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
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import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 82 | 89 | theorem continuous_left_toIocMod : ContinuousWithinAt (toIocMod hp a) (Iic x) x := by |
rw [(funext fun y => Eq.trans (by rw [neg_neg]) <| toIocMod_neg _ _ _ :
toIocMod hp a = (fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg)]
-- Porting note: added
have : ContinuousNeg 𝕜 := TopologicalAddGroup.toContinuousNeg
exact
(continuous_sub_left _).continuousAt.comp_continuousWithinAt <|
(co... | false |
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import... | Mathlib/Analysis/Calculus/MeanValue.lean | 92 | 124 | theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Ic... |
change Icc a b ⊆ { x | f x ≤ B x }
set s := { x | f x ≤ B x } ∩ Icc a b
have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
have : IsClosed s := by
simp only [s, inter_comm]
exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
apply this.Icc_subset_of_fo... | false |
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 27 | 72 | theorem exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [Nonempty M] [Semigroup M]
[TopologicalSpace M] [CompactSpace M] [T2Space M]
(continuous_mul_left : ∀ r : M, Continuous (· * r)) : ∃ m : M, m * m = m := by |
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`.
It will turn out that any minimal element is `{m}` for an idempotent `m : M`. -/
let S : Set (Set M) :=
{ N | IsClosed N ∧ N.Nonempty ∧ ∀ (m) (_ : m ∈ N) (m') (_ : m' ∈ N), m * m' ∈ N }
rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul... | false |
import Mathlib.MeasureTheory.Measure.Dirac
set_option autoImplicit true
open Set
open scoped ENNReal Classical
variable [MeasurableSpace α] [MeasurableSpace β] {s : Set α}
noncomputable section
namespace MeasureTheory.Measure
def count : Measure α :=
sum dirac
#align measure_theory.measure.count MeasureTheo... | Mathlib/MeasureTheory/Measure/Count.lean | 39 | 40 | theorem count_apply (hs : MeasurableSet s) : count s = ∑' i : s, 1 := by |
simp only [count, sum_apply, hs, dirac_apply', ← tsum_subtype s (1 : α → ℝ≥0∞), Pi.one_apply]
| false |
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 117 | 122 | theorem moveRight_card {b : Board} {m : ℤ × ℤ} (h : m ∈ right b) :
Finset.card (moveRight b m) + 2 = Finset.card b := by |
dsimp [moveRight]
rw [Finset.card_erase_of_mem (fst_pred_mem_erase_of_mem_right h)]
rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]
exact tsub_add_cancel_of_le (card_of_mem_right h)
| false |
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 133 | 152 | theorem circleTransformDeriv_bound {R : ℝ} (hR : 0 < R) {z x : ℂ} {f : ℂ → ℂ} (hx : x ∈ ball z R)
(hf : ContinuousOn f (sphere z R)) : ∃ B ε : ℝ, 0 < ε ∧
ball x ε ⊆ ball z R ∧ ∀ (t : ℝ), ∀ y ∈ ball x ε, ‖circleTransformDeriv R z y f t‖ ≤ B := by |
obtain ⟨r, hr, hrx⟩ := exists_lt_mem_ball_of_mem_ball hx
obtain ⟨ε', hε', H⟩ := exists_ball_subset_ball hrx
obtain ⟨⟨⟨a, b⟩, ⟨ha, hb⟩⟩, hab⟩ :=
abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z
let V : ℝ → ℂ → ℂ := fun θ w => circleTransformDeriv R z w (fun _ => 1) θ
obtain ⟨X, -, HX2⟩... | false |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 70 | 72 | theorem gold_add_goldConj : φ + ψ = 1 := by |
rw [goldenRatio, goldenConj]
ring
| false |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 49 | 52 | theorem trinomial_leading_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff n = w := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]
| false |
import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 116 | 117 | theorem second_iso_of_eq {x y a b : X} (hm : IsMaximal x a) (ha : x ⊔ y = a) (hb : x ⊓ y = b) :
Iso (x, a) (b, y) := by | substs a b; exact second_iso hm
| false |
import Mathlib.RepresentationTheory.Rep
import Mathlib.Algebra.Category.FGModuleCat.Limits
import Mathlib.CategoryTheory.Preadditive.Schur
import Mathlib.RepresentationTheory.Basic
#align_import representation_theory.fdRep from "leanprover-community/mathlib"@"19a70dceb9dff0994b92d2dd049de7d84d28112b"
suppress_comp... | Mathlib/RepresentationTheory/FdRep.lean | 95 | 100 | theorem Iso.conj_ρ {V W : FdRep k G} (i : V ≅ W) (g : G) :
W.ρ g = (FdRep.isoToLinearEquiv i).conj (V.ρ g) := by |
-- Porting note: Changed `rw` to `erw`
erw [FdRep.isoToLinearEquiv, ← FGModuleCat.Iso.conj_eq_conj, Iso.conj_apply]
