Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
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import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 95 | 101 | theorem comp_differentiableWithinAt_iff {f : G → E} {s : Set G} {x : G} :
DifferentiableWithinAt 𝕜 (iso ∘ f) s x ↔ DifferentiableWithinAt 𝕜 f s x := by
refine |
refine
⟨fun H => ?_, fun H => iso.differentiable.differentiableAt.comp_differentiableWithinAt x H⟩
have : DifferentiableWithinAt 𝕜 (iso.symm ∘ iso ∘ f) s x :=
iso.symm.differentiable.differentiableAt.comp_differentiableWithinAt x H
rwa [← Function.comp.assoc iso.symm iso f, iso.symm_comp_self] at this
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.compare_exp from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Asympto... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 127 | 151 | theorem isLittleO_log_abs_re (hl : IsExpCmpFilter l) : (fun z => Real.log (abs z)) =o[l] re :=
calc
(fun z => Real.log (abs z)) =O[l] fun z => Real.log (√2) + Real.log (max z.re |z.im|) :=
IsBigO.of_bound 1 <|
(hl.tendsto_re.eventually_ge_atTop 1).mono fun z hz => by
have h2 : 0 < √2 := by... | simp
have hz' : 1 ≤ abs z := hz.trans (re_le_abs z)
have hm₀ : 0 < max z.re |z.im| := lt_max_iff.2 (Or.inl <| one_pos.trans_le hz)
rw [one_mul, Real.norm_eq_abs, _root_.abs_of_nonneg (Real.log_nonneg hz')]
refine le_trans ?_ (le_abs_self _)
rw [← Real.log_mul, Real.log... | true |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 45 | 48 | theorem diagonal (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((𝟙 X, g) : (X, Y) ⟶ (X, Y')) ≫ F.map ((f, 𝟙 Y') : (X, Y') ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by |
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
| true |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 113 | 115 | theorem hasFiniteIntegral_iff_norm (f : α → β) :
HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) < ∞ := by |
simp only [HasFiniteIntegral, ofReal_norm_eq_coe_nnnorm]
| true |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 118 | 129 | theorem T_neg (n : ℤ) : T R (-n) = T R n := by
induction n using Polynomial.Chebyshev.induct with |
induction n using Polynomial.Chebyshev.induct with
| zero => rfl
| one => show 2 * X * 1 - X = X; ring
| add_two n ih1 ih2 =>
have h₁ := T_add_two R n
have h₂ := T_sub_two R (-n)
linear_combination (norm := ring_nf) (2 * (X:R[X])) * ih1 - ih2 - h₁ + h₂
| neg_add_one n ih1 ih2 =>
have h₁ := T_... | true |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 328 | 329 | theorem imageSubobject_arrow_comp : factorThruImageSubobject f ≫ (imageSubobject f).arrow = f := by |
simp [factorThruImageSubobject, imageSubobject_arrow]
| true |
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.PSeriesComplex
#align_import number_theory.zeta_function from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
o... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 179 | 189 | theorem completedZeta_eq_tsum_of_one_lt_re {s : ℂ} (hs : 1 < re s) :
completedRiemannZeta s =
(π : ℂ) ^ (-s / 2) * Gamma (s / 2) * ∑' n : ℕ, 1 / (n : ℂ) ^ s := by
have := (hasSum_nat_completedCosZeta 0 hs).tsum_eq.symm |
have := (hasSum_nat_completedCosZeta 0 hs).tsum_eq.symm
simp only [QuotientAddGroup.mk_zero, completedCosZeta_zero] at this
simp only [this, Gammaℝ_def, mul_zero, zero_mul, Real.cos_zero, ofReal_one, mul_one, mul_one_div,
← tsum_mul_left]
congr 1 with n
split_ifs with h
· simp only [h, Nat.cast_zero, z... | true |
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 111 | 111 | theorem moveLeft_nim {o : Ordinal} (i) : (nim o).moveLeft (toLeftMovesNim i) = nim i := by | simp
| true |
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
variable {F : Type*} [Field F] (S : Set F)
structure IsSubfield extends IsSubring S : Prop where
inv_mem : ∀ {x : F}, x ∈ S → x⁻... | Mathlib/Deprecated/Subfield.lean | 134 | 137 | theorem closure_subset {T : Set F} (hT : IsSubfield T) (H : S ⊆ T) : closure S ⊆ T := by
rintro _ ⟨p, hp, q, hq, hq0, rfl⟩ |
rintro _ ⟨p, hp, q, hq, hq0, rfl⟩
exact hT.div_mem (Ring.closure_subset hT.toIsSubring H hp)
(Ring.closure_subset hT.toIsSubring H hq)
| true |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 95 | 104 | theorem WithTop.coe_sInf' [InfSet α] {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) :
↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithTop α) := by
obtain ⟨x, hx⟩ := hs |
obtain ⟨x, hx⟩ := hs
change _ = ite _ _ _
split_ifs with h
· rcases h with h1 | h2
· cases h1 (mem_image_of_mem _ hx)
· exact (h2 (Monotone.map_bddBelow coe_mono h's)).elim
· rw [preimage_image_eq]
exact Option.some_injective _
| true |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
set_option autoImplicit true
open Set
namespace OrderIso
variable [Preorder α] [Preorder β] (f : α ≃o β)
theorem upperBounds_image {... | Mathlib/Order/Bounds/OrderIso.lean | 55 | 56 | theorem isLUB_preimage {s : Set β} {x : α} : IsLUB (f ⁻¹' s) x ↔ IsLUB s (f x) := by |
rw [← f.symm_symm, ← image_eq_preimage, isLUB_image]
| true |
import Mathlib.Algebra.Lie.Abelian
#align_import algebra.lie.tensor_product from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
universe u v w w₁ w₂ w₃
variable {R : Type u} [CommRing R]
open LieModule
namespace TensorProduct
open scoped TensorProduct
namespace... | Mathlib/Algebra/Lie/TensorProduct.lean | 125 | 127 | theorem liftLie_apply (f : M →ₗ⁅R,L⁆ N →ₗ[R] P) (m : M) (n : N) :
liftLie R L M N P f (m ⊗ₜ n) = f m n := by |
simp only [coe_liftLie_eq_lift_coe, LieModuleHom.coe_toLinearMap, lift_apply]
| true |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 83 | 90 | theorem measurable_lineDeriv_uncurry [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun (p : E × E) ↦ lineDeriv 𝕜 f p.1 p.2) := by
