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 |
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import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 59 | 61 | theorem einfsep_pos : 0 < s.einfsep ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y := by |
rw [pos_iff_ne_zero, Ne, einfsep_zero]
simp only [not_forall, not_exists, not_lt, exists_prop, not_and]
|
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open List Function Nat
namespace List
namespace Nat
def antidiagonal (n : ℕ) : List (ℕ × ℕ) :=
(range (n + 1)).map fun i ↦ (i,... | Mathlib/Data/List/NatAntidiagonal.lean | 76 | 82 | theorem antidiagonal_succ' {n : ℕ} :
antidiagonal (n + 1) = (antidiagonal n).map (Prod.map id Nat.succ) ++ [(n + 1, 0)] := by |
simp only [antidiagonal, range_succ, Nat.add_sub_cancel_left, map_append, append_assoc,
Nat.sub_self, singleton_append, map_map, map]
congr 1
apply map_congr
simp (config := { contextual := true }) [le_of_lt, Nat.succ_eq_add_one, Nat.sub_add_comm]
|
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 | 84 | 86 | theorem thickenedIndicatorAux_one (δ : ℝ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
thickenedIndicatorAux δ E x = 1 := by |
simp [thickenedIndicatorAux, infEdist_zero_of_mem x_in_E, tsub_zero]
|
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 149 | 162 | theorem contMDiffWithinAt_fst {s : Set (M × N)} {p : M × N} :
ContMDiffWithinAt (I.prod J) I n Prod.fst s p := by |
/- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
rw [contMDiffWithinAt_iff']
refine' ⟨continuousWithinAt_fst, _⟩
refine' contDiffWithinAt_fst.congr (fun y hy => _) _
· simp only [mfld_simps] at hy
simp only [hy, mfld_simps]
· simp only [mfld_simps]
-/
rw [contMDiffWithinAt_iff']
... |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap... | Mathlib/RingTheory/AdicCompletion/Algebra.lean | 87 | 89 | theorem evalₐ_mk (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by |
simp [evalₐ]
|
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 67 | 80 | theorem hasseDeriv_coeff (n : ℕ) :
(hasseDeriv k f).coeff n = (n + k).choose k * f.coeff (n + k) := by |
rw [hasseDeriv_apply, coeff_sum, sum_def, Finset.sum_eq_single (n + k), coeff_monomial]
· simp only [if_true, add_tsub_cancel_right, eq_self_iff_true]
· intro i _hi hink
rw [coeff_monomial]
by_cases hik : i < k
· simp only [Nat.choose_eq_zero_of_lt hik, ite_self, Nat.cast_zero, zero_mul]
· push_n... |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antid... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 88 | 95 | theorem natDegree_trunc_lt (f : R⟦X⟧) (n) : (trunc (n + 1) f).natDegree < n + 1 := by |
rw [Nat.lt_succ_iff, natDegree_le_iff_coeff_eq_zero]
intros
rw [coeff_trunc]
split_ifs with h
· rw [lt_succ, ← not_lt] at h
contradiction
· rfl
|
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 | 103 | 108 | theorem ker_eval {s : S} (hs : IsIntegral R s) :
RingHom.ker ((Polynomial.aeval s).toRingHom : R[X] →+* S) =
Ideal.span ({minpoly R s} : Set R[X]) := by |
ext p
simp_rw [RingHom.mem_ker, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
isIntegrallyClosed_dvd_iff hs, ← Ideal.mem_span_singleton]
|
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
#align_import order.hom.complete_lattice from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function OrderDual Set
variable {F α β γ δ : Type*} {ι : Sort*} {κ : ι → Sort*}
-- Porting note: mathport made this & sInf... | Mathlib/Order/Hom/CompleteLattice.lean | 134 | 135 | theorem map_iSup₂ [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
f (⨆ (i) (j), g i j) = ⨆ (i) (j), f (g i j) := by | simp_rw [map_iSup]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 142 | 146 | theorem contDiffOn_of_differentiableOn_deriv {n : ℕ∞}
(h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) :
ContDiffOn 𝕜 n f s := by |
apply contDiffOn_of_differentiableOn
simpa only [iteratedFDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_differentiableOn_iff]
|
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 84 | 95 | theorem IsSRGWith.card_neighborFinset_union_eq {v w : V} (h : G.IsSRGWith n k ℓ μ) :
(G.neighborFinset v ∪ G.neighborFinset w).card =
2 * k - Fintype.card (G.commonNeighbors v w) := by |
apply Nat.add_right_cancel (m := Fintype.card (G.commonNeighbors v w))
rw [Nat.sub_add_cancel, ← Set.toFinset_card]
-- Porting note: Set.toFinset_inter needs workaround to use unification to solve for one of the
-- instance arguments:
· simp [commonNeighbors, @Set.toFinset_inter _ _ _ _ _ _ (_),
← neig... |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 187 | 189 | theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by |
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, ← mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
|
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 88 | 89 | theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
|
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 259 | 266 | theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r := by |
apply IsBigOWith.of_bound
filter_upwards [hg, h.bound] with x hgx hx
calc
|f x ^ r| ≤ |f x| ^ r := abs_rpow_le_abs_rpow _ _
_ ≤ (c * |g x|) ^ r := rpow_le_rpow (abs_nonneg _) hx hr
_ = c ^ r * |g x ^ r| := by rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx]