rw [Iso.eq_inv_comp ((Action.forget (FGModuleCat k) (MonCat.of G)).mapIso i)]
exact (i.hom.comm g).symm
| false |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 120 | 125 | theorem expChar_one_iff_char_zero (p q : ℕ) [CharP R p] [ExpChar R q] : q = 1 ↔ p = 0 := by |
constructor
· rintro rfl
exact char_zero_of_expChar_one R p
· rintro rfl
exact expChar_one_of_char_zero R q
| false |
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 | 81 | 85 | theorem LocallyIntegrableOn.integrableOn_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
(hf : LocallyIntegrableOn f (Iic a) μ) (ho : f =O[atBot] g)
(hg : IntegrableAtFilter g atBot μ) : IntegrableOn f (Iic a) μ := by |
refine integrableOn_Iic_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact ⟨Iic a, Iic_mem_atBot a, hf.aestronglyMeasurable⟩
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 57 | 59 | theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero] |
rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
exact Real.rpow_eq_zero_iff_of_nonneg x.2
| true |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 103 | 114 | theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
(ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
Continuous (gaugeRescale s t) := by
have hta : Absorbent ℝ t := absorbent_nhds_zero ht₀ |
have hta : Absorbent ℝ t := absorbent_nhds_zero ht₀
refine continuous_iff_continuousAt.2 fun x ↦ ?_
rcases eq_or_ne x 0 with rfl | hx
· rw [ContinuousAt, gaugeRescale_zero]
nth_rewrite 2 [← comap_gauge_nhds_zero htb ht₀]
simp only [tendsto_comap_iff, (· ∘ ·), gauge_gaugeRescale _ hta htb]
exact ten... | true |
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
#align_import analysis.inner_product_space.orientation from "leanprover-community/mathlib"@"bd65478311e4dfd41f48bf38c7e3b02fb75d0163"
noncomputable section
variable {E : Type*} [NormedAddCommGroup E] [InnerProduct... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 91 | 96 | theorem det_eq_neg_det_of_opposite_orientation (h : e.toBasis.orientation ≠ f.toBasis.orientation) :
e.toBasis.det = -f.toBasis.det := by
rw [e.toBasis.det.eq_smul_basis_det f.toBasis] |
rw [e.toBasis.det.eq_smul_basis_det f.toBasis]
-- Porting note: added `neg_one_smul` with explicit type
simp [e.det_to_matrix_orthonormalBasis_of_opposite_orientation f h,
neg_one_smul ℝ (M := E [⋀^ι]→ₗ[ℝ] ℝ)]
| true |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 128 | 132 | theorem vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan {s : Set P} {p : P} {v : V}
(hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s) : v +ᵥ p ∈ spanPoints k s := by
rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩ |
rcases hp with ⟨p2, ⟨hp2, ⟨v2, ⟨hv2, hv2p⟩⟩⟩⟩
rw [hv2p, vadd_vadd]
exact ⟨p2, hp2, v + v2, (vectorSpan k s).add_mem hv hv2, rfl⟩
| true |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Hom
open Set Pointwise
| Mathlib/GroupTheory/GroupAction/Pointwise.lean | 33 | 41 | theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) := by
lift m to Mˣ using hm |
lift m to Mˣ using hm
rw [Function.bijective_iff_has_inverse]
use fun a ↦ m⁻¹ • a
constructor
· intro x; simp [← Units.smul_def]
· intro x; simp [← Units.smul_def]
| true |
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
universe u v
open Nat Function
variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α ... | Mathlib/Data/List/Nodup.lean | 39 | 40 | theorem nodup_cons {a : α} {l : List α} : Nodup (a :: l) ↔ a ∉ l ∧ Nodup l := by |
simp only [Nodup, pairwise_cons, forall_mem_ne]
| true |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 67 | 68 | theorem unitors_inv_equal : (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv := by |
coherence
| true |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
noncomputable section
open Function Polynomial Finsupp Finset
open scoped Polynomial
namespace ... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 111 | 114 | theorem trailingDegree_eq_iff_natTrailingDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) :
p.trailingDegree = n ↔ p.natTrailingDegree = n := by
rw [trailingDegree_eq_natTrailingDegree hp] |
rw [trailingDegree_eq_natTrailingDegree hp]
exact WithTop.coe_eq_coe
| true |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 67 | 77 | theorem ultrafilter_converges_iff {u : Ultrafilter (Ultrafilter α)} {x : Ultrafilter α} :
↑u ≤ 𝓝 x ↔ x = joinM u := by
