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 Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 49 | 55 | theorem list_reverse (n) : (list n).reverse = (list n).map rev := by
induction n with |
induction n with
| zero => rfl
| succ n ih =>
conv => lhs; rw [list_succ_last]
conv => rhs; rw [list_succ]
simp [List.reverse_map, ih, Function.comp_def, rev_succ]
| true |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 283 | 283 | theorem self_act_act_eq {x y : R} : (x ◃ x) ◃ y = x ◃ y := by | rw [← right_inv x y, ← self_distrib]
| true |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 136 | 137 | theorem map_inv (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
I⁻¹.map (h : K →ₐ[R₁] K') = (I.map h)⁻¹ := by | rw [inv_eq, map_div, map_one, inv_eq]
| true |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 53 | 60 | theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
{a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) := by
intro p hp |
intro p hp
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')
· exact ⟨a, left_mem_Icc.mpr hab, rfl⟩
· exact ⟨b, right_mem_Icc.mpr hab, rfl⟩
· have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
exact image_subset f Ioo_subset_Icc_self this
| true |
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.LinearAlgebra... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 66 | 74 | theorem Matrix.toLinearMap₂'Aux_stdBasis (f : Matrix n m R) (i : n) (j : m) :
f.toLinearMap₂'Aux σ₁ σ₂ (LinearMap.stdBasis R₁ (fun _ => R₁) i 1)
(LinearMap.stdBasis R₂ (fun _ => R₂) j 1) = f i j := by
rw [Matrix.toLinearMap₂'Aux, mk₂'ₛₗ_apply] |
rw [Matrix.toLinearMap₂'Aux, mk₂'ₛₗ_apply]
have : (∑ i', ∑ j', (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j := by
simp_rw [mul_assoc, ← Finset.mul_sum]
simp only [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true, mul_comm (f _ _)]
rw [← this]
exact Finset.sum_congr rfl f... | true |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 241 | 242 | theorem preimage_neg_Ico : -Ico a b = Ioc (-b) (-a) := by |
simp [← Ici_inter_Iio, ← Ioi_inter_Iic, inter_comm]
| true |
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 | 64 | 64 | theorem logb_abs (x : ℝ) : logb b |x| = logb b x := by | rw [logb, logb, log_abs]
| true |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 54 | 73 | theorem adic_basis (I : Ideal R) : SubmodulesRingBasis fun n : ℕ => (I ^ n • ⊤ : Ideal R) :=
{ inter := by
suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by |
suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by
simpa only [smul_eq_mul, mul_top, Algebra.id.map_eq_id, map_id, le_inf_iff] using this
intro i j
exact ⟨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)⟩
leftMul := by
suffices ∀ (a : R) (i : ℕ... | true |
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Init.Control.Lawful
#align_import control.monad.cont from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
universe u v w u₀ u₁ v₀ v₁
structure MonadCont.Label (α : Type w) (m : Type u → Type v) (β : Typ... | Mathlib/Control/Monad/Cont.lean | 193 | 194 | theorem WriterT.goto_mkLabel {α β ω : Type _} [EmptyCollection ω] (x : Label (α × ω) m β) (i : α) :
goto (WriterT.mkLabel x) i = monadLift (goto x (i, ∅)) := by | cases x; rfl
| true |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 104 | 106 | theorem kernelSubobject_arrow' :
(kernelSubobjectIso f).inv ≫ (kernelSubobject f).arrow = kernel.ι f := by |
simp [kernelSubobjectIso]
| true |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.discrete_valuation_ring.tfae from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variab... | Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean | 37 | 89 | theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
(h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
∃ n : ℕ, I = maximalIdeal R ^ n := by
by_cases h : IsField R; |
by_cases h : IsField R;
· exact ⟨0, by simp [letI := h.toField; (eq_bot_or_eq_top I).resolve_left hI]⟩
classical
obtain ⟨x, hx : _ = Ideal.span _⟩ := h'
by_cases hI' : I = ⊤
· use 0; rw [pow_zero, hI', Ideal.one_eq_top]
have H : ∀ r : R, ¬IsUnit r ↔ x ∣ r := fun r =>
(SetLike.ext_iff.mp hx r).trans I... | true |
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
import Mathlib.Data.List.Basic
#align_import data.list.pairwise from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Nat Function
namespace List
variable {α β : Type*} {R S T : α → α → Prop} {a : α} {l : List α}
mk_iff_o... | Mathlib/Data/List/Pairwise.lean | 81 | 86 | theorem Pairwise.forall (hR : Symmetric R) (hl : l.Pairwise R) :
∀ ⦃a⦄, a ∈ l → ∀ ⦃b⦄, b ∈ l → a ≠ b → R a b := by
apply Pairwise.forall_of_forall |
apply Pairwise.forall_of_forall
· exact fun a b h hne => hR (h hne.symm)
· exact fun _ _ hx => (hx rfl).elim
· exact hl.imp (@fun a b h _ => by exact h)
| true |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 293 | 294 | theorem map_smulₛₗ₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (c : R) (x : M) (y : F) :
f (c • x) y = ρ₁₂ c • f x y := by | rw [f.map_smulₛₗ, smul_apply]
| true |
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {α β γ ι ι' : Type*} {κ : Sort*} {r p q : α → α → Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 27 | 36 | theorem pairwise_iUnion {f : κ → Set α} (h : Directed (· ⊆ ·) f) :