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 74 | 106 | theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E}
(hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) (h't : ∀ᵐ x ∂μ, f x ∈ t) : f =ᵐ[μ] 0 := by |
rcases ht with ⟨d, d_count, hd⟩
haveI : Encodable d := d_count.toEncodable
have : ∀ x : d, ∃ g : E →L[𝕜] 𝕜, ‖g‖ ≤ 1 ∧ g x = ‖(x : E)‖ :=
fun x => exists_dual_vector'' 𝕜 (x : E)
choose s hs using this
have A : ∀ a : E, a ∈ t → (∀ x, ⟪a, s x⟫ = (0 : 𝕜)) → a = 0 := by
intro a hat ha
contrapose! ... |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 116 | 117 | theorem divX_C_mul_X_pow : divX (C a * X ^ n) = if n = 0 then 0 else C a * X ^ (n - 1) := by |
simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]
|
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 118 | 119 | theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by |
rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
|
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacobi_theta.basic from "leanprover-community/mathlib"@"57f9349f2fe19d2de7207e99b0341808d977cdcf"
open Complex Real Asymptotics Filter Topology
open scope... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 45 | 54 | theorem jacobiTheta_S_smul (τ : ℍ) :
jacobiTheta ↑(ModularGroup.S • τ) = (-I * τ) ^ (1 / 2 : ℂ) * jacobiTheta τ := by |
have h0 : (τ : ℂ) ≠ 0 := ne_of_apply_ne im (zero_im.symm ▸ ne_of_gt τ.2)
have h1 : (-I * τ) ^ (1 / 2 : ℂ) ≠ 0 := by
rw [Ne, cpow_eq_zero_iff, not_and_or]
exact Or.inl <| mul_ne_zero (neg_ne_zero.mpr I_ne_zero) h0
simp_rw [UpperHalfPlane.modular_S_smul, jacobiTheta_eq_jacobiTheta₂]
conv_rhs => erw [← of... |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 796 | 797 | theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by | simp [h]
|
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 72 | 78 | theorem map_snd_prod (I : Ideal R) (J : Ideal S) : map (RingHom.snd R S) (prod I J) = J := by |
ext x
rw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
exact
⟨by
rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2, fun h => ⟨⟨0, x⟩, ⟨⟨Ideal.zero_mem _, h⟩, rfl⟩⟩⟩
|
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
namespace CategoryTheory
namespace Bicategory
open Category
open scoped Bicategory
open Mathlib.Tactic.BicategoryCoherence (bicategoricalComp bicategoricalIsoComp)
universe w v u
variable {B : Type u} [Bicategory... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 151 | 164 | theorem comp_right_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
rightZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (ρ_ _).hom ≫ (λ_ _).inv := by |
calc
_ = 𝟙 _ ⊗≫
(g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫
g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
(rightZigz... |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 64 | 66 | theorem goldConj_mul_gold : ψ * φ = -1 := by |
rw [mul_comm]
exact gold_mul_goldConj
|
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 | 145 | 147 | theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by |
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Algebra.MvPolynomial.Polynomial
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.RingTheory.Polynomial.Basic
#align_import data.mv_polynomial.funext from "leanprover-community/mathlib"@"0b89934139d3be96f9dab477f10c20f9f93da580"
namespace MvPoly... | Mathlib/Algebra/MvPolynomial/Funext.lean | 46 | 59 | theorem funext {σ : Type*} {p q : MvPolynomial σ R} (h : ∀ x : σ → R, eval x p = eval x q) :
p = q := by |
suffices ∀ p, (∀ x : σ → R, eval x p = 0) → p = 0 by
rw [← sub_eq_zero, this (p - q)]
simp only [h, RingHom.map_sub, forall_const, sub_self]
clear h p q
intro p h
obtain ⟨n, f, hf, p, rfl⟩ := exists_fin_rename p
suffices p = 0 by rw [this, AlgHom.map_zero]
apply funext_fin
intro x
classical
... |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section Order
variable {s : Se... | Mathlib/Data/Set/Function.lean | 274 | 278 | theorem _root_.StrictMonoOn.congr (h₁ : StrictMonoOn f₁ s) (h : s.EqOn f₁ f₂) :
StrictMonoOn f₂ s := by |
intro a ha b hb hab
rw [← h ha, ← h hb]
exact h₁ ha hb hab
|
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Range
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
#align_import linear_algebra.basic from "leanprover-c... | Mathlib/LinearAlgebra/Basic.lean | 169 | 173 | theorem ofSubmodule'_toLinearMap [Module R M] [Module R₂ M₂] (f : M ≃ₛₗ[σ₁₂] M₂)
(U : Submodule R₂ M₂) :
(f.ofSubmodule' U).toLinearMap = (f.toLinearMap.domRestrict _).codRestrict _ Subtype.prop := by |
ext
rfl
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 256 | 256 | theorem descPochhammer_one : descPochhammer R 1 = X := by | simp [descPochhammer]
|
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 137 | 160 | theorem aux_calc (n : ℕ) {m : ℝ} (hm : 2 ≤ m) :
(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) ≤ 1 / (m ^ n !) ^ n :=
calc
(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) ≤ 2 * (1 / m ^ (n + 1)!) :=
-- the second factors coincide (and are non-negative),
-- the first factors satisfy the inequality `sub_one_div_inv_le_two`
... |
-- [NB: in this block, I do not follow the brace convention for subgoals -- I wait until
-- I solve all extraneous goals at once with `exact pow_pos (zero_lt_two.trans_le hm) _`.]