rw [eq_comm, ← Ultrafilter.coe_le_coe] |
rw [eq_comm, ← Ultrafilter.coe_le_coe]
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, { v : Ultrafilter α | s ∈ v } ∈ u
simp only [TopologicalSpace.nhds_generateFrom, le_iInf_iff, ultrafilterBasis, le_principal_iff,
mem_setOf_eq]
constructor
· intro h a ha
exact h _ ⟨ha, a, rfl⟩
· rintro h a ⟨xi, a, rfl⟩
exact h ... | true |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.function.special_functions.inner from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
variable {α : Type*} {𝕜 : Type*} {E : Type*}
variable [RCLike ... | Mathlib/MeasureTheory/Function/SpecialFunctions/Inner.lean | 41 | 47 | theorem AEMeasurable.inner {m : MeasurableSpace α} [MeasurableSpace E] [OpensMeasurableSpace E]
[SecondCountableTopology E] {μ : MeasureTheory.Measure α} {f g : α → E}
(hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : AEMeasurable (fun x => ⟪f x, g x⟫) μ := by
refine ⟨fun x => ⟪hf.mk f x, hg.mk g x⟫, hf.measu... |
refine ⟨fun x => ⟪hf.mk f x, hg.mk g x⟫, hf.measurable_mk.inner hg.measurable_mk, ?_⟩
refine hf.ae_eq_mk.mp (hg.ae_eq_mk.mono fun x hxg hxf => ?_)
dsimp only
congr
| 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 | 69 | 78 | theorem hyperoperation_two : hyperoperation 2 = (· * ·) := by
ext m k |
ext m k
induction' k with bn bih
· rw [hyperoperation]
exact (Nat.mul_zero m).symm
· rw [hyperoperation_recursion, hyperoperation_one, bih]
-- Porting note: was `ring`
dsimp only
nth_rewrite 1 [← mul_one m]
rw [← mul_add, add_comm]
| true |
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
#align_import model_theory.elementary_maps from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open FirstOrder
namespace FirstOrder
namespace Language
open Structure
variable (L : Language) (M : Type*) (N : T... | Mathlib/ModelTheory/ElementaryMaps.lean | 78 | 94 | theorem map_boundedFormula (f : M ↪ₑ[L] N) {α : Type*} {n : ℕ} (φ : L.BoundedFormula α n)
(v : α → M) (xs : Fin n → M) : φ.Realize (f ∘ v) (f ∘ xs) ↔ φ.Realize v xs := by
classical |
classical
rw [← BoundedFormula.realize_restrictFreeVar Set.Subset.rfl, Set.inclusion_eq_id, iff_eq_eq]
have h :=
f.map_formula' ((φ.restrictFreeVar id).toFormula.relabel (Fintype.equivFin _))
(Sum.elim (v ∘ (↑)) xs ∘ (Fintype.equivFin _).symm)
simp only [Formula.realize_relabel, BoundedForm... | 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 | 80 | 83 | theorem affineSegment_image (f : P →ᵃ[R] P') (x y : P) :
f '' affineSegment R x y = affineSegment R (f x) (f y) := by
rw [affineSegment, affineSegment, Set.image_image, ← comp_lineMap] |
rw [affineSegment, affineSegment, Set.image_image, ← comp_lineMap]
rfl
| true |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 148 | 149 | theorem catalan_three : catalan 3 = 5 := by |
norm_num [catalan_eq_centralBinom_div, Nat.centralBinom, Nat.choose]
| true |
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Int.Order.Lemmas
#align_import group_theory.submonoid.membership fro... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 262 | 265 | theorem mem_sSup_of_mem {S : Set (Submonoid M)} {s : Submonoid M} (hs : s ∈ S) :
∀ {x : M}, x ∈ s → x ∈ sSup S := by
rw [← SetLike.le_def] |
rw [← SetLike.le_def]
exact le_sSup hs
| true |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 312 | 320 | theorem next_toList_eq_apply (p : Perm α) (x y : α) (hy : y ∈ toList p x) :
next (toList p x) y hy = p y := by
rw [mem_toList_iff] at hy |
rw [mem_toList_iff] at hy
obtain ⟨k, hk, hk'⟩ := hy.left.exists_pow_eq_of_mem_support hy.right
rw [← nthLe_toList p x k (by simpa using hk)] at hk'
simp_rw [← hk']
rw [next_nthLe _ (nodup_toList _ _), nthLe_toList, nthLe_toList, ← mul_apply, ← pow_succ',
length_toList, ← pow_mod_orderOf_cycleOf_apply p (... | true |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.arctan from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Real
open Set Filter
open scoped Topology Real
theorem tan_add {x y : ℝ}
... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 47 | 49 | theorem tan_two_mul {x : ℝ} : tan (2 * x) = 2 * tan x / (1 - tan x ^ 2) := by
have := @Complex.tan_two_mul x |
have := @Complex.tan_two_mul x
norm_cast at *
| true |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.ContinuedFractions.Basic
#align_import algebra.continued_fractions.computation.basic from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
-- Fix a carrier `K`.
