Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {x y : ℝ}
theorem log_mul_self_monotoneOn... | Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 85 | 88 | theorem log_div_sqrt_antitoneOn : AntitoneOn (fun x : ℝ => log x / √x) { x | exp 2 ≤ x } := by |
simp_rw [sqrt_eq_rpow]
convert @log_div_self_rpow_antitoneOn (1 / 2) (by norm_num)
norm_num
| 0 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 114 | 123 | theorem strongEpi_of_strongEpi [StrongEpi (f ≫ g)] : StrongEpi g :=
{ epi := epi_of_epi f g
llp := fun {X Y} z _ => by
constructor
intro u v sq
have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v := by | simp only [Category.assoc, sq.w]
exact
CommSq.HasLift.mk'
⟨(CommSq.mk h₀).lift, by
simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w], by simp⟩ }
| 0 |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 159 | 165 | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by |
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
| 0 |
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 | 98 | 103 | theorem exp_of_re_eq_zero (q : Quaternion ℝ) (hq : q.re = 0) :
exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q := by |
rw [exp_eq_tsum]
refine HasSum.tsum_eq ?_
simp_rw [← expSeries_apply_eq]
exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)
| 0 |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 69 | 80 | theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ := by |
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv
have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1
simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢
refine hp.mono fun x hx => ?_
by_cases h : x = 0
· convert hasSum_single (α := E) 0 _ <;> intros <;> simp [*]
· have hxx : ∀ n : ℕ,... | 0 |
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 | 114 | 123 | theorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by |
dsimp [destruct]
induction' f0 : s.1 0 with _ <;> intro h
· contradiction
· apply Subtype.eq
funext n
induction' n with n IH
· injection h with h'
rwa [h'] at f0
· exact s.2 IH
| 0 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.FiniteDimension
variable {𝕜 E A B : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
[CommSemiring A] {z : E} {... | Mathlib/Analysis/Analytic/Polynomial.lean | 26 | 32 | theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt 𝕜 f z) (p : A[X]) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by |
refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_
· simp_rw [aeval_C]; apply analyticAt_const
· simp_rw [aeval_add]; exact hp.add hq
· convert hp.mul hf
simp_rw [pow_succ, aeval_mul, ← mul_assoc, aeval_X]
| 0 |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 56 | 72 | theorem isFiltered_structuredArrow_of_isFiltered_of_exists [IsFilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) (h₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),
∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t) (d : D) :
IsFiltered (StructuredArrow d F) := by |
have : Nonempty (StructuredArrow d F) := by
obtain ⟨c, ⟨f⟩⟩ := h₁ d
exact ⟨.mk f⟩
suffices IsFilteredOrEmpty (StructuredArrow d F) from IsFiltered.mk
refine ⟨fun f g => ?_, fun f g η μ => ?_⟩
· obtain ⟨c, ⟨t, ht⟩⟩ := h₂ (f.hom ≫ F.map (IsFiltered.leftToMax f.right g.right))
(g.hom ≫ F.map (IsFi... | 0 |
import Mathlib.Data.Finsupp.Defs
#align_import data.list.to_finsupp from "leanprover-community/mathlib"@"06a655b5fcfbda03502f9158bbf6c0f1400886f9"
namespace List
variable {M : Type*} [Zero M] (l : List M) [DecidablePred (getD l · 0 ≠ 0)] (n : ℕ)
def toFinsupp : ℕ →₀ M where
toFun i := getD l i 0
support := ... | Mathlib/Data/List/ToFinsupp.lean | 147 | 156 | theorem toFinsupp_eq_sum_map_enum_single {R : Type*} [AddMonoid R] (l : List R)
[DecidablePred (getD l · 0 ≠ 0)] :
toFinsupp l = (l.enum.map fun nr : ℕ × R => Finsupp.single nr.1 nr.2).sum := by |
/- Porting note (#11215): TODO: `induction` fails to substitute `l = []` in
`[DecidablePred (getD l · 0 ≠ 0)]`, so we manually do some `revert`/`intro` as a workaround -/
revert l; intro l
induction l using List.reverseRecOn with
| nil => exact toFinsupp_nil
| append_singleton x xs ih =>
classical simp... | 0 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 113 | 115 | theorem HasEigenvector.pow_apply {f : End R M} {μ : R} {v : M} (hv : f.HasEigenvector μ v) (n : ℕ) :
(f ^ n) v = μ ^ n • v := by |
induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
| 0 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.PNat.Basic
import Mathlib.GroupTheory.GroupAction.Prod
variable {M : Type*}
class PNatPowAssoc (M : Type*) [Mul M] [Pow M ℕ+] : Prop where
protected ppow_add : ∀ (k n : ℕ+) (x : M), x ^ (k + n) = x ^ k * x ^ n
prote... | Mathlib/Algebra/Group/PNatPowAssoc.lean | 67 | 70 | theorem ppow_mul (x : M) (m n : ℕ+) : x ^ (m * n) = (x ^ m) ^ n := by |
refine PNat.recOn n ?_ fun k hk ↦ ?_
· rw [ppow_one, mul_one]
· rw [ppow_add, ppow_one, mul_add, ppow_add, mul_one, hk]
| 0 |
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.cont_mdiff_mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners Bundle
open sc... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 227 | 280 | theorem ContMDiffOn.continuousOn_tangentMapWithin_aux {f : H → H'} {s : Set H}
(hf : ContMDiffOn I I' n f s) (hn : 1 ≤ n) (hs : UniqueMDiffOn I s) :
ContinuousOn (tangentMapWithin I I' f s) (π E (TangentSpace I) ⁻¹' s) := by |
suffices h :
ContinuousOn
(fun p : H × E =>
(f p.fst,
(fderivWithin 𝕜 (writtenInExtChartAt I I' p.fst f) (I.symm ⁻¹' s ∩ range I)
((extChartAt I p.fst) p.fst) : E →L[𝕜] E') p.snd)) (Prod.fst ⁻¹' s) by
have A := (tangentBundleModelSpaceHomeomorph H I).continuous
r... | 0 |
import Mathlib.RingTheory.Flat.Basic
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.LinearAlgebra.TensorProduct.Tower
universe u v w t
open Function (Injective Surjective)
open LinearMap (lsmul rTensor lTensor)
open TensorProduct
namespace Module.Flat
section Composition
variable (R : Type u) (S :... | Mathlib/RingTheory/Flat/Stability.lean | 86 | 94 | theorem comp [Module.Flat R S] [Module.Flat S M] : Module.Flat R M := by |
rw [Module.Flat.iff_lTensor_injective']
intro I
rw [← EquivLike.comp_injective _ (TensorProduct.rid R M)]
haveI h : TensorProduct.rid R M ∘ lTensor M (Submodule.subtype I) =
TensorProduct.rid R M ∘ₗ lTensor M I.subtype := rfl
simp only [h, ← auxLTensor_eq R S M, LinearMap.coe_restrictScalars, LinearMap.c... | 0 |
