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 |
|---|---|---|---|---|---|---|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 94 | 95 | theorem interior_setOf_re_le (a : ℝ) : interior { z : ℂ | z.re ≤ a } = { z | z.re < a } := by |
simpa only [interior_Iic] using interior_preimage_re (Iic a)
| true |
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Logic.Pairwise
#align_import data.set.intervals.group from "lean... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 212 | 214 | theorem pairwise_disjoint_Ioc_zpow :
Pairwise (Disjoint on fun n : ℤ => Ioc (b ^ n) (b ^ (n + 1))) := by |
simpa only [one_mul] using pairwise_disjoint_Ioc_mul_zpow 1 b
| true |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 176 | 180 | theorem fourier_norm [Fact (0 < T)] (n : ℤ) : ‖@fourier T n‖ = 1 := by
rw [ContinuousMap.norm_eq_iSup_norm] |
rw [ContinuousMap.norm_eq_iSup_norm]
have : ∀ x : AddCircle T, ‖fourier n x‖ = 1 := fun x => abs_coe_circle _
simp_rw [this]
exact @ciSup_const _ _ _ Zero.instNonempty _
| true |
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.presented_group from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {α : Type*}
def PresentedGroup (rels : Set (FreeGroup α)) :=
FreeGroup α ⧸ Subgroup.normalClosu... | Mathlib/GroupTheory/PresentedGroup.lean | 53 | 58 | theorem closure_range_of (rels : Set (FreeGroup α)) :
Subgroup.closure (Set.range (PresentedGroup.of : α → PresentedGroup rels)) = ⊤ := by
have : (PresentedGroup.of : α → PresentedGroup rels) = QuotientGroup.mk' _ ∘ FreeGroup.of := rfl |
have : (PresentedGroup.of : α → PresentedGroup rels) = QuotientGroup.mk' _ ∘ FreeGroup.of := rfl
rw [this, Set.range_comp, ← MonoidHom.map_closure (QuotientGroup.mk' _),
FreeGroup.closure_range_of, ← MonoidHom.range_eq_map]
exact MonoidHom.range_top_of_surjective _ (QuotientGroup.mk'_surjective _)
| true |
import Mathlib.Data.List.Duplicate
import Mathlib.Data.List.Sort
#align_import data.list.nodup_equiv_fin from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace List
variable {α : Type*}
section Sublist
theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
... | Mathlib/Data/List/NodupEquivFin.lean | 211 | 232 | theorem duplicate_iff_exists_distinct_get {l : List α} {x : α} :
l.Duplicate x ↔
∃ (n m : Fin l.length) (_ : n < m),
x = l.get n ∧ x = l.get m := by
classical |
classical
rw [duplicate_iff_two_le_count, le_count_iff_replicate_sublist,
sublist_iff_exists_fin_orderEmbedding_get_eq]
constructor
· rintro ⟨f, hf⟩
refine ⟨f ⟨0, by simp⟩, f ⟨1, by simp⟩, f.lt_iff_lt.2 (Nat.zero_lt_one), ?_⟩
rw [← hf, ← hf]; simp
· rintro ⟨n, m, hnm, h, h'⟩
r... | true |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
| Mathlib/Data/List/Sym.lean | 40 | 43 | theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs).sym2 ↔ z = s(x, x) ∨ (∃ y, y ∈ xs ∧ z = s(x, y)) ∨ z ∈ xs.sym2 := by
simp only [List.sym2, map_cons, cons_append, mem_cons, mem_append, mem_map] |
simp only [List.sym2, map_cons, cons_append, mem_cons, mem_append, mem_map]
simp only [eq_comm]
| true |
import Mathlib.Data.Fintype.Card
import Mathlib.Computability.Language
import Mathlib.Tactic.NormNum
#align_import computability.DFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Computability
universe u v
-- Porting note: Required as `DFA` is used in mathlib3
set_option li... | Mathlib/Computability/DFA.lean | 151 | 166 | theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
(hlen : Fintype.card σ ≤ List.length x) :
∃ a b c,
x = a ++ b ++ c ∧
a.length + b.length ≤ Fintype.card σ ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts := by
obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M... |
obtain ⟨_, a, b, c, hx, hlen, hnil, rfl, hb, hc⟩ := M.evalFrom_split (s := M.start) hlen rfl
use a, b, c, hx, hlen, hnil
intro y hy
rw [Language.mem_mul] at hy
rcases hy with ⟨ab, hab, c', hc', rfl⟩
rw [Language.mem_mul] at hab
rcases hab with ⟨a', ha', b', hb', rfl⟩
rw [Set.mem_singleton_iff] at ha' h... | true |
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 78 | 82 | theorem measurable_map (f : α → β) (hf : Measurable f) :
Measurable fun μ : Measure α => map f μ := by
refine measurable_of_measurable_coe _ fun s hs => ?_ |
refine measurable_of_measurable_coe _ fun s hs => ?_
simp_rw [map_apply hf hs]
exact measurable_coe (hf hs)
| true |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.normed_space.extend from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable {𝕜 : Ty... | Mathlib/Analysis/NormedSpace/Extend.lean | 88 | 90 | theorem extendTo𝕜'_apply_re (fr : F →ₗ[ℝ] ℝ) (x : F) : re (fr.extendTo𝕜' x : 𝕜) = fr x := by
simp only [extendTo𝕜'_apply, map_sub, zero_mul, mul_zero, sub_zero, |
simp only [extendTo𝕜'_apply, map_sub, zero_mul, mul_zero, sub_zero,
rclike_simps]
| true |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uM₁ uM₂ uM₃ uM₄
variable {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 37 | 46 | theorem tmul_comp_tensorMap
{Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂}
{Q₃ : QuadraticForm R M₃} {Q₄ : QuadraticForm R M₄}
(f : Q₁ →qᵢ Q₂) (g : Q₃ →qᵢ Q₄) :
(Q₂.tmul Q₄).comp (TensorProduct.map f.toLinearMap g.toLinearMap) = Q₁.tmul Q₃ := by