(⋃ n, f n).Pairwise r ↔ ∀ n, (f n).Pairwise r := by
constructor |
constructor
· intro H n
exact Pairwise.mono (subset_iUnion _ _) H
· intro H i hi j hj hij
rcases mem_iUnion.1 hi with ⟨m, hm⟩
rcases mem_iUnion.1 hj with ⟨n, hn⟩
rcases h m n with ⟨p, mp, np⟩
exact H p (mp hm) (np hn) hij
| true |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 93 | 97 | theorem sub_mem_orthogonal_of_inner_right {x y : E} (h : ∀ v : K, ⟪(v : E), x⟫ = ⟪(v : E), y⟫) :
x - y ∈ Kᗮ := by
intro u hu |
intro u hu
rw [inner_sub_right, sub_eq_zero]
exact h ⟨u, hu⟩
| true |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 108 | 130 | theorem convexBodyLT_volume :
volume (convexBodyLT K f) = (convexBodyLTFactor K) * ∏ w, (f w) ^ (mult w) := by
calc |
calc
_ = (∏ x : {w // InfinitePlace.IsReal w}, ENNReal.ofReal (2 * (f x.val))) *
∏ x : {w // InfinitePlace.IsComplex w}, ENNReal.ofReal (f x.val) ^ 2 * NNReal.pi := by
simp_rw [volume_eq_prod, prod_prod, volume_pi, pi_pi, Real.volume_ball, Complex.volume_ball]
_ = ((2:ℝ≥0) ^ NrRealPlaces K * ... | true |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 55 | 59 | theorem self_subset_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
s ⊆ generateMeasurableRec s i := by
unfold generateMeasurableRec |
unfold generateMeasurableRec
apply_rules [subset_union_of_subset_left]
exact subset_rfl
| true |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 144 | 151 | theorem norm_max_aux₂ {f : ℂ → F} {z w : ℂ} (hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ := by
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL |
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
have he : ∀ x, ‖e x‖ = ‖x‖ := UniformSpace.Completion.norm_coe
replace hz : IsMaxOn (norm ∘ e ∘ f) (closedBall z (dist w z)) z := by
simpa only [IsMaxOn, (· ∘ ·), he] using hz
simpa only [he, (· ∘ ·)]
using norm_max_aux₁ (e.differentiable.comp_diff... | true |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 183 | 183 | theorem id_eval (v) : id.eval v = pure v := by | simp [id]
| true |
import Mathlib.Data.Multiset.Powerset
#align_import data.multiset.antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists Ring
universe u
namespace Multiset
open List
variable {α β : Type*}
def antidiagonal (s : Multiset α) : Multiset (Multiset α × Multis... | Mathlib/Data/Multiset/Antidiagonal.lean | 90 | 99 | theorem antidiagonal_eq_map_powerset [DecidableEq α] (s : Multiset α) :
s.antidiagonal = s.powerset.map fun t ↦ (s - t, t) := by
induction' s using Multiset.induction_on with a s hs |
induction' s using Multiset.induction_on with a s hs
· simp only [antidiagonal_zero, powerset_zero, zero_tsub, map_singleton]
· simp_rw [antidiagonal_cons, powerset_cons, map_add, hs, map_map, Function.comp, Prod.map_mk,
id, sub_cons, erase_cons_head]
rw [add_comm]
congr 1
refine Multiset.map_c... | true |
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 | 433 | 447 | theorem beth_strictMono : StrictMono beth := by
intro a b |
intro a b
induction' b using Ordinal.induction with b IH generalizing a
intro h
rcases zero_or_succ_or_limit b with (rfl | ⟨c, rfl⟩ | hb)
· exact (Ordinal.not_lt_zero a h).elim
· rw [lt_succ_iff] at h
rw [beth_succ]
apply lt_of_le_of_lt _ (cantor _)
rcases eq_or_lt_of_le h with (rfl | h)
· ... | true |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
#align_import linear_algebra.vandermonde from "leanprover-community/mathlib"@"70fd9563a21e7b963887c93... | Mathlib/LinearAlgebra/Vandermonde.lean | 49 | 56 | theorem vandermonde_cons {n : ℕ} (v0 : R) (v : Fin n → R) :
vandermonde (Fin.cons v0 v : Fin n.succ → R) =
Fin.cons (fun (j : Fin n.succ) => v0 ^ (j : ℕ)) fun i => Fin.cons 1
fun j => v i * vandermonde v i j := by
ext i j |
ext i j
refine Fin.cases (by simp) (fun i => ?_) i
refine Fin.cases (by simp) (fun j => ?_) j
simp [pow_succ']
| true |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 219 | 223 | theorem fderivWithin_zero_of_nmem_closure (h : x ∉ closure s) : fderivWithin 𝕜 f s x = 0 := by
apply fderivWithin_zero_of_isolated |
apply fderivWithin_zero_of_isolated
simp only [mem_closure_iff_nhdsWithin_neBot, neBot_iff, Ne, Classical.not_not] at h
rw [eq_bot_iff, ← h]
exact nhdsWithin_mono _ diff_subset
| true |
import Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots
import Mathlib.FieldTheory.Finite.Trace
import Mathlib.Algebra.Group.AddChar
import Mathlib.Data.ZMod.Units
import Mathlib.Analysis.Complex.Polynomial
#align_import number_theory.legendre_symbol.add_character from "leanprover-community/mathlib"@"0723536a0522d24fc2... | Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean | 169 | 171 | theorem zmodChar_apply' {n : ℕ+} {ζ : C} (hζ : ζ ^ (n : ℕ) = 1) (a : ℕ) :
zmodChar n hζ a = ζ ^ a := by |
rw [pow_eq_pow_mod a hζ, zmodChar_apply, ZMod.val_natCast a]
| true |
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
open Set Filter Bornology
open scoped ENNReal Uniformity Topology Pointwise
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
namespace Metric
#align metric.bounded Bornology.I... | Mathlib/Topology/MetricSpace/Bounded.lean | 142 | 144 | theorem tendsto_dist_left_atTop_iff (c : α) {f : β → α} {l : Filter β} :