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.monoidal.free.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786cadc4c35ff"
universe u
namespace CategoryTheory
open Mo... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 184 | 191 | theorem tensorFunc_obj_map (Z : F C) {n n' : N C} (f : n ⟶ n') :
((tensorFunc C).obj Z).map f = inclusion.map f ▷ Z := by |
cases n
cases n'
rcases f with ⟨⟨h⟩⟩
dsimp at h
subst h
simp
| 0 |
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.ZMod.Algebra
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
import Mathlib.FieldTheory.Perfect
#align_import ring_theory.witt_vector.frobenius from "leanprover-community/mathlib"@"0723536a0522d24fc2f159a096fb3304bef77472"... | Mathlib/RingTheory/WittVector/Frobenius.lean | 123 | 127 | theorem map_frobeniusPoly.key₁ (n j : ℕ) (hj : j < p ^ n) :
p ^ (n - v p ⟨j + 1, j.succ_pos⟩) ∣ (p ^ n).choose (j + 1) := by |
apply multiplicity.pow_dvd_of_le_multiplicity
rw [hp.out.multiplicity_choose_prime_pow hj j.succ_ne_zero]
rfl
| 0 |
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
#align_import analysis.normed_space.M_structure from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
variable (X : Type*) [NormedAddCommGroup X]
... | Mathlib/Analysis/NormedSpace/MStructure.lean | 147 | 162 | theorem mul [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
IsLprojection X (P * Q) := by |
refine ⟨IsIdempotentElem.mul_of_commute (h₁.commute h₂) h₁.proj h₂.proj, ?_⟩
intro x
refine le_antisymm ?_ ?_
· calc
‖x‖ = ‖(P * Q) • x + (x - (P * Q) • x)‖ := by rw [add_sub_cancel ((P * Q) • x) x]
_ ≤ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖ := by apply norm_add_le
_ = ‖(P * Q) • x‖ + ‖(1 - P * Q)... | 0 |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
section Module
variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGroup V] [Module K V]
noncomputable def Basis.ofRankEqZero [Mo... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 46 | 60 | theorem le_rank_iff_exists_linearIndependent [Module.Free K V] {c : Cardinal} :
c ≤ Module.rank K V ↔ ∃ s : Set V, #s = c ∧ LinearIndependent K ((↑) : s → V) := by |
haveI := nontrivial_of_invariantBasisNumber K
constructor
· intro h
obtain ⟨κ, t'⟩ := Module.Free.exists_basis (R := K) (M := V)
let t := t'.reindexRange
have : LinearIndependent K ((↑) : Set.range t' → V) := by
convert t.linearIndependent
ext; exact (Basis.reindexRange_apply _ _).symm
... | 0 |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 124 | 128 | theorem isComplement_singleton_left {g : G} : IsComplement {g} S ↔ S = univ := by |
refine
⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => (congr_arg _ h).mpr isComplement_singleton_univ⟩
obtain ⟨⟨⟨z, rfl : z = g⟩, y, _⟩, hy⟩ := h.2 (g * x)
rwa [← mul_left_cancel hy]
| 0 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 172 | 180 | theorem areaForm_comp_linearIsometryEquiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < LinearMap.det (φ.toLinearEquiv : E →ₗ[ℝ] E)) (x y : E) :
o.areaForm (φ x) (φ y) = o.areaForm x y := by |
convert o.areaForm_map φ (φ x) (φ y)
· symm
rwa [← o.map_eq_iff_det_pos φ.toLinearEquiv] at hφ
rw [@Fact.out (finrank ℝ E = 2), Fintype.card_fin]
· simp
· simp
| 0 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 86 | 86 | theorem vectorSpan_singleton (p : P) : vectorSpan k ({p} : Set P) = ⊥ := by | simp [vectorSpan_def]
| 0 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 339 | 385 | theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R → S}
(f_decr : Antitone f) (f_cont : ContinuousAt f F.limsSup)
(bdd_above : F.IsBounded (· ≤ ·) := by | isBoundedDefault)
(bdd_below : F.IsBounded (· ≥ ·) := by isBoundedDefault) :
f F.limsSup = F.liminf f := by
have cobdd : F.IsCobounded (· ≤ ·) := bdd_below.isCobounded_flip
apply le_antisymm
· rw [limsSup, f_decr.map_sInf_of_continuousAt' f_cont bdd_above cobdd]
apply le_of_forall_lt
intro c hc
... | 0 |
import Mathlib.Topology.Separation
#align_import topology.shrinking_lemma from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Function
open scoped Classical
noncomputable section
variable {ι X : Type*} [TopologicalSpace X] [NormalSpace X]
namespace ShrinkingLemma
-- the tr... | Mathlib/Topology/ShrinkingLemma.lean | 124 | 132 | theorem mem_find_carrier_iff {c : Set (PartialRefinement u s)} {i : ι} (ne : c.Nonempty) :
i ∈ (find c ne i).carrier ↔ i ∈ chainSupCarrier c := by |
rw [find]
split_ifs with h
· have := h.choose_spec
exact iff_of_true this.2 (mem_iUnion₂.2 ⟨_, this.1, this.2⟩)
· push_neg at h
refine iff_of_false (h _ ne.some_mem) ?_
simpa only [chainSupCarrier, mem_iUnion₂, not_exists]
| 0 |
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import measure_theory.measure.sub from "leanprover-community/mathlib"@"562bbf524c595c153470e53d36c57b6f891cc480"
open Set
namespace MeasureTheory
namespace Measure
noncomputable instance instSub {α : Type*} [MeasurableSpace α] : Sub (Measure α) :=
⟨fun ... | Mathlib/MeasureTheory/Measure/Sub.lean | 137 | 139 | theorem sub_apply_eq_zero_of_restrict_le_restrict (h_le : μ.restrict s ≤ ν.restrict s)
(h_meas_s : MeasurableSet s) : (μ - ν) s = 0 := by |
rw [← restrict_apply_self, restrict_sub_eq_restrict_sub_restrict, sub_eq_zero_of_le] <;> simp [*]
| 0 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 132 | 138 | theorem fold_image_idem [DecidableEq α] {g : γ → α} {s : Finset γ} [hi : Std.IdempotentOp op] :
(image g s).fold op b f = s.fold op b (f ∘ g) := by |
induction' s using Finset.cons_induction with x xs hx ih
· rw [fold_empty, image_empty, fold_empty]
· haveI := Classical.decEq γ
rw [fold_cons, cons_eq_insert, image_insert, fold_insert_idem, ih]
simp only [Function.comp_apply]
| 0 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.analytic.radius_liminf from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpa... | Mathlib/Analysis/Analytic/RadiusLiminf.lean | 35 | 61 | theorem radius_eq_liminf :
p.radius = liminf (fun n => (1 / (‖p n‖₊ ^ (1 / (n : ℝ)) : ℝ≥0) : ℝ≥0∞)) atTop := by |
-- Porting note: added type ascription to make elaborated statement match Lean 3 version
have :
∀ (r : ℝ≥0) {n : ℕ},
0 < n → ((r : ℝ≥0∞) ≤ 1 / ↑(‖p n‖₊ ^ (1 / (n : ℝ))) ↔ ‖p n‖₊ * r ^ n ≤ 1) := by
intro r n hn
have : 0 < (n : ℝ) := Nat.cast_pos.2 hn
conv_lhs =>
rw [one_div, ENNReal.le_i... | 0 |