have h₁ : Q₁ = Q₂.comp f.toLinearMap := QuadraticFo... |
have h₁ : Q₁ = Q₂.comp f.toLinearMap := QuadraticForm.ext fun x => (f.map_app x).symm
have h₃ : Q₃ = Q₄.comp g.toLinearMap := QuadraticForm.ext fun x => (g.map_app x).symm
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₁ m₃ m₁' m₃'
simp [-associated_apply, h₁, h₃, associated_tmul]
| true |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section TietzeExten... | Mathlib/Topology/TietzeExtension.lean | 73 | 77 | theorem ContinuousMap.exists_extension (f : C(X₁, Y)) :
∃ (g : C(X, Y)), g.comp ⟨e, he.continuous⟩ = f := by
let e' : X₁ ≃ₜ Set.range e := Homeomorph.ofEmbedding _ he.toEmbedding |
let e' : X₁ ≃ₜ Set.range e := Homeomorph.ofEmbedding _ he.toEmbedding
obtain ⟨g, hg⟩ := (f.comp e'.symm).exists_restrict_eq he.isClosed_range
exact ⟨g, by ext x; simpa using congr($(hg) ⟨e' x, x, rfl⟩)⟩
| true |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 166 | 168 | theorem one_eq_bit1 {a : R} : 1 = bit1 a ↔ a = 0 := by
rw [eq_comm] |
rw [eq_comm]
exact bit1_eq_one
| true |
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 81 | 81 | theorem one_add (n : PosNum) : 1 + n = succ n := by | cases n <;> rfl
| true |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 145 | 160 | theorem hasFDerivAt_stereoInvFunAux (v : E) :
HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) 0 := by
have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : E →L[ℝ] ℝ) 0 := by |
have h₀ : HasFDerivAt (fun w : E => ‖w‖ ^ 2) (0 : E →L[ℝ] ℝ) 0 := by
convert (hasStrictFDerivAt_norm_sq (0 : E)).hasFDerivAt
simp
have h₁ : HasFDerivAt (fun w : E => (‖w‖ ^ 2 + 4)⁻¹) (0 : E →L[ℝ] ℝ) 0 := by
convert (hasFDerivAt_inv _).comp _ (h₀.add (hasFDerivAt_const 4 0)) <;> simp
have h₂ : HasFDer... | true |
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE
#align_import algebraic_geometry.morphisms.basic from "leanprover-community/mathlib"@"434e2fd21c1900747afc6d13d8be7f4eedba7218"
set_option lin... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 131 | 141 | theorem targetAffineLocally_respectsIso {P : AffineTargetMorphismProperty}
(hP : P.toProperty.RespectsIso) : (targetAffineLocally P).RespectsIso := by
constructor |
constructor
· introv H U
rw [morphismRestrict_comp, affine_cancel_left_isIso hP]
exact H U
· introv H
rintro ⟨U, hU : IsAffineOpen U⟩; dsimp
haveI : IsAffine _ := hU.map_isIso e.hom
rw [morphismRestrict_comp, affine_cancel_right_isIso hP]
exact H ⟨(Opens.map e.hom.val.base).obj U, hU.map_... | true |
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 65 | 73 | theorem Monic.map [Semiring S] (f : R →+* S) (hp : Monic p) : Monic (p.map f) := by
unfold Monic |
unfold Monic
nontriviality
have : f p.leadingCoeff ≠ 0 := by
rw [show _ = _ from hp, f.map_one]
exact one_ne_zero
rw [Polynomial.leadingCoeff, coeff_map]
suffices p.coeff (p.map f).natDegree = 1 by simp [this]
rwa [natDegree_eq_of_degree_eq (degree_map_eq_of_leadingCoeff_ne_zero f this)]
| true |
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
universe w' w v₁ v₂ u₁ u₂
noncomputable section
namespace CategoryTheory
open Category Limits
variable {C : Type u₁} [Catego... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 142 | 146 | theorem preservesColimitsIso_inv_comp_desc (t : Cocone F) :
(preservesColimitIso G F).inv ≫ G.map (colimit.desc _ t) =
colimit.desc _ (G.mapCocone t) := by
ext |
ext
simp [← G.map_comp]
| true |
import Mathlib.RingTheory.RingHomProperties
import Mathlib.RingTheory.IntegralClosure
#align_import ring_theory.ring_hom.integral from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
| Mathlib/RingTheory/RingHom/Integral.lean | 24 | 25 | theorem isIntegral_stableUnderComposition : StableUnderComposition fun f => f.IsIntegral := by |
introv R hf hg; exact hf.trans _ _ hg
| true |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 613 | 615 | theorem frobenius_norm_row (v : m → α) : ‖row v‖ = ‖(WithLp.equiv 2 _).symm v‖ := by
rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow] |
rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]
simp only [row_apply, Real.rpow_two, WithLp.equiv_symm_pi_apply]
| true |
import Mathlib.CategoryTheory.Equivalence
#align_import algebraic_topology.dold_kan.compatibility from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category
namespace AlgebraicTopology
namespace DoldKan
namespace Compatibility
variable {A A' B B'... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 86 | 88 | theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF := by
ext Y |
ext Y
simp [equivalence₁, equivalence₀]
| true |
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 287 | 289 | theorem span_single_image (s : Set M) (a : α) :
Submodule.span R (single a '' s) = (Submodule.span R s).map (lsingle a : M →ₗ[R] α →₀ M) := by |
rw [← span_image]; rfl
| true |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 161 | 169 | theorem of_algebraRat [Algebra ℚ R] : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I) := by
intro I hI |
intro I hI
constructor
intro a b h_ab
contrapose! hI
-- `↑a - ↑b` is a unit contained in `I`, which contradicts `I ≠ ⊤`.
