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.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 76 | 78 | theorem cobounded_eq : Bornology.cobounded ℤ = atBot ⊔ atTop := by |
simp_rw [← comap_dist_right_atTop (0 : ℤ), dist_eq', sub_zero,
← comap_abs_atTop, ← @Int.comap_cast_atTop ℝ, comap_comap]; rfl
| 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 | 162 | 165 | theorem mk_coe_def (p : K[X]) (q : K[X]⁰) :
-- Porting note: filled in `(FractionRing K[X])` that was an underscore.
RatFunc.mk p q = ofFractionRing (IsLocalization.mk' (FractionRing K[X]) p q) := by |
simp only [mk_eq_div', ← Localization.mk_eq_mk', FractionRing.mk_eq_div]
| false |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section FoldrIdx
-- Porting... | Mathlib/Data/List/Indexes.lean | 246 | 250 | theorem foldrIdx_eq_foldrIdxSpec (f : ℕ → α → β → β) (b as start) :
foldrIdx f b as start = foldrIdxSpec f b as start := by |
induction as generalizing start
· rfl
· simp only [foldrIdx, foldrIdxSpec_cons, *]
| false |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 59 | 61 | theorem intDegree_X : intDegree (X : RatFunc K) = 1 := by |
rw [intDegree, num_X, Polynomial.natDegree_X, denom_X, Polynomial.natDegree_one,
Int.ofNat_one, Int.ofNat_zero, sub_zero]
| false |
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
attribute [local instance] Classical.propDecidable
open F... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 136 | 138 | theorem extend_coe [T2Space β] (hf : UniformContinuous f) (a : α) : (pkg.extend f) (ι a) = f a := by |
rw [pkg.extend_def hf]
exact pkg.denseInducing.extend_eq hf.continuous a
| false |
import Mathlib.Algebra.Group.Support
import Mathlib.Data.Set.Pointwise.SMul
#align_import data.set.pointwise.support from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Pointwise
open Function Set
section Group
variable {α β γ : Type*} [Group α] [MulAction α β]
| Mathlib/Data/Set/Pointwise/Support.lean | 26 | 29 | theorem mulSupport_comp_inv_smul [One γ] (c : α) (f : β → γ) :
(mulSupport fun x ↦ f (c⁻¹ • x)) = c • mulSupport f := by |
ext x
simp only [mem_smul_set_iff_inv_smul_mem, mem_mulSupport]
| false |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 116 | 118 | theorem card_eq_zero_iff [Finite α] : Nat.card α = 0 ↔ IsEmpty α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_eq_zero_iff]
| false |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finite.Card
#align_import group_theory.subgroup.finite from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
variable {G : Type*} [Group G]
variable {A : Type*} [AddGroup A]
n... | Mathlib/Algebra/Group/Subgroup/Finite.lean | 231 | 234 | theorem pi_mem_of_mulSingle_mem [Finite η] [DecidableEq η] {H : Subgroup (∀ i, f i)} (x : ∀ i, f i)
(h : ∀ i, Pi.mulSingle i (x i) ∈ H) : x ∈ H := by |
cases nonempty_fintype η
exact pi_mem_of_mulSingle_mem_aux Finset.univ x (by simp) fun i _ => h i
| false |
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 79 | 79 | theorem compl_mem_ae_iff {s : Set α} : sᶜ ∈ ae μ ↔ μ s = 0 := by | simp only [mem_ae_iff, compl_compl]
| false |
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.Algebraic
#align_import field_theory.minpoly.field from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
open scoped Classical
open Polynomial Set Function minpoly
namespace... | Mathlib/FieldTheory/Minpoly/Field.lean | 53 | 62 | theorem unique {p : A[X]} (pmonic : p.Monic) (hp : Polynomial.aeval x p = 0)
(pmin : ∀ q : A[X], q.Monic → Polynomial.aeval x q = 0 → degree p ≤ degree q) :
p = minpoly A x := by |
have hx : IsIntegral A x := ⟨p, pmonic, hp⟩
symm; apply eq_of_sub_eq_zero
by_contra hnz
apply degree_le_of_ne_zero A x hnz (by simp [hp]) |>.not_lt
apply degree_sub_lt _ (minpoly.ne_zero hx)
· rw [(monic hx).leadingCoeff, pmonic.leadingCoeff]
· exact le_antisymm (min A x pmonic hp) (pmin (minpoly A x) (m... | false |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 112 | 115 | theorem right_of_add (hn : IsPeriodicPt f (n + m) x) (hm : IsPeriodicPt f n x) :
IsPeriodicPt f m x := by |
rw [add_comm] at hn
exact hn.left_of_add hm
| false |
import Mathlib.Data.PNat.Prime
import Mathlib.Algebra.IsPrimePow
import Mathlib.NumberTheory.Cyclotomic.Basic
import Mathlib.RingTheory.Adjoin.PowerBasis
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Polynomial.Cyclotomic.Expand
#align_import number_theo... | Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean | 289 | 291 | theorem norm_eq_neg_one_pow (hζ : IsPrimitiveRoot ζ 2) [IsDomain L] :
norm K ζ = (-1 : K) ^ finrank K L := by |
rw [hζ.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]
| false |