-- Clear denominators and massage*
apply (div_le_div_iff _ _).mpr
focus
conv_rhs => rw [one_mul, mul_... |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 172 | 174 | theorem mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
f ∈ vanishingIdeal t ↔ ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal := by |
rw [← SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
|
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 135 | 137 | theorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by |
rw [toComplex_def₂, toComplex_def₂]
exact congr_arg₂ _ rfl (Int.cast_neg _)
|
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 68 | 69 | theorem tsub_tsub_tsub_cancel_right (h : c ≤ b) : a - c - (b - c) = a - b := by |
rw [tsub_tsub, add_tsub_cancel_of_le h]
|
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 52 | 53 | theorem factorThruImage_comp_hom {X Y : A} (f : X ⟶ Y) :
factorThruImage (L.map f) ≫ (iso L f).hom = L.map (factorThruImage f) := by | simp
|
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α M N P : Type*}
namespace Finsupp
variable [DecidableEq α]
section NHasZero
variable [DecidableEq N] [Zero N] (f g : α →₀ N)
def neLocus (f g : α →₀ ... | Mathlib/Data/Finsupp/NeLocus.lean | 42 | 44 | theorem mem_neLocus {f g : α →₀ N} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a := by |
simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,
and_iff_right_iff_imp] using Ne.ne_or_ne _
|
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 86 | 88 | theorem isSimpleModule_iff_isAtom : IsSimpleModule R m ↔ IsAtom m := by |
rw [← Set.isSimpleOrder_Iic_iff_isAtom]
exact m.mapIic.isSimpleOrder_iff
|
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 119 | 123 | theorem imageToKernel_comp_left {Z : V} (h : Z ⟶ A) (w : f ≫ g = 0) :
imageToKernel (h ≫ f) g (by simp [w]) =
Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫ imageToKernel f g w := by |
ext
simp
|
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 161 | 167 | theorem discr_powerBasis_eq_prod (e : Fin pb.dim ≃ (L →ₐ[K] E)) [IsSeparable K L] :
algebraMap K E (discr K pb.basis) =
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, (e j pb.gen - e i pb.gen) ^ 2 := by |
rw [discr_eq_det_embeddingsMatrixReindex_pow_two K E pb.basis e,
embeddingsMatrixReindex_eq_vandermonde, det_transpose, det_vandermonde, ← prod_pow]
congr; ext i
rw [← prod_pow]
|
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 244 | 258 | theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
(v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι := by |
classical
by_contra h
rw [not_le, ← Cardinal.mk_finset_of_infinite ι] at h
let Φ := fun k : κ => (b.repr (v k)).support
obtain ⟨s, w : Infinite ↑(Φ ⁻¹' {s})⟩ := Cardinal.exists_infinite_fiber Φ h (by infer_instance)
let v' := fun k : Φ ⁻¹' {s} => v k
have i' : LinearIndependent R v' := i.comp _ Subtype.v... |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
open s... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 115 | 140 | theorem condexp_restrict_ae_eq_restrict (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
(hs_m : MeasurableSet[m] s) (hf_int : Integrable f μ) :
(μ.restrict s)[f|m] =ᵐ[μ.restrict s] μ[f|m] := by |
have : SigmaFinite ((μ.restrict s).trim hm) := by rw [← restrict_trim hm _ hs_m]; infer_instance
rw [ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m)]
refine EventuallyEq.trans ?_ (condexp_indicator hf_int hs_m)
refine ae_eq_condexp_of_forall_setIntegral_eq hm (hf_int.indicator (hm _ hs_m)) ?_ ?_ ?_
· intro t ... |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 134 | 140 | theorem vars_pow (φ : MvPolynomial σ R) (n : ℕ) : (φ ^ n).vars ⊆ φ.vars := by |
classical
induction' n with n ih
· simp
· rw [pow_succ']
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset (Finset.Subset.refl _) ih
|
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/mathlib"@"1c4e18434eeb5546b212e830b2b39de6a83c473c"
-- Porting note: the lemmas `mk_complex` and `mk_univ_complex` should be in the namespace `Cardinal`
-- like their real counter... | Mathlib/Data/Complex/Cardinality.lean | 35 | 37 | theorem not_countable_complex : ¬(Set.univ : Set ℂ).Countable := by |
rw [← le_aleph0_iff_set_countable, not_le, mk_univ_complex]
apply cantor
|
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 104 | 106 | theorem IsUpperSet.div_right (hs : IsUpperSet s) : IsUpperSet (s / t) := by |
rw [div_eq_mul_inv]
exact hs.mul_right
|
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 112 | 112 | theorem intrinsicInterior_empty : intrinsicInterior 𝕜 (∅ : Set P) = ∅ := by | simp [intrinsicInterior]
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 151 | 154 | theorem mem_thickening_iff {E : Set X} {x : X} : x ∈ thickening δ E ↔ ∃ z ∈ E, dist x z < δ := by |