refine I.eq_top_of_isUnit_mem ?_ (IsUnit.map (algebraMap ℚ R) (IsUnit.mk0 (a - b : ℚ) ?_))
· simpa only [← Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]
simpa on... | true |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 132 | 133 | theorem nat_mul_inj' {n : ℕ} {a b : R} (h : (n : R) * a = (n : R) * b) (w : n ≠ 0) : a = b := by |
simpa [w] using nat_mul_inj h
| true |
import Mathlib.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import number_theory.number_field.units from "leanprover-community/mathlib"@"00f91228655eecdcd3ac... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 128 | 151 | theorem log_le_of_logEmbedding_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
(w : InfinitePlace K) : |Real.log (w x)| ≤ (Fintype.card (InfinitePlace K)) * r := by
have tool : ∀ x : ℝ, 0 ≤ x → x ≤ mult w * x := fun x hx => by |
have tool : ∀ x : ℝ, 0 ≤ x → x ≤ mult w * x := fun x hx => by
nth_rw 1 [← one_mul x]
refine mul_le_mul ?_ le_rfl hx ?_
all_goals { rw [mult]; split_ifs <;> norm_num }
by_cases hw : w = w₀
· have hyp := congr_arg (‖·‖) (sum_logEmbedding_component x).symm
replace hyp := (le_of_eq hyp).trans (norm_s... | true |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 197 | 199 | theorem self_mem_span_image [Nontrivial R] {i : ι} {s : Set ι} :
b i ∈ span R (b '' s) ↔ i ∈ s := by |
simp [mem_span_image, Finsupp.support_single_ne_zero]
| false |
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.properties from "leanprover-community/mathlib"@"88... | Mathlib/AlgebraicGeometry/Properties.lean | 105 | 112 | theorem affine_isReduced_iff (R : CommRingCat) :
IsReduced (Scheme.Spec.obj <| op R) ↔ _root_.IsReduced R := by |
refine ⟨?_, fun h => inferInstance⟩
intro h
have : _root_.IsReduced
(LocallyRingedSpace.Γ.obj (op <| Spec.toLocallyRingedSpace.obj <| op R)) := by
change _root_.IsReduced ((Scheme.Spec.obj <| op R).presheaf.obj <| op ⊤); infer_instance
exact isReduced_of_injective (toSpecΓ R) (asIso <| toSpecΓ R).com... | false |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
variable {α β... | Mathlib/RingTheory/Multiplicity.lean | 65 | 73 | theorem Int.natCast_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by |
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 54 | 66 | theorem liouvilleWith_one (x : ℝ) : LiouvilleWith 1 x := by |
use 2
refine ((eventually_gt_atTop 0).mono fun n hn => ?_).frequently
have hn' : (0 : ℝ) < n := by simpa
have : x < ↑(⌊x * ↑n⌋ + 1) / ↑n := by
rw [lt_div_iff hn', Int.cast_add, Int.cast_one];
exact Int.lt_floor_add_one _
refine ⟨⌊x * n⌋ + 1, this.ne, ?_⟩
rw [abs_sub_comm, abs_of_pos (sub_pos.2 this... | false |
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import probability.martingale.borel_cantelli from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Filter
open scoped NNRea... | Mathlib/Probability/Martingale/BorelCantelli.lean | 93 | 98 | theorem stoppedValue_stoppedValue_leastGE (f : ℕ → Ω → ℝ) (π : Ω → ℕ) (r : ℝ) {n : ℕ}
(hπn : ∀ ω, π ω ≤ n) : stoppedValue (fun i => stoppedValue f (leastGE f r i)) π =
stoppedValue (stoppedProcess f (leastGE f r n)) π := by |
ext1 ω
simp (config := { unfoldPartialApp := true }) only [stoppedProcess, stoppedValue]
rw [leastGE_eq_min _ _ _ hπn]
| false |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.Combinatorics.Pigeonhole
#align_import dynamics.ergodic.conservative from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open scoped Classi... | Mathlib/Dynamics/Ergodic/Conservative.lean | 83 | 106 | theorem frequently_measure_inter_ne_zero (hf : Conservative f μ) (hs : MeasurableSet s)
(h0 : μ s ≠ 0) : ∃ᶠ m in atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0 := by |
by_contra H
simp only [not_frequently, eventually_atTop, Ne, Classical.not_not] at H
rcases H with ⟨N, hN⟩
induction' N with N ihN
· apply h0
simpa using hN 0 le_rfl
rw [imp_false] at ihN
push_neg at ihN
rcases ihN with ⟨n, hn, hμn⟩
set T := s ∩ ⋃ n ≥ N + 1, f^[n] ⁻¹' s
have hT : MeasurableSet ... | false |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 106 | 116 | theorem cramer_transpose_row_self (i : n) : Aᵀ.cramer (A i) = Pi.single i A.det := by |
ext j
rw [cramer_apply, Pi.single_apply]
split_ifs with h
· -- i = j: this entry should be `A.det`
subst h
simp only [updateColumn_transpose, det_transpose, updateRow_eq_self]
· -- i ≠ j: this entry should be 0
rw [updateColumn_transpose, det_transpose]
apply det_zero_of_row_eq h
rw [upda... | false |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
| Mathlib/Data/Real/Sign.lean | 36 | 36 | theorem sign_of_neg {r : ℝ} (hr : r < 0) : sign r = -1 := by | rw [sign, if_pos hr]
| false |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 90 | 94 | theorem lt_or_lt_of_mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ := by |
contrapose!