borelize 𝕜 |
borelize 𝕜
let g : (E × E) → 𝕜 → F := fun p t ↦ f (p.1 + t • p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
exact (measurable_deriv_with_param this).comp measurable_prod_mk_right
| true |
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 106 | 107 | theorem involute_eq_id :
(involute : CliffordAlgebra (0 : QuadraticForm R Unit) →ₐ[R] _) = AlgHom.id R _ := by | ext; simp
| true |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 130 | 143 | theorem FiniteField.isSquare_odd_prime_iff (hF : ringChar F ≠ 2) {p : ℕ} [Fact p.Prime]
(hp : p ≠ 2) :
IsSquare (p : F) ↔ quadraticChar (ZMod p) (χ₄ (Fintype.card F) * Fintype.card F) ≠ -1 := by
classical |
classical
by_cases hFp : ringChar F = p
· rw [show (p : F) = 0 by rw [← hFp]; exact ringChar.Nat.cast_ringChar]
simp only [isSquare_zero, Ne, true_iff_iff, map_mul]
obtain ⟨n, _, hc⟩ := FiniteField.card F (ringChar F)
have hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n ... | true |
import Mathlib.Data.Set.Basic
open Function
universe u v
namespace Set
section Nontrivial
variable {α : Type u} {a : α} {s t : Set α}
protected def Nontrivial (s : Set α) : Prop :=
∃ x ∈ s, ∃ y ∈ s, x ≠ y
#align set.nontrivial Set.Nontrivial
theorem nontrivial_of_mem_mem_ne {x y} (hx : x ∈ s) (hy : y ∈... | Mathlib/Data/Set/Subsingleton.lean | 194 | 198 | theorem Nontrivial.exists_ne (hs : s.Nontrivial) (z) : ∃ x ∈ s, x ≠ z := by
by_contra! H |
by_contra! H
rcases hs with ⟨x, hx, y, hy, hxy⟩
rw [H x hx, H y hy] at hxy
exact hxy rfl
| true |
import Mathlib.GroupTheory.Perm.Cycle.Basic
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {ι α β : Type*}
namespace Equiv.Perm
section Generation
variable [Finite β]
open Subgroup
theorem closure... | Mathlib/GroupTheory/Perm/Closure.lean | 96 | 108 | theorem closure_cycle_coprime_swap {n : ℕ} {σ : Perm α} (h0 : Nat.Coprime n (Fintype.card α))
(h1 : IsCycle σ) (h2 : σ.support = Finset.univ) (x : α) :
closure ({σ, swap x ((σ ^ n) x)} : Set (Perm α)) = ⊤ := by
rw [← Finset.card_univ, ← h2, ← h1.orderOf] at h0 |
rw [← Finset.card_univ, ← h2, ← h1.orderOf] at h0
cases' exists_pow_eq_self_of_coprime h0 with m hm
have h2' : (σ ^ n).support = ⊤ := Eq.trans (support_pow_coprime h0) h2
have h1' : IsCycle ((σ ^ n) ^ (m : ℤ)) := by rwa [← hm] at h1
replace h1' : IsCycle (σ ^ n) :=
h1'.of_pow (le_trans (support_pow_le σ ... | true |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 149 | 151 | theorem Ioc_disjoint_Ioc : Disjoint (Ioc a₁ a₂) (Ioc b₁ b₂) ↔ min a₂ b₂ ≤ max a₁ b₁ := by
have h : _ ↔ min (toDual a₁) (toDual b₁) ≤ max (toDual a₂) (toDual b₂) := Ico_disjoint_Ico |
have h : _ ↔ min (toDual a₁) (toDual b₁) ≤ max (toDual a₂) (toDual b₂) := Ico_disjoint_Ico
simpa only [dual_Ico] using h
| true |
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 186 | 186 | theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by | simp [le_refl]
| true |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 133 | 136 | theorem HasDerivAt.scomp_hasDerivWithinAt_of_eq (hg : HasDerivAt g₁ g₁' y)
(hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x := by |
rw [hy] at hg; exact hg.scomp_hasDerivWithinAt x hh
| true |
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 46 | 48 | theorem length (n m : ℕ) : length (Ico n m) = m - n := by
dsimp [Ico] |
dsimp [Ico]
simp [length_range', autoParam]
| true |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 166 | 167 | theorem first_denominator_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.denominators 1 = gp.b := by | simp [denom_eq_conts_b, first_continuant_eq zeroth_s_eq]
| true |
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 | 102 | 137 | theorem HasConstantSpeedOnWith.union {t : Set ℝ} (hfs : HasConstantSpeedOnWith f s l)
(hft : HasConstantSpeedOnWith f t l) {x : ℝ} (hs : IsGreatest s x) (ht : IsLeast t x) :
HasConstantSpeedOnWith f (s ∪ t) l := by
rw [hasConstantSpeedOnWith_iff_ordered] at hfs hft ⊢ |
rw [hasConstantSpeedOnWith_iff_ordered] at hfs hft ⊢
rintro z (zs | zt) y (ys | yt) zy
· have : (s ∪ t) ∩ Icc z y = s ∩ Icc z y := by
ext w; constructor
· rintro ⟨ws | wt, zw, wy⟩
· exact ⟨ws, zw, wy⟩
· exact ⟨(le_antisymm (wy.trans (hs.2 ys)) (ht.2 wt)).symm ▸ hs.1, zw, wy⟩
· r... | true |
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Semicontinuous
import Mathlib.Topology.Baire.Lemmas
open Filter Topology Set ContinuousLinearMap
section defs
class BarrelledSpace (𝕜 E : Type*) [SeminormedRing 𝕜] [AddGroup E] [SMul 𝕜 E]
[TopologicalSpace E] : Prop where
con... | Mathlib/Analysis/LocallyConvex/Barrelled.lean | 93 | 103 | theorem Seminorm.continuous_iSup
{ι : Sort*} {𝕜 E : Type*} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [BarrelledSpace 𝕜 E] (p : ι → Seminorm 𝕜 E)
(hp : ∀ i, Continuous (p i)) (bdd : BddAbove (range p)) :
Continuous (⨆ i, p i) := by
rw [← Seminorm.coe_iSup_eq bdd] |
rw [← Seminorm.coe_iSup_eq bdd]
refine Seminorm.continuous_of_lowerSemicontinuous _ ?_
rw [Seminorm.coe_iSup_eq bdd]
rw [Seminorm.bddAbove_range_iff] at bdd
convert lowerSemicontinuous_ciSup (f := fun i x ↦ p i x) bdd (fun i ↦ (hp i).lowerSemicontinuous)
exact iSup_apply
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 276 | 285 | theorem hasStrictFDerivAt_rpow_of_pos (p : ℝ × ℝ) (hp : 0 < p.1) :
HasStrictFDerivAt (fun x : ℝ × ℝ => x.1 ^ x.2)
((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +
(p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ) p := by
have : (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x... |
have : (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) :=
(continuousAt_fst.eventually (lt_mem_nhds hp)).mono fun p hp => rpow_def_of_pos hp _
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