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 MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 109 | 129 | theorem mapIdxGo_append : ∀ (f : ℕ → α → β) (l₁ l₂ : List α) (arr : Array β),
mapIdx.go f (l₁ ++ l₂) arr = mapIdx.go f l₂ (List.toArray (mapIdx.go f l₁ arr)) := by |
intros f l₁ l₂ arr
generalize e : (l₁ ++ l₂).length = len
revert l₁ l₂ arr
induction' len with len ih <;> intros l₁ l₂ arr h
· have l₁_nil : l₁ = [] := by
cases l₁
· rfl
· contradiction
have l₂_nil : l₂ = [] := by
cases l₂
· rfl
· rw [List.length_append] at h; contradi... | 0 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 156 | 160 | theorem IntValuation.map_one' : v.intValuationDef 1 = 1 := by |
rw [v.intValuationDef_if_neg (zero_ne_one.symm : (1 : R) ≠ 0), Ideal.span_singleton_one, ←
Ideal.one_eq_top, Associates.mk_one, Associates.factors_one,
Associates.count_zero (by apply v.associates_irreducible), Int.ofNat_zero, neg_zero, ofAdd_zero,
WithZero.coe_one]
| 0 |
import Mathlib.ModelTheory.ElementarySubstructures
#align_import model_theory.skolem from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
universe u v w w'
namespace FirstOrder
namespace Language
open Structure Cardinal
open Cardinal
variable (L : Language.{u, v}) {M : Type w} [None... | Mathlib/ModelTheory/Skolem.lean | 50 | 62 | theorem card_functions_sum_skolem₁ :
#(Σ n, (L.sum L.skolem₁).Functions n) = #(Σ n, L.BoundedFormula Empty (n + 1)) := by |
simp only [card_functions_sum, skolem₁_Functions, mk_sigma, sum_add_distrib']
conv_lhs => enter [2, 1, i]; rw [lift_id'.{u, v}]
rw [add_comm, add_eq_max, max_eq_left]
· refine sum_le_sum _ _ fun n => ?_
rw [← lift_le.{_, max u v}, lift_lift, lift_mk_le.{v}]
refine ⟨⟨fun f => (func f default).bdEqual (f... | 0 |
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 | 59 | 78 | theorem expSeries_odd_of_imaginary {q : Quaternion ℝ} (hq : q.re = 0) (n : ℕ) :
expSeries ℝ (Quaternion ℝ) (2 * n + 1) (fun _ => q) =
(((-1 : ℝ) ^ n * ‖q‖ ^ (2 * n + 1) / (2 * n + 1)!) / ‖q‖) • q := by |
rw [expSeries_apply_eq]
obtain rfl | hq0 := eq_or_ne q 0
· simp
have hq2 : q ^ 2 = -normSq q := sq_eq_neg_normSq.mpr hq
have hqn := norm_ne_zero_iff.mpr hq0
let k : ℝ := ↑(2 * n + 1)!
calc
k⁻¹ • q ^ (2 * n + 1) = k⁻¹ • ((-normSq q) ^ n * q) := by rw [pow_succ, pow_mul, hq2]
_ = k⁻¹ • ((-1 : ℝ) ^ ... | 0 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 133 | 143 | theorem BlankRel.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃ := by |
rintro (h₁ | h₁) (h₂ | h₂)
· exact Or.inl (h₁.trans h₂)
· rcases le_total l₁.length l₃.length with h | h
· exact Or.inl (h₁.above_of_le h₂ h)
· exact Or.inr (h₂.above_of_le h₁ h)
· rcases le_total l₁.length l₃.length with h | h
· exact Or.inl (h₁.below_of_le h₂ h)
· exact Or.inr (h₂.below_of_le... | 0 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 309 | 316 | theorem toDual_total_left (f : ι →₀ R) (i : ι) :
b.toDual (Finsupp.total ι M R b f) (b i) = f i := by |
rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
Finset.sum_ite_eq']
split_ifs with h
· rfl
· rw [Finsupp.not_mem_support_iff.mp h]
| 0 |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : ℕ} {x : ℝ}
theorem cosZeta_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 76 | 97 | theorem sinZeta_two_mul_nat_add_one (hk : k ≠ 0) (hx : x ∈ Icc 0 1) :
sinZeta x (2 * k + 1) = (-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by |
rw [← (hasSum_nat_sinZeta x (?_ : 1 < re (2 * k + 1))).tsum_eq]
refine Eq.trans ?_ <| (congr_arg ofReal' (hasSum_one_div_nat_pow_mul_sin hk hx).tsum_eq).trans ?_
· rw [ofReal_tsum]
refine tsum_congr fun n ↦ ?_
rw [mul_comm (1 / _), mul_one_div, ofReal_div, mul_assoc (2 * π), mul_comm x n, ← mul_assoc]
... | 0 |
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.normed_space.continuous_affine_map from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
namespace Con... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 148 | 151 | theorem decomp (f : V →ᴬ[R] W) : (f : V → W) = f.contLinear + Function.const V (f 0) := by |
rcases f with ⟨f, h⟩
rw [coe_mk_const_linear_eq_linear, coe_mk, f.decomp, Pi.add_apply, LinearMap.map_zero, zero_add,
← Function.const_def]
| 0 |
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 | 175 | 195 | theorem lt_three {p q r : ℕ+} (hpq : p ≤ q) (hqr : q ≤ r) (H : 1 < sumInv {p, q, r}) : p < 3 := by |
have h3 : (0 : ℚ) < 3 := by norm_num
contrapose! H
rw [sumInv_pqr]
have h3q := H.trans hpq
have h3r := h3q.trans hqr
have hp: (p : ℚ)⁻¹ ≤ 3⁻¹ := by
rw [inv_le_inv _ h3]
· assumption_mod_cast
· norm_num
have hq: (q : ℚ)⁻¹ ≤ 3⁻¹ := by
rw [inv_le_inv _ h3]
· assumption_mod_cast
· nor... | 0 |
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Analysis.Convex.PartitionOfUnity
#align_import topology.metric_space.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal NNReal Filter Set Fu... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 42 | 61 | theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
(hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i := by |
suffices ∀ i, x ∈ K i → ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, closedBall p.2 p.1 ⊆ U i by
apply mp_mem ((eventually_all_finite (hfin.point_finite x)).2 this)
(mp_mem (@tendsto_snd ℝ≥0∞ _ (𝓝 0) _ _ (hfin.iInter_compl_mem_nhds hK x)) _)
apply univ_mem'
rintro ⟨r, y⟩ hxy hyU i hi
simp only [mem_iInter... | 0 |
import Mathlib.Topology.LocalAtTarget
import Mathlib.AlgebraicGeometry.Morphisms.Basic
#align_import algebraic_geometry.morphisms.open_immersion from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace... | Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean | 53 | 74 | theorem isOpenImmersion_is_local_at_target : PropertyIsLocalAtTarget @IsOpenImmersion := by |
constructor
· exact isOpenImmersion_respectsIso
· intros; infer_instance
· intro X Y f 𝒰 H
rw [isOpenImmersion_iff_stalk]
constructor
· apply (openEmbedding_iff_openEmbedding_of_iSup_eq_top 𝒰.iSup_opensRange f.1.base.2).mpr
intro i
have := ((isOpenImmersion_respectsIso.arrow_iso_iff
... | 0 |
import Batteries.Data.RBMap.WF
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] Path.fill
def OnRoot (p : α → Prop) : RBNode α → Prop
| nil => True
| node _ _ x _ => p x
namespace Path
@[inline] def fill' : RBNode α × Path α → RBNode α := fun (t, path) => path.fill t
| .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean | 34 | 38 | theorem zoom_fill' (cut : α → Ordering) (t : RBNode α) (path : Path α) :
fill' (zoom cut t path) = path.fill t := by |
induction t generalizing path with