exact fun h => mul_le_mul' h.1 h.2
| false |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 76 | 84 | theorem exists_ord_compl_eq_one_iff_isPrimePow {n : ℕ} (hn : n ≠ 1) :
IsPrimePow n ↔ ∃ p : ℕ, p.Prime ∧ ord_compl[p] n = 1 := by |
refine ⟨fun h => IsPrimePow.exists_ord_compl_eq_one h, fun h => ?_⟩
rcases h with ⟨p, pp, h⟩
rw [isPrimePow_nat_iff]
rw [← Nat.eq_of_dvd_of_div_eq_one (Nat.ord_proj_dvd n p) h] at hn ⊢
refine ⟨p, n.factorization p, pp, ?_, by simp⟩
contrapose! hn
simp [Nat.le_zero.1 hn]
| false |
import Mathlib.Algebra.TrivSqZeroExt
#align_import algebra.dual_number from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
variable {R A B : Type*}
abbrev DualNumber (R : Type*) : Type _ :=
TrivSqZeroExt R R
#align dual_number DualNumber
def DualNumber.eps [Zero R] [One R] : DualN... | Mathlib/Algebra/DualNumber.lean | 96 | 97 | theorem commute_eps_left [Semiring R] (x : DualNumber R) : Commute ε x := by |
ext <;> simp
| false |
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Vector
variable {α β : Type*} {n : ℕ} (a a' : α)
@[simp]
theorem get_mem (i : Fin n) (v : Vector α n) : v.get i ∈ v.toList := by
rw [get_eq_get]
exact List.... | Mathlib/Data/Vector/Mem.lean | 38 | 41 | theorem not_mem_nil : a ∉ (Vector.nil : Vector α 0).toList := by |
unfold Vector.nil
dsimp
simp
| false |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 299 | 301 | theorem pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by |
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H
| false |
import Mathlib.Data.Int.GCD
import Mathlib.Tactic.NormNum
namespace Tactic
namespace NormNum
theorem int_gcd_helper' {d : ℕ} {x y : ℤ} (a b : ℤ) (h₁ : (d : ℤ) ∣ x) (h₂ : (d : ℤ) ∣ y)
(h₃ : x * a + y * b = d) : Int.gcd x y = d := by
refine Nat.dvd_antisymm ?_ (Int.natCast_dvd_natCast.1 (Int.dvd_gcd h₁ h₂))
... | Mathlib/Tactic/NormNum/GCD.lean | 36 | 43 | theorem nat_gcd_helper_2 (d x y a b : ℕ) (hu : x % d = 0) (hv : y % d = 0)
(h : x * a = y * b + d) : Nat.gcd x y = d := by |
rw [← Int.gcd_natCast_natCast]
apply int_gcd_helper' a (-b)
(Int.natCast_dvd_natCast.mpr (Nat.dvd_of_mod_eq_zero hu))
(Int.natCast_dvd_natCast.mpr (Nat.dvd_of_mod_eq_zero hv))
rw [mul_neg, ← sub_eq_add_neg, sub_eq_iff_eq_add']
exact mod_cast h
| false |
import Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive
import Mathlib.AlgebraicTopology.DoldKan.Compatibility
import Mathlib.CategoryTheory.Idempotents.SimplicialObject
#align_import algebraic_topology.dold_kan.equivalence_pseudoabelian from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b5... | Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean | 108 | 114 | theorem hη :
Compatibility.τ₀ =
Compatibility.τ₁ isoN₁ isoΓ₀
(N₁Γ₀ : Γ ⋙ N₁ ≅ (toKaroubiEquivalence (ChainComplex C ℕ)).functor) := by |
ext K : 3
simp only [Compatibility.τ₀_hom_app, Compatibility.τ₁_hom_app]
exact (N₂Γ₂_compatible_with_N₁Γ₀ K).trans (by simp )
| false |
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 | 225 | 229 | theorem left_cancel (x : R) {y y' : R} : x ◃ y = x ◃ y' ↔ y = y' := by |
constructor
· apply (act' x).injective
rintro rfl
rfl
| false |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 119 | 121 | theorem isBounded_convexHull {s : Set E} :
Bornology.IsBounded (convexHull ℝ s) ↔ Bornology.IsBounded s := by |
simp only [Metric.isBounded_iff_ediam_ne_top, convexHull_ediam]
| false |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 140 | 142 | theorem zero_of_comp_mono {X Y Z : C} {f : X ⟶ Y} (g : Y ⟶ Z) [Mono g] (h : f ≫ g = 0) : f = 0 := by |
rw [← zero_comp, cancel_mono] at h
exact h
| false |
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Order.Filter.ModEq
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.List.TFAE
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.specific_lim... | Mathlib/Analysis/SpecificLimits/Normed.lean | 62 | 68 | theorem tendsto_norm_zpow_nhdsWithin_0_atTop {𝕜 : Type*} [NormedDivisionRing 𝕜] {m : ℤ}
(hm : m < 0) :
Tendsto (fun x : 𝕜 ↦ ‖x ^ m‖) (𝓝[≠] 0) atTop := by |
rcases neg_surjective m with ⟨m, rfl⟩
rw [neg_lt_zero] at hm; lift m to ℕ using hm.le; rw [Int.natCast_pos] at hm
simp only [norm_pow, zpow_neg, zpow_natCast, ← inv_pow]
exact (tendsto_pow_atTop hm.ne').comp NormedField.tendsto_norm_inverse_nhdsWithin_0_atTop
| false |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 106 | 107 | theorem taylor_mul {R} [CommSemiring R] (r : R) (p q : R[X]) :
taylor r (p * q) = taylor r p * taylor r q := by | simp only [taylor_apply, mul_comp]
| false |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open scoped Int
variable {M G : Type*}
namespace Sem... | Mathlib/Algebra/Group/Semiconj/Units.lean | 64 | 67 | theorem units_inv_symm_left {a : Mˣ} {x y : M} (h : SemiconjBy (↑a) x y) : SemiconjBy (↑a⁻¹) y x :=
calc
↑a⁻¹ * y = ↑a⁻¹ * (y * a * ↑a⁻¹) := by | rw [Units.mul_inv_cancel_right]
_ = x * ↑a⁻¹ := by rw [← h.eq, ← mul_assoc, Units.inv_mul_cancel_left]
| false |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 115 | 120 | theorem compress_idem (u v a : α) : compress u v (compress u v a) = compress u v a := by |
unfold compress
split_ifs with h h'
· rw [le_sdiff_iff.1 h'.2, sdiff_bot, sdiff_bot, sup_assoc, sup_idem]
· rfl
· rfl
| false |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 84 | 86 | theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by |