convert ((hasStrictFDerivAt_fst.log hp.ne').mul hasStrictFDerivAt_snd).exp using 1
rw... | true |
import Mathlib.RingTheory.RootsOfUnity.Basic
universe u
variable {L : Type u} [CommRing L] [IsDomain L]
variable (n : ℕ+)
theorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :
∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by
obtain ⟨m, hm⟩ := MonoidHom.map_cyclic ((g : L ≃* L).re... | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 120 | 125 | theorem toFun_unique (g : L ≃+* L) (c : ZMod (Fintype.card (rootsOfUnity n L)))
(hc : ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ c.val : Lˣ)) : c = χ₀ n g := by
apply IsCyclic.ext rfl (fun ζ ↦ ?_) |
apply IsCyclic.ext rfl (fun ζ ↦ ?_)
specialize hc ζ
suffices ((ζ ^ c.val : Lˣ) : L) = (ζ ^ (χ₀ n g).val : Lˣ) by exact_mod_cast this
rw [← toFun_spec g ζ, hc]
| true |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 256 | 256 | theorem top_bihimp : ⊤ ⇔ a = a := by | rw [bihimp_comm, bihimp_top]
| true |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 96 | 99 | theorem integral_comp_neg_Ioi {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(c : ℝ) (f : ℝ → E) : (∫ x in Ioi c, f (-x)) = ∫ x in Iic (-c), f x := by
rw [← neg_neg c, ← integral_comp_neg_Iic] |
rw [← neg_neg c, ← integral_comp_neg_Iic]
simp only [neg_neg]
| true |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 306 | 310 | theorem convexBodySumFun_add_le (x y : E K) :
convexBodySumFun (x + y) ≤ convexBodySumFun x + convexBodySumFun y := by
simp_rw [convexBodySumFun, ← Finset.sum_add_distrib, ← mul_add] |
simp_rw [convexBodySumFun, ← Finset.sum_add_distrib, ← mul_add]
exact Finset.sum_le_sum
fun _ _ ↦ mul_le_mul_of_nonneg_left (normAtPlace_add_le _ x y) (Nat.cast_pos.mpr mult_pos).le
| true |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set Function Filter
open scoped NNReal Topology
instance Real.punctured_nhds_module_neBot {E ... | Mathlib/Analysis/NormedSpace/Real.lean | 76 | 78 | theorem frontier_ball (x : E) {r : ℝ} (hr : r ≠ 0) :
frontier (ball x r) = sphere x r := by |
rw [frontier, closure_ball x hr, isOpen_ball.interior_eq, closedBall_diff_ball]
| true |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | Mathlib/Order/Filter/Lift.lean | 45 | 53 | theorem HasBasis.mem_lift_iff {ι} {p : ι → Prop} {s : ι → Set α} {f : Filter α}
(hf : f.HasBasis p s) {β : ι → Type*} {pg : ∀ i, β i → Prop} {sg : ∀ i, β i → Set γ}
{g : Set α → Filter γ} (hg : ∀ i, (g <| s i).HasBasis (pg i) (sg i)) (gm : Monotone g)
{s : Set γ} : s ∈ f.lift g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg... |
refine (mem_biInf_of_directed ?_ ⟨univ, univ_sets _⟩).trans ?_
· intro t₁ ht₁ t₂ ht₂
exact ⟨t₁ ∩ t₂, inter_mem ht₁ ht₂, gm inter_subset_left, gm inter_subset_right⟩
· simp only [← (hg _).mem_iff]
exact hf.exists_iff fun t₁ t₂ ht H => gm ht H
| true |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 185 | 187 | theorem _root_.NumberField.mixedEmbedding_injective [NumberField K] :
Function.Injective (NumberField.mixedEmbedding K) := by |
exact RingHom.injective _
| true |
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.GaloisConnection
import Mathlib.Order.Hom.Basic
#align_import order.hom.order from "leanprover-community/mathlib"@"ba2245edf0c8bb155f1569fd9b9492a9b384cde6"
namespace OrderHom
variable {α β : Type*}
section Preorder
variable [Preorder α]
instance [Sem... | Mathlib/Order/Hom/Order.lean | 117 | 119 | theorem coe_iSup {ι : Sort*} [CompleteLattice β] (f : ι → α →o β) :
((⨆ i, f i : α →o β) : α → β) = ⨆ i, (f i : α → β) := by |
funext x; simp [iSup_apply]
| true |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 136 | 138 | theorem prod_univ_four [CommMonoid β] (f : Fin 4 → β) : ∏ i, f i = f 0 * f 1 * f 2 * f 3 := by
rw [prod_univ_castSucc, prod_univ_three] |
rw [prod_univ_castSucc, prod_univ_three]
rfl
| true |
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.GroupTheory.GroupAction.ConjAct
#align_import group_theory.subgroup.pointwise from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
open Set
open Pointwise
variable {α G A S... | Mathlib/Algebra/Group/Subgroup/Pointwise.lean | 89 | 102 | theorem closure_induction_left {p : (x : G) → x ∈ closure s → Prop} (one : p 1 (one_mem _))
(mul_left : ∀ x (hx : x ∈ s), ∀ (y) hy, p y hy → p (x * y) (mul_mem (subset_closure hx) hy))
(mul_left_inv : ∀ x (hx : x ∈ s), ∀ (y) hy, p y hy →
p (x⁻¹ * y) (mul_mem (inv_mem (subset_closure hx)) hy))
{x : G} ... |
revert h
simp_rw [← mem_toSubmonoid, closure_toSubmonoid] at *
intro h
induction h using Submonoid.closure_induction_left with
| one => exact one
| mul_left x hx y hy ih =>
cases hx with
| inl hx => exact mul_left _ hx _ hy ih
| inr hx => simpa only [inv_inv] using mul_left_inv _ hx _ hy ih
| true |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topology Classical NNReal
n... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean | 74 | 83 | theorem approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : HasStrictFDerivAt f f' a) {c : ℝ≥0} (hc : Subsingleton E ∨ 0 < c) :
∃ s ∈ 𝓝 a, ApproximatesLinearOn f f' s c := by
cases' hc with hE hc |
cases' hc with hE hc
· refine ⟨univ, IsOpen.mem_nhds isOpen_univ trivial, fun x _ y _ => ?_⟩
simp [@Subsingleton.elim E hE x y]
have := hf.def hc
rw [nhds_prod_eq, Filter.Eventually, mem_prod_same_iff] at this
rcases this with ⟨s, has, hs⟩
exact ⟨s, has, fun x hx y hy => hs (mk_mem_prod hx hy)⟩
| true |
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 | 151 | 153 | theorem mul_right_not_lt {a : R} (b) (h : a ≠ 0) : ¬a * b ≺ b := by
rw [mul_comm] |
rw [mul_comm]
exact mul_left_not_lt b h
| true |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 44 | 45 | theorem intDegree_zero : intDegree (0 : RatFunc K) = 0 := by |
rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]
| true |
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 115 | 119 | theorem Algebra.discr_localizationLocalization (b : Basis ι R S) :
Algebra.discr Rₘ (b.localizationLocalization Rₘ M Sₘ) =
algebraMap R Rₘ (Algebra.discr R b) := by
rw [Algebra.discr_def, Algebra.discr_def, RingHom.map_det, |
rw [Algebra.discr_def, Algebra.discr_def, RingHom.map_det,
Algebra.traceMatrix_localizationLocalization]
| true |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 90 | 97 | theorem mul_mem_cancel_left {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hf : f ∈ S.arrows c d) :
f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e := by
constructor |
constructor
· rintro h
suffices Groupoid.inv f ≫ f ≫ g ∈ S.arrows d e by
simpa only [inv_eq_inv, IsIso.inv_hom_id_assoc] using this
apply S.mul (S.inv hf) h
· apply S.mul hf
| true |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 63 | 64 | theorem foldl_loop_eq (f : α → Fin n → α) (x) : foldl.loop n f x n = x := by |
rw [foldl.loop, dif_neg (Nat.lt_irrefl _)]
| true |
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.Submonoid.Centralizer
import Mathlib.RingTheory.NonUnitalSubsem... | Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 39 | 40 | theorem natCast_mem [AddSubmonoidWithOneClass S R] (n : ℕ) : (n : R) ∈ s := by |
induction n <;> simp [zero_mem, add_mem, one_mem, *]
| true |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Multiset.Functor
#align_import data.finset.functor from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
universe u
open Function
namespace Finset
protected instance pure : Pure Finset :=
⟨fun... | Mathlib/Data/Finset/Functor.lean | 200 | 202 | theorem id_traverse [DecidableEq α] (s : Finset α) : traverse (pure : α → Id α) s = s := by
rw [traverse, Multiset.id_traverse] |
rw [traverse, Multiset.id_traverse]
exact s.val_toFinset
| true |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 98 | 100 | theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a := by
rw [← mul_le_mul_iff_left a] |
rw [← mul_le_mul_iff_left a]
simp
| true |
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 127 | 132 | theorem coe_rootSpaceWeightSpaceProduct_tmul (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃)
(x : rootSpace H χ₁) (m : weightSpace M χ₂) :
(rootSpaceWeightSpaceProduct R L H M χ₁ χ₂ χ₃ hχ (x ⊗ₜ m) : M) = ⁅(x : L), (m : M)⁆ := by
simp only [rootSpaceWeightSpaceProduct, rootSpaceWeightSpaceProductAux, coe_liftLie_eq_lif... |
simp only [rootSpaceWeightSpaceProduct, rootSpaceWeightSpaceProductAux, coe_liftLie_eq_lift_coe,
AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, lift_apply, LinearMap.coe_mk, AddHom.coe_mk,
Submodule.coe_mk]
| true |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.Topology.LocalAtTarget
#align_import algebraic_geometry.morphisms.universally_closed from "leanprover-community/mathlib"@"a8ae1b3f7979249a0af6bc7cf20c1f6bf656ca73"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalS... | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | 88 | 94 | theorem universallyClosed_is_local_at_target : PropertyIsLocalAtTarget @UniversallyClosed := by
rw [universallyClosed_eq] |
rw [universallyClosed_eq]
apply universallyIsLocalAtTargetOfMorphismRestrict
· exact topologically_isClosedMap_respectsIso
· intro X Y f ι U hU H
simp_rw [topologically, morphismRestrict_base] at H
exact (isClosedMap_iff_isClosedMap_of_iSup_eq_top hU).mpr H
| true |
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 97 | 98 | theorem card_eq_of_equiv_fin {α : Type*} {n : ℕ} (f : α ≃ Fin n) : Nat.card α = n := by |
simpa only [card_eq_fintype_card, Fintype.card_fin] using card_congr f
| true |
import Mathlib.Analysis.Calculus.Deriv.Basic
#align_import analysis.calculus.deriv.support from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v
variable {𝕜 : Type u} [NontriviallyNormedField 𝕜]
variable {E : Type v} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
variable {f : ... | Mathlib/Analysis/Calculus/Deriv/Support.lean | 36 | 41 | theorem support_deriv_subset : support (deriv f) ⊆ tsupport f := by
intro x |
intro x
rw [← not_imp_not]
intro h2x
rw [not_mem_tsupport_iff_eventuallyEq] at h2x
exact nmem_support.mpr (h2x.deriv_eq.trans (deriv_const x 0))
| true |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Tactic.Ring
#align_import data.nat.hyperoperation from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
def hyperoperation : ℕ → ℕ → ℕ → ℕ
| 0, _, k => k + 1
| 1, m, 0 => m
| 2, _, 0 => 0
| _ + 3, _, 0 => 1
| n + 1, m, k + 1 ... | Mathlib/Data/Nat/Hyperoperation.lean | 49 | 50 | theorem hyperoperation_ge_three_eq_one (n m : ℕ) : hyperoperation (n + 3) m 0 = 1 := by |
rw [hyperoperation]
| true |
import Mathlib.Data.ZMod.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Tactic.IntervalCases
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.GroupTheory.SpecificGroups.Cyclic
#align_import group_theory.specific_groups.quaternion from "leanprover-community/mathlib"@"879155bff5af618b9062cbb2915... | Mathlib/GroupTheory/SpecificGroups/Quaternion.lean | 174 | 176 | theorem card [NeZero n] : Fintype.card (QuaternionGroup n) = 4 * n := by
rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul] |
rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul]
ring
| true |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
universe uD uE uF uG
variable {𝕜 : Type*} [NontriviallyNormedField ... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 35 | 43 | theorem contDiffOn_clm_apply {n : ℕ∞} {f : E → F →L[𝕜] G} {s : Set E} [FiniteDimensional 𝕜 F] :
ContDiffOn 𝕜 n f s ↔ ∀ y, ContDiffOn 𝕜 n (fun x => f x y) s := by
refine ⟨fun h y => h.clm_apply contDiffOn_const, fun h => ?_⟩ |