have key_iff : ∀ z : X, edist x z < ENNReal.ofReal δ ↔ dist x z < δ := fun z ↦ by
rw [dist_edist, lt_ofReal_iff_toReal_lt (edist_ne_top _ _)]
simp_rw [mem_thickening_iff_exists_edist_lt, key_iff]
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 132 | 133 | theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by |
rw [Fintype.card_ofFinset, card_Ioo]
|
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic
namespace CategoryTheory
variable {C : Type*} [Category C] [Precoherent C]
universe w in
lemma isSheaf_coherent (P : Cᵒᵖ ⥤ Type w) :
Presieve.IsSheaf (coheren... | Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean | 44 | 58 | theorem isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (coherentTopology C) (yoneda.obj W) := by |
rw [isSheaf_coherent]
intro X α _ Y π H
have h_colim := isColimitOfEffectiveEpiFamilyStruct Y π H.effectiveEpiFamily.some
rw [← Sieve.generateFamily_eq] at h_colim
intro x hx
let x_ext := Presieve.FamilyOfElements.sieveExtend x
have hx_ext := Presieve.FamilyOfElements.Compatible.sieveExtend hx
let S :=... |
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 | 450 | 451 | theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by |
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
|
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 91 | 94 | theorem card_singleton_inter [DecidableEq α] : ({a} ∩ s).card ≤ 1 := by |
cases' Finset.decidableMem a s with h h
· simp [Finset.singleton_inter_of_not_mem h]
· simp [Finset.singleton_inter_of_mem h]
|
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 167 | 175 | theorem forall₂_comp_perm_eq_perm_comp_forall₂ : Forall₂ r ∘r Perm = Perm ∘r Forall₂ r := by |
funext l₁ l₃; apply propext
constructor
· intro h
rcases h with ⟨l₂, h₁₂, h₂₃⟩
have : Forall₂ (flip r) l₂ l₁ := h₁₂.flip
rcases perm_comp_forall₂ h₂₃.symm this with ⟨l', h₁, h₂⟩
exact ⟨l', h₂.symm, h₁.flip⟩
· exact fun ⟨l₂, h₁₂, h₂₃⟩ => perm_comp_forall₂ h₁₂ h₂₃
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Set.Pointwise.Iterate
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Group.AddCircle
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import dynamics.ergodic.add_circle from "lea... | Mathlib/Dynamics/Ergodic/AddCircle.lean | 104 | 120 | theorem ergodic_zsmul {n : ℤ} (hn : 1 < |n|) : Ergodic fun y : AddCircle T => n • y :=
{ measurePreserving_zsmul volume (abs_pos.mp <| lt_trans zero_lt_one hn) with
ae_empty_or_univ := fun s hs hs' => by
let u : ℕ → AddCircle T := fun j => ↑((↑1 : ℝ) / ↑(n.natAbs ^ j) * T)
replace hn : 1 < n.natAbs :=... | rwa [Int.abs_eq_natAbs, Nat.one_lt_cast] at hn
have hu₀ : ∀ j, addOrderOf (u j) = n.natAbs ^ j := fun j => by
convert addOrderOf_div_of_gcd_eq_one (p := T) (m := 1)
(pow_pos (pos_of_gt hn) j) (gcd_one_left _)
norm_cast
have hnu : ∀ j, n ^ j • u j = 0 := fun j => by
rw [← a... |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 130 | 131 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a := by |
cases a <;> cases b <;> simp [h_comm]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 167 | 167 | theorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by | rw [add_comm, map_add_one, add_comm 1]
|
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 EDist
variable [EDist α] [EDist β]
open scope... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 161 | 164 | theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
edist f g =
(if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) := by |
convert if_pos rfl
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 73 | 74 | theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := by |
rw [← coe_nonempty, coe_Ioo, Set.nonempty_Ioo]
|
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace List
@[simp]
theorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l := by
-- Porting ... | Mathlib/Data/Bool/Count.lean | 60 | 71 | theorem count_not_eq_count (hl : Chain' (· ≠ ·) l) (h2 : Even (length l)) (b : Bool) :
count (!b) l = count b l := by |
cases' l with x l
· rfl
rw [length_cons, Nat.even_add_one, Nat.not_even_iff] at h2
suffices count (!x) (x :: l) = count x (x :: l) by
-- Porting note: old proof is
-- cases b <;> cases x <;> try exact this;
cases b <;> cases x <;>
revert this <;> simp only [Bool.not_false, Bool.not_true] <;> in... |
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 | 338 | 340 | theorem hasStrictDerivAt_const_rpow {a : ℝ} (ha : 0 < a) (x : ℝ) :
HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a) x := by |
simpa using (hasStrictDerivAt_const _ _).rpow (hasStrictDerivAt_id x) ha
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 179 | 180 | theorem natCast_get {x : PartENat} (h : x.Dom) : (x.get h : PartENat) = x := by |