variable (K : Type*)
structu... | Mathlib/Algebra/ContinuedFractions/Computation/Basic.lean | 159 | 161 | theorem stream_isSeq (v : K) : (IntFractPair.stream v).IsSeq := by
intro _ hyp |
intro _ hyp
simp [IntFractPair.stream, hyp]
| true |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.Linarith
#align_import set_theory.cardinal.ordinal from "leanprover-community/mathlib"@"7c2ce0c2da15516b4e65d0c9e254bb6dc93abd1f"
noncomputa... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 500 | 543 | theorem mul_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : c * c = c := by
refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c) |
refine le_antisymm ?_ (by simpa only [mul_one] using mul_le_mul_left' (one_le_aleph0.trans h) c)
-- the only nontrivial part is `c * c ≤ c`. We prove it inductively.
refine Acc.recOn (Cardinal.lt_wf.apply c) (fun c _ => Quotient.inductionOn c fun α IH ol => ?_) h
-- consider the minimal well-order `r` on `α` (... | true |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 188 | 188 | theorem half_sub (a : R) : a / 2 - a = -(a / 2) := by | rw [← neg_sub, sub_half]
| true |
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {R M M₂ M₃ : Type*}
variable [Ring R] [AddCommGroup M] [AddCommGroup M₂] [AddCommGroup M₃]
variable [Module R M] [Module R M₂] [Modul... | Mathlib/LinearAlgebra/Isomorphisms.lean | 81 | 85 | theorem quotientInfEquivSupQuotient_injective (p p' : Submodule R M) :
Function.Injective (quotientInfToSupQuotient p p') := by
rw [← ker_eq_bot, quotientInfToSupQuotient, ker_liftQ_eq_bot] |
rw [← ker_eq_bot, quotientInfToSupQuotient, ker_liftQ_eq_bot]
rw [ker_comp, ker_mkQ]
exact fun ⟨x, hx1⟩ hx2 => ⟨hx1, hx2⟩
| true |
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function Set
variable {α : Type*} (r r₁ r₂ : α → α → Prop) (s... | Mathlib/Order/Minimal.lean | 121 | 128 | theorem minimals_eq_minimals_of_subset_of_forall [IsTrans α r] (hts : t ⊆ s)
(h : ∀ x ∈ s, ∃ y ∈ t, r y x) : minimals r s = minimals r t := by
refine Set.ext fun a ↦ ⟨fun ⟨has, hmin⟩ ↦ ⟨?_,fun b hbt ↦ hmin (hts hbt)⟩, |
refine Set.ext fun a ↦ ⟨fun ⟨has, hmin⟩ ↦ ⟨?_,fun b hbt ↦ hmin (hts hbt)⟩,
fun ⟨hat, hmin⟩ ↦ ⟨hts hat, fun b hbs hba ↦ ?_⟩⟩
· obtain ⟨a', ha', haa'⟩ := h _ has
rwa [antisymm (hmin (hts ha') haa') haa']
obtain ⟨b', hb't, hb'b⟩ := h b hbs
rwa [antisymm (hmin hb't (Trans.trans hb'b hba)) (Trans.trans hb'b... | true |
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Data.Real.Sqrt
#align_import analysis.normed_space.basic from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set Metric Pointwise
var... | Mathlib/Analysis/NormedSpace/HomeomorphBall.lean | 130 | 131 | theorem univBall_target (c : P) {r : ℝ} (hr : 0 < r) : (univBall c r).target = ball c r := by |
rw [univBall, dif_pos hr]; rfl
| true |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 256 | 261 | theorem generatePiSystem_measurableSet {α} [M : MeasurableSpace α] {S : Set (Set α)}
(h_meas_S : ∀ s ∈ S, MeasurableSet s) (t : Set α) (h_in_pi : t ∈ generatePiSystem S) :
MeasurableSet t := by
induction' h_in_pi with s h_s s u _ _ _ h_s h_u |
induction' h_in_pi with s h_s s u _ _ _ h_s h_u
· apply h_meas_S _ h_s
· apply MeasurableSet.inter h_s h_u
| true |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 220 | 262 | theorem exists_extension_norm_eq_of_closedEmbedding' (f : X →ᵇ ℝ) (e : C(X, Y))
(he : ClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g.compContinuous e = f := by
/- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference |
/- For the proof, we iterate `tietze_extension_step`. Each time we apply it to the difference
between the previous approximation and `f`. -/
choose F hF_norm hF_dist using fun f : X →ᵇ ℝ => tietze_extension_step f e he
set g : ℕ → Y →ᵇ ℝ := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0
have g0 :... | true |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.function.special_functions.is_R_or_C from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
noncomputable section
open NNReal ENNReal
namespace RCLike
variabl... | Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean | 80 | 84 | theorem aemeasurable_of_re