refine ⟨fun h y => h.clm_apply contDiffOn_const, fun h => ?_⟩
let d := finrank 𝕜 F
have hd : d = finrank 𝕜 (Fin d → 𝕜) := (finrank_fin_fun 𝕜).symm
let e₁ := ContinuousLinearEquiv.ofFinrankEq hd
let e₂ := (e₁.arrowCongr (1 : G ≃L[𝕜] G)).trans (ContinuousLinearEquiv.piRing (Fin d))
rw [← id_comp f, ← e₂... | true |
import Mathlib.MeasureTheory.Measure.AEMeasurable
#align_import dynamics.ergodic.measure_preserving from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
variable {α β γ δ : Type*} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ]
[MeasurableSpace δ]
namespace MeasureTheory
... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 87 | 89 | theorem restrict_image_emb {f : α → β} (hf : MeasurePreserving f μa μb) (h₂ : MeasurableEmbedding f)
(s : Set α) : MeasurePreserving f (μa.restrict s) (μb.restrict (f '' s)) := by |
simpa only [Set.preimage_image_eq _ h₂.injective] using hf.restrict_preimage_emb h₂ (f '' s)
| true |
import Mathlib.Algebra.Ring.Regular
import Mathlib.Data.Int.GCD
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.modeq from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
assert_not_exists Function.support
namespace Nat
def ModEq (n a b :... | Mathlib/Data/Nat/ModEq.lean | 89 | 91 | theorem modEq_iff_dvd : a ≡ b [MOD n] ↔ (n : ℤ) ∣ b - a := by
rw [ModEq, eq_comm, ← Int.natCast_inj, Int.natCast_mod, Int.natCast_mod, |
rw [ModEq, eq_comm, ← Int.natCast_inj, Int.natCast_mod, Int.natCast_mod,
Int.emod_eq_emod_iff_emod_sub_eq_zero, Int.dvd_iff_emod_eq_zero]
| true |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
th... | Mathlib/GroupTheory/Archimedean.lean | 91 | 95 | theorem AddSubgroup.cyclic_of_isolated_zero {H : AddSubgroup G} {a : G} (h₀ : 0 < a)
(hd : Disjoint (H : Set G) (Ioo 0 a)) : ∃ b, H = closure {b} := by
rcases eq_or_ne H ⊥ with rfl | hbot |
rcases eq_or_ne H ⊥ with rfl | hbot
· exact ⟨0, closure_singleton_zero.symm⟩
· exact (exists_isLeast_pos hbot h₀ hd).imp fun _ => cyclic_of_min
| true |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749fc4661828cba18cd0f4e3c5eb66a8e80598"
open scoped Pointwise
universe u₁ u₂ u₃
namespace MonoidAlgebra
open Finset Finsupp
variable {k : Type u₁} ... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 95 | 97 | theorem mem_span_support (f : MonoidAlgebra k G) :
f ∈ Submodule.span k (of k G '' (f.support : Set G)) := by |
erw [of, MonoidHom.coe_mk, ← supported_eq_span_single, Finsupp.mem_supported]
| true |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 94 | 97 | theorem eqRec_heq_self {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') :
HEq (@Eq.rec α a motive x a' e) x := by |
subst e; rfl
| true |
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Data.PNat.Defs
#align_import data.pnat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Function PNat
namespace PNat
variable (a b : ℕ+)
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ+ := Subtype.... | Mathlib/Data/PNat/Interval.lean | 76 | 81 | theorem card_Ico : (Ico a b).card = b - a := by
rw [← Nat.card_Ico] |
rw [← Nat.card_Ico]
-- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-- https://github.com/leanprover-community/mathlib4/issues/5164
erw [← Finset.map_subtype_embedding_Ico _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
rw [card_map]
| true |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section B... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 158 | 196 | theorem snorm_le_snorm_top_mul_snorm (p : ℝ≥0∞) (f : α → E) {g : α → F}
(hg : AEStronglyMeasurable g μ) (b : E → F → G)
(h : ∀ᵐ x ∂μ, ‖b (f x) (g x)‖₊ ≤ ‖f x‖₊ * ‖g x‖₊) :
snorm (fun x => b (f x) (g x)) p μ ≤ snorm f ∞ μ * snorm g p μ := by
by_cases hp_top : p = ∞ |
by_cases hp_top : p = ∞
· simp_rw [hp_top, snorm_exponent_top]
refine le_trans (essSup_mono_ae <| h.mono fun a ha => ?_) (ENNReal.essSup_mul_le _ _)
simp_rw [Pi.mul_apply, ← ENNReal.coe_mul, ENNReal.coe_le_coe]
exact ha
by_cases hp_zero : p = 0
· simp only [hp_zero, snorm_exponent_zero, mul_zero, l... | true |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 148 | 150 | theorem spanSingleton_div_spanSingleton (x y : K) :
spanSingleton R₁⁰ x / spanSingleton R₁⁰ y = spanSingleton R₁⁰ (x / y) := by |
rw [div_spanSingleton, mul_comm, spanSingleton_mul_spanSingleton, div_eq_mul_inv]
| true |
import Mathlib.Probability.Martingale.BorelCantelli
import Mathlib.Probability.ConditionalExpectation
import Mathlib.Probability.Independence.Basic
#align_import probability.borel_cantelli from "leanprover-community/mathlib"@"2f8347015b12b0864dfaf366ec4909eb70c78740"
open scoped MeasureTheory ProbabilityTheory EN... | Mathlib/Probability/BorelCantelli.lean | 60 | 66 | theorem iIndepSet.condexp_indicator_filtrationOfSet_ae_eq (hsm : ∀ n, MeasurableSet (s n))
(hs : iIndepSet s μ) (hij : i < j) :
μ[(s j).indicator (fun _ => 1 : Ω → ℝ)|filtrationOfSet hsm i] =ᵐ[μ]
fun _ => (μ (s j)).toReal := by
rw [Filtration.filtrationOfSet_eq_natural (β := ℝ) hsm] |
rw [Filtration.filtrationOfSet_eq_natural (β := ℝ) hsm]
refine (iIndepFun.condexp_natural_ae_eq_of_lt _ hs.iIndepFun_indicator hij).trans ?_
simp only [integral_indicator_const _ (hsm _), Algebra.id.smul_eq_mul, mul_one]; rfl
| true |
import Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
noncomputable section
universe w v₁ v₂ u₁ u₂
open Cate... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean | 215 | 218 | theorem map_π_preserves_coequalizer_inv_desc {W : D} (k : G.obj Y ⟶ W)
(wk : G.map f ≫ k = G.map g ≫ k) : G.map (coequalizer.π f g) ≫
(PreservesCoequalizer.iso G f g).inv ≫ coequalizer.desc k wk = k := by |
rw [← Category.assoc, map_π_preserves_coequalizer_inv, coequalizer.π_desc]
| true |
