem Memℒp.uniformIntegrable_of_identDistrib_aux {ι : Type*} {f : ι → α → E} {j : ι} {p : ℝ≥0∞}
(hp : 1 ≤ p) (hp' : p ≠ ∞) (hℒp : Memℒp (f j) p μ) (hfmeas : ∀ i, StronglyMeasurable (f i))
(hf : ∀ i, IdentDistrib (f i) (f j) μ μ) : UniformIntegrable f p μ := by |
refine uniformIntegrable_of' hp hp' hfmeas fun ε hε => ?_
by_cases hι : Nonempty ι
swap; · exact ⟨0, fun i => False.elim (hι <| Nonempty.intro i)⟩
obtain ⟨C, hC₁, hC₂⟩ := hℒp.snorm_indicator_norm_ge_pos_le (hfmeas _) hε
refine ⟨⟨C, hC₁.le⟩, fun i => le_trans (le_of_eq ?_) hC₂⟩
have : {x | (⟨C, hC₁.le⟩ : ℝ≥... | false |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | Mathlib/Topology/Order/LeftRightNhds.lean | 131 | 138 | theorem TFAE_mem_nhdsWithin_Iio {a b : α} (h : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[<] b,-- 0 : `s` is a neighborhood of `b` within `(-∞, b)`
s ∈ 𝓝[Ico a b] b,-- 1 : `s` is a neighborhood of `b` within `[a, b)`
s ∈ 𝓝[Ioo a b] b,-- 2 : `s` is a neighborhood of `b` within `(a, b)`
∃ l ∈ Ico a b... | -- 4 : `s` includes `(l, b)` for some `l < b`
simpa only [exists_prop, OrderDual.exists, dual_Ioi, dual_Ioc, dual_Ioo] using
TFAE_mem_nhdsWithin_Ioi h.dual (ofDual ⁻¹' s)
| false |
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.small_sets from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Filter
open Filter Set
variable {α β : Type*} {ι : Sort*}
namespace Filter
variable {l l' la : Filter α} {lb : Filter ... | Mathlib/Order/Filter/SmallSets.lean | 40 | 42 | theorem smallSets_eq_generate {f : Filter α} : f.smallSets = generate (powerset '' f.sets) := by |
simp_rw [generate_eq_biInf, smallSets, iInf_image]
rfl
| false |
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
universe u v
namespace MvQPF
open TypeVec
open MvFunctor (LiftP LiftR)
open MvFunctor
var... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 120 | 121 | theorem wEquiv.refl {α : TypeVec n} (x : q.P.W α) : WEquiv x x := by |
apply q.P.w_cases _ x; intro a f' f; exact WEquiv.abs a f' f a f' f rfl
| false |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 98 | 125 | theorem parallelepiped_orthonormalBasis_one_dim (b : OrthonormalBasis ι ℝ ℝ) :
parallelepiped b = Icc 0 1 ∨ parallelepiped b = Icc (-1) 0 := by |
have e : ι ≃ Fin 1 := by
apply Fintype.equivFinOfCardEq
simp only [← finrank_eq_card_basis b.toBasis, finrank_self]
have B : parallelepiped (b.reindex e) = parallelepiped b := by
convert parallelepiped_comp_equiv b e.symm
ext i
simp only [OrthonormalBasis.coe_reindex]
rw [← B]
let F : ℝ → F... | false |
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 | 129 | 131 | theorem intervalIntegrable_const_iff {c : E} :
IntervalIntegrable (fun _ => c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ∞ := by |
simp only [intervalIntegrable_iff, integrableOn_const]
| false |
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Set Filter Function Metric MeasureTheory
open scoped MeasureTheory Topology Pointwise
@[... | Mathlib/NumberTheory/WellApproximable.lean | 108 | 116 | theorem image_pow_subset_of_coprime (hm : 0 < m) (hmn : n.Coprime m) :
(fun (y : A) => y ^ m) '' approxOrderOf A n δ ⊆ approxOrderOf A n (m * δ) := by |
rintro - ⟨a, ha, rfl⟩
obtain ⟨b, hb, hab⟩ := mem_approxOrderOf_iff.mp ha
replace hb : b ^ m ∈ {u : A | orderOf u = n} := by
rw [← hb] at hmn ⊢; exact hmn.orderOf_pow
apply ball_subset_thickening hb ((m : ℝ) • δ)
convert pow_mem_ball hm hab using 1
simp only [nsmul_eq_mul, Algebra.id.smul_eq_mul]
| false |
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingTheory.Localization.NormTrace