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
| false |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 121 | 123 | theorem coeff_expand_mul {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (n * p) = f.coeff n := by |
rw [coeff_expand hp, if_pos (dvd_mul_left _ _), Nat.mul_div_cancel _ hp]
| false |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 85 | 92 | theorem transpose_dualTensorHom (f : Module.Dual R M) (m : M) :
Dual.transpose (R := R) (dualTensorHom R M M (f ⊗ₜ m)) =
dualTensorHom R _ _ (Dual.eval R M m ⊗ₜ f) := by |
ext f' m'
simp only [Dual.transpose_apply, coe_comp, Function.comp_apply, dualTensorHom_apply,
LinearMap.map_smulₛₗ, RingHom.id_apply, Algebra.id.smul_eq_mul, Dual.eval_apply,
LinearMap.smul_apply]
exact mul_comm _ _
| false |
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 | 85 | 91 | theorem toFunLinear_mul_tmul_mul_aux_2 (k : ℕ) (a₁ a₂ : A) (p₁ p₂ : R[X]) :
a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) =
(Finset.antidiagonal k).sum fun x =>
a₁ * (algebraMap R A) (coeff p₁ x.1) * (a₂ * (algebraMap R A) (coeff p₂ x.2)) := by |
simp_rw [mul_assoc, Algebra.commutes, ← Finset.mul_sum, mul_assoc, ← Finset.mul_sum]
congr
simp_rw [Algebra.commutes (coeff p₂ _), coeff_mul, map_sum, RingHom.map_mul]
| false |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ι : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 132 | 137 | theorem iSup_range_stdBasis [Finite ι] : ⨆ i, range (stdBasis R φ i) = ⊤ := by |
cases nonempty_fintype ι
convert top_unique (iInf_emptyset.ge.trans <| iInf_ker_proj_le_iSup_range_stdBasis R φ _)
· rename_i i
exact ((@iSup_pos _ _ _ fun _ => range <| stdBasis R φ i) <| Finset.mem_univ i).symm
· rw [Finset.coe_univ, Set.union_empty]
| false |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 30 | 38 | theorem iteratedDerivWithin_congr (hfg : Set.EqOn f g s) :
Set.EqOn (iteratedDerivWithin n f s) (iteratedDerivWithin n g s) s := by |
induction n generalizing f g with
| zero => rwa [iteratedDerivWithin_zero]
| succ n IH =>
intro y hy
have : UniqueDiffWithinAt 𝕜 s y := h.uniqueDiffWithinAt hy
rw [iteratedDerivWithin_succ this, iteratedDerivWithin_succ this]
exact derivWithin_congr (IH hfg) (IH hfg hy)
| false |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Inf
-- can be defined with just `[Top α]` where some lemmas hold with... | Mathlib/Data/Multiset/Lattice.lean | 168 | 169 | theorem inf_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).inf = s₁.inf ⊓ s₂.inf := by |
rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_add]; simp
| false |
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 | 91 | 101 | theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r := by |
refine .congr (f := fun z ↦ continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin (z - x) 1)) ?_
fun z hz ↦ ?_
· refine continuousMultilinearCurryFin1 𝕜 E F
|>.toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesOnBall ?_
simpa using ((p.hasFPowerSeriesOnBall_changeOrigin 1
(h.r_... | false |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 84 | 87 | theorem existsUnique_zsmul_near_of_pos' {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, 0 ≤ g - k • a ∧ g - k • a < a := by |
simpa only [sub_nonneg, add_zsmul, one_zsmul, sub_lt_iff_lt_add'] using
existsUnique_zsmul_near_of_pos ha g
| false |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 102 | 105 | theorem orthogonalComplement_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
T v ∈ (⨆ μ, eigenspace T μ)ᗮ := by |
rw [← Submodule.iInf_orthogonal] at hv ⊢
exact T.iInf_invariant hT.invariant_orthogonalComplement_eigenspace v hv
| false |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 291 | 302 | theorem ae_nonneg_of_forall_setIntegral_nonneg (hf : Integrable f μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f x ∂μ) : 0 ≤ᵐ[μ] f := by |
rcases hf.1 with ⟨f', hf'_meas, hf_ae⟩
have hf'_integrable : Integrable f' μ := Integrable.congr hf hf_ae
have hf'_zero : ∀ s, MeasurableSet s → μ s < ∞ → 0 ≤ ∫ x in s, f' x ∂μ := by
intro s hs h's
rw [setIntegral_congr_ae hs (hf_ae.mono fun x hx _ => hx.symm)]
exact hf_zero s hs h's
exact
(ae_... | false |
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 133 | 133 | theorem inv_ne_top : a⁻¹ ≠ ∞ ↔ a ≠ 0 := by | simp
| false |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 121 | 135 | theorem normSq_exp (q : ℍ[ℝ]) : normSq (exp ℝ q) = exp ℝ q.re ^ 2 :=
calc
normSq (exp ℝ q) =
normSq (exp ℝ q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im)) := by |
rw [exp_eq]
_ = exp ℝ q.re ^ 2 * normSq (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im) := by
rw [normSq_smul]
_ = exp ℝ q.re ^ 2 * (Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2) := by
congr 1
obtain hv | hv := eq_or_ne ‖q.im‖ 0
· simp [hv]
rw [normSq_add, normSq_smul,... | false |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 161 | 162 | theorem bit1_eq_one {a : R} : bit1 a = 1 ↔ a = 0 := by |
rw [show (1 : R) = bit1 0 by simp, bit1_eq_bit1]
| false |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 94 | 96 | theorem of_algebraMap_smul [SMul R M] (h : ∀ (r : R) (x : M), algebraMap R A r • x = r • x) :
IsScalarTower R A M where
smul_assoc r a x := by | rw [Algebra.smul_def, mul_smul, h]
| false |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
namespace Algebra