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 209 | 215 | theorem isBasis_affine_open (X : Scheme) : Opens.IsBasis X.affineOpens := by
rw [Opens.isBasis_iff_nbhd] |
rw [Opens.isBasis_iff_nbhd]
rintro U x (hU : x ∈ (U : Set X))
obtain ⟨S, hS, hxS, hSU⟩ := X.affineBasisCover_is_basis.exists_subset_of_mem_open hU U.isOpen
refine ⟨⟨S, X.affineBasisCover_is_basis.isOpen hS⟩, ?_, hxS, hSU⟩
rcases hS with ⟨i, rfl⟩
exact rangeIsAffineOpenOfOpenImmersion _
| true |
import Mathlib.Data.Matrix.Basic
variable {l m n o : Type*}
universe u v w
variable {R : Type*} {α : Type v} {β : Type w}
namespace Matrix
def col (w : m → α) : Matrix m Unit α :=
of fun x _ => w x
#align matrix.col Matrix.col
-- TODO: set as an equation lemma for `col`, see mathlib4#3024
@[simp]
theorem col... | Mathlib/Data/Matrix/RowCol.lean | 123 | 126 | theorem col_vecMul [Fintype m] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : m → α) :
Matrix.col (v ᵥ* M) = (Matrix.row v * M)ᵀ := by
ext |
ext
rfl
| true |
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.injective_resolution from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
noncomputable s... | Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean | 104 | 106 | theorem ι_f_zero_comp_complex_d :
I.ι.f 0 ≫ I.cocomplex.d 0 1 = 0 := by |
simp
| true |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Analysis.Normed.Field.UnitBall
#align_import analysis.complex.circle from "leanprover-community/mathlib"@"ad3dfaca9ea2465198bcf58aa114401c324e29d1"
noncomputable section
open Complex Metric
open ComplexC... | Mathlib/Analysis/Complex/Circle.lean | 66 | 66 | theorem normSq_eq_of_mem_circle (z : circle) : normSq z = 1 := by | simp [normSq_eq_abs]
| true |
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed_space.int from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
namespace Int
theorem nnnorm_coe_units (e : ℤˣ) : ‖(e : ℤ)‖₊ = 1 := by
obtain rfl | rfl := units_eq_one_or e <;>
simp only [Units.coe_neg_one, Un... | Mathlib/Analysis/NormedSpace/Int.lean | 41 | 42 | theorem toNat_add_toNat_neg_eq_nnnorm (n : ℤ) : ↑n.toNat + ↑(-n).toNat = ‖n‖₊ := by |
rw [← Nat.cast_add, toNat_add_toNat_neg_eq_natAbs, NNReal.natCast_natAbs]
| true |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 67 | 74 | theorem affineSegment_same (x : P) : affineSegment R x x = {x} := by
-- Porting note: added as this doesn't do anything in `simp_rw` any more |
-- Porting note: added as this doesn't do anything in `simp_rw` any more
rw [affineSegment]
-- Note: when adding "simp made no progress" in lean4#2336,
-- had to change `lineMap_same` to `lineMap_same _`. Not sure why?
-- Porting note: added `_ _` and `Function.const`
simp_rw [lineMap_same _, AffineMap.coe... | true |
import Mathlib.Algebra.Group.Center
#align_import group_theory.subsemigroup.centralizer from "leanprover-community/mathlib"@"cc67cd75b4e54191e13c2e8d722289a89e67e4fa"
variable {M : Type*} {S T : Set M}
namespace Set
variable (S)
@[to_additive addCentralizer " The centralizer of a subset of an additive magma. ... | Mathlib/Algebra/Group/Centralizer.lean | 64 | 65 | theorem zero_mem_centralizer [MulZeroClass M] : (0 : M) ∈ centralizer S := by |
simp [mem_centralizer_iff]
| true |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 89 | 91 | theorem ofFractionRing_add (p q : FractionRing K[X]) :
ofFractionRing (p + q) = ofFractionRing p + ofFractionRing q := by |
simp only [HAdd.hAdd, Add.add, RatFunc.add]
| true |
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 130 | 136 | theorem splitUpper_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, update I.lower i x j < I.upper j :=
(forall_update_iff I.lower fun j y => y < I.upper j).2
⟨h.2, fun j _ => I.lower_lt_upper _⟩) :
I.splitUpper i x = (⟨update I.lower i x, I.upper, h'⟩ : Box ι) := by
si... |
simp (config := { unfoldPartialApp := true }) only [splitUpper, mk'_eq_coe, max_eq_left h.1.le,
update, and_self]
| true |
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 75 | 77 | theorem sInf_mem {s : Set ℕ} (h : s.Nonempty) : sInf s ∈ s := by
rw [Nat.sInf_def h] |
rw [Nat.sInf_def h]
exact Nat.find_spec h
| true |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section Dist
variable [Dist α] [Dist β]
open scoped C... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 231 | 233 | theorem prod_dist_eq_card (f g : WithLp 0 (α × β)) : dist f g =
(if dist f.fst g.fst = 0 then 0 else 1) + (if dist f.snd g.snd = 0 then 0 else 1) := by |
convert if_pos rfl
| true |
import Mathlib.MeasureTheory.SetSemiring
open MeasurableSpace Set
namespace MeasureTheory
variable {α : Type*} {𝒜 : Set (Set α)} {s t : Set α}
structure IsSetAlgebra (𝒜 : Set (Set α)) : Prop where
empty_mem : ∅ ∈ 𝒜
compl_mem : ∀ ⦃s⦄, s ∈ 𝒜 → sᶜ ∈ 𝒜
union_mem : ∀ ⦃s t⦄, s ∈ 𝒜 → t ∈ 𝒜 → s ∪ t ∈ 𝒜
... | Mathlib/MeasureTheory/SetAlgebra.lean | 151 | 158 | theorem generateSetAlgebra_subset {ℬ : Set (Set α)} (h : 𝒜 ⊆ ℬ)
(hℬ : IsSetAlgebra ℬ) : generateSetAlgebra 𝒜 ⊆ ℬ := by
intro s hs |
intro s hs
induction hs with
| base t t_mem => exact h t_mem
| empty => exact hℬ.empty_mem
| compl t _ t_mem => exact hℬ.compl_mem t_mem
| union t u _ _ t_mem u_mem => exact hℬ.union_mem t_mem u_mem
| true |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) ≤ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : ℕ) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 117 | 126 | theorem coeff_pow_of_natDegree_le_of_eq_ite' {m n o : ℕ} {a : R} {p : R[X]}
(h_pow : natDegree p ≤ n) (h_exp : m * n ≤ o) (h_pow_bas : coeff p n = a) :
coeff (p ^ m) o = if o = m * n then a ^ m else 0 := by
split_ifs with h |
split_ifs with h
· subst h h_pow_bas
exact coeff_pow_of_natDegree_le ‹_›
· apply coeff_eq_zero_of_natDegree_lt
apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ‹_› ?_)