exact Part.ext' (iff_of_true trivial h) fun _ _ => rfl
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 116 | 116 | theorem factorization_zero : factorization 0 = 0 := by | ext; simp [factorization]
|
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Binary
variable (xs : Vector α n) (ys : Vector β n)
@[simp]
theorem mapAccumr₂_mapAccumr_left (f₁ : γ → β → σ₁ → σ₁ × ζ) (f₂ : α → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr f₂... | Mathlib/Data/Vector/MapLemmas.lean | 120 | 130 | theorem mapAccumr₂_mapAccumr₂_left_right
(f₁ : γ → β → σ₁ → σ₁ × φ) (f₂ : α → β → σ₂ → σ₂ × γ) :
(mapAccumr₂ f₁ (mapAccumr₂ f₂ xs ys s₂).snd ys s₁)
= let m := mapAccumr₂ (fun x y (s₁, s₂) =>
let r₂ := f₂ x y s₂
let r₁ := f₁ r₂.snd y s₁
((r₁.fst, r₂.fst), r₁.sn... |
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
|
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 57 | 59 | theorem coprime_div_gcd_div_gcd
(H : 0 < gcd m n) : Coprime (m / gcd m n) (n / gcd m n) := by |
rw [coprime_iff_gcd_eq_one, gcd_div (gcd_dvd_left m n) (gcd_dvd_right m n), Nat.div_self H]
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 99 | 100 | theorem interior_setOf_im_le (a : ℝ) : interior { z : ℂ | z.im ≤ a } = { z | z.im < a } := by |
simpa only [interior_Iic] using interior_preimage_im (Iic a)
|
import Mathlib.MeasureTheory.Integral.Lebesgue
open Set hiding restrict restrict_apply
open Filter ENNReal NNReal MeasureTheory.Measure
namespace MeasureTheory
variable {α : Type*} {m0 : MeasurableSpace α} {μ : Measure α}
noncomputable
def Measure.withDensity {m : MeasurableSpace α} (μ : Measure α) (f : α → ℝ≥... | Mathlib/MeasureTheory/Measure/WithDensity.lean | 122 | 127 | theorem withDensity_smul (r : ℝ≥0∞) {f : α → ℝ≥0∞} (hf : Measurable f) :
μ.withDensity (r • f) = r • μ.withDensity f := by |
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf]
simp only [Pi.smul_apply, smul_eq_mul]
|
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 87 | 101 | theorem noetherianSpace_TFAE :
TFAE [NoetherianSpace α,
WellFounded fun s t : Closeds α => s < t,
∀ s : Set α, IsCompact s,
∀ s : Opens α, IsCompact (s : Set α)] := by |
tfae_have 1 ↔ 2
· refine (noetherianSpace_iff α).trans (Opens.compl_bijective.2.wellFounded_iff ?_)
exact (@OrderIso.compl (Set α)).lt_iff_lt.symm
tfae_have 1 ↔ 4
· exact noetherianSpace_iff_opens α
tfae_have 1 → 3
· exact @NoetherianSpace.isCompact α _
tfae_have 3 → 4
· exact fun h s => h s
tfae... |
import Mathlib.Mathport.Rename
set_option autoImplicit true
namespace Thunk
#align thunk.mk Thunk.mk
-- Porting note: Added `Thunk.ext` to get `ext` tactic to work.
@[ext]
| Mathlib/Lean/Thunk.lean | 20 | 24 | theorem ext {α : Type u} {a b : Thunk α} (eq : a.get = b.get) : a = b := by |
have ⟨_⟩ := a
have ⟨_⟩ := b
congr
exact funext fun _ ↦ eq
|
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 93 | 132 | theorem exists_int_int_abs_mul_sub_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ j k : ℤ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - j| ≤ 1 / (n + 1) := by |
let f : ℤ → ℤ := fun m => ⌊fract (ξ * m) * (n + 1)⌋
have hn : 0 < (n : ℝ) + 1 := mod_cast Nat.succ_pos _
have hfu := fun m : ℤ => mul_lt_of_lt_one_left hn <| fract_lt_one (ξ * ↑m)
conv in |_| ≤ _ => rw [mul_comm, le_div_iff hn, ← abs_of_pos hn, ← abs_mul]
let D := Icc (0 : ℤ) n
by_cases H : ∃ m ∈ D, f m = ... |
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.Lattice
import Mathlib.Order.CompleteLattice
#align_import category_theory.sites.sieves from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef... | Mathlib/CategoryTheory/Sites/Sieves.lean | 164 | 176 | theorem ofArrows_bind {ι : Type*} (Z : ι → C) (g : ∀ i : ι, Z i ⟶ X)
(j : ∀ ⦃Y⦄ (f : Y ⟶ X), ofArrows Z g f → Type*) (W : ∀ ⦃Y⦄ (f : Y ⟶ X) (H), j f H → C)
(k : ∀ ⦃Y⦄ (f : Y ⟶ X) (H i), W f H i ⟶ Y) :
((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) =
ofArrows (fun i : Σi, j _ (ofArrows.mk ... |
funext Y
ext f
constructor
· rintro ⟨_, _, _, ⟨i⟩, ⟨i'⟩, rfl⟩
exact ofArrows.mk (Sigma.mk _ _)
· rintro ⟨i⟩
exact bind_comp _ (ofArrows.mk _) (ofArrows.mk _)
|
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 | 143 | 148 | theorem chain'_succ (n m : ℕ) : Chain' (fun a b => b = succ a) (Ico n m) := by |
by_cases h : n < m
· rw [eq_cons h]
exact chain_succ_range' _ _ 1
· rw [eq_nil_of_le (le_of_not_gt h)]
trivial
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
#align_import measure_theory.function.locally_integrable from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49de8c0d677b1b"
open MeasureTheory MeasureTheory.Measure Set Function TopologicalSpace Bornology
open scoped Topology Interval ENNReal