Tendsto (fun x ↦ dist c (f x)) l atTop ↔ Tendsto f l (cobounded α) := by |
simp only [dist_comm c, tendsto_dist_right_atTop_iff]
| true |
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 | 141 | 143 | theorem comp_left_top (r : Rel α β) : (⊤ : Rel γ α) • r = fun _ y ↦ y ∈ r.codom := by
ext x z |
ext x z
simp [comp, Top.top, codom]
| true |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section Group
variable [Group α] (e : α) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 137 | 137 | theorem mulETransformRight_one : mulETransformRight 1 x = x := by | simp [mulETransformRight]
| true |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 149 | 150 | theorem continuous_univBall (c : P) (r : ℝ) : Continuous (univBall c r) := by |
simpa [continuous_iff_continuousOn_univ] using (univBall c r).continuousOn
| true |
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.Convex.Deriv
#align_import analysis.convex.specific_functions.deriv from "leanprover-communi... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 88 | 94 | theorem int_prod_range_pos {m : ℤ} {n : ℕ} (hn : Even n) (hm : m ∉ Ico (0 : ℤ) n) :
0 < ∏ k ∈ Finset.range n, (m - k) := by
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_ |
refine (int_prod_range_nonneg m n hn).lt_of_ne fun h => hm ?_
rw [eq_comm, Finset.prod_eq_zero_iff] at h
obtain ⟨a, ha, h⟩ := h
rw [sub_eq_zero.1 h]
exact ⟨Int.ofNat_zero_le _, Int.ofNat_lt.2 <| Finset.mem_range.1 ha⟩
| true |
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.optional_stopping from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheory
vari... | Mathlib/Probability/Martingale/OptionalStopping.lean | 42 | 63 | theorem Submartingale.expected_stoppedValue_mono [SigmaFiniteFiltration μ 𝒢]
(hf : Submartingale f 𝒢 μ) (hτ : IsStoppingTime 𝒢 τ) (hπ : IsStoppingTime 𝒢 π) (hle : τ ≤ π)
{N : ℕ} (hbdd : ∀ ω, π ω ≤ N) : μ[stoppedValue f τ] ≤ μ[stoppedValue f π] := by
rw [← sub_nonneg, ← integral_sub', stoppedValue_sub_eq_s... |
rw [← sub_nonneg, ← integral_sub', stoppedValue_sub_eq_sum' hle hbdd]
· simp only [Finset.sum_apply]
have : ∀ i, MeasurableSet[𝒢 i] {ω : Ω | τ ω ≤ i ∧ i < π ω} := by
intro i
refine (hτ i).inter ?_
convert (hπ i).compl using 1
ext x
simp; rfl
rw [integral_finset_sum]
· ref... | true |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 122 | 125 | theorem foldr_eq_foldr_list (f : Fin n → α → α) (x) : foldr n f x = (list n).foldr f x := by
induction n with |
induction n with
| zero => rw [foldr_zero, list_zero, List.foldr_nil]
| succ n ih => rw [foldr_succ, ih, list_succ, List.foldr_cons, List.foldr_map]
| true |
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 27 | 30 | theorem exists_sum {γ : α ⊕ β → Sort*} (p : (∀ ab, γ ab) → Prop) :
(∃ fab, p fab) ↔ (∃ fa fb, p (Sum.rec fa fb)) := by
rw [← not_forall_not, forall_sum] |
rw [← not_forall_not, forall_sum]
simp
| true |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 77 | 83 | theorem natAbs_euclideanDomain_gcd (a b : ℤ) :
Int.natAbs (EuclideanDomain.gcd a b) = Int.gcd a b := by
apply Nat.dvd_antisymm <;> rw [← Int.natCast_dvd_natCast] |
apply Nat.dvd_antisymm <;> rw [← Int.natCast_dvd_natCast]
· rw [Int.natAbs_dvd]
exact Int.dvd_gcd (EuclideanDomain.gcd_dvd_left _ _) (EuclideanDomain.gcd_dvd_right _ _)
· rw [Int.dvd_natAbs]
exact EuclideanDomain.dvd_gcd Int.gcd_dvd_left Int.gcd_dvd_right
| true |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section LT
variable [LT α] {a b : α}
def CovBy (a b :... | Mathlib/Order/Cover.lean | 233 | 234 | theorem not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b := by |
simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
| true |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 133 | 138 | theorem definable_finset_sup {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.sup f) := by
classical |
classical
refine Finset.induction definable_empty (fun i s _ h => ?_) s
rw [Finset.sup_insert]
exact (hf i).union h
| true |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 140 | 148 | theorem moment_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f μ) {A : ℝ} (hA : 0 ≤ A)
{n : ℕ} (hn : n ≠ 0) : ∫ x, truncation f A x ^ n ∂μ = ∫ y in -A..A, y ^ n ∂Measure.map f μ := by
have M : MeasurableSet (Set.Ioc (-A) A) := measurableSet_Ioc |
have M : MeasurableSet (Set.Ioc (-A) A) := measurableSet_Ioc
change ∫ x, (fun z => indicator (Set.Ioc (-A) A) id z ^ n) (f x) ∂μ = _
rw [← integral_map (f := fun z => _ ^ n) hf.aemeasurable, intervalIntegral.integral_of_le,
← integral_indicator M]
· simp only [indicator, zero_pow hn, id, ite_pow]
· linar... | true |
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.wilson from "leanprover-community/mathlib"@"c471da714c044131b90c133701e51b877c246677"
open Finset Nat FiniteField ZMod
open scoped Nat
namespace ZMod
variable (p : ℕ) [Fact p.Prime]
@[simp]
theorem wilsons_lemma : ((p - 1)! : ZMod p) = -1 :=... | Mathlib/NumberTheory/Wilson.lean | 73 | 79 | theorem prod_Ico_one_prime : ∏ x ∈ Ico 1 p, (x : ZMod p) = -1 := by
-- Porting note: was `conv in Ico 1 p =>` |
-- Porting note: was `conv in Ico 1 p =>`
conv =>
congr
congr
rw [← Nat.add_one_sub_one p, succ_sub (Fact.out (p := p.Prime)).pos]
rw [← prod_natCast, Finset.prod_Ico_id_eq_factorial, wilsons_lemma]
| true |