| nil => rfl
| node _ _ _ _ iha ihb => unfold zoom; split <;> [apply iha; apply ihb; rfl]
| 0 |
import Mathlib.Analysis.Calculus.FDeriv.Prod
#align_import analysis.calculus.fderiv.bilinear from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Asymptotics ENNReal
noncomputable section
... | Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean | 51 | 74 | theorem IsBoundedBilinearMap.hasStrictFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
HasStrictFDerivAt b (h.deriv p) p := by |
simp only [HasStrictFDerivAt]
simp only [← map_add_left_nhds_zero (p, p), isLittleO_map]
set T := (E × F) × E × F
calc
_ = fun x ↦ h.deriv (x.1 - x.2) (x.2.1, x.1.2) := by
ext ⟨⟨x₁, y₁⟩, ⟨x₂, y₂⟩⟩
rcases p with ⟨x, y⟩
simp only [map_sub, deriv_apply, Function.comp_apply, Prod.mk_add_mk, h... | 0 |
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 186 | 191 | theorem Filter.Tendsto.inseparable_iff_uniformity {l : Filter β} [NeBot l] {f g : β → α} {a b : α}
(ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) :
Inseparable a b ↔ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α) := by |
refine ⟨fun h ↦ (ha.prod_mk_nhds hb).mono_right h.nhds_le_uniformity, fun h ↦ ?_⟩
rw [inseparable_iff_clusterPt_uniformity]
exact (ClusterPt.of_le_nhds (ha.prod_mk_nhds hb)).mono h
| 0 |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 125 | 135 | theorem _root_.IsIntegrallyClosed.minpoly.unique {s : S} {P : R[X]} (hmo : P.Monic)
(hP : Polynomial.aeval s P = 0)
(Pmin : ∀ Q : R[X], Q.Monic → Polynomial.aeval s Q = 0 → degree P ≤ degree Q) :
P = minpoly R s := by |
have hs : IsIntegral R s := ⟨P, hmo, hP⟩
symm; apply eq_of_sub_eq_zero
by_contra hnz
refine IsIntegrallyClosed.degree_le_of_ne_zero hs hnz (by simp [hP]) |>.not_lt ?_
refine degree_sub_lt ?_ (ne_zero hs) ?_
· exact le_antisymm (min R s hmo hP) (Pmin (minpoly R s) (monic hs) (aeval R s))
· rw [(monic hs).... | 0 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : ∀ {b₁ b₂ : Buckets α β}, b₁.1.data = b₂.1.data → b₁ = b₂
| ⟨⟨_⟩, _⟩, ⟨⟨_⟩, _⟩, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 23 | 27 | theorem exists_of_update (self : Buckets α β) (i d h) :
∃ l₁ l₂, self.1.data = l₁ ++ self.1[i] :: l₂ ∧ List.length l₁ = i.toNat ∧
(self.update i d h).1.data = l₁ ++ d :: l₂ := by |
simp only [Array.data_length, Array.ugetElem_eq_getElem, Array.getElem_eq_data_get]
exact List.exists_of_set' h
| 0 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Geometry.Manifold.ChartedSpace
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.Calculus.ContDiff.Basic
... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 256 | 258 | theorem target_eq : I.target = range (I : H → E) := by |
rw [← image_univ, ← I.source_eq]
exact I.image_source_eq_target.symm
| 0 |
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 | 221 | 266 | theorem convexBodyLT'_volume :
volume (convexBodyLT' K f w₀) = convexBodyLT'Factor K * ∏ w, (f w) ^ (mult w) := by |
have vol_box : ∀ B : ℝ≥0, volume {x : ℂ | |x.re| < 1 ∧ |x.im| < B^2} = 4*B^2 := by
intro B
rw [← (Complex.volume_preserving_equiv_real_prod.symm).measure_preimage]
· simp_rw [Set.preimage_setOf_eq, Complex.measurableEquivRealProd_symm_apply]
rw [show {a : ℝ × ℝ | |a.1| < 1 ∧ |a.2| < B ^ 2} =
... | 0 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.mean_inequalities_pow from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
universe u... | Mathlib/Analysis/MeanInequalitiesPow.lean | 72 | 86 | theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
(∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by |
rcases s.eq_empty_or_nonempty with (rfl | hs)
· simp_rw [Finset.sum_empty, zero_pow n.succ_ne_zero, zero_div]; rfl
· have hs0 : 0 < (s.card : ℝ) := Nat.cast_pos.2 hs.card_pos
suffices (∑ x ∈ s, f x / s.card) ^ (n + 1) ≤ ∑ x ∈ s, f x ^ (n + 1) / s.card by
rwa [← Finset.sum_div, ← Finset.sum_div, div_pow... | 0 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 240 | 254 | theorem extract_gcd (s : Multiset α) (hs : s ≠ 0) :
∃ t : Multiset α, s = t.map (s.gcd * ·) ∧ t.gcd = 1 := by |
classical
by_cases h : ∀ x ∈ s, x = (0 : α)
· use replicate (card s) 1
rw [map_replicate, eq_replicate, mul_one, s.gcd_eq_zero_iff.2 h, ← nsmul_singleton,
← gcd_dedup, dedup_nsmul (card_pos.2 hs).ne', dedup_singleton, gcd_singleton]
exact ⟨⟨rfl, h⟩, normalize_one⟩
· choose f hf using @gcd... | 0 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analysis.special_functions.japanese_bracket from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
op... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 79 | 95 | theorem finite_integral_rpow_sub_one_pow_aux {r : ℝ} (n : ℕ) (hnr : (n : ℝ) < r) :
(∫⁻ x : ℝ in Ioc 0 1, ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n)) < ∞ := by |
have hr : 0 < r := lt_of_le_of_lt n.cast_nonneg hnr
have h_int : ∀ x : ℝ, x ∈ Ioc (0 : ℝ) 1 →
ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n) ≤ ENNReal.ofReal (x ^ (-(r⁻¹ * n))) := fun x hx ↦ by
apply ENNReal.ofReal_le_ofReal
rw [← neg_mul, rpow_mul hx.1.le, rpow_natCast]
refine pow_le_pow_left ?_ (by simp... | 0 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Pi
#align_import data.finset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Finset
open Multiset
section Pi
variable {α : Type*}
def Pi.empty (β : α → Sort*) (a : α) (h : a ∈ (∅ : Finset α)) : β a :=... | Mathlib/Data/Finset/Pi.lean | 96 | 112 | theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
(ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) := by |
apply eq_of_veq
rw [← (pi (insert a s) t).2.dedup]
refine
(fun s' (h : s' = a ::ₘ s.1) =>
(?_ :
dedup (Multiset.pi s' fun a => (t a).1) =
dedup
((t a).1.bind fun b =>
dedup <|
(Multiset.pi s.1 fun a : α => (t a).val).map fun f a' h... | 0 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 86 | 92 | theorem bernoulli_eval_one (n : ℕ) : (bernoulli n).eval 1 = bernoulli' n := by |
simp only [bernoulli, eval_finset_sum]
simp only [← succ_eq_add_one, sum_range_succ, mul_one, cast_one, choose_self,
(_root_.bernoulli _).mul_comm, sum_bernoulli, one_pow, mul_one, eval_C, eval_monomial, one_mul]
by_cases h : n = 1
· norm_num [h]
· simp [h, bernoulli_eq_bernoulli'_of_ne_one h]
| 0 |
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.RingTheory.IntegralDomain
#align_import field_theory.primitive_element from "leanprover-community/mathlib"@"df76f43357840485b9d04ed5dee5ab115d420e87"