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_import set_theory.ordinal.basic from "leanprover-community/mathlib"@"8ea5598db6caeddde6cb734aa179cc2408dbd345"
assert_not_exists Module
assert_not_exists Field
noncomputabl... | Mathlib/SetTheory/Ordinal/Basic.lean | 137 | 139 | theorem eta (o : WellOrder) : mk o.α o.r o.wo = o := by |
cases o
rfl
| false |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 129 | 138 | theorem isCodetecting_op_iff (𝒢 : Set C) : IsCodetecting 𝒢.op ↔ IsDetecting 𝒢 := by |
refine ⟨fun h𝒢 X Y f hf => ?_, fun h𝒢 X Y f hf => ?_⟩
· refine (isIso_op_iff _).1 (h𝒢 _ fun G hG h => ?_)
obtain ⟨t, ht, ht'⟩ := hf (unop G) (Set.mem_op.1 hG) h.unop
exact
⟨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))⟩
· refine (isIso_unop_iff _).1... | false |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 65 | 69 | theorem isRoot_of_hasEigenvalue {f : End K V} {μ : K} (h : f.HasEigenvalue μ) :
(minpoly K f).IsRoot μ := by |
rcases (Submodule.ne_bot_iff _).1 h with ⟨w, ⟨H, ne0⟩⟩
refine Or.resolve_right (smul_eq_zero.1 ?_) ne0
simp [← aeval_apply_of_hasEigenvector ⟨H, ne0⟩, minpoly.aeval K f]
| false |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | Mathlib/Data/Nat/Choose/Multinomial.lean | 88 | 92 | theorem multinomial_congr {f g : α → ℕ} (h : ∀ a ∈ s, f a = g a) :
multinomial s f = multinomial s g := by |
simp only [multinomial]; congr 1
· rw [Finset.sum_congr rfl h]
· exact Finset.prod_congr rfl fun a ha => by rw [h a ha]
| false |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 46 | 46 | theorem mem_map {f : α → β} {y : β} {o : Option α} : y ∈ o.map f ↔ ∃ x ∈ o, f x = y := by | simp
| false |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.List.Perm
#align_import data.list.prime from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
open List
section CommMonoidWithZero
variable {M : Type*} [CommMonoidWithZero M]
| Mathlib/Data/List/Prime.lean | 27 | 38 | theorem Prime.dvd_prod_iff {p : M} {L : List M} (pp : Prime p) : p ∣ L.prod ↔ ∃ a ∈ L, p ∣ a := by |
constructor
· intro h
induction' L with L_hd L_tl L_ih
· rw [prod_nil] at h
exact absurd h pp.not_dvd_one
· rw [prod_cons] at h
cases' pp.dvd_or_dvd h with hd hd
· exact ⟨L_hd, mem_cons_self L_hd L_tl, hd⟩
· obtain ⟨x, hx1, hx2⟩ := L_ih hd
exact ⟨x, mem_cons_of_mem L_hd ... | false |
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Cases
import Mathlib.Algebra.NeZero
import Mathlib.Logic.Function.Basic
#align_import algebra.char_zero.defs from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
class CharZero (R) [AddMonoidWithOne R] : Prop where
cast_injecti... | Mathlib/Algebra/CharZero/Defs.lean | 79 | 79 | theorem cast_eq_zero {n : ℕ} : (n : R) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
| false |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 606 | 607 | theorem rec_heq_iff_heq {C : α → Sort*} {x : C a} {y : β} {e : a = b} :
HEq (e ▸ x) y ↔ HEq x y := by | subst e; rfl
| false |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 620 | 625 | theorem oangle_right_eq_arctan_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arctan (dist p₁ p₂ / dist p₃ p₂) := by |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arctan_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(right_ne_of_oangle_eq_pi_div_two h)]
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 120 | 121 | theorem rpow_pos_of_pos {x : ℝ} (hx : 0 < x) (y : ℝ) : 0 < x ^ y := by |
rw [rpow_def_of_pos hx]; apply exp_pos
| false |
import Mathlib.Data.Set.Lattice
#align_import order.concept from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Function OrderDual Set
variable {ι : Sort*} {α β γ : Type*} {κ : ι → Sort*} (r : α → β → Prop) {s s₁ s₂ : Set α}
{t t₁ t₂ : Set β}
def intentClosure (s : Set α) :... | Mathlib/Order/Concept.lean | 180 | 185 | theorem ext (h : c.fst = d.fst) : c = d := by |
obtain ⟨⟨s₁, t₁⟩, h₁, _⟩ := c
obtain ⟨⟨s₂, t₂⟩, h₂, _⟩ := d
dsimp at h₁ h₂ h
substs h h₁ h₂
rfl
| false |
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Init.Data.Bool.Lemmas
#align_import analysis.box_integral.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open... | Mathlib/Analysis/BoxIntegral/Basic.lean | 143 | 145 | theorem integralSum_neg (f : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (-f) vol π = -integralSum f vol π := by |
simp only [integralSum, Pi.neg_apply, (vol _).map_neg, Finset.sum_neg_distrib]
| false |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 116 | 125 | theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
(hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)] <;>
simp_rw [hXint, sub_zero]
· rfl
· convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top with ω
simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toRe... | 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 | 196 | 197 | theorem mul_mul_mul_comm (a b c d : G) : a * b * (c * d) = a * c * (b * d) := by |
simp only [mul_left_comm, mul_assoc]
| false |
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee352b648ea"
noncomputable section
open scoped Classical
namespace WittVector
open Function
variable {p : ℕ} {R : Type*}
local notation "𝕎" => WittVe... | Mathlib/RingTheory/WittVector/Domain.lean | 69 | 76 | theorem verschiebung_shift (x : 𝕎 R) (k : ℕ) (h : ∀ i < k + 1, x.coeff i = 0) :
verschiebung (x.shift k.succ) = x.shift k := by |
ext ⟨j⟩
· rw [verschiebung_coeff_zero, shift_coeff, h]
apply Nat.lt_succ_self
· simp only [verschiebung_coeff_succ, shift]
congr 1
rw [Nat.add_succ, add_comm, Nat.add_succ, add_comm]
| false |
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 167 | 167 | theorem one_minus_nonneg (x : I) : 0 ≤ 1 - (x : ℝ) := by | simpa using x.2.2
| 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 | 75 | 78 | theorem rank_range_add_rank_ker (f : M →ₗ[R] M₁) :
Module.rank R (LinearMap.range f) + Module.rank R (LinearMap.ker f) = Module.rank R M := by |
haveI := fun p : Submodule R M => Classical.decEq (M ⧸ p)
rw [← f.quotKerEquivRange.rank_eq, rank_quotient_add_rank]
| false |
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
op... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 107 | 120 | theorem hasFiniteIntegral_compProd_iff' ⦃f : β × γ → E⦄
(h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
(∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by |
rw [hasFiniteIntegral_congr h1f.ae_eq_mk,
hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk]
apply and_congr
· apply eventually_congr
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with x hx using
hasFiniteIntegral_congr hx
· apply hasFiniteIntegral_congr
filter_upwards [ae_ae... | false |
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 68 | 73 | theorem mem_reesAlgebra_iff_support (f : R[X]) :
f ∈ reesAlgebra I ↔ ∀ i ∈ f.support, f.coeff i ∈ I ^ i := by |
apply forall_congr'
intro a
rw [mem_support_iff, Iff.comm, Classical.imp_iff_right_iff, Ne, ← imp_iff_not_or]
exact fun e => e.symm ▸ (I ^ a).zero_mem
| false |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.measure.lebesgue.complex from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
noncomputable section
namespace Complex... | Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean | 53 | 59 | theorem volume_preserving_equiv_pi : MeasurePreserving measurableEquivPi := by |
convert (measurableEquivPi.symm.measurable.measurePreserving volume).symm
rw [← addHaarMeasure_eq_volume_pi, ← Basis.parallelepiped_basisFun, ← Basis.addHaar,
measurableEquivPi, Homeomorph.toMeasurableEquiv_symm_coe,
ContinuousLinearEquiv.symm_toHomeomorph, ContinuousLinearEquiv.coe_toHomeomorph,
Basis... | false |
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 | 169 | 186 | theorem convexBodyLT'_mem {x : K} :
mixedEmbedding K x ∈ convexBodyLT' K f w₀ ↔
(∀ w : InfinitePlace K, w ≠ w₀ → w x < f w) ∧
|(w₀.val.embedding x).re| < 1 ∧ |(w₀.val.embedding x).im| < (f w₀: ℝ) ^ 2 := by |
simp_rw [mixedEmbedding, RingHom.prod_apply, Set.mem_prod, Set.mem_pi, Set.mem_univ,
forall_true_left, Pi.ringHom_apply, apply_ite, mem_ball_zero_iff, ← Complex.norm_real,
embedding_of_isReal_apply, norm_embedding_eq, Subtype.forall, Set.mem_setOf_eq]
refine ⟨fun ⟨h₁, h₂⟩ ↦ ⟨fun w h_ne ↦ ?_, ?_⟩, fun ⟨h₁, ... | false |
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 | 65 | 67 | theorem continuous_cos : Continuous cos := by |
change Continuous fun z => (exp (z * I) + exp (-z * I)) / 2
continuity
| false |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 178 | 181 | theorem fibRec_charPoly_eq {β : Type*} [CommRing β] :
fibRec.charPoly = X ^ 2 - (X + (1 : β[X])) := by |
rw [fibRec, LinearRecurrence.charPoly]
simp [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ', ← smul_X_eq_monomial]
| false |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.Cyclotomic.Rat
section case1
open ZMod
private lemma cube_of_castHom_ne_zero {n : ZMod 9} :
castHom (show 3 ∣ 9 by norm_num) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 := by
revert n; decide
private lemma cube_of_n... | Mathlib/NumberTheory/FLT/Three.lean | 36 | 44 | theorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :
a ^ 3 + b ^ 3 ≠ c ^ 3 := by |
simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd
apply mt (congrArg (Int.cast : ℤ → ZMod 9))
simp_rw [Int.cast_add, Int.cast_pow]
rcases cube_of_not_dvd hdvd.1.1 with ha | ha <;>
rcases cube_of_not_dvd hdvd.1.2 with hb | hb <;>
rcases cube_of_not_dvd hdvd.2 with hc | hc <;>
rw [ha, hb, hc] <;> decide