theorem adjoin_restrictScalars (C D E : Typ... | Mathlib/RingTheory/Adjoin/Tower.lean | 49 | 58 | theorem adjoin_res_eq_adjoin_res (C D E F : Type*) [CommSemiring C] [CommSemiring D]
[CommSemiring E] [CommSemiring F] [Algebra C D] [Algebra C E] [Algebra C F] [Algebra D F]
[Algebra E F] [IsScalarTower C D F] [IsScalarTower C E F] {S : Set D} {T : Set E}
(hS : Algebra.adjoin C S = ⊤) (hT : Algebra.adjoin ... |
rw [adjoin_restrictScalars C E, adjoin_restrictScalars C D, ← hS, ← hT, ← Algebra.adjoin_image,
← Algebra.adjoin_image, ← AlgHom.coe_toRingHom, ← AlgHom.coe_toRingHom,
IsScalarTower.coe_toAlgHom, IsScalarTower.coe_toAlgHom, ← adjoin_union_eq_adjoin_adjoin, ←
adjoin_union_eq_adjoin_adjoin, Set.union_comm]... | false |
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 135 | 137 | theorem measure_union_add_inter' (hs : MeasurableSet s) (t : Set α) :
μ (s ∪ t) + μ (s ∩ t) = μ s + μ t := by |
rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]
| false |
import Mathlib.Data.Finset.Grade
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finset.interval from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
variable {α β : Type*}
namespace Finset
section Decidable
variable [DecidableEq α] (s t : Finset α)
instance instLocally... | Mathlib/Data/Finset/Interval.lean | 80 | 87 | theorem Icc_eq_image_powerset (h : s ⊆ t) : Icc s t = (t \ s).powerset.image (s ∪ ·) := by |
ext u
simp_rw [mem_Icc, mem_image, mem_powerset]
constructor
· rintro ⟨hs, ht⟩
exact ⟨u \ s, sdiff_le_sdiff_right ht, sup_sdiff_cancel_right hs⟩
· rintro ⟨v, hv, rfl⟩
exact ⟨le_sup_left, union_subset h <| hv.trans sdiff_subset⟩
| false |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Card
#align_import algebra.order.field.pi from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {α ι : Type*} [LinearOrderedSemifield α]
| Mathlib/Algebra/Order/Field/Pi.lean | 21 | 31 | theorem Pi.exists_forall_pos_add_lt [ExistsAddOfLE α] [Finite ι] {x y : ι → α}
(h : ∀ i, x i < y i) : ∃ ε, 0 < ε ∧ ∀ i, x i + ε < y i := by |
cases nonempty_fintype ι
cases isEmpty_or_nonempty ι
· exact ⟨1, zero_lt_one, isEmptyElim⟩
choose ε hε hxε using fun i => exists_pos_add_of_lt' (h i)
obtain rfl : x + ε = y := funext hxε
have hε : 0 < Finset.univ.inf' Finset.univ_nonempty ε := (Finset.lt_inf'_iff _).2 fun i _ => hε _
exact
⟨_, half_p... | false |
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame → Prop
| ⟨_, _, L, R⟩ => (... | Mathlib/SetTheory/Surreal/Basic.lean | 85 | 86 | theorem left_lt_right {x : PGame} (o : Numeric x) (i : x.LeftMoves) (j : x.RightMoves) :
x.moveLeft i < x.moveRight j := by | cases x; exact o.1 i j
| false |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.nonarchimedean.basic from "leanprover-community/mathlib"@"83f81aea33931a1edb94ce0f32b9a5d484de6978"
open scoped Pointwise Topology
class Nonarchimede... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 84 | 93 | theorem prod_subset {U} (hU : U ∈ 𝓝 (1 : G × K)) :
∃ (V : OpenSubgroup G) (W : OpenSubgroup K), (V : Set G) ×ˢ (W : Set K) ⊆ U := by |
erw [nhds_prod_eq, Filter.mem_prod_iff] at hU
rcases hU with ⟨U₁, hU₁, U₂, hU₂, h⟩
cases' is_nonarchimedean _ hU₁ with V hV
cases' is_nonarchimedean _ hU₂ with W hW
use V; use W
rw [Set.prod_subset_iff]
intro x hX y hY
exact Set.Subset.trans (Set.prod_mono hV hW) h (Set.mem_sep hX hY)
| false |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 58 | 60 | theorem continuum_lt_lift {c : Cardinal.{u}} : 𝔠 < lift.{v} c ↔ 𝔠 < c := by |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_lt]
| false |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 126 | 129 | theorem polar_comp {F : Type*} [CommRing S] [FunLike F R S] [AddMonoidHomClass F R S]
(f : M → R) (g : F) (x y : M) :
polar (g ∘ f) x y = g (polar f x y) := by |
simp only [polar, Pi.smul_apply, Function.comp_apply, map_sub]
| false |
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Order.Hom.CompleteLattice
namespace Submodule
variable (S : Type*) {R M : Type*} [Semiring R] [AddCommMonoid M] [Semiring S]
[Module S M] [Module R M] [SMul S R] [IsScalarTower S R M]
def restrictScalars (V : Submodule R M) : Submodule S M where
... | Mathlib/Algebra/Module/Submodule/RestrictScalars.lean | 106 | 107 | theorem restrictScalars_eq_bot_iff {p : Submodule R M} : restrictScalars S p = ⊥ ↔ p = ⊥ := by |
simp [SetLike.ext_iff]
| false |
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 72 | 72 | theorem toList_singleton (x : α) : toList (singleton x) = [x] := by | simp
| false |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
| .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 14 | 29 | theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by |
let rec loop : ∀ {i h b j}, j + i = as.size →
Array.forIn.loop as f i h b = forIn (as.data.drop j) b f
| 0, _, _, _, rfl => by rw [List.drop_length]; rfl
| i+1, _, _, j, ij => by
simp only [forIn.loop, Nat.add]
have j_eq : j = size as - 1 - i := by simp [← ij, ← Nat.add_assoc]
have : ... | false |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 106 | 112 | theorem gcd_self_sub_left {m n : ℕ} (h : m ≤ n) : gcd (n - m) n = gcd m n := by |
have := Nat.sub_add_cancel h
rw [gcd_comm m n, ← this, gcd_add_self_left (n - m) m]
have : gcd (n - m) n = gcd (n - m) m := by
nth_rw 2 [← Nat.add_sub_cancel' h]
rw [gcd_add_self_right, gcd_comm]
convert this