noncomputable section
open scoped Classical Polynomial
open FiniteDimensional Polynomial In... | Mathlib/FieldTheory/PrimitiveElement.lean | 246 | 275 | theorem isAlgebraic_of_adjoin_eq_adjoin {α : E} {m n : ℕ} (hneq : m ≠ n)
(heq : F⟮α ^ m⟯ = F⟮α ^ n⟯) : IsAlgebraic F α := by |
wlog hmn : m < n
· exact this F E hneq.symm heq.symm (hneq.lt_or_lt.resolve_left hmn)
by_cases hm : m = 0
· rw [hm] at heq hmn
simp only [pow_zero, adjoin_one] at heq
obtain ⟨y, h⟩ := mem_bot.1 (heq.symm ▸ mem_adjoin_simple_self F (α ^ n))
refine ⟨X ^ n - C y, X_pow_sub_C_ne_zero hmn y, ?_⟩
sim... | 0 |
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 | 83 | 91 | theorem orthogonalFamily_eigenspaces :
OrthogonalFamily 𝕜 (fun μ => eigenspace T μ) fun μ => (eigenspace T μ).subtypeₗᵢ := by |
rintro μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩
by_cases hv' : v = 0
· simp [hv']
have H := hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector ⟨hv, hv'⟩)
rw [mem_eigenspace_iff] at hv hw
refine Or.resolve_left ?_ hμν.symm
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
| 0 |
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine
open Set
universe u₁ u₂ u₃ u₄
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V ... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 174 | 179 | theorem coord_apply_ne (h : i ≠ j) : b.coord i (b j) = 0 := by |
-- Porting note:
-- in mathlib3 we didn't need to given the `fun j => j ≠ i` argument to `Subtype.coe_mk`,
-- but I don't think we can complain: this proof was over-golfed.
rw [coord, AffineMap.coe_mk, ← @Subtype.coe_mk _ (fun j => j ≠ i) j h.symm, ← b.basisOf_apply,
Basis.sumCoords_self_apply, sub_self]
| 0 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.Spectral.Hom
import Mathlib.AlgebraicGeometry.Limits
#align_import algebraic_geometry.morphisms.quasi_compact from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
noncomputable section
open CategoryTheory CategoryT... | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | 114 | 120 | theorem quasiCompact_iff_affineProperty :
QuasiCompact f ↔ targetAffineLocally QuasiCompact.affineProperty f := by |
rw [quasiCompact_iff_forall_affine]
trans ∀ U : Y.affineOpens, IsCompact (f.1.base ⁻¹' (U : Set Y.carrier))
· exact ⟨fun h U => h U U.prop, fun h U hU => h ⟨U, hU⟩⟩
apply forall_congr'
exact fun _ => isCompact_iff_compactSpace
| 0 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set α)) :
{ t | GenerateMeasurable s t } =
⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i := by |
ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
· inhabit ω₁
induction' ht with u hu u _ IH f _ IH
· exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
· exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
· rcases mem_iUnion.1 IH with ⟨i, hi⟩
obtain ⟨j, hj⟩ := ex... | 0 |
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 | 39 | 44 | theorem scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : A[X]} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) : aeval (algebraMap A S r) (scaleRoots p s) = 0 := by |
convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr
-- Porting note: added
funext
rw [aeval_def, mk'_spec' _ r s]
| 0 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 115 | 116 | theorem addHaarMeasure_eq_volume : addHaarMeasure Icc01 = volume := by |
convert (addHaarMeasure_unique volume Icc01).symm; simp [Icc01]
| 0 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 126 | 144 | theorem ClassGroup.mk_eq_mk_of_coe_ideal {I J : (FractionalIdeal R⁰ <| FractionRing R)ˣ}
{I' J' : Ideal R} (hI : (I : FractionalIdeal R⁰ <| FractionRing R) = I')
(hJ : (J : FractionalIdeal R⁰ <| FractionRing R) = J') :
ClassGroup.mk I = ClassGroup.mk J ↔
∃ x y : R, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' ... |
rw [ClassGroup.mk_eq_mk]
constructor
· rintro ⟨x, rfl⟩
rw [Units.val_mul, hI, coe_toPrincipalIdeal, mul_comm,
spanSingleton_mul_coeIdeal_eq_coeIdeal] at hJ
exact ⟨_, _, sec_fst_ne_zero (R := R) le_rfl x.ne_zero,
sec_snd_ne_zero (R := R) le_rfl (x : FractionRing R), hJ⟩
· rintro ⟨x, y, hx, h... | 0 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 136 | 141 | theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
(H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ := by |
classical
have := card_subgroup_dvd_card H
rwa [Nat.card_eq_fintype_card (α := G), Nat.dvd_prime hp.1, ← Nat.card_eq_fintype_card,
← eq_bot_iff_card, card_eq_iff_eq_top] at this
| 0 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 114 | 119 | theorem isPerfectMatching_iff : M.IsPerfectMatching ↔ ∀ v, ∃! w, M.Adj v w := by |
refine ⟨?_, fun hm => ⟨fun v _ => hm v, fun v => ?_⟩⟩
· rintro ⟨hm, hs⟩ v
exact hm (hs v)
· obtain ⟨w, hw, -⟩ := hm v
exact M.edge_vert hw
| 0 |
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 128 | 130 | theorem const_add_term {α} [AddCommMonoid α] (k n x a a') (h : k + a = a') :
k + @term α _ n x a = term n x a' := by |
simp [h.symm, term, add_comm, add_assoc]
| 0 |
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 | 215 | 219 | theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
μ U = ⨆ (K) (_ : K ⊆ U) (_ : p K), μ K := by |
refine
le_antisymm (le_of_forall_lt fun r hr => ?_) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
simpa only [lt_iSup_iff, exists_prop] using H hU r hr
| 0 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 31 | 34 | theorem Algebra.trace_complex_apply (z : ℂ) : Algebra.trace ℝ ℂ z = 2 * z.re := by |
rw [Algebra.trace_eq_matrix_trace Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.trace_fin_two]
exact (two_mul _).symm
| 0 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
nam... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 98 | 106 | theorem dart_card_eq_twice_card_edges : Fintype.card G.Dart = 2 * G.edgeFinset.card := by |
classical
rw [← card_univ]
rw [@card_eq_sum_card_fiberwise _ _ _ Dart.edge _ G.edgeFinset fun d _h =>
by rw [mem_edgeFinset]; apply Dart.edge_mem]
rw [← mul_comm, sum_const_nat]
intro e h
apply G.dart_edge_fiber_card e
rwa [← mem_edgeFinset]
| 0 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import algebra.char_p.local_ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/Algebra/CharP/LocalRing.lean | 25 | 67 | theorem charP_zero_or_prime_power (R : Type*) [CommRing R] [LocalRing R] (q : ℕ)
[char_R_q : CharP R q] : q = 0 ∨ IsPrimePow q := by |
-- Assume `q := char(R)` is not zero.