variabl... | Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | 57 | 62 | theorem LocallyIntegrableOn.mono (hf : LocallyIntegrableOn f s μ) {g : X → F}
(hg : AEStronglyMeasurable g μ) (h : ∀ᵐ x ∂μ, ‖g x‖ ≤ ‖f x‖) :
LocallyIntegrableOn g s μ := by |
intro x hx
rcases hf x hx with ⟨t, t_mem, ht⟩
exact ⟨t, t_mem, Integrable.mono ht hg.restrict (ae_restrict_of_ae h)⟩
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 132 | 135 | theorem generateFrom_eq_prod {C : Set (Set α)} {D : Set (Set β)} (hC : generateFrom C = ‹_›)
(hD : generateFrom D = ‹_›) (h2C : IsCountablySpanning C) (h2D : IsCountablySpanning D) :
generateFrom (image2 (· ×ˢ ·) C D) = Prod.instMeasurableSpace := by |
rw [← hC, ← hD, generateFrom_prod_eq h2C h2D]
|
import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
universe u v w x
structure PEquiv (α : Type u) (β : Type v) where
toFun : α → Option β
invFun : β → Option α
inv : ∀ (a : α) (b :... | Mathlib/Data/PEquiv.lean | 169 | 170 | theorem refl_trans (f : α ≃. β) : (PEquiv.refl α).trans f = f := by |
ext; dsimp [PEquiv.trans]; rfl
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variab... | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 43 | 49 | theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
(hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
HasStrictFDerivAt f f' x := by |
rcases hf 1 hn with ⟨u, H, p, hp⟩
simp only [nhdsWithin_univ, mem_univ, insert_eq_of_mem] at H
have := hp.hasStrictFDerivAt le_rfl H
rwa [hf'.unique this.hasFDerivAt]
|
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.Module.StrongTopology
#align_import analysis.normed_space.compact_operator from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Function Set Filter Bornology Metric Pointwise Topology
def IsCompactOperat... | Mathlib/Analysis/NormedSpace/CompactOperator.lean | 336 | 365 | theorem IsCompactOperator.continuous {f : M₁ →ₛₗ[σ₁₂] M₂} (hf : IsCompactOperator f) :
Continuous f := by |
letI : UniformSpace M₂ := TopologicalAddGroup.toUniformSpace _
haveI : UniformAddGroup M₂ := comm_topologicalAddGroup_is_uniform
-- Since `f` is linear, we only need to show that it is continuous at zero.
-- Let `U` be a neighborhood of `0` in `M₂`.
refine continuous_of_continuousAt_zero f fun U hU => ?_
r... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 73 | 75 | theorem lifts_iff_coeff_lifts (p : S[X]) : p ∈ lifts f ↔ ∀ n : ℕ, p.coeff n ∈ Set.range f := by |
rw [lifts_iff_ringHom_rangeS, mem_map_rangeS f]
rfl
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 88 | 92 | theorem degrees_monomial (s : σ →₀ ℕ) (a : R) : degrees (monomial s a) ≤ toMultiset s := by |
classical
refine (supDegree_single s a).trans_le ?_
split_ifs
exacts [bot_le, le_rfl]
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 154 | 155 | theorem edgeDensity_empty_left (t : Finset β) : edgeDensity r ∅ t = 0 := by |
rw [edgeDensity, Finset.card_empty, Nat.cast_zero, zero_mul, div_zero]
|
import Mathlib.CategoryTheory.Adjunction.Basic
open CategoryTheory
variable {C D : Type*} [Category C] [Category D]
namespace CategoryTheory.Adjunction
@[simps]
def natTransEquiv {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G') :
(G ⟶ G') ≃ (F' ⟶ F) where
toFun f := {
app := fun X ↦ F'.map... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 117 | 119 | theorem homEquiv_leftAdjointUniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
(x : C) : adj1.homEquiv _ _ ((leftAdjointUniq adj1 adj2).hom.app x) = adj2.unit.app x := by |
simp [leftAdjointUniq]
|
import Mathlib.CategoryTheory.Iso
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.comma from "leanprover-community/mathlib"@"8a318021995877a44630c898d0b2bc376fceef3b"
namespace CategoryTheory
open Category
-- declare the `v`'s first; see `Catego... | Mathlib/CategoryTheory/Comma/Basic.lean | 173 | 176 | theorem eqToHom_right (X Y : Comma L R) (H : X = Y) :
CommaMorphism.right (eqToHom H) = eqToHom (by cases H; rfl) := by |
cases H
rfl
|
import Mathlib.Data.Vector.Basic
set_option autoImplicit true
namespace Vector
def snoc : Vector α n → α → Vector α (n+1) :=
fun xs x => append xs (x ::ᵥ Vector.nil)
section Simp
variable (xs : Vector α n)
@[simp]
theorem snoc_cons : (x ::ᵥ xs).snoc y = x ::ᵥ (xs.snoc y) :=
rfl
@[simp]
theorem snoc_nil... | Mathlib/Data/Vector/Snoc.lean | 42 | 45 | theorem reverse_cons : reverse (x ::ᵥ xs) = (reverse xs).snoc x := by |
cases xs
simp only [reverse, cons, toList_mk, List.reverse_cons, snoc]
congr
|
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.Analysis.Analytic.RadiusLiminf
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.Expon... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 176 | 178 | theorem spectralRadius_le_nnnorm [NormOneClass A] (a : A) : spectralRadius 𝕜 a ≤ ‖a‖₊ := by |