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 116 | 126 | theorem exists_nonzero_mem_of_ne_bot {P : Ideal R[X]} (Pb : P ≠ ⊥) (hP : ∀ x : R, C x ∈ P → x = 0) :
∃ p : R[X], p ∈ P ∧ Polynomial.map (Quotient.mk (P.comap (C : R →+* R[X]))) p ≠ 0 := by
obtain ⟨m, hm⟩ := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb) |
obtain ⟨m, hm⟩ := Submodule.nonzero_mem_of_bot_lt (bot_lt_iff_ne_bot.mpr Pb)
refine ⟨m, Submodule.coe_mem m, fun pp0 => hm (Submodule.coe_eq_zero.mp ?_)⟩
refine
(injective_iff_map_eq_zero (Polynomial.mapRingHom (Ideal.Quotient.mk
(P.comap (C : R →+* R[X]))))).mp
?_ _ pp0
refine map_injective _ ... | true |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 119 | 123 | theorem lift_base (f : ∀ i, G i →* K) (k : H →* K)
(hf : ∀ i, (f i).comp (φ i) = k)
(g : H) : (lift f k hf) (base φ g : PushoutI φ) = k g := by
delta PushoutI lift base |
delta PushoutI lift base
simp only [MonoidHom.coe_comp, Con.coe_mk', comp_apply, Con.lift_coe, lift_apply_inr]
| true |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 118 | 145 | theorem iff_pid_with_one_nonzero_prime (R : Type u) [CommRing R] [IsDomain R] :
DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P : Ideal R, P ≠ ⊥ ∧ IsPrime P := by
constructor |
constructor
· intro RDVR
rcases id RDVR with ⟨Rlocal⟩
constructor
· assumption
use LocalRing.maximalIdeal R
constructor
· exact ⟨Rlocal, inferInstance⟩
· rintro Q ⟨hQ1, hQ2⟩
obtain ⟨q, rfl⟩ := (IsPrincipalIdealRing.principal Q).1
have hq : q ≠ 0 := by
rintro rfl
... | true |
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
universe v u
namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
def endofunctorMonoidalCategory : MonoidalCategory (C ⥤ C) where... | Mathlib/CategoryTheory/Monoidal/End.lean | 129 | 131 | theorem ε_inv_naturality {X Y : C} (f : X ⟶ Y) :
(MonoidalFunctor.εIso F).inv.app X ≫ (𝟙_ (C ⥤ C)).map f = F.εIso.inv.app X ≫ f := by |
aesop_cat
| true |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.cover from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open Set OrderDual
variable {α β : Type*}
section WeaklyCovers
section Preorder
variable [Preorder α] [Preorder β] {a ... | Mathlib/Order/Cover.lean | 122 | 126 | theorem WCovBy.image (f : α ↪o β) (hab : a ⩿ b) (h : (range f).OrdConnected) : f a ⩿ f b := by
refine ⟨f.monotone hab.le, fun c ha hb => ?_⟩ |
refine ⟨f.monotone hab.le, fun c ha hb => ?_⟩
obtain ⟨c, rfl⟩ := h.out (mem_range_self _) (mem_range_self _) ⟨ha.le, hb.le⟩
rw [f.lt_iff_lt] at ha hb
exact hab.2 ha hb
| true |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Combinatorics.SetFamily.Compression.Down
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Fintype.Powerset
#align_import combinatorics.set_family.harris_kleitman from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
open Finset... | Mathlib/Combinatorics/SetFamily/HarrisKleitman.lean | 41 | 45 | theorem IsLowerSet.memberSubfamily (h : IsLowerSet (𝒜 : Set (Finset α))) :
IsLowerSet (𝒜.memberSubfamily a : Set (Finset α)) := by
rintro s t hts |
rintro s t hts
simp_rw [mem_coe, mem_memberSubfamily]
exact And.imp (h <| insert_subset_insert _ hts) (mt <| @hts _)
| true |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.constructions.prod.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable s... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 127 | 129 | theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
(hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν := by |
rw [← uncurry_curry f] at hf; exact hf.integral_prod_right
| true |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 119 | 120 | theorem card_Ioo : (Ioo a b).card = b - a - 1 := by |
rw [← Nat.card_Ioo, ← map_valEmbedding_Ioo, card_map]
| true |
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"... | Mathlib/RingTheory/WittVector/Frobenius.lean | 71 | 74 | theorem bind₁_frobeniusPolyRat_wittPolynomial (n : ℕ) :
bind₁ (frobeniusPolyRat p) (wittPolynomial p ℚ n) = wittPolynomial p ℚ (n + 1) := by
delta frobeniusPolyRat |
delta frobeniusPolyRat
rw [← bind₁_bind₁, bind₁_xInTermsOfW_wittPolynomial, bind₁_X_right, Function.comp_apply]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 56 | 61 | theorem toFunBilinear_apply_eq_sum (a : A) (p : R[X]) :
toFunBilinear R A a p = p.sum fun n r => monomial n (a * algebraMap R A r) := by
simp only [toFunBilinear_apply_apply, aeval_def, eval₂_eq_sum, Polynomial.sum, Finset.smul_sum] |
simp only [toFunBilinear_apply_apply, aeval_def, eval₂_eq_sum, Polynomial.sum, Finset.smul_sum]
congr with i : 1
rw [← Algebra.smul_def, ← C_mul', mul_smul_comm, C_mul_X_pow_eq_monomial, ← Algebra.commutes,
← Algebra.smul_def, smul_monomial]
| true |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 63 | 65 | theorem IsUpperSet.mul_left (ht : IsUpperSet t) : IsUpperSet (s * t) := by
rw [← smul_eq_mul, ← Set.iUnion_smul_set] |
rw [← smul_eq_mul, ← Set.iUnion_smul_set]
exact isUpperSet_iUnion₂ fun x _ ↦ ht.smul
| true |
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 | 116 | 118 | theorem elim_eq_iff {u u' : α → γ} {v v' : β → γ} :
Sum.elim u v = Sum.elim u' v' ↔ u = u' ∧ v = v' := by |
simp [funext_iff]
| true |
import Batteries.Data.List.Basic
namespace Batteries
inductive AssocList (α : Type u) (β : Type v) where