| false |
import Mathlib.CategoryTheory.Idempotents.Karoubi
#align_import category_theory.idempotents.functor_extension from "leanprover-community/mathlib"@"5f68029a863bdf76029fa0f7a519e6163c14152e"
namespace CategoryTheory
namespace Idempotents
open Category Karoubi
variable {C D E : Type*} [Category C] [Category D] [Ca... | Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | 35 | 40 | theorem natTrans_eq {F G : Karoubi C ⥤ D} (φ : F ⟶ G) (P : Karoubi C) :
φ.app P = F.map (decompId_i P) ≫ φ.app P.X ≫ G.map (decompId_p P) := by |
rw [← φ.naturality, ← assoc, ← F.map_comp]
conv_lhs => rw [← id_comp (φ.app P), ← F.map_id]
congr
apply decompId
| false |
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y X₁... | Mathlib/CategoryTheory/Conj.lean | 50 | 52 | theorem homCongr_apply {X Y X₁ Y₁ : C} (α : X ≅ X₁) (β : Y ≅ Y₁) (f : X ⟶ Y) :
α.homCongr β f = α.inv ≫ f ≫ β.hom := by |
rfl
| 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 | 46 | 52 | theorem ContinuousLinearMap.dslope_comp {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
(f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → DifferentiableAt 𝕜 g a) :
dslope (f ∘ g) a b = f (dslope g a b) := by |
rcases eq_or_ne b a with (rfl | hne)
· simp only [dslope_same]
exact (f.hasFDerivAt.comp_hasDerivAt b (H rfl).hasDerivAt).deriv
· simpa only [dslope_of_ne _ hne] using f.toLinearMap.slope_comp g a b
| false |
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 95 | 103 | theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
HasFDerivAt polarCoord.symm
(LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
!![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])) p := by |
rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
convert HasFDerivAt.prod (𝕜 := ℝ)
(hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
(hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>
simp [smul_smul, add_comm, neg_mul, smul_neg, n... | false |
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.special_functions.improper_inte... | Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean | 41 | 46 | theorem integral_exp_Iic (c : ℝ) : ∫ x : ℝ in Iic c, exp x = exp c := by |
refine
tendsto_nhds_unique
(intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_
simp_rw [integral_exp, show 𝓝 (exp c) = 𝓝 (exp c - 0) by rw [sub_zero]]
exact tendsto_exp_atBot.const_sub _
| false |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 36 | 41 | theorem AffineSubspace.isClosed_direction_iff (s : AffineSubspace 𝕜 Q) :
IsClosed (s.direction : Set W) ↔ IsClosed (s : Set Q) := by |
rcases s.eq_bot_or_nonempty with (rfl | ⟨x, hx⟩); · simp [isClosed_singleton]
rw [← (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image,
AffineSubspace.coe_direction_eq_vsub_set_right hx]
rfl
| false |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 254 | 257 | theorem smul (H : InnerRegularWRT μ p q) (c : ℝ≥0∞) : InnerRegularWRT (c • μ) p q := by |
intro U hU r hr
rw [smul_apply, H.measure_eq_iSup hU, smul_eq_mul] at hr
simpa only [ENNReal.mul_iSup, lt_iSup_iff, exists_prop] using hr
| false |
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 60 | 61 | theorem ker_pi (f : (i : ι) → M₂ →ₗ[R] φ i) : ker (pi f) = ⨅ i : ι, ker (f i) := by |
ext c; simp [funext_iff]
| false |
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.Localization.Integral
import Mathlib.RingTheory.IntegrallyClosed
#align_import ring_theory.polynomial.gauss_lemma from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
open... | Mathlib/RingTheory/Polynomial/GaussLemma.lean | 124 | 130 | theorem IsPrimitive.irreducible_of_irreducible_map_of_injective (h_irr : Irreducible (map φ f)) :
Irreducible f := by |
refine
⟨fun h => h_irr.not_unit (IsUnit.map (mapRingHom φ) h), fun a b h =>
(h_irr.isUnit_or_isUnit <| by rw [h, Polynomial.map_mul]).imp ?_ ?_⟩
all_goals apply ((isPrimitive_of_dvd hf _).isUnit_iff_isUnit_map_of_injective hinj).mpr
exacts [Dvd.intro _ h.symm, Dvd.intro_left _ h.symm]
| 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 | 171 | 178 | theorem trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)
(hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
IntervalIntegrable f μ (a m) (a n) := by |
revert hint
refine Nat.le_induction ?_ ?_ n hmn
· simp
· intro p hp IH h
exact (IH fun k hk => h k (Ico_subset_Ico_right p.le_succ hk)).trans (h p (by simp [hp]))
| false |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 158 | 159 | theorem circleMap_eq_center_iff {c : ℂ} {R : ℝ} {θ : ℝ} : circleMap c R θ = c ↔ R = 0 := by |
simp [circleMap, exp_ne_zero]
| false |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 126 | 128 | theorem dedup_nsmul {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : (n • s).dedup = s.dedup := by |