refine iSup₂_le fun k hk => ?_
exact mod_cast norm_le_norm_of_mem hk
|
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 43 | 57 | theorem mem_sumLift₂ :
c ∈ sumLift₂ f g a b ↔
(∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f a₁ b₁) ∨
∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g a₂ b₂ := by |
constructor
· cases' a with a a <;> cases' b with b b
· rw [sumLift₂, mem_map]
rintro ⟨c, hc, rfl⟩
exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩
· refine fun h ↦ (not_mem_empty _ h).elim
· refine fun h ↦ (not_mem_empty _ h).elim
· rw [sumLift₂, mem_map]
rintro ⟨c, hc, rfl⟩
exact... |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 140 | 146 | theorem permutations'Aux_eq_permutationsAux2 (t : α) (ts : List α) :
permutations'Aux t ts = (permutationsAux2 t [] [ts ++ [t]] ts id).2 := by |
induction' ts with a ts ih; · rfl
simp only [permutations'Aux, ih, cons_append, permutationsAux2_snd_cons, append_nil, id_eq,
cons.injEq, true_and]
simp (config := { singlePass := true }) only [← permutationsAux2_append]
simp [map_permutationsAux2]
|
import Mathlib.Algebra.Order.Nonneg.Ring
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Int.Lemmas
#align_import data.rat.nnrat from "leanprover-community/mathlib"@"b3f4f007a962e3787aa0f3b5c7942a1317f7d88e"
open Function
deriving instance CanonicallyOrderedCommSemiring for NNRat
deriving instance Cano... | Mathlib/Data/NNRat/Defs.lean | 142 | 142 | theorem coe_eq_zero : (q : ℚ) = 0 ↔ q = 0 := by | norm_cast
|
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.Finiteness
open scoped TensorProduct
open Submodule
variable {R M N : Type*}
variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N]
variable {M₁ M₂ : Submodule R M} {N₁ N₂ : Submodule R N}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean | 65 | 75 | theorem exists_finsupp_left (x : M ⊗[R] N) :
∃ S : M →₀ N, x = S.sum fun m n ↦ m ⊗ₜ[R] n := by |
induction x using TensorProduct.induction_on with
| zero => exact ⟨0, by simp⟩
| tmul x y => exact ⟨Finsupp.single x y, by simp⟩
| add x y hx hy =>
obtain ⟨Sx, hx⟩ := hx
obtain ⟨Sy, hy⟩ := hy
use Sx + Sy
rw [hx, hy]
exact (Finsupp.sum_add_index' (by simp) TensorProduct.tmul_add).symm
|
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 227 | 234 | theorem widePullback_ext {B : C} {ι : Type w} {X : ι → C} (f : ∀ j : ι, X j ⟶ B)
[HasWidePullback B X f] [PreservesLimit (wideCospan B X f) (forget C)]
(x y : ↑(widePullback B X f)) (h₀ : base f x = base f y) (h : ∀ j, π f j x = π f j y) :
x = y := by |
apply Concrete.limit_ext
rintro (_ | j)
· exact h₀
· apply h
|
import Mathlib.Data.PNat.Basic
#align_import data.pnat.find from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
namespace PNat
variable {p q : ℕ+ → Prop} [DecidablePred p] [DecidablePred q] (h : ∃ n, p n)
instance decidablePredExistsNat : DecidablePred fun n' : ℕ => ∃ (n : ℕ+) (_ : n'... | Mathlib/Data/PNat/Find.lean | 86 | 87 | theorem find_le_iff (n : ℕ+) : PNat.find h ≤ n ↔ ∃ m ≤ n, p m := by |
simp only [exists_prop, ← lt_add_one_iff, find_lt_iff]
|
import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
∃ r, r ≠ 0 ∧ ∀ x ∈ s, ∀ y ∈... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 115 | 117 | theorem finite_iUnion_right_iff {ι : Type*} {I : Set ι} (hI : I.Finite) {s : Set X}
{t : ι → Set X} : IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i) := by |
simpa only [@comm _ _ s] using finite_iUnion_left_iff hI
|
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 UnionIxx
variable [LinearOrder α] {s ... | Mathlib/Order/Interval/Set/Disjoint.lean | 201 | 205 | theorem IsGLB.biUnion_Ioi_eq (h : IsGLB s a) : ⋃ x ∈ s, Ioi x = Ioi a := by |
refine (iUnion₂_subset fun x hx => ?_).antisymm fun x hx => ?_
· exact Ioi_subset_Ioi (h.1 hx)
· rcases h.exists_between hx with ⟨y, hys, _, hyx⟩
exact mem_biUnion hys hyx
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
variable {α : Type*}
-- Porting note: mathlib3 had a `@[protect_proj]` here, but adding `protect... | Mathlib/Algebra/GCDMonoid/Basic.lean | 162 | 166 | theorem normUnit_mul_normUnit (a : α) : normUnit (a * normUnit a) = 1 := by |
nontriviality α using Subsingleton.elim a 0
obtain rfl | h := eq_or_ne a 0
· rw [normUnit_zero, zero_mul, normUnit_zero]
· rw [normUnit_mul h (Units.ne_zero _), normUnit_coe_units, mul_inv_eq_one]
|
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE
#align_import algebraic_geometry.morphisms.basic from "leanprover-community/mathlib"@"434e2fd21c1900747afc6d13d8be7f4eedba7218"
set_option lin... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 99 | 101 | theorem affine_cancel_left_isIso {P : AffineTargetMorphismProperty} (hP : P.toProperty.RespectsIso)