apply or_iff_not_imp_left.2
intro q_pos
let K := LocalRing.ResidueField R
haveI RM_char := ringChar.charP K
let r := ringChar K
let n := q.factorization r
-- `r := char(R/m)` is either prime or zero:
cases' CharP.char_is_prime_or_zero K r with r_prime r_zero
· ... | 0 |
import Mathlib.Topology.VectorBundle.Basic
#align_import topology.vector_bundle.hom from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
noncomputable section
open scoped Bundle
open Bundle Set ContinuousLinearMap
variable {𝕜₁ : Type*} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type*} [Non... | Mathlib/Topology/VectorBundle/Hom.lean | 92 | 112 | theorem continuousOn_continuousLinearMapCoordChange [VectorBundle 𝕜₁ F₁ E₁] [VectorBundle 𝕜₂ F₂ E₂]
[MemTrivializationAtlas e₁] [MemTrivializationAtlas e₁'] [MemTrivializationAtlas e₂]
[MemTrivializationAtlas e₂'] :
ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂')
(e₁.baseSet ∩ e₂.baseS... |
have h₁ := (compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂)).continuous
have h₂ := (ContinuousLinearMap.flip (compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ)).continuous
have h₃ := continuousOn_coordChange 𝕜₁ e₁' e₁
have h₄ := continuousOn_coordChange 𝕜₂ e₂ e₂'
refine ((h₁.comp_continuousOn (h₄.mono ?_)).clm_comp (h₂.comp_continuo... | 0 |
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Tactic.Nontriviality
#align_import algebra.group_with_zero.commute from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
#align_import algebra.group_with_zero.power from "leanprover-communi... | Mathlib/Algebra/GroupWithZero/Commute.lean | 27 | 34 | theorem mul_inverse_rev' {a b : M₀} (h : Commute a b) :
inverse (a * b) = inverse b * inverse a := by |
by_cases hab : IsUnit (a * b)
· obtain ⟨⟨a, rfl⟩, b, rfl⟩ := h.isUnit_mul_iff.mp hab
rw [← Units.val_mul, inverse_unit, inverse_unit, inverse_unit, ← Units.val_mul, mul_inv_rev]
obtain ha | hb := not_and_or.mp (mt h.isUnit_mul_iff.mpr hab)
· rw [inverse_non_unit _ hab, inverse_non_unit _ ha, mul_zero]
· ... | 0 |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 45 | 58 | theorem convergents'Aux_stable_step_of_terminated {s : Stream'.Seq <| Pair K}
(terminated_at_n : s.TerminatedAt n) : convergents'Aux s (n + 1) = convergents'Aux s n := by |
change s.get? n = none at terminated_at_n
induction n generalizing s with
| zero => simp only [convergents'Aux, terminated_at_n, Stream'.Seq.head]
| succ n IH =>
cases s_head_eq : s.head with
| none => simp only [convergents'Aux, s_head_eq]
| some gp_head =>
have : s.tail.TerminatedAt n := by... | 0 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 84 | 89 | theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ι ℝ F) :
MeasurePreserving b.measurableEquiv volume volume := by |
convert (b.measurableEquiv.symm.measurable.measurePreserving _).symm
rw [← (EuclideanSpace.basisFun ι ℝ).addHaar_eq_volume]
erw [MeasurableEquiv.coe_toEquiv_symm, Basis.map_addHaar _ b.repr.symm.toContinuousLinearEquiv]
exact b.addHaar_eq_volume.symm
| 0 |
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.SchwartzSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5... | Mathlib/Analysis/Fourier/PoissonSummation.lean | 131 | 157 | theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
(fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by |
-- First establish an explicit estimate on decay of inverse powers.
-- This is logically independent of the rest of the proof, but of no mathematical interest in
-- itself, so it is proved in-line rather than being formulated as a separate lemma.
have claim : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y →
... | 0 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Tactic.LiftLets
#align_import linear_algebra.clifford_algebra.even_equiv from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36d... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 82 | 86 | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by |
refine neg_eq_of_add_eq_zero_right ((ι_mul_ι_add_swap _ _).trans ?_)
dsimp [QuadraticForm.polar]
simp only [add_zero, mul_zero, mul_one, zero_add, neg_zero, QuadraticForm.map_zero,
add_sub_cancel_right, sub_self, map_zero, zero_sub]
| 0 |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
theorem bir... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 57 | 61 | theorem map_birkhoffAverage (S : Type*) {F N : Type*}
[DivisionSemiring S] [AddCommMonoid N] [Module S N] [FunLike F M N]
[AddMonoidHomClass F M N] (g' : F) (f : α → α) (g : α → M) (n : ℕ) (x : α) :
g' (birkhoffAverage R f g n x) = birkhoffAverage S f (g' ∘ g) n x := by |
simp only [birkhoffAverage, map_inv_natCast_smul g' R S, map_birkhoffSum]
| 0 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 68 | 75 | theorem PiToModule.fromEnd_injective (hb : Submodule.span R (Set.range b) = ⊤) :
Function.Injective (PiToModule.fromEnd R b) := by |
intro x y e
ext m
obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.total R R b) := by
rw [(Fintype.range_total R b).trans hb]
exact Submodule.mem_top
exact (LinearMap.congr_fun e m : _)
| 0 |
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.cocompact_map from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
universe u v w
open Filter Set
structure CocompactMap (α : Type u) (β : Type v) [TopologicalSpace α] [TopologicalSpace β] e... | Mathlib/Topology/ContinuousFunction/CocompactMap.lean | 185 | 195 | theorem isCompact_preimage [T2Space β] (f : CocompactMap α β) ⦃s : Set β⦄ (hs : IsCompact s) :
IsCompact (f ⁻¹' s) := by |
obtain ⟨t, ht, hts⟩ :=
mem_cocompact'.mp
(by
simpa only [preimage_image_preimage, preimage_compl] using
mem_map.mp
(cocompact_tendsto f <|
mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩))
exact
ht.of_isClosed_subset (hs.isClos... | 0 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.lagrange from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Polynomial
section PolynomialDetermination
namespace Poly... | Mathlib/LinearAlgebra/Lagrange.lean | 63 | 67 | theorem eq_of_degrees_lt_of_eval_finset_eq (degree_f_lt : f.degree < s.card)
(degree_g_lt : g.degree < s.card) (eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g := by |
rw [← mem_degreeLT] at degree_f_lt degree_g_lt
refine eq_of_degree_sub_lt_of_eval_finset_eq _ ?_ eval_fg
rw [← mem_degreeLT]; exact Submodule.sub_mem _ degree_f_lt degree_g_lt
| 0 |
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import category_theory.sites.induced_topology from "leanprover-community/mathlib"@"ba43124c37cfe0009bbfc57505f9503ae0e8c1af"