ext a
by_cases h : a ∈ s <;> simp [h, h0]
| false |
import Mathlib.CategoryTheory.Limits.Preserves.Opposites
import Mathlib.Topology.Category.TopCat.Yoneda
import Mathlib.Condensed.Explicit
universe w w' v u
open CategoryTheory Opposite Limits regularTopology ContinuousMap
variable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})
(X : Type w') [TopologicalSpac... | Mathlib/Condensed/TopComparison.lean | 40 | 58 | theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
[PreservesLimit (cospan π π) G]
{a : C(G.obj Z, X)}
(ha : a ∘ (G.map pullback.fst) = a ∘ (G.map (pullback.snd (f := π) (g := π)))) :
Function.FactorsThrough a (G.map π) := by |
intro x y hxy
let xy : G.obj (pullback π π) := (PreservesPullback.iso G π π).inv <|
(TopCat.pullbackIsoProdSubtype (G.map π) (G.map π)).inv ⟨(x, y), hxy⟩
have ha' := congr_fun ha xy
dsimp at ha'
have h₁ : ∀ y, G.map pullback.fst ((PreservesPullback.iso G π π).inv y) =
pullback.fst (f := G.map π) (g... | false |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 141 | 145 | theorem StrictConvexSpace.of_norm_add
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y) : StrictConvexSpace ℝ E := by |
refine StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun h₂ => ?_
rw [mem_sphere_zero_iff_norm] at hx hy
exact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h₂)
| false |
import Mathlib.Data.Matroid.Dual
open Set
namespace Matroid
variable {α : Type*} {M : Matroid α} {R I J X Y : Set α}
section restrict
@[simps] def restrictIndepMatroid (M : Matroid α) (R : Set α) : IndepMatroid α where
E := R
Indep I := M.Indep I ∧ I ⊆ R
indep_empty := ⟨M.empty_indep, empty_subset _⟩
i... | Mathlib/Data/Matroid/Restrict.lean | 156 | 157 | theorem base_restrict_iff' : (M ↾ X).Base I ↔ M.Basis' I X := by |
simp_rw [Basis', base_iff_maximal_indep, mem_maximals_setOf_iff, restrict_indep_iff]
| false |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 175 | 175 | theorem add_top (x : PartENat) : x + ⊤ = ⊤ := by | rw [add_comm, top_add]
| false |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.IntervalCases
#align_import number_theory.ADE_inequality from "leanprover-community/math... | Mathlib/NumberTheory/ADEInequality.lean | 117 | 119 | theorem sumInv_pqr (p q r : ℕ+) : sumInv {p, q, r} = (p : ℚ)⁻¹ + (q : ℚ)⁻¹ + (r : ℚ)⁻¹ := by |
simp only [sumInv, add_zero, insert_eq_cons, add_assoc, map_cons, sum_cons,
map_singleton, sum_singleton]
| false |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 80 | 80 | theorem isReflection_inv : cs.IsReflection t⁻¹ := by | rwa [ht.inv]
| false |
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 | 125 | 130 | theorem definable_finset_inf {ι : Type*} {f : ι → Set (α → M)} (hf : ∀ i, A.Definable L (f i))
(s : Finset ι) : A.Definable L (s.inf f) := by |
classical
refine Finset.induction definable_univ (fun i s _ h => ?_) s
rw [Finset.inf_insert]
exact (hf i).inter h
| false |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 150 | 152 | theorem second_continuant_aux_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.continuantsAux 2 = ⟨gp.b * g.h + gp.a, gp.b⟩ := by |
simp [zeroth_s_eq, continuantsAux, nextContinuants, nextDenominator, nextNumerator]
| false |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 51 | 53 | theorem pairwise_lt (n m : ℕ) : Pairwise (· < ·) (Ico n m) := by |
dsimp [Ico]
simp [pairwise_lt_range', autoParam]
| false |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 170 | 171 | theorem lineDerivWithin_univ : lineDerivWithin 𝕜 f univ x v = lineDeriv 𝕜 f x v := by |
simp [lineDerivWithin, lineDeriv]
| false |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 38 | 41 | theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
F.map ((f ≫ g, 𝟙 W) : (X, W) ⟶ (Z, W)) =
F.map ((f, 𝟙 W) : (X, W) ⟶ (Y, W)) ≫ F.map ((g, 𝟙 W) : (Y, W) ⟶ (Z, W)) := by |
rw [← Functor.map_comp, prod_comp, Category.comp_id]
| false |
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 78 | 81 | theorem isOpen_connectedComponent [LocallyConnectedSpace α] {x : α} :
IsOpen (connectedComponent x) := by |
rw [← connectedComponentIn_univ]
exact isOpen_univ.connectedComponentIn
| false |
import Mathlib.Logic.Basic
import Mathlib.Init.ZeroOne
import Mathlib.Init.Order.Defs
#align_import algebra.ne_zero from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
variable {R : Type*} [Zero R]
class NeZero (n : R) : Prop where
out : n ≠ 0
#align ne_zero NeZero
theorem NeZero... | Mathlib/Algebra/NeZero.lean | 45 | 45 | theorem not_neZero {n : R} : ¬NeZero n ↔ n = 0 := by | simp [neZero_iff]
| false |