#align_import number_theory.number_field.norm from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
open scoped NumberField
open Finset NumberField Algebra FiniteDimensional
namespace RingOfIn... | Mathlib/NumberTheory/NumberField/Norm.lean | 111 | 126 | theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x := by |
letI : Algebra K (AlgebraicClosure K) := AlgebraicClosure.instAlgebra K
let L := normalClosure K F (AlgebraicClosure F)
haveI : FiniteDimensional F L := FiniteDimensional.right K F L
haveI : IsAlgClosure K (AlgebraicClosure F) :=
IsAlgClosure.ofAlgebraic K F (AlgebraicClosure F)
haveI : IsGalois F L := I... | false |
import Mathlib.Topology.MetricSpace.Isometry
#align_import topology.metric_space.gluing from "leanprover-community/mathlib"@"e1a7bdeb4fd826b7e71d130d34988f0a2d26a177"
noncomputable section
universe u v w
open Function Set Uniformity Topology
namespace Metric
--section
section InductiveLimit
open Nat
variab... | Mathlib/Topology/MetricSpace/Gluing.lean | 648 | 658 | theorem toInductiveLimit_commute (I : ∀ n, Isometry (f n)) (n : ℕ) :
toInductiveLimit I n.succ ∘ f n = toInductiveLimit I n := by |
let h := inductivePremetric I
let _ := h.toUniformSpace.toTopologicalSpace
funext x
simp only [comp, toInductiveLimit]
refine SeparationQuotient.mk_eq_mk.2 (Metric.inseparable_iff.2 ?_)
show inductiveLimitDist f ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0
rw [inductiveLimitDist_eq_dist I ⟨n.succ, f n x⟩ ⟨n, x⟩ n.succ, le... | false |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 56 | 57 | theorem hammingDist_comm (x y : ∀ i, β i) : hammingDist x y = hammingDist y x := by |
simp_rw [hammingDist, ne_comm]
| false |
import Mathlib.Control.Bifunctor
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.functor from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
universe u v w
variable {α β : Type u}
open Equiv
namespace Functor
variable (f : Type u → Type v) [Functor f] [LawfulFunctor f]
d... | Mathlib/Logic/Equiv/Functor.lean | 57 | 60 | theorem mapEquiv_refl : mapEquiv f (Equiv.refl α) = Equiv.refl (f α) := by |
ext x
simp only [mapEquiv_apply, refl_apply]
exact LawfulFunctor.id_map x
| false |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 113 | 113 | theorem projIci_of_mem (hx : x ∈ Ici a) : projIci a x = ⟨x, hx⟩ := by | simpa [projIci]
| false |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Localization.AsSubring
#align_import algebraic_geometry.prime_spectrum.maximal from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
noncomputable section
open scoped Classical
universe u v
variable (R : Typ... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Maximal.lean | 92 | 117 | theorem iInf_localization_eq_bot : (⨅ v : MaximalSpectrum R,
Localization.subalgebra.ofField K _ v.asIdeal.primeCompl_le_nonZeroDivisors) = ⊥ := by |
ext x
rw [Algebra.mem_bot, Algebra.mem_iInf]
constructor
· contrapose
intro hrange hlocal
let denom : Ideal R := (Submodule.span R {1} : Submodule R K).colon (Submodule.span R {x})
have hdenom : (1 : R) ∉ denom := by
intro hdenom
rcases Submodule.mem_span_singleton.mp
(Submodule... | false |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-community/mathlib"@"ba2245edf0c8bb155f1569fd9b9492a9b384cde6"
namespace OrderHom
variable {α β : Type*}
section Preorder
variable [Preorder α]
instance [Sem... | Mathlib/Order/Hom/Order.lean | 133 | 154 | theorem iterate_sup_le_sup_iff {α : Type*} [SemilatticeSup α] (f : α →o α) :
(∀ n₁ n₂ a₁ a₂, f^[n₁ + n₂] (a₁ ⊔ a₂) ≤ f^[n₁] a₁ ⊔ f^[n₂] a₂) ↔
∀ a₁ a₂, f (a₁ ⊔ a₂) ≤ f a₁ ⊔ a₂ := by |