{X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] [IsAffine Z] : P (f ≫ g) ↔ P g := by |
rw [← P.toProperty_apply, ← P.toProperty_apply, hP.cancel_left_isIso]
|
import Batteries.Data.List.Lemmas
import Batteries.Tactic.Classical
import Mathlib.Tactic.TypeStar
import Mathlib.Mathport.Rename
#align_import data.list.tfae from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
namespace List
def TFAE (l : List Prop) : Prop :=
∀ x ∈ l, ∀ y ∈ l, x ↔ ... | Mathlib/Data/List/TFAE.lean | 91 | 96 | theorem forall_tfae {α : Type*} (l : List (α → Prop)) (H : ∀ a : α, (l.map (fun p ↦ p a)).TFAE) :
(l.map (fun p ↦ ∀ a, p a)).TFAE := by |
simp only [TFAE, List.forall_mem_map_iff]
intros p₁ hp₁ p₂ hp₂
exact forall_congr' fun a ↦ H a (p₁ a) (mem_map_of_mem (fun p ↦ p a) hp₁)
(p₂ a) (mem_map_of_mem (fun p ↦ p a) hp₂)
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 46 | 48 | theorem eraseLead_coeff (i : ℕ) :
f.eraseLead.coeff i = if i = f.natDegree then 0 else f.coeff i := by |
simp only [eraseLead, coeff_erase]
|
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 146 | 156 | theorem sup_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A.topologicalClosure := by |
rw [sup_eq_half_smul_add_add_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.add_mem
(A.topologicalClosure.add_mem (A.le_topologicalClosure f.property)
(A.le_topologicalClosure g.property))
?_)
_
exact mod_cast abs_mem_subalgebra_closure A _
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 46 | 47 | theorem mk_mem_sym2_iff : s(a, b) ∈ s.sym2 ↔ a ∈ s ∧ b ∈ s := by |
rw [mem_mk, sym2_val, Multiset.mk_mem_sym2_iff, mem_mk, mem_mk]
|
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
open Asymptotics MeasureTheory Set Filter
variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
{f : α → E} {g : α → F} {a b :... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 47 | 50 | theorem _root_.Asymptotics.IsBigO.integrable (hfm : AEStronglyMeasurable f μ)
(hf : f =O[⊤] g) (hg : Integrable g μ) : Integrable f μ := by |
rewrite [← integrableAtFilter_top] at *
exact hf.integrableAtFilter ⟨univ, univ_mem, hfm.restrict⟩ hg
|
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 53 | 54 | theorem normed_neg (f : ContDiffBump (0 : E)) (x : E) : f.normed μ (-x) = f.normed μ x := by |
simp_rw [f.normed_def, f.neg]
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 107 | 109 | theorem norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin :
‖iteratedFDerivWithin 𝕜 n f s x‖ = ‖iteratedDerivWithin n f s x‖ := by |
rw [iteratedDerivWithin_eq_equiv_comp, Function.comp_apply, LinearIsometryEquiv.norm_map]
|
import Mathlib.Analysis.Calculus.Deriv.Add
#align_import analysis.calculus.local_extr from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v
open Filter Set
open scoped Topology Classical
section Module
variable {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E ... | Mathlib/Analysis/Calculus/LocalExtr/Basic.lean | 99 | 101 | theorem mem_posTangentConeAt_of_segment_subset' {s : Set E} {x y : E}
(h : segment ℝ x (x + y) ⊆ s) : y ∈ posTangentConeAt s x := by |
simpa only [add_sub_cancel_left] using mem_posTangentConeAt_of_segment_subset h
|
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 114 | 120 | theorem div_dvd_of_dvd {p q : R} (hpq : q ∣ p) : p / q ∣ p := by |
by_cases hq : q = 0
· rw [hq, zero_dvd_iff] at hpq
rw [hpq]
exact dvd_zero _
use q
rw [mul_comm, ← EuclideanDomain.mul_div_assoc _ hpq, mul_comm, mul_div_cancel_right₀ _ hq]
|
import Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves
namespace CategoryTheory.regularTopology
open Limits
variable {C : Type*} [Category C] [Preregular C] {X : C}
| Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean | 30 | 41 | theorem mem_sieves_of_hasEffectiveEpi (S : Sieve X) :
(∃ (Y : C) (π : Y ⟶ X), EffectiveEpi π ∧ S.arrows π) → (S ∈ (regularTopology C).sieves X) := by |
rintro ⟨Y, π, h⟩
have h_le : Sieve.generate (Presieve.ofArrows (fun () ↦ Y) (fun _ ↦ π)) ≤ S := by
rw [Sieve.sets_iff_generate (Presieve.ofArrows _ _) S]
apply Presieve.le_of_factorsThru_sieve (Presieve.ofArrows _ _) S _
intro W g f
refine ⟨W, 𝟙 W, ?_⟩
cases f
exact ⟨π, ⟨h.2, Category.id_c... |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 60 | 65 | theorem inl_mem_sumLift₂ {c₁ : γ₁} :
inl c₁ ∈ sumLift₂ f g a b ↔ ∃ a₁ b₁, a = inl a₁ ∧ b = inl b₁ ∧ c₁ ∈ f a₁ b₁ := by |
rw [mem_sumLift₂, or_iff_left]
· simp only [inl.injEq, exists_and_left, exists_eq_left']
rintro ⟨_, _, c₂, _, _, h, _⟩
exact inl_ne_inr h
|
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 222 | 223 | theorem vars_map_of_injective (hf : Injective f) : (map f p).vars = p.vars := by |
simp [vars, degrees_map_of_injective _ hf]
|
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