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {α : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 110 | 121 | theorem nhds_mkOfNhds_of_hasBasis {n : α → Filter α} {ι : α → Sort*} {p : ∀ a, ι a → Prop}
{s : ∀ a, ι a → Set α} (hb : ∀ a, (n a).HasBasis (p a) (s a))
(hpure : ∀ a i, p a i → a ∈ s a i) (hopen : ∀ a i, p a i → ∀ᶠ x in n a, s a i ∈ n x) (a : α) :
@nhds α (.mkOfNhds n) a = n a := by |
let t : TopologicalSpace α := .mkOfNhds n
apply le_antisymm
· intro U hU
replace hpure : pure ≤ n := fun x ↦ (hb x).ge_iff.2 (hpure x)
refine mem_nhds_iff.2 ⟨{x | U ∈ n x}, fun x hx ↦ hpure x hx, fun x hx ↦ ?_, hU⟩
rcases (hb x).mem_iff.1 hx with ⟨i, hpi, hi⟩
exact (hopen x i hpi).mono fun y hy ↦... | false |
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 84 | 87 | theorem natDegree_add_le_iff_right {n : ℕ} (p q : R[X]) (pn : p.natDegree ≤ n) :
(p + q).natDegree ≤ n ↔ q.natDegree ≤ n := by |
rw [add_comm]
exact natDegree_add_le_iff_left _ _ pn
| false |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 151 | 153 | theorem mirror_trailingCoeff : p.mirror.trailingCoeff = p.leadingCoeff := by |
rw [leadingCoeff, trailingCoeff, mirror_natTrailingDegree, coeff_mirror,
revAt_le (Nat.le_add_left _ _), add_tsub_cancel_right]
| false |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 293 | 295 | theorem finsuppTensorFinsuppLid_apply_apply (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ) :
finsuppTensorFinsuppLid R N ι κ (f ⊗ₜ[R] g) (a, b) = f a • g b := by |
simp [finsuppTensorFinsuppLid]
| false |
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 64 | 66 | theorem toNat_strictMonoOn : StrictMonoOn toNat (Iio ℵ₀) := by |
simp only [← range_natCast, StrictMonoOn, forall_mem_range, toNat_natCast, Nat.cast_lt]
exact fun _ _ ↦ id
| false |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 68 | 73 | theorem inr_mem_sumLift₂ {c₂ : γ₂} :
inr c₂ ∈ sumLift₂ f g a b ↔ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ g a₂ b₂ := by |
rw [mem_sumLift₂, or_iff_right]
· simp only [inr.injEq, exists_and_left, exists_eq_left']
rintro ⟨_, _, c₂, _, _, h, _⟩
exact inr_ne_inl h
| 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
| Mathlib/Topology/ContinuousFunction/Weierstrass.lean | 32 | 44 | theorem polynomialFunctions_closure_eq_top' : (polynomialFunctions I).topologicalClosure = ⊤ := by |
rw [eq_top_iff]
rintro f -
refine Filter.Frequently.mem_closure ?_
refine Filter.Tendsto.frequently (bernsteinApproximation_uniform f) ?_
apply frequently_of_forall
intro n
simp only [SetLike.mem_coe]
apply Subalgebra.sum_mem
rintro n -
apply Subalgebra.smul_mem
dsimp [bernstein, polynomialFuncti... | false |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 102 | 103 | theorem supported_univ : supported R (Set.univ : Set σ) = ⊤ := by |
simp [Algebra.eq_top_iff, mem_supported]
| false |
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
namespace PowerSeries
open Polynomial Derivation Nat
section CommutativeSemiring
variable {R} [CommSemiring R]
noncomputable def derivativeFun (f : R⟦X⟧) : R⟦X⟧ := mk fun n ↦ coef... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 127 | 133 | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by |
cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative]
| false |
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 | 108 | 112 | theorem vars_add_subset [DecidableEq σ] (p q : MvPolynomial σ R) :
(p + q).vars ⊆ p.vars ∪ q.vars := by |
intro x hx
simp only [vars_def, Finset.mem_union, Multiset.mem_toFinset] at hx ⊢
simpa using Multiset.mem_of_le (degrees_add _ _) hx
| false |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 95 | 97 | theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by | simp [hfp.2]
| false |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
variable {K n : Type*} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K}
| Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 26 | 56 | theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) := by |
cases isEmpty_or_nonempty n
· exfalso
exact hμ Submodule.eq_bot_of_subsingleton
· obtain ⟨v, h_eg, h_nz⟩ := hμ.exists_hasEigenvector
obtain ⟨i, -, h_i⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun i => ‖v i‖)
have h_nz : v i ≠ 0 := by
contrapose! h_nz
ext j
rw [Pi.zero_app... | false |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 93 | 102 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by |
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul,... | false |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
section
variable {R S : Type*} [CommSemiring R] [CommSemiring S] (I J ... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 91 | 101 | theorem Ideal.sInf_minimalPrimes : sInf I.minimalPrimes = I.radical := by |