namespace CategoryTheory
universe v u
open Limits Opposite Presieve
section
variable {C : Type*} [Category C] {D : Type*} [Catego... | Mathlib/CategoryTheory/Sites/InducedTopology.lean | 59 | 65 | theorem pushforward_cover_iff_cover_pullback {X : C} (S : Sieve X) :
K _ (S.functorPushforward G) ↔ ∃ T : K (G.obj X), T.val.functorPullback G = S := by |
constructor
· intro hS
exact ⟨⟨_, hS⟩, (Sieve.fullyFaithfulFunctorGaloisCoinsertion G X).u_l_eq S⟩
· rintro ⟨T, rfl⟩
exact Hld T
| 0 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 87 | 89 | theorem coeff_coe_powerSeries (x : PowerSeries R) (n : ℕ) :
HahnSeries.coeff (x : LaurentSeries R) n = PowerSeries.coeff R n x := by |
rw [ofPowerSeries_apply_coeff]
| 0 |
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 96 | 107 | theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) := by |
/- We have already proved the theorem around the basepoint of the orbit, in
`smul_singleton_mem_nhds_of_sigmaCompact`. The general statement follows around an arbitrary
point by changing basepoints. -/
simp_rw [isOpenMap_iff_nhds_le, Filter.le_map_iff]
intro g U hU
have : (· • x) = (· • (g • x)) ∘ (· * g⁻¹... | 0 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 160 | 162 | theorem star_exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (x : 𝔸) :
star (exp 𝕂 x) = exp 𝕂 (star x) := by |
simp_rw [exp_eq_tsum, ← star_pow, ← star_inv_natCast_smul, ← tsum_star]
| 0 |
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 30 | 34 | theorem rtendsto'_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
RTendsto' r l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → r.preimage s ∈ l := by |
rw [rtendsto'_def]
apply all_mem_nhds_filter
apply Rel.preimage_mono
| 0 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 34 | 39 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
tensorHom ℬ f g ≫ (Limits.BinaryFan.braiding (ℬ Y Y').isLimit (ℬ Y' Y).isLimit).hom =
(Limits.BinaryFan.braiding (ℬ X X').isLimit (ℬ X' X).isLimit).hom ≫ tensorHom ℬ g f := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| 0 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Topology.TietzeExtension
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analysis.NormedSpace.RCLike
universe u u₁ v w
-- this is not an instance because Lean cannot determine `𝕜`.
theorem TietzeExtension.o... | Mathlib/Analysis/Complex/Tietze.lean | 105 | 118 | theorem exists_norm_eq_restrict_eq (f : s →ᵇ E) :
∃ g : X →ᵇ E, ‖g‖ = ‖f‖ ∧ g.restrict s = f := by |
by_cases hf : ‖f‖ = 0; · exact ⟨0, by aesop⟩
have := Metric.instTietzeExtensionClosedBall.{u, v} 𝕜 (0 : E) (by aesop : 0 < ‖f‖)
have hf' x : f x ∈ Metric.closedBall 0 ‖f‖ := by simpa using f.norm_coe_le_norm x
obtain ⟨g, hg_mem, hg⟩ := (f : C(s, E)).exists_forall_mem_restrict_eq hs hf'
simp only [Metric.mem... | 0 |
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 | 134 | 143 | theorem TietzeExtension.of_retract {Y : Type v} {Z : Type w} [TopologicalSpace Y]
[TopologicalSpace Z] [TietzeExtension.{u, w} Z] (ι : C(Y, Z)) (r : C(Z, Y))
(h : r.comp ι = .id Y) : TietzeExtension.{u, v} Y where
exists_restrict_eq' s hs f := by |
obtain ⟨g, hg⟩ := (ι.comp f).exists_restrict_eq hs
use r.comp g
ext1 x
have := congr(r.comp $(hg))
rw [← r.comp_assoc ι, h, f.id_comp] at this
congrm($this x)
| 0 |
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 | 96 | 99 | theorem gcd_sub_self_left {m n : ℕ} (h : m ≤ n) : gcd (n - m) m = gcd n m := by |
calc
gcd (n - m) m = gcd (n - m + m) m := by rw [← gcd_add_self_left (n - m) m]
_ = gcd n m := by rw [Nat.sub_add_cancel h]
| 0 |
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
open MeasureTheory ProbabilityTheory
open scoped MeasureTheory ENNReal NNReal
namesp... | Mathlib/Probability/Kernel/WithDensity.lean | 135 | 144 | theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
(f : α → β → ℝ≥0∞) :
@withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
kernel.sum fun i => withDensity (κ i) f := by |
by_cases hf : Measurable (Function.uncurry f)
· ext1 a
simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,
withDensity_sum (fun n => κ n a) (f a)]
· simp_rw [withDensity_of_not_measurable _ hf]
exact sum_zero.symm
| 0 |
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 | 251 | 253 | theorem ad_conj {R : Type*} [Rack R] (x y : R) : act' (x ◃ y) = act' x * act' y * (act' x)⁻¹ := by |
rw [eq_mul_inv_iff_mul_eq]; ext z
apply self_distrib.symm
| 0 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {𝕜 E F : Type*}
[NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 106 | 109 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar (h : Function.Surjective L) :
∃ (c : ℝ≥0∞), 0 < c ∧ μ.map L = c • ν := by |
rcases L.exists_map_addHaar_eq_smul_addHaar' μ ν h with ⟨c, c_pos, -, hc⟩
exact ⟨_, by simp [c_pos, NeZero.ne addHaar], hc⟩
| 0 |
import Mathlib.Algebra.Group.Fin
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β m n R : Type*}
namespace Matrix
open Function
open Matrix
def circulant [Sub n] (v : n → α)... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 126 | 132 | theorem circulant_mul [Semiring α] [Fintype n] [AddGroup n] (v w : n → α) :
circulant v * circulant w = circulant (circulant v *ᵥ w) := by |
ext i j
simp only [mul_apply, mulVec, circulant_apply, dotProduct]
refine Fintype.sum_equiv (Equiv.subRight j) _ _ ?_
intro x
simp only [Equiv.subRight_apply, sub_sub_sub_cancel_right]
| 0 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 122 | 126 | theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) := by |
intro x y h
have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure
rw [h] at this
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
| 0 |
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 77 | 86 | theorem hallMatchingsOn.nonempty {ι : Type u} {α : Type v} [DecidableEq α] (t : ι → Finset α)
(h : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) (ι' : Finset ι) :
Nonempty (hallMatchingsOn t ι') := by |
classical
refine ⟨Classical.indefiniteDescription _ ?_⟩
apply (all_card_le_biUnion_card_iff_existsInjective' fun i : ι' => t i).mp
intro s'
convert h (s'.image (↑)) using 1
· simp only [card_image_of_injective s' Subtype.coe_injective]
· rw [image_biUnion]
| 0 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 138 | 148 | theorem condCount_inter (hs : s.Finite) :