| nil
| cons (key : α) (value : β) (tail : AssocList α β)
deriving Inhabited
namespace AssocList
@[simp] def toList : AssocList α β → List (α × β)
| nil => []
| cons a b es => (a, b) :: es.toL... | .lake/packages/batteries/Batteries/Data/AssocList.lean | 55 | 56 | theorem length_toList (l : AssocList α β) : l.toList.length = l.length := by |
induction l <;> simp_all
| true |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 240 | 250 | theorem hasFTaylorSeriesUpToOn_top_iff :
HasFTaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFTaylorSeriesUpToOn n f p s := by
constructor |
constructor
· intro H n; exact H.of_le le_top
· intro H
constructor
· exact (H 0).zero_eq
· intro m _
apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))
· intro m _
apply (H m).cont m le_rfl
| true |
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 | 195 | 201 | theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by
ext1 x |
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
| true |
import Mathlib.Data.Set.Basic
open Function
universe u v
namespace Set
section Subsingleton
variable {α : Type u} {a : α} {s t : Set α}
protected def Subsingleton (s : Set α) : Prop :=
∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), x = y
#align set.subsingleton Set.Subsingleton
theorem Subsingleton.anti (ht : t.Subs... | Mathlib/Data/Set/Subsingleton.lean | 109 | 113 | theorem subsingleton_coe (s : Set α) : Subsingleton s ↔ s.Subsingleton := by
constructor |
constructor
· refine fun h => fun a ha b hb => ?_
exact SetCoe.ext_iff.2 (@Subsingleton.elim s h ⟨a, ha⟩ ⟨b, hb⟩)
· exact fun h => Subsingleton.intro fun a b => SetCoe.ext (h a.property b.property)
| true |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 164 | 165 | theorem one_add_mul [RightDistribClass α] (a b : α) : (1 + a) * b = b + a * b := by |
rw [add_mul, one_mul]
| true |
import Mathlib.CategoryTheory.SingleObj
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.CategoryTheory.Limits.IsLimit
#align_import category_theory.category.Groupoid from "leanprover-community/mathlib"@"c9c9fa15fec7ca18e9ec97306fb8764bfe988a7e"
universe v... | Mathlib/CategoryTheory/Category/Grpd.lean | 152 | 155 | theorem piIsoPi_hom_π (J : Type u) (f : J → Grpd.{u, u}) (j : J) :
(piIsoPi J f).hom ≫ Limits.Pi.π f j = CategoryTheory.Pi.eval _ j := by
simp [piIsoPi] |
simp [piIsoPi]
rfl
| true |
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 126 | 128 | theorem pullbackIsoProdSubtype_hom_fst (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullbackIsoProdSubtype f g).hom ≫ pullbackFst f g = pullback.fst := by |
rw [← Iso.eq_inv_comp, pullbackIsoProdSubtype_inv_fst]
| true |
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {α β γ ι ι' : Type*}
namespace Filter
section Relation
... | Mathlib/Order/LiminfLimsup.lean | 103 | 106 | theorem IsBoundedUnder.mono_le [Preorder β] {l : Filter α} {u v : α → β}
(hu : IsBoundedUnder (· ≤ ·) l u) (hv : v ≤ᶠ[l] u) : IsBoundedUnder (· ≤ ·) l v := by
apply hu.imp |
apply hu.imp
exact fun b hb => (eventually_map.1 hb).mp <| hv.mono fun x => le_trans
| true |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.complete_lattice_intervals from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
open scoped Classical
open Set
variable {ι : ... | Mathlib/Order/CompleteLatticeIntervals.lean | 102 | 104 | theorem subset_sInf_emptyset [Inhabited s] :
sInf (∅ : Set s) = default := by |
simp [sInf]
| true |
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 | 80 | 84 | theorem exists_finsupp_right (x : M ⊗[R] N) :
∃ S : N →₀ M, x = S.sum fun n m ↦ m ⊗ₜ[R] n := by
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x) |
obtain ⟨S, h⟩ := exists_finsupp_left (TensorProduct.comm R M N x)
refine ⟨S, (TensorProduct.comm R M N).injective ?_⟩
simp_rw [h, Finsupp.sum, map_sum, comm_tmul]
| true |
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 | 87 | 88 | theorem Rel.addLeft (a : lib R X) {b c : lib R X} (h : Rel R X b c) : Rel R X (a + b) (a + c) := by |
rw [add_comm _ b, add_comm _ c]; exact h.add_right _
| true |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 231 | 234 | theorem vars_eq_support_biUnion_support [DecidableEq σ] :
p.vars = p.support.biUnion Finsupp.support := by
ext i |
ext i
rw [mem_vars, Finset.mem_biUnion]
| true |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 107 | 124 | theorem sum_le_of_monotoneOn_Icc (f : α → E) {s : Set α} {m n : ℕ} {u : ℕ → α}
(hu : MonotoneOn u (Icc m n)) (us : ∀ i ∈ Icc m n, u i ∈ s) :
(∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))) ≤ eVariationOn f s := by
rcases le_total n m with hnm | hmn |
rcases le_total n m with hnm | hmn
· simp [Finset.Ico_eq_empty_of_le hnm]
let π := projIcc m n hmn
let v i := u (π i)
calc
∑ i ∈ Finset.Ico m n, edist (f (u (i + 1))) (f (u i))
= ∑ i ∈ Finset.Ico m n, edist (f (v (i + 1))) (f (v i)) :=
Finset.sum_congr rfl fun i hi ↦ by
rw [Finset.m... | true |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 314 | 346 | theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
continuousMultilinearCurryFin1 𝕜 (∀ i, E i) F (f.toFormalMultilinearSeries.changeOrigin x 1) =
f.linearDeriv x := by
ext y |
ext y
rw [continuousMultilinearCurryFin1_apply, linearDeriv_apply,