| false |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 159 | 162 | theorem asModuleEquiv_symm_map_smul (r : k) (x : V) :
ρ.asModuleEquiv.symm (r • x) = algebraMap k (MonoidAlgebra k G) r • ρ.asModuleEquiv.symm x := by |
apply_fun ρ.asModuleEquiv
simp
| false |
import Mathlib.Data.List.Forall2
#align_import data.list.sections from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat Function
namespace List
variable {α β : Type*}
| Mathlib/Data/List/Sections.lean | 23 | 34 | theorem mem_sections {L : List (List α)} {f} : f ∈ sections L ↔ Forall₂ (· ∈ ·) f L := by |
refine ⟨fun h => ?_, fun h => ?_⟩
· induction L generalizing f
· cases mem_singleton.1 h
exact Forall₂.nil
simp only [sections, bind_eq_bind, mem_bind, mem_map] at h
rcases h with ⟨_, _, _, _, rfl⟩
simp only [*, forall₂_cons, true_and_iff]
· induction' h with a l f L al fL fs
· simp onl... | false |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 137 | 140 | theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
Gᶜ.IsRegularOfDegree (n - k - 1) := by |
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
exact h.regular.compl
| false |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 125 | 128 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
(f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by |
rw [tensorHom_def' f g, tensorHom_def g f]
simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
| false |
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.polynomial.algebra_map from "leanprover-community/mathlib"@"e064a7bf82ad94c3c17b5128bbd860d1ec34874e"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
univer... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 131 | 136 | theorem eval₂_algebraMap_X {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A] (p : R[X])
(f : R[X] →ₐ[R] A) : eval₂ (algebraMap R A) (f X) p = f p := by |
conv_rhs => rw [← Polynomial.sum_C_mul_X_pow_eq p]
simp only [eval₂_eq_sum, sum_def]
simp only [f.map_sum, f.map_mul, f.map_pow, eq_intCast, map_intCast]
simp [Polynomial.C_eq_algebraMap]
| false |
import Mathlib.Data.Fin.Tuple.Sort
import Mathlib.Order.WellFounded
#align_import data.fin.tuple.bubble_sort_induction from "leanprover-community/mathlib"@"bf2428c9486c407ca38b5b3fb10b87dad0bc99fa"
namespace Tuple
| Mathlib/Data/Fin/Tuple/BubbleSortInduction.lean | 34 | 44 | theorem bubble_sort_induction' {n : ℕ} {α : Type*} [LinearOrder α] {f : Fin n → α}
{P : (Fin n → α) → Prop} (hf : P f)
(h : ∀ (σ : Equiv.Perm (Fin n)) (i j : Fin n),
i < j → (f ∘ σ) j < (f ∘ σ) i → P (f ∘ σ) → P (f ∘ σ ∘ Equiv.swap i j)) :
P (f ∘ sort f) := by |
letI := @Preorder.lift _ (Lex (Fin n → α)) _ fun σ : Equiv.Perm (Fin n) => toLex (f ∘ σ)
refine
@WellFounded.induction_bot' _ _ _ (IsWellFounded.wf : WellFounded (· < ·))
(Equiv.refl _) (sort f) P (fun σ => f ∘ σ) (fun σ hσ hfσ => ?_) hf
obtain ⟨i, j, hij₁, hij₂⟩ := antitone_pair_of_not_sorted' hσ
ex... | false |
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 139 | 142 | theorem noetherianSpace_set_iff (s : Set α) :
NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t := by |
simp only [noetherianSpace_iff_isCompact, embedding_subtype_val.isCompact_iff,
Subtype.forall_set_subtype]
| false |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 75 | 76 | theorem volume_val (s) : volume s = StieltjesFunction.id.measure s := by |
simp [volume_eq_stieltjes_id]
| false |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 90 | 92 | theorem baseChange_zero : baseChange A (0 : M →ₗ[R] N) = 0 := by |
ext
simp [baseChange_eq_ltensor]
| false |
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 | 124 | 132 | theorem tprod_iSup_decode₂ [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (s : β → α) :
∏' i : ℕ, m (⨆ b ∈ decode₂ β i, s b) = ∏' b : β, m (s b) := by |
rw [← tprod_extend_one (@encode_injective β _)]
refine tprod_congr fun n ↦ ?_
rcases em (n ∈ Set.range (encode : β → ℕ)) with ⟨a, rfl⟩ | hn
· simp [encode_injective.extend_apply]
· rw [extend_apply' _ _ _ hn]
rw [← decode₂_ne_none_iff, ne_eq, not_not] at hn
simp [hn, m0]
| false |
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 | 406 | 413 | theorem det_fromBlocks₂₂ (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α)
(D : Matrix n n α) [Invertible D] :
(Matrix.fromBlocks A B C D).det = det D * det (A - B * ⅟ D * C) := by |
have : fromBlocks A B C D =
(fromBlocks D C B A).submatrix (Equiv.sumComm _ _) (Equiv.sumComm _ _) := by
ext (i j)
cases i <;> cases j <;> rfl
rw [this, det_submatrix_equiv_self, det_fromBlocks₁₁]
| false |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 | theorem gcd_zero_left (a : R) : gcd 0 a = a := by |
rw [gcd]
exact if_pos rfl
| false |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 136 | 139 | theorem discr_not_zero_of_basis [IsSeparable K L] (b : Basis ι K L) :
discr K b ≠ 0 := by |
rw [discr_def, traceMatrix_of_basis, ← LinearMap.BilinForm.nondegenerate_iff_det_ne_zero]
exact traceForm_nondegenerate _ _
| false |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
namespace List
variable {α : Type u}
@[simp]
theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = ... | Mathlib/Data/List/FinRange.lean | 54 | 55 | theorem ofFn_eq_map {n} {f : Fin n → α} : ofFn f = (finRange n).map f := by |
rw [← ofFn_id, map_ofFn, Function.comp_id]
| false |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 116 | 119 | theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by |