constructor <;> intro h
· exact h 1 0
· intro n₁ n₂ a₁ a₂
have h' : ∀ n a₁ a₂, f^[n] (a₁ ⊔ a₂) ≤ f^[n] a₁ ⊔ a₂ := by
intro n
induction' n with n ih <;> intro a₁ a₂
· rfl
· calc
f^[n + 1] (a₁ ⊔ a₂) = f^[n] (f (a₁ ⊔ a₂)) := Function.iterate_succ_apply f n _
_ ≤ f^[n]... | false |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_limits_deriv from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Filter
open scoped uniformity Filter Topology
section d... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 455 | 474 | theorem UniformCauchySeqOnFilter.one_smulRight {l' : Filter 𝕜}
(hf' : UniformCauchySeqOnFilter f' l l') :
UniformCauchySeqOnFilter (fun n => fun z => (1 : 𝕜 →L[𝕜] 𝕜).smulRight (f' n z)) l l' := by |
-- The tricky part of this proof is that operator norms are written in terms of `≤` whereas
-- metrics are written in terms of `<`. So we need to shrink `ε` utilizing the archimedean
-- property of `ℝ`
rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero,
Metric.tendstoUniforml... | false |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 317 | 332 | theorem isBigO_rpow_zero_log_smul [NormedSpace ℝ E] {a b : ℝ} {f : ℝ → E} (hab : a < b)
(hf : f =O[𝓝[>] 0] (· ^ (-a))) :
(fun t : ℝ => log t • f t) =O[𝓝[>] 0] (· ^ (-b)) := by |
have : log =o[𝓝[>] 0] fun t : ℝ => t ^ (a - b) := by
refine ((isLittleO_log_rpow_atTop (sub_pos.mpr hab)).neg_left.comp_tendsto
tendsto_inv_zero_atTop).congr'
(eventually_nhdsWithin_iff.mpr <| eventually_of_forall fun t ht => ?_)
(eventually_nhdsWithin_iff.mpr <| eventually_of_forall fun t... | false |
import Mathlib.Algebra.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupTheory.GroupAction.Ring
#align_import algebra.lie.free from "leanprover-community/mathlib"@"841ac1a3d9162bf51c6327812ecb6e5e71883ac4"
universe ... | Mathlib/Algebra/Lie/Free.lean | 95 | 96 | theorem Rel.subLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a - b) (a - c) := by |
simpa only [sub_eq_add_neg] using h.neg.addLeft a
| false |
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 | 106 | 113 | theorem ext (I J : HasZeroMorphisms C) : I = J := by |
apply ext_aux
intro X Y
have : (I.zero X Y).zero ≫ (J.zero Y Y).zero = (I.zero X Y).zero := by
apply I.zero_comp X (J.zero Y Y).zero
have that : (I.zero X Y).zero ≫ (J.zero Y Y).zero = (J.zero X Y).zero := by
apply J.comp_zero (I.zero X Y).zero Y
rw [← this, ← that]
| false |
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 | 105 | 110 | theorem zero_prodMap_dualTensorHom (g : Module.Dual R N) (q : Q) :
(0 : M →ₗ[R] P).prodMap ((dualTensorHom R N Q) (g ⊗ₜ[R] q)) =
dualTensorHom R (M × N) (P × Q) ((g ∘ₗ snd R M N) ⊗ₜ inr R P Q q) := by |
ext <;>
simp only [coe_comp, coe_inr, Function.comp_apply, prodMap_apply, dualTensorHom_apply,
snd_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
| false |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 77 | 80 | theorem directedOn_image {s : Set β} {f : β → α} :
DirectedOn r (f '' s) ↔ DirectedOn (f ⁻¹'o r) s := by |
simp only [DirectedOn, Set.mem_image, exists_exists_and_eq_and, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂, Order.Preimage]
| false |
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open FirstOrder
namespace FirstOrder
namespace Language
open Structure
variable (L : Language) (M : Type*) (N : T... | Mathlib/ModelTheory/ElementaryMaps.lean | 272 | 302 | theorem isElementary_of_exists (f : M ↪[L] N)
(htv :
∀ (n : ℕ) (φ : L.BoundedFormula Empty (n + 1)) (x : Fin n → M) (a : N),
φ.Realize default (Fin.snoc (f ∘ x) a : _ → N) →