rw [I.radical_eq_sInf]
apply le_antisymm
· intro x hx
rw [Ideal.mem_sInf] at hx ⊢
rintro J ⟨e, hJ⟩
obtain ⟨p, hp, hp'⟩ := Ideal.exists_minimalPrimes_le e
exact hp' (hx hp)
· apply sInf_le_sInf _
intro I hI
exact hI.1.symm
| false |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 87 | 92 | theorem Invariant.comp [IsSFiniteKernel κ] (hκ : Invariant κ μ) (hη : Invariant η μ) :
Invariant (κ ∘ₖ η) μ := by |
cases' isEmpty_or_nonempty α with _ hα
· exact Subsingleton.elim _ _
· simp_rw [Invariant, ← comp_const_apply_eq_bind (κ ∘ₖ η) μ hα.some, comp_assoc, hη.comp_const,
hκ.comp_const, const_apply]
| false |
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import order.succ_pred.linear_locally_finite from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Order
variable {ι : Type*}... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 72 | 74 | theorem le_succFn (i : ι) : i ≤ succFn i := by |
rw [le_isGLB_iff (succFn_spec i), mem_lowerBounds]
exact fun x hx ↦ le_of_lt hx
| false |
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
#align_import cat... | Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean | 51 | 54 | theorem pullbackFst_eq_pullback_snd (F : WalkingParallelPair ⥤ C) :
pullbackFst F = pullback.snd := by |
convert (eq_whisker pullback.condition Limits.prod.fst :
(_ : constructEqualizer F ⟶ F.obj WalkingParallelPair.zero) = _) <;> simp
| false |
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u w v' u' w'
open ... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 113 | 116 | theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (forget A)) := by |
simp only [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top]
rfl
| false |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Basis
#align_import linear_algebra.determinant from "lea... | Mathlib/LinearAlgebra/Determinant.lean | 96 | 99 | theorem det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) :
det (M * N * M') = det N := by |
rw [← det_comm' hM'M hMM', ← Matrix.mul_assoc, hM'M, Matrix.one_mul]
| false |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
namespace SimpleGraph
universe u v
variable {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'}
namespace Subgraph
protected structure Preconnected (H : G.Subgraph) : Prop where
protected coe : H.coe.Preconnected
instance {H : G.Subgraph}... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 64 | 69 | theorem singletonSubgraph_connected {v : V} : (G.singletonSubgraph v).Connected := by |
refine ⟨⟨?_⟩⟩
rintro ⟨a, ha⟩ ⟨b, hb⟩
simp only [singletonSubgraph_verts, Set.mem_singleton_iff] at ha hb
subst_vars
rfl
| false |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 48 | 49 | theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iic]
| false |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 59 | 60 | theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 - e2) = t1 - t2 := by | simp [left_distrib, *, sub_eq_add_neg]
| false |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.euclidean_domain from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
section
open Euclidean... | Mathlib/RingTheory/EuclideanDomain.lean | 42 | 47 | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by |
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, mul_div_cancel_right₀ _ pq0]
exact r0
| false |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
namespace CategoryTheory
namespace Bicategory
open Category
open scoped Bicategory
open Mathlib.Tactic.BicategoryCoherence (bicategoricalComp bicategoricalIsoComp)
universe w v u
variable {B : Type u} [Bicategory... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 232 | 247 | theorem adjointifyCounit_left_triangle (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :
leftZigzagIso η (adjointifyCounit η ε) = λ_ f ≪≫ (ρ_ f).symm := by |
apply Iso.ext
dsimp [adjointifyCounit, bicategoricalIsoComp]
calc
_ = 𝟙 _ ⊗≫ (η.hom ▷ (f ≫ 𝟙 b) ≫ (f ≫ g) ◁ f ◁ ε.inv) ⊗≫
f ◁ g ◁ η.inv ▷ f ⊗≫ f ◁ ε.hom := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫ f ◁ ε.inv ⊗≫ (η.hom ▷ (f ≫ g) ≫ (f ≫ g) ◁ η.inv) ▷ f ⊗≫ f ◁ ε.hom := by
rw... | false |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 81 | 85 | theorem add_one_le_of_lt {n m : WithBot ℕ} (h : n < m) : n + 1 ≤ m := by |
cases n
· exact bot_le
cases m
exacts [(not_lt_bot h).elim, WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 h)]
| 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 IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 73 | 76 | theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
| false |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 126 | 136 | theorem destruct_eq_think {s : Computation α} {s'} : destruct s = Sum.inr s' → s = think s' := by |