· exact natDegree_pow_le_of_le m ‹_›
· exact Iff.mp ne_comm h
| true |
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Group.Invertible.Basic
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.GroupTheory.GroupAction.Units
#align_import group_theory.group_action.group from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
universe u v w
... | Mathlib/GroupTheory/GroupAction/Group.lean | 35 | 36 | theorem smul_inv_smul (c : α) (x : β) : c • c⁻¹ • x = x := by |
rw [smul_smul, mul_right_inv, one_smul]
| true |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 148 | 153 | theorem of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor |
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
| true |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 74 | 74 | theorem zero_eq_dist {x y : γ} : 0 = dist x y ↔ x = y := by | rw [eq_comm, dist_eq_zero]
| true |
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.NormedSpace.Pointwise
#align_import analysis.normed_space.is_R_or_C from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric
variable {𝕜 : Type*} [RCLike 𝕜] {E :... | Mathlib/Analysis/NormedSpace/RCLike.lean | 36 | 36 | theorem RCLike.norm_coe_norm {z : E} : ‖(‖z‖ : 𝕜)‖ = ‖z‖ := by | simp
| true |
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Sum
import Mathlib.Logic.Embedding.Set
#align_import data.fintype.sum from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v
variable {α β : Type*}
open Finset
instance (α : Type u) (β : Type v) [Fintype α] [Fintyp... | Mathlib/Data/Fintype/Sum.lean | 47 | 57 | theorem image_subtype_ne_univ_eq_image_erase [Fintype α] [DecidableEq β] (k : β) (b : α → β) :
image (fun i : { a // b a ≠ k } => b ↑i) univ = (image b univ).erase k := by
apply subset_antisymm |
apply subset_antisymm
· rw [image_subset_iff]
intro i _
apply mem_erase_of_ne_of_mem i.2 (mem_image_of_mem _ (mem_univ _))
· intro i hi
rw [mem_image]
rcases mem_image.1 (erase_subset _ _ hi) with ⟨a, _, ha⟩
subst ha
exact ⟨⟨a, ne_of_mem_erase hi⟩, mem_univ _, rfl⟩
| true |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 255 | 258 | theorem mul_self_lt_top_iff {a : ℝ≥0∞} : a * a < ⊤ ↔ a < ⊤ := by
rw [ENNReal.mul_lt_top_iff, and_self, or_self, or_iff_left_iff_imp] |
rw [ENNReal.mul_lt_top_iff, and_self, or_self, or_iff_left_iff_imp]
rintro rfl
exact zero_lt_top
| true |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 124 | 148 | theorem p_pow_smul_lift {x y : M} {k : ℕ} (hM' : Module.IsTorsionBy R M (p ^ pOrder hM y))
(h : p ^ k • x ∈ R ∙ y) : ∃ a : R, p ^ k • x = p ^ k • a • y := by
-- Porting note: needed to make `smul_smul` work below. |
-- Porting note: needed to make `smul_smul` work below.
letI : MulAction R M := MulActionWithZero.toMulAction
by_cases hk : k ≤ pOrder hM y
· let f :=
((R ∙ p ^ (pOrder hM y - k) * p ^ k).quotEquivOfEq _ ?_).trans
(quotTorsionOfEquivSpanSingleton R M y)
· have : f.symm ⟨p ^ k • x, h⟩ ∈
... | true |
import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
#align_import topology.homotopy.contractible from "leanprover-community/mathlib"@"16728b3064a1751103e1dc2815ed8d00560e0d87"
noncomputable section
namespace ContinuousMap
variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y]... | Mathlib/Topology/Homotopy/Contractible.lean | 39 | 43 | theorem Nullhomotopic.comp_left {f : C(Y, Z)} (hf : f.Nullhomotopic) (g : C(X, Y)) :
(f.comp g).Nullhomotopic := by
cases' hf with y hy |
cases' hf with y hy
use y
exact Homotopic.hcomp (Homotopic.refl g) hy
| true |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 33 | 38 | theorem den_dvd (a b : ℤ) : ((a /. b).den : ℤ) ∣ b := by
by_cases b0 : b = 0; · simp [b0] |
by_cases b0 : b = 0; · simp [b0]
cases' e : a /. b with n d h c
rw [mk'_eq_divInt, divInt_eq_iff b0 (ne_of_gt (Int.natCast_pos.2 (Nat.pos_of_ne_zero h)))] at e
refine Int.dvd_natAbs.1 <| Int.natCast_dvd_natCast.2 <| c.symm.dvd_of_dvd_mul_left ?_
rw [← Int.natAbs_mul, ← Int.natCast_dvd_natCast, Int.dvd_natAbs... | true |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 146 | 161 | theorem iterate_derivative_at_0_eq_zero_of_lt (n : ℕ) {ν k : ℕ} :
k < ν → (Polynomial.derivative^[k] (bernsteinPolynomial R n ν)).eval 0 = 0 := by
cases' ν with ν |
cases' ν with ν
· rintro ⟨⟩
· rw [Nat.lt_succ_iff]
induction' k with k ih generalizing n ν
· simp [eval_at_0]
· simp only [derivative_succ, Int.natCast_eq_zero, mul_eq_zero, Function.comp_apply,
Function.iterate_succ, Polynomial.iterate_derivative_sub,
Polynomial.iterate_derivative_na... | true |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 116 | 126 | theorem isClosedMap_iff_isClosedMap_of_iSup_eq_top :
IsClosedMap f ↔ ∀ i, IsClosedMap ((U i).1.restrictPreimage f) := by
refine ⟨fun h i => h.restrictPreimage _, ?_⟩ |
refine ⟨fun h i => h.restrictPreimage _, ?_⟩
rintro H s hs
rw [isClosed_iff_coe_preimage_of_iSup_eq_top hU]
intro i
convert H i _ ⟨⟨_, hs.1, eq_compl_comm.mpr rfl⟩⟩
ext ⟨x, hx⟩
suffices (∃ y, y ∈ s ∧ f y = x) ↔ ∃ y, y ∈ s ∧ f y ∈ U i ∧ f y = x by
simpa [Set.restrictPreimage, ← Subtype.coe_inj]
exac... | true |
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 108 | 116 | theorem commutator_commutator_eq_bot_of_rotate (h1 : ⁅⁅H₂, H₃⁆, H₁⁆ = ⊥) (h2 : ⁅⁅H₃, H₁⁆, H₂⁆ = ⊥) :
⁅⁅H₁, H₂⁆, H₃⁆ = ⊥ := by
simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le, |
simp_rw [commutator_eq_bot_iff_le_centralizer, commutator_le,
mem_centralizer_iff_commutator_eq_one, ← commutatorElement_def] at h1 h2 ⊢
intro x hx y hy z hz
trans x * z * ⁅y, ⁅z⁻¹, x⁻¹⁆⁆⁻¹ * z⁻¹ * y * ⁅x⁻¹, ⁅y⁻¹, z⁆⁆⁻¹ * y⁻¹ * x⁻¹
· group
· rw [h1 _ (H₂.inv_mem hy) _ hz _ (H₁.inv_mem hx), h2 _ (H₃.inv_m... | true |
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 | 115 | 119 | theorem isReducedOfIsAffineIsReduced [IsAffine X] [h : _root_.IsReduced (X.presheaf.obj (op ⊤))] :