changeOrigin, FormalMultilinearSeries.sum]
cases isEmpty_or_nonempty ι
· have (l) : 1 + l ≠ Fintype.card ι := by
rw [add_comm, Fintype.card_eq_zero]; exact Nat.succ_ne_zero _
simp_rw [Fintype.sum_empty, changeOriginSeries_suppor... | true |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 290 | 292 | theorem multipliable_iff_nat_tprod_vanishing {f : ℕ → G} : Multipliable f ↔
∀ e ∈ 𝓝 1, ∃ N : ℕ, ∀ t ⊆ {n | N ≤ n}, (∏' n : t, f n) ∈ e := by |
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_nat_tprod_vanishing]
| true |
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.Algebra.CharP.Algebra
#align_import field_theory.splitting_field.construction from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial
universe u v w
variable {... | Mathlib/FieldTheory/SplittingField/Construction.lean | 97 | 100 | theorem natDegree_removeFactor (f : K[X]) : f.removeFactor.natDegree = f.natDegree - 1 := by
-- Porting note: `(map (AdjoinRoot.of f.factor) f)` was `_` |
-- Porting note: `(map (AdjoinRoot.of f.factor) f)` was `_`
rw [removeFactor, natDegree_divByMonic (map (AdjoinRoot.of f.factor) f) (monic_X_sub_C _),
natDegree_map, natDegree_X_sub_C]
| true |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 116 | 120 | theorem foldr_succ_last (f : Fin (n+1) → α → α) (x) :
foldr (n+1) f x = foldr n (f ·.castSucc) (f (last n) x) := by
induction n generalizing x with |
induction n generalizing x with
| zero => simp [foldr_succ, Fin.last]
| succ n ih => rw [foldr_succ, ih (f ·.succ), foldr_succ]; simp [succ_castSucc]
| true |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 162 | 166 | theorem quotient_norm_mk_eq (S : AddSubgroup M) (m : M) :
‖mk' S m‖ = sInf ((‖m + ·‖) '' S) := by
rw [mk'_apply, norm_mk, sInf_image', ← infDist_image isometry_neg, image_neg, |
rw [mk'_apply, norm_mk, sInf_image', ← infDist_image isometry_neg, image_neg,
neg_coe_set (H := S), infDist_eq_iInf]
simp only [dist_eq_norm', sub_neg_eq_add, add_comm]
| true |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 70 | 73 | theorem one_div_le_one_div {n m : ℕ} (h : n ≤ m) : 1 / ((m : α) + 1) ≤ 1 / ((n : α) + 1) := by
refine one_div_le_one_div_of_le ?_ ?_ |
refine one_div_le_one_div_of_le ?_ ?_
· exact Nat.cast_add_one_pos _
· simpa
| true |
import Mathlib.Probability.Kernel.Disintegration.Basic
open MeasureTheory ProbabilityTheory MeasurableSpace
open scoped ENNReal
namespace ProbabilityTheory
variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
[MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω]
namespace MeasureTheory... | Mathlib/Probability/Kernel/Disintegration/Integral.lean | 261 | 267 | theorem AEStronglyMeasurable.ae_integrable_condKernel_iff {f : α × Ω → F}
(hf : AEStronglyMeasurable f ρ) :
(∀ᵐ a ∂ρ.fst, Integrable (fun ω ↦ f (a, ω)) (ρ.condKernel a)) ∧
Integrable (fun a ↦ ∫ ω, ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst ↔ Integrable f ρ := by
rw [← ρ.compProd_fst_condKernel] at hf |
rw [← ρ.compProd_fst_condKernel] at hf
conv_rhs => rw [← ρ.compProd_fst_condKernel]
rw [Measure.integrable_compProd_iff hf]
| true |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 321 | 322 | theorem rowLen_eq_card (μ : YoungDiagram) {i : ℕ} : μ.rowLen i = (μ.row i).card := by |
simp [row_eq_prod]
| true |
import Mathlib.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
#align_import computability.halting from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
open Encodable Denumerable
namespace Nat.Partrec
open Computable Part
| Mathlib/Computability/Halting.lean | 28 | 60 | theorem merge' {f g} (hf : Nat.Partrec f) (hg : Nat.Partrec g) :
∃ h, Nat.Partrec h ∧
∀ a, (∀ x ∈ h a, x ∈ f a ∨ x ∈ g a) ∧ ((h a).Dom ↔ (f a).Dom ∨ (g a).Dom) := by
obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf |
obtain ⟨cf, rfl⟩ := Code.exists_code.1 hf
obtain ⟨cg, rfl⟩ := Code.exists_code.1 hg
have : Nat.Partrec fun n => Nat.rfindOpt fun k => cf.evaln k n <|> cg.evaln k n :=
Partrec.nat_iff.1
(Partrec.rfindOpt <|
Primrec.option_orElse.to_comp.comp
(Code.evaln_prim.to_comp.comp <| (snd.pair (... | true |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 76 | 78 | theorem continuous_sinh : Continuous sinh := by
change Continuous fun z => (exp z - exp (-z)) / 2 |
change Continuous fun z => (exp z - exp (-z)) / 2
continuity
| true |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 93 | 95 | theorem Subsingleton.einfsep (hs : s.Subsingleton) : s.einfsep = ∞ := by
rw [einfsep_top] |
rw [einfsep_top]
exact fun _ hx _ hy hxy => (hxy <| hs hx hy).elim
| true |
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
import Mathlib.Topology.Category.Profinite.Limits
import Mathlib.Topology.Category.Stonean.Basic
universe u
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTh... | Mathlib/Topology/Category/Profinite/EffectiveEpi.lean | 69 | 82 | theorem effectiveEpi_tfae
{B X : Profinite.{u}} (π : X ⟶ B) :
TFAE
[ EffectiveEpi π
, Epi π
, Function.Surjective π
] := by
tfae_have 1 → 2 |
tfae_have 1 → 2
· intro; infer_instance
tfae_have 2 ↔ 3
· exact epi_iff_surjective π
tfae_have 3 → 1
· exact fun hπ ↦ ⟨⟨struct π hπ⟩⟩
tfae_finish
| true |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 113 | 115 | theorem image_coe_Ico : (some : α → WithTop α) '' Ico a b = Ico (a : WithTop α) b := by