dsimp [destruct]
induction' f0 : s.1 0 with a' <;> intro h
· injection h with h'
rw [← h']
cases' s with f al
apply Subtype.eq
dsimp [think, tail]
rw [← f0]
exact (Stream'.eta f).symm
· contradiction
| false |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 95 | 100 | theorem length_mul_right_ne (w : W) : ℓ (t * w) ≠ ℓ w := by |
suffices cs.lengthParity (t * w) ≠ cs.lengthParity w by
contrapose! this
simp only [lengthParity_eq_ofAdd_length, this]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| false |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 46 | 47 | theorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0 := by |
rwa [zero_opow', Ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero]
| false |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Rat.BigOperators
#align_import combinatorics.simple_graph.regularity.energy from "leanprover-community/mathlib"@"bf7ef0... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 61 | 63 | theorem coe_energy {𝕜 : Type*} [LinearOrderedField 𝕜] : (P.energy G : 𝕜) =
(∑ uv ∈ P.parts.offDiag, (G.edgeDensity uv.1 uv.2 : 𝕜) ^ 2) / (P.parts.card : 𝕜) ^ 2 := by |
rw [energy]; norm_cast
| false |
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 215 | 216 | theorem symmetric_symmetrizeRel (V : Set (α × α)) : SymmetricRel (symmetrizeRel V) := by |
simp [SymmetricRel, symmetrizeRel, preimage_inter, inter_comm, ← preimage_comp]
| false |
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 | 213 | 214 | theorem units_inv_apply_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
(f⁻¹ : CircleDeg1Liftˣ) (f x) = x := by | simp only [← mul_apply, f.inv_mul, coe_one, id]
| false |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.isometric_smul from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set
open ENNReal Pointwise
universe u v w
vari... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 121 | 123 | theorem edist_div_right [DivInvMonoid M] [PseudoEMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
(a b c : M) : edist (a / c) (b / c) = edist a b := by |
simp only [div_eq_mul_inv, edist_mul_right]
| false |
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous
import Mathlib.Algebra.Polynomial.Roots
#align_i... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 124 | 127 | theorem totalDegree_zero_iff_isHomogeneous {p : MvPolynomial σ R} :
p.totalDegree = 0 ↔ IsHomogeneous p 0 := by |
rw [← weightedTotalDegree_one,
← isWeightedHomogeneous_zero_iff_weightedTotalDegree_eq_zero, IsHomogeneous]
| false |
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import category_theory.preadditive.schur from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 114 | 125 | theorem finrank_endomorphism_eq_one {X : C} (isIso_iff_nonzero : ∀ f : X ⟶ X, IsIso f ↔ f ≠ 0)
[I : FiniteDimensional 𝕜 (X ⟶ X)] : finrank 𝕜 (X ⟶ X) = 1 := by |
have id_nonzero := (isIso_iff_nonzero (𝟙 X)).mp (by infer_instance)
refine finrank_eq_one (𝟙 X) id_nonzero ?_
intro f
have : Nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero
have : FiniteDimensional 𝕜 (End X) := I
obtain ⟨c, nu⟩ := spectrum.nonempty_of_isAlgClosed_of_finiteDimensional 𝕜 (End.of f)... | false |
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 | 142 | 143 | theorem card_fintypeIoc : Fintype.card (Set.Ioc a b) = b - a := by |
rw [← card_Ioc, Fintype.card_ofFinset]
| false |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ι κ α β R M : Type*}
| Mathlib/Algebra/Module/BigOperators.lean | 50 | 51 | theorem Finset.cast_card [CommSemiring R] (s : Finset α) : (s.card : R) = ∑ a ∈ s, 1 := by |
rw [Finset.sum_const, Nat.smul_one_eq_cast]
| false |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.Antichain
import Mathlib.Order.Interval.Finset.Nat
#align_import data.finset.slice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Finset Nat
variable {α : Type*} {ι : Sort*} {κ : ι → Sort*}
namespace Set
... | Mathlib/Data/Finset/Slice.lean | 64 | 66 | theorem sized_iUnion {f : ι → Set (Finset α)} : (⋃ i, f i).Sized r ↔ ∀ i, (f i).Sized r := by |
simp_rw [Set.Sized, Set.mem_iUnion, forall_exists_index]
exact forall_swap
| false |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 106 | 112 | theorem summable_inner (f g : lp G 2) : Summable fun i => ⟪f i, g i⟫ := by |
-- Apply the Direct Comparison Test, comparing with ∑' i, ‖f i‖ * ‖g i‖ (summable by Hölder)
refine .of_norm_bounded (fun i => ‖f i‖ * ‖g i‖) (lp.summable_mul ?_ f g) ?_
· rw [Real.isConjExponent_iff]; norm_num
intro i
-- Then apply Cauchy-Schwarz pointwise
exact norm_inner_le_norm (𝕜 := 𝕜) _ _
| false |
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