cases x
rfl
| false |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 142 | 145 | theorem IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ∪ t) := by |
rcases H with ⟨x, hxs, hxt⟩
exact hs.union x hxs hxt ht
| false |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.fin_range from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u
namespace List
variable {α : Type u}
@[simp]
theorem map_coe_finRange (n : ℕ) : ((finRange n) : List (Fin n)).map (Fin.val) = ... | Mathlib/Data/List/FinRange.lean | 30 | 34 | theorem finRange_succ_eq_map (n : ℕ) : finRange n.succ = 0 :: (finRange n).map Fin.succ := by |
apply map_injective_iff.mpr Fin.val_injective
rw [map_cons, map_coe_finRange, range_succ_eq_map, Fin.val_zero, ← map_coe_finRange, map_map,
map_map]
simp only [Function.comp, Fin.val_succ]
| false |
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 | 53 | 53 | theorem rotate'_zero (l : List α) : l.rotate' 0 = l := by | cases l <;> rfl
| false |
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 63 | 71 | theorem tfae_of_cycle {a b} {l : List Prop} (h_chain : List.Chain (· → ·) a (b :: l))
(h_last : getLastD l b → a) : TFAE (a :: b :: l) := by |
induction l generalizing a b with
| nil => simp_all [tfae_cons_cons, iff_def]
| cons c l IH =>
simp only [tfae_cons_cons, getLastD_cons, tfae_singleton, and_true, chain_cons, Chain.nil] at *
rcases h_chain with ⟨ab, ⟨bc, ch⟩⟩
have := IH ⟨bc, ch⟩ (ab ∘ h_last)
exact ⟨⟨ab, h_last ∘ (this.2 c (.head... | false |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.rational_root from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
open scoped Polynomial
section ScaleRoots
var... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 49 | 54 | theorem num_isRoot_scaleRoots_of_aeval_eq_zero [UniqueFactorizationMonoid A] {p : A[X]} {x : K}
(hr : aeval x p = 0) : IsRoot (scaleRoots p (den A x)) (num A x) := by |
apply isRoot_of_eval₂_map_eq_zero (IsFractionRing.injective A K)
refine scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero ?_
rw [mk'_num_den]
exact hr
| false |
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 165 | 173 | theorem compression_self (u : α) (s : Finset α) : 𝓒 u u s = s := by |
unfold compression
convert union_empty s
· ext a
rw [mem_filter, compress_self, and_self_iff]
· refine eq_empty_of_forall_not_mem fun a ha ↦ ?_
simp_rw [mem_filter, mem_image, compress_self] at ha
obtain ⟨⟨b, hb, rfl⟩, hb'⟩ := ha
exact hb' hb
| false |
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compacts
import Mathlib.Analysis.Normed.Group.InfiniteSum
#align_import topology.continuous_function.compact from "leanprover-community/mathlib"@"d3af0609f6d... | Mathlib/Topology/ContinuousFunction/Compact.lean | 146 | 147 | theorem dist_lt_iff_of_nonempty [Nonempty α] : dist f g < C ↔ ∀ x : α, dist (f x) (g x) < C := by |
simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]
| false |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 56 | 66 | theorem Gamma_mem (N : ℕ) (γ : SL(2, ℤ)) : γ ∈ Gamma N ↔ ((↑ₘγ 0 0 : ℤ) : ZMod N) = 1 ∧
((↑ₘγ 0 1 : ℤ) : ZMod N) = 0 ∧ ((↑ₘγ 1 0 : ℤ) : ZMod N) = 0 ∧ ((↑ₘγ 1 1 : ℤ) : ZMod N) = 1 := by |
rw [Gamma_mem']
constructor
· intro h
simp [← SL_reduction_mod_hom_val N γ, h]
· intro h
ext i j
rw [SL_reduction_mod_hom_val N γ]
fin_cases i <;> fin_cases j <;> simp only [h]
exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]
| false |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 91 | 91 | theorem count_one : count p 1 = if p 0 then 1 else 0 := by | simp [count_succ]
| false |
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Order.Hom.CompleteLattice
namespace Submodule
variable (S : Type*) {R M : Type*} [Semiring R] [AddCommMonoid M] [Semiring S]
[Module S M] [Module R M] [SMul S R] [IsScalarTower S R M]
def restrictScalars (V : Submodule R M) : Submodule S M where
... | Mathlib/Algebra/Module/Submodule/RestrictScalars.lean | 116 | 117 | theorem restrictScalars_eq_top_iff {p : Submodule R M} : restrictScalars S p = ⊤ ↔ p = ⊤ := by |
simp [SetLike.ext_iff]
| false |
import Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence
import Mathlib.Analysis.Complex.UpperHalfPlane.Manifold
import Mathlib.Analysis.Complex.LocallyUniformLimit
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open ModularForm EisensteinSeries UpperHalfPlane Set Filter... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/MDifferentiable.lean | 54 | 65 | theorem eisensteinSeries_SIF_MDifferentiable {k : ℤ} {N : ℕ} (hk : 3 ≤ k) (a : Fin 2 → ZMod N) :
MDifferentiable 𝓘(ℂ) 𝓘(ℂ) (eisensteinSeries_SIF a k) := by |
intro τ
suffices DifferentiableAt ℂ (↑ₕeisensteinSeries_SIF a k) τ.1 by
convert MDifferentiableAt.comp τ (DifferentiableAt.mdifferentiableAt this) τ.mdifferentiable_coe
exact funext fun z ↦ (comp_ofComplex (eisensteinSeries_SIF a k) z).symm
refine DifferentiableOn.differentiableAt ?_
((isOpen_lt cont... | false |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 85 | 89 | theorem closure_mul_image_eq_top (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g =>
⟨g * (toFun hR g : G)⁻¹, mul_inv_toFun_mem hR g⟩ : Set H) = ⊤ := by |
rw [eq_top_iff, ← map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]
exact (map_subtype_le ⊤).trans (ge_of_eq (closure_mul_image_eq hR hR1 hS))
| false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.