∃ b : M, φ.Realize default (Fin.snoc (f ∘ x) (f b) : _ → N)) :
∀ {n} (φ : L.Formula (Fin n)) (x : Fin n → M), φ.Re... |
suffices h : ∀ (n : ℕ) (φ : L.BoundedFormula Empty n) (xs : Fin n → M),
φ.Realize (f ∘ default) (f ∘ xs) ↔ φ.Realize default xs by
intro n φ x
exact φ.realize_relabel_sum_inr.symm.trans (_root_.trans (h n _ _) φ.realize_relabel_sum_inr)
refine fun n φ => φ.recOn ?_ ?_ ?_ ?_ ?_
· exact fun {_} _ => ... | false |
import Mathlib.ModelTheory.Substructures
#align_import model_theory.finitely_generated from "leanprover-community/mathlib"@"0602c59878ff3d5f71dea69c2d32ccf2e93e5398"
open FirstOrder Set
namespace FirstOrder
namespace Language
open Structure
variable {L : Language} {M : Type*} [L.Structure M]
namespace Substru... | Mathlib/ModelTheory/FinitelyGenerated.lean | 116 | 135 | theorem cg_iff_empty_or_exists_nat_generating_family {N : L.Substructure M} :
N.CG ↔ N = (∅ : Set M) ∨ ∃ s : ℕ → M, closure L (range s) = N := by |
rw [cg_def]
constructor
· rintro ⟨S, Scount, hS⟩
rcases eq_empty_or_nonempty (N : Set M) with h | h
· exact Or.intro_left _ h
obtain ⟨f, h'⟩ :=
(Scount.union (Set.countable_singleton h.some)).exists_eq_range
(singleton_nonempty h.some).inr
refine Or.intro_right _ ⟨f, ?_⟩
rw [← h... | false |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Homeomorph
#align_import topology.algebra.group_with_zero from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862b9fb9a992fef2"
open Topology Filter Function
variable {α β G₀ : Type*}
section DivConst... | Mathlib/Topology/Algebra/GroupWithZero.lean | 75 | 76 | theorem Continuous.div_const (hf : Continuous f) (y : G₀) : Continuous fun x => f x / y := by |
simpa only [div_eq_mul_inv] using hf.mul continuous_const
| false |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 70 | 71 | theorem convexJoin_singleton_right (s : Set E) (y : E) :
convexJoin 𝕜 s {y} = ⋃ x ∈ s, segment 𝕜 x y := by | simp [convexJoin]
| false |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 132 | 135 | theorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :
μ (f ⁻¹' s) = ν (g ⁻¹' s) := by |
rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ←
Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]
| false |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 274 | 279 | theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : ℝ) :
normAtPlace w (c • x) = |c| * normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs
· rw [Prod.smul_fst, Pi.smul_apply, norm_smul, Real.norm_eq_abs]
· rw [Prod.smul_snd, Pi.smul_apply, norm_smul, Real.norm_eq_abs, Complex.norm_eq_abs]
| false |
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 285 | 288 | theorem tangentMap_prod_fst {p : TangentBundle (I.prod I') (M × M')} :
tangentMap (I.prod I') I Prod.fst p = ⟨p.proj.1, p.2.1⟩ := by |
-- Porting note: `rfl` wasn't needed
simp [tangentMap]; rfl
| false |
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u
open Function
namespace Set
variable {α β : Type u} {s : Set α} ... | Mathlib/Data/Set/Functor.lean | 135 | 139 | theorem coe_eq_image_val (t : Set s) :
@Lean.Internal.coeM Set s α _ Set.monad t = (t : Set α) := by |
change ⋃ (x ∈ t), {x.1} = _
ext
simp
| false |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 155 | 159 | theorem Odd.zpow_neg_iff (hn : Odd n) : a ^ n < 0 ↔ a < 0 := by |
refine ⟨lt_imp_lt_of_le_imp_le (zpow_nonneg · _), fun ha ↦ ?_⟩
obtain ⟨k, rfl⟩ := hn
rw [zpow_add_one₀ ha.ne]
exact mul_neg_of_pos_of_neg (Even.zpow_pos (even_two_mul _) ha.ne) ha
| false |
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