IsReduced X :=
haveI : IsReduced (Scheme.Spec.obj (op (Scheme.Γ.obj (op X)))) := by
rw [affine_isReduced_iff]; exact h |
rw [affine_isReduced_iff]; exact h
isReducedOfOpenImmersion X.isoSpec.hom
| true |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 76 | 76 | theorem one_geom_sum (n : ℕ) : ∑ i ∈ range n, (1 : α) ^ i = n := by | simp
| true |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 146 | 148 | theorem ite_mul_one {P : Prop} [Decidable P] {a b : M} :
ite P (a * b) 1 = ite P a 1 * ite P b 1 := by |
by_cases h:P <;> simp [h]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.RingTheory.Derivation.Basic
noncomputable section
namespace Polynomial
section CommSemiring
variable {R A : Type*} [CommSemiring R]
@[simps]
def derivative' : D... | Mathlib/Algebra/Polynomial/Derivation.lean | 43 | 46 | theorem C_smul_derivation_apply (D : Derivation R R[X] A) (a : R) (f : R[X]) :
C a • D f = a • D f := by
have : C a • D f = D (C a * f) := by simp |
have : C a • D f = D (C a * f) := by simp
rw [this, C_mul', D.map_smul]
| true |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 214 | 215 | theorem mem_transpose {μ : YoungDiagram} {c : ℕ × ℕ} : c ∈ μ.transpose ↔ c.swap ∈ μ := by |
simp [transpose]
| true |
import Mathlib.CategoryTheory.GlueData
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.Tactic.Generalize
import Mathlib.CategoryTheory.Elementwise
#align_import topology.gluing from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d... | Mathlib/Topology/Gluing.lean | 164 | 201 | theorem eqvGen_of_π_eq
-- Porting note: was `{x y : ∐ D.U} (h : 𝖣.π x = 𝖣.π y)`
{x y : sigmaObj (β := D.toGlueData.J) (C := TopCat) D.toGlueData.U}
(h : 𝖣.π x = 𝖣.π y) :
EqvGen
-- Porting note: was (Types.CoequalizerRel 𝖣.diagram.fstSigmaMap 𝖣.diagram.sndSigmaMap)
(Types.CoequalizerRel... |
delta GlueData.π Multicoequalizer.sigmaπ at h
-- Porting note: inlined `inferInstance` instead of leaving as a side goal.
replace h := (TopCat.mono_iff_injective (Multicoequalizer.isoCoequalizer 𝖣.diagram).inv).mp
inferInstance h
let diagram := parallelPair 𝖣.diagram.fstSigmaMap 𝖣.diagram.sndSigmaMap ⋙ ... | true |
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 75 | 77 | theorem nthLe_ofFn {n} (f : Fin n → α) (i : Fin n) :
nthLe (ofFn f) i ((length_ofFn f).symm ▸ i.2) = f i := by |
simp [nthLe]
| true |
import Mathlib.Data.ENNReal.Real
#align_import data.real.conjugate_exponents from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open scoped ENNReal
namespace Real
@[mk_iff]
structure IsConjExponent (p q : ℝ) : Prop where
one_lt : 1 < p
inv_add_inv_conj : p⁻... | Mathlib/Data/Real/ConjExponents.lean | 110 | 112 | theorem div_conj_eq_sub_one : p / q = p - 1 := by
field_simp [h.symm.ne_zero] |
field_simp [h.symm.ne_zero]
rw [h.sub_one_mul_conj]
| true |
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 | 72 | 74 | theorem ppow_mul' (x : M) (m n : ℕ+) : x ^ (m * n) = (x ^ n) ^ m := by
rw [mul_comm] |
rw [mul_comm]
exact ppow_mul x n m
| true |
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Combinatorics.Derangements.Finite
import Mathlib.Order.Filter.Basic
#align_import combinatorics.derangements.exponential from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter NormedSpace
open scoped Topology
... | Mathlib/Combinatorics/Derangements/Exponential.lean | 24 | 52 | theorem numDerangements_tendsto_inv_e :
Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) := by
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1. |
-- we show that d(n)/n! is the partial sum of exp(-1), but offset by 1.
-- this isn't entirely obvious, since we have to ensure that asc_factorial and
-- factorial interact in the right way, e.g., that k ≤ n always
let s : ℕ → ℝ := fun n => ∑ k ∈ Finset.range n, (-1 : ℝ) ^ k / k.factorial
suffices ∀ n : ℕ, (... | true |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {α : Type u} {r : α → α → Prop} {a : α} {l... | Mathlib/Data/List/Sort.lean | 104 | 120 | theorem eq_of_perm_of_sorted [IsAntisymm α r] {l₁ l₂ : List α} (hp : l₁ ~ l₂) (hs₁ : Sorted r l₁)
(hs₂ : Sorted r l₂) : l₁ = l₂ := by
induction' hs₁ with a l₁ h₁ hs₁ IH generalizing l₂ |
induction' hs₁ with a l₁ h₁ hs₁ IH generalizing l₂
· exact hp.nil_eq
· have : a ∈ l₂ := hp.subset (mem_cons_self _ _)
rcases append_of_mem this with ⟨u₂, v₂, rfl⟩
have hp' := (perm_cons a).1 (hp.trans perm_middle)
obtain rfl := IH hp' (hs₂.sublist <| by simp)
change a :: u₂ ++ v₂ = u₂ ++ ([a] ++ ... | true |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 69 | 73 | theorem arsinh_neg (x : ℝ) : arsinh (-x) = -arsinh x := by
rw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh] |
rw [← exp_eq_exp, exp_arsinh, exp_neg, exp_arsinh]
apply eq_inv_of_mul_eq_one_left
rw [neg_sq, neg_add_eq_sub, add_comm x, mul_comm, ← sq_sub_sq, sq_sqrt, add_sub_cancel_right]
exact add_nonneg zero_le_one (sq_nonneg _)
| true |
import Mathlib.Algebra.Order.Floor
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
#align_import data.rat.floor from "leanprover-community/mathlib"@"e1bccd6e40ae78370f01659715d3c948716e3b7e"
open Int
namespace Rat
variable {α : Type*} [LinearOrderedField α] [FloorRi... | Mathlib/Data/Rat/Floor.lean | 80 | 82 | theorem round_cast (x : ℚ) : round (x : α) = round x := by
have : ((x + 1 / 2 : ℚ) : α) = x + 1 / 2 := by simp |
have : ((x + 1 / 2 : ℚ) : α) = x + 1 / 2 := by simp
rw [round_eq, round_eq, ← this, floor_cast]
| true |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 68 | 70 | theorem weightedDegree_apply (w : σ → M) (f : σ →₀ ℕ):
weightedDegree w f = Finsupp.sum f (fun i c => c • w i) := by |
rfl
| true |
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