rw [← preimage_coe_Ico, image_preimage_eq_inter_range, range_coe, |
rw [← preimage_coe_Ico, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Subset.trans Ico_subset_Iio_self <| Iio_subset_Iio le_top)]
| true |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 133 | 141 | theorem isOpen_A (L : E →L[𝕜] F) (r ε : ℝ) : IsOpen (A f L r ε) := by
rw [Metric.isOpen_iff] |
rw [Metric.isOpen_iff]
rintro x ⟨r', r'_mem, hr'⟩
obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between r'_mem.1
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩
refine ⟨r' - s, by linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
have B : ball x' s ⊆ ball x r' := ball_subset (le... | true |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iSup
@[simp]
theorem iSup_eq_zero {ι : Sort*} {f : ι → ℝ≥0∞} : ⨆ i, f i = 0 ↔ ∀ i, f i = 0 :=
iSup_eq_bot
#align ennr... | Mathlib/Data/ENNReal/Real.lean | 676 | 676 | theorem iSup_zero_eq_zero {ι : Sort*} : ⨆ _ : ι, (0 : ℝ≥0∞) = 0 := by | simp
| true |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 337 | 339 | theorem coe_toDual_self (i : ι) : b.toDual (b i) = b.coord i := by
ext |
ext
apply toDual_apply_right
| true |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Finset
open Function Multiset
variable {α : Type*} {s t : Finset α}
section Powerset
def powerset (s : Finset... | Mathlib/Data/Finset/Powerset.lean | 83 | 84 | theorem powerset_eq_singleton_empty : s.powerset = {∅} ↔ s = ∅ := by |
rw [← powerset_empty, powerset_inj]
| true |
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.absolute_value from "leanprover-community/mathlib"@"ab0a2959c83b06280ef576bc830d4aa5fe8c8e61"
open Matrix
namespace Matrix
open Equiv Finset
variable {R S : Type*} [CommRing R] [Nontr... | Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean | 37 | 49 | theorem det_le {A : Matrix n n R} {abv : AbsoluteValue R S} {x : S} (hx : ∀ i j, abv (A i j) ≤ x) :
abv A.det ≤ Nat.factorial (Fintype.card n) • x ^ Fintype.card n :=
calc
abv A.det = abv (∑ σ : Perm n, Perm.sign σ • ∏ i, A (σ i) i) := congr_arg abv (det_apply _)
_ ≤ ∑ σ : Perm n, abv (Perm.sign σ • ∏ i, ... |
rw [sum_const, Finset.card_univ, Fintype.card_perm]
| true |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 162 | 166 | theorem eq_of_Ico_disjoint {x₁ x₂ y₁ y₂ : α} (h : Disjoint (Ico x₁ x₂) (Ico y₁ y₂)) (hx : x₁ < x₂)
(h2 : x₂ ∈ Ico y₁ y₂) : y₁ = x₂ := by
rw [Ico_disjoint_Ico, min_eq_left (le_of_lt h2.2), le_max_iff] at h |
rw [Ico_disjoint_Ico, min_eq_left (le_of_lt h2.2), le_max_iff] at h
apply le_antisymm h2.1
exact h.elim (fun h => absurd hx (not_lt_of_le h)) id
| true |
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 93 | 101 | theorem tendsto_support_normed_smallSets {ι} {φ : ι → ContDiffBump c} {l : Filter ι}
(hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)) :
Tendsto (fun i => Function.support fun x => (φ i).normed μ x) l (𝓝 c).smallSets := by
simp_rw [NormedAddCommGroup.tendsto_nhds_zero, Real.norm_eq_abs, |
simp_rw [NormedAddCommGroup.tendsto_nhds_zero, Real.norm_eq_abs,
abs_eq_self.mpr (φ _).rOut_pos.le] at hφ
rw [nhds_basis_ball.smallSets.tendsto_right_iff]
refine fun ε hε ↦ (hφ ε hε).mono fun i hi ↦ ?_
rw [(φ i).support_normed_eq]
exact ball_subset_ball hi.le
| true |
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 | 96 | 97 | theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by |
simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
| true |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 159 | 165 | theorem IsDiag.fromBlocks [Zero α] {A : Matrix m m α} {D : Matrix n n α} (ha : A.IsDiag)
(hd : D.IsDiag) : (A.fromBlocks 0 0 D).IsDiag := by
rintro (i | i) (j | j) hij |
rintro (i | i) (j | j) hij
· exact ha (ne_of_apply_ne _ hij)
· rfl
· rfl
· exact hd (ne_of_apply_ne _ hij)
| true |
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 | 290 | 291 | theorem hom_inv_id_assoc (f : X ⟶ Y) [I : IsIso f] {Z} (g : X ⟶ Z) : f ≫ inv f ≫ g = g := by |
simp [← Category.assoc]
| true |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 68 | 71 | theorem Icc_mul_Ico_subset' (a b c d : α) : Icc a b * Ico c d ⊆ Ico (a * c) (b * d) := by
haveI := covariantClass_le_of_lt |
haveI := covariantClass_le_of_lt
rintro x ⟨y, ⟨hya, hyb⟩, z, ⟨hzc, hzd⟩, rfl⟩
exact ⟨mul_le_mul' hya hzc, mul_lt_mul_of_le_of_lt hyb hzd⟩
| true |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 132 | 133 | theorem length_rotate (l : List α) (n : ℕ) : (l.rotate n).length = l.length := by |
rw [rotate_eq_rotate', length_rotate']
| true |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 140 | 148 | theorem isEquivalent_const_iff_tendsto {c : β} (h : c ≠ 0) :
u ~[l] const _ c ↔ Tendsto u l (𝓝 c) := by
simp (config := { unfoldPartialApp := true }) only [IsEquivalent, const, isLittleO_const_iff h] |
simp (config := { unfoldPartialApp := true }) only [IsEquivalent, const, isLittleO_const_iff h]
constructor <;> intro h
· have := h.sub (tendsto_const_nhds (x := -c))
simp only [Pi.sub_apply, sub_neg_eq_add, sub_add_cancel, zero_add] at this
exact this
· have := h.sub (tendsto_const_nhds (x := c))
... | true |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 151 | 153 | theorem frobeniusEquiv_symm_comp_frobenius :