condCount s (t ∩ u) = condCount (s ∩ t) u * condCount s t := by |
by_cases hst : s ∩ t = ∅
· rw [hst, condCount_empty_meas, Measure.coe_zero, Pi.zero_apply, zero_mul,
condCount_eq_zero_iff hs, ← Set.inter_assoc, hst, Set.empty_inter]
rw [condCount, condCount, cond_apply _ hs.measurableSet, cond_apply _ hs.measurableSet,
cond_apply _ (hs.inter_of_left _).measurableSet... | 0 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Int.Sqrt
#align_import data.rat.sqrt from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
namespace Rat
-- @[pp_nodot] porting note: unknown attribute
def sqrt... | Mathlib/Data/Rat/Sqrt.lean | 30 | 31 | theorem sqrt_eq (q : ℚ) : Rat.sqrt (q * q) = |q| := by |
rw [sqrt, mul_self_num, mul_self_den, Int.sqrt_eq, Nat.sqrt_eq, abs_def, divInt_ofNat]
| 0 |
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 64 | 68 | theorem hasConstantSpeedOnWith_of_subsingleton (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton)
(l : ℝ≥0) : HasConstantSpeedOnWith f s l := by |
rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero]
| 0 |
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 133 | 134 | theorem unzip_zip_right {l₁ : List α} {l₂ : List β} (h : length l₂ ≤ length l₁) :
(unzip (zip l₁ l₂)).2 = l₂ := by | rw [← zip_swap, unzip_swap]; exact unzip_zip_left h
| 0 |
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.integrally_closed from "leanprover-community/mathlib"@"d35b4ff446f1421bd551fafa4b8efd98ac3ac408"
open scoped nonZeroDivisors Polynomial
open Polynomial
abbrev IsIntegrallyClosedIn (R A : Type*) [... | Mathlib/RingTheory/IntegrallyClosed.lean | 110 | 120 | theorem isIntegrallyClosedIn_iff {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
IsIntegrallyClosedIn R A ↔
Function.Injective (algebraMap R A) ∧
∀ {x : A}, IsIntegral R x → ∃ y, algebraMap R A y = x := by |
constructor
· rintro ⟨_, cl⟩
aesop
· rintro ⟨inj, cl⟩
refine ⟨inj, by aesop, ?_⟩
rintro ⟨y, rfl⟩
apply isIntegral_algebraMap
| 0 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 130 | 153 | theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
(δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
Tendsto (fun n => thickenedIndicatorAux (δseq n) E) atTop
(𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0∞))) := by |
rw [tendsto_pi_nhds]
intro x
by_cases x_mem_closure : x ∈ closure E
· simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds
· rw [show (closure E)... | 0 |
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.AdaptationNote
open Metric Function AffineMap Set AffineSubspace
open scoped Topology RealInnerProductSpace
variable {E F : Type*} [NormedAddCommGrou... | Mathlib/Geometry/Euclidean/Inversion/Calculus.lean | 87 | 108 | theorem hasFDerivAt_inversion (hx : x ≠ c) :
HasFDerivAt (inversion c R)
((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x := by |
rcases add_left_surjective c x with ⟨x, rfl⟩
have : HasFDerivAt (inversion c R) (?_ : F →L[ℝ] F) (c + x) := by
#adaptation_note /-- nightly-2024-03-16: simp was
simp (config := { unfoldPartialApp := true }) only [inversion] -/
simp only [inversion_def]
simp_rw [dist_eq_norm, div_pow, div_eq_mul_inv... | 0 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed.group.controlled_closure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset
open Topology
variable {G : Type*} [NormedAddCommGroup G] [CompleteSpace... | Mathlib/Analysis/Normed/Group/ControlledClosure.lean | 32 | 106 | theorem controlled_closure_of_complete {f : NormedAddGroupHom G H} {K : AddSubgroup H} {C ε : ℝ}
(hC : 0 < C) (hε : 0 < ε) (hyp : f.SurjectiveOnWith K C) :
f.SurjectiveOnWith K.topologicalClosure (C + ε) := by |
rintro (h : H) (h_in : h ∈ K.topologicalClosure)
-- We first get rid of the easy case where `h = 0`.
by_cases hyp_h : h = 0
· rw [hyp_h]
use 0
simp
/- The desired preimage will be constructed as the sum of a series. Convergence of
the series will be guaranteed by completeness of `G`. We first wri... | 0 |
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower
#align_import linear_algebra.tensor_algebra.to_tensor_power from "leanprover-community/mathlib"@"d97a0c9f7a7efe6d76d652c5a6b7c9c634b70e0a"
suppress_compilation
open scoped DirectSum TensorProduct
variable {R M : Type*} [Com... | Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 68 | 72 | theorem toTensorAlgebra_galgebra_toFun (r : R) :
TensorPower.toTensorAlgebra (DirectSum.GAlgebra.toFun (R := R) (A := fun n => ⨂[R]^n M) r) =
algebraMap _ _ r := by |
rw [TensorPower.galgebra_toFun_def, TensorPower.algebraMap₀_eq_smul_one, LinearMap.map_smul,
TensorPower.toTensorAlgebra_gOne, Algebra.algebraMap_eq_smul_one]
| 0 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 56 | 79 | theorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by |
dsimp only [reflTransSymmAux]
split_ifs
· constructor
· apply mul_nonneg
· apply mul_nonneg
· unit_interval
· norm_num
· unit_interval
· rw [mul_assoc]
apply mul_le_one
· unit_interval
· apply mul_nonneg
· norm_num
· unit_interval
· lina... | 0 |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Tactic.ComputeDegree
#align_import linear_algebra.matrix.polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 89 | 102 | theorem leadingCoeff_det_X_one_add_C (A : Matrix n n α) :
leadingCoeff (det ((X : α[X]) • (1 : Matrix n n α[X]) + A.map C)) = 1 := by |
cases subsingleton_or_nontrivial α
· simp [eq_iff_true_of_subsingleton]
rw [← @det_one n, ← coeff_det_X_add_C_card _ A, leadingCoeff]
simp only [Matrix.map_one, C_eq_zero, RingHom.map_one]
rcases (natDegree_det_X_add_C_le 1 A).eq_or_lt with h | h
· simp only [RingHom.map_one, Matrix.map_one, C_eq_zero] at ... | 0 |
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
#align_import analysis.convex.specific_functions.basic from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open Real Set NNReal
theorem strictConvexOn_exp : St... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 138 | 163 | theorem rpow_one_add_lt_one_add_mul_self {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠ 0) {p : ℝ} (hp1 : 0 < p)
(hp2 : p < 1) : (1 + s) ^ p < 1 + p * s := by |
rcases eq_or_lt_of_le hs with rfl | hs
· rwa [add_right_neg, zero_rpow hp1.ne', mul_neg_one, lt_add_neg_iff_add_lt, zero_add]
have hs1 : 0 < 1 + s := neg_lt_iff_pos_add'.mp hs
have hs2 : 0 < 1 + p * s := by
rw [← neg_lt_iff_pos_add']
rcases lt_or_gt_of_ne hs' with h | h
· exact hs.trans (lt_mul_of_... | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.