((frobeniusEquiv R p).symm : R →+* R).comp (frobenius R p) = RingHom.id R := by |
ext; simp
| true |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 390 | 394 | theorem det_fromBlocks₁₁ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [Invertible A] :
(Matrix.fromBlocks A B C D).det = det A * det (D - C * ⅟ A * B) := by
rw [fromBlocks_eq_of_invertible₁₁ (A := A), det_mul, det_mul, det_fromBlocks_zero₂₁, |
rw [fromBlocks_eq_of_invertible₁₁ (A := A), det_mul, det_mul, det_fromBlocks_zero₂₁,
det_fromBlocks_zero₂₁, det_fromBlocks_zero₁₂, det_one, det_one, one_mul, one_mul, mul_one]
| true |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 147 | 147 | theorem map_zero (d : ℕ) (a : Fin 0 → ℕ) : map d a = 0 := by | simp [map]
| true |
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 | 52 | 53 | theorem mul_eq {α} [OrderedSemiring α] {a b : α} (ha : a = 0) (_ : 0 < b) : b * a = 0 := by |
simp [*]
| true |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
noncomputable section
variable {𝕜 : Type*} {E F G H : Type*}
open Filter List
open scoped Topol... | Mathlib/Analysis/Analytic/Composition.lean | 140 | 162 | theorem applyComposition_update (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n)
(j : Fin n) (v : Fin n → E) (z : E) :
p.applyComposition c (Function.update v j z) =
Function.update (p.applyComposition c v) (c.index j)
(p (c.blocksFun (c.index j))
(Function.update (v ∘ c.... |
ext k
by_cases h : k = c.index j
· rw [h]
let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
simp only [Function.update_same]
change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
let j' := c.invEmbedding j
suffices B : Function.update v j z ∘ r = Function.u... | true |
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.CategoryTheory.Sites.LocallyBijective
import Mathlib.CategoryTheory.Sites.PreservesLocallyBijective
import Mathlib.CategoryTheory.Sites.Whiskering
universe u
namespace CategoryTheory
open Functor Limits GrothendieckTopology
variable {C : Type*} [C... | Mathlib/CategoryTheory/Sites/Equivalence.lean | 67 | 82 | theorem coverPreserving : CoverPreserving J (e.locallyCoverDense J).inducedTopology e.functor where
cover_preserve {U S} h := by
change _ ∈ J.sieves (e.inverse.obj (e.functor.obj U)) |
change _ ∈ J.sieves (e.inverse.obj (e.functor.obj U))
convert J.pullback_stable (e.unitInv.app U) h
ext Z f
rw [← Sieve.functorPushforward_comp]
simp only [Sieve.functorPushforward_apply, Presieve.functorPushforward, exists_and_left, id_obj,
comp_obj, Sieve.pullback_apply]
constructor
... | true |
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Tactic.FinCases
#align_import linear_algebra.matrix.block from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Finset Function OrderDual
open Matrix
universe v
v... | Mathlib/LinearAlgebra/Matrix/Block.lean | 63 | 69 | theorem blockTriangular_reindex_iff {b : n → α} {e : m ≃ n} :
(reindex e e M).BlockTriangular b ↔ M.BlockTriangular (b ∘ e) := by
refine ⟨fun h => ?_, fun h => ?_⟩ |
refine ⟨fun h => ?_, fun h => ?_⟩
· convert h.submatrix
simp only [reindex_apply, submatrix_submatrix, submatrix_id_id, Equiv.symm_comp_self]
· convert h.submatrix
simp only [comp.assoc b e e.symm, Equiv.self_comp_symm, comp_id]
| true |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 95 | 96 | theorem map₂_map_left (f : γ → β → δ) (g : α → γ) :
map₂ f (a.map g) b = map₂ (fun a b => f (g a) b) a b := by | cases a <;> rfl
| true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 218 | 219 | theorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by | simp only [← mul_apply, f.mul_inv, coe_one, id]
| true |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 111 | 118 | theorem nonempty_linearEquiv_of_lift_rank_eq
(cnd : Cardinal.lift.{v'} (Module.rank R M) = Cardinal.lift.{v} (Module.rank R M')) :
Nonempty (M ≃ₗ[R] M') := by
obtain ⟨⟨α, B⟩⟩ := Module.Free.exists_basis (R := R) (M := M) |
obtain ⟨⟨α, B⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
obtain ⟨⟨β, B'⟩⟩ := Module.Free.exists_basis (R := R) (M := M')
have : Cardinal.lift.{v', v} #α = Cardinal.lift.{v, v'} #β := by
rw [B.mk_eq_rank'', cnd, B'.mk_eq_rank'']
exact (Cardinal.lift_mk_eq.{v, v', 0}.1 this).map (B.equiv B')
| true |
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.quotient_pi from "leanprover-community/mathlib"@"398f60f60b43ef42154bd2bdadf5133daf1577a4"
namespace Submodule
open LinearMap
variable {ι R : Type*} [CommRing R]
variable {Ms : ι → Type*} [∀ i, AddCommGroup (Ms i)... | Mathlib/LinearAlgebra/QuotientPi.lean | 42 | 46 | theorem piQuotientLift_mk [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
(q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (x : ∀ i, Ms i) :
(piQuotientLift p q f hf fun i => Quotient.mk (x i)) = Quotient.mk (lsum _ _ R f x) := by
rw [piQuotientLift, lsum_apply, sum_apply,... |
rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]
simp only [coe_proj, mapQ_apply, mkQ_apply, comp_apply]
| true |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 52 | 54 | theorem coprime_fintype_prod_left_iff [Fintype ι] {s : ι → ℕ} {x : ℕ} :
Coprime (∏ i, s i) x ↔ ∀ i, Coprime (s i) x := by |
simp [coprime_prod_left_iff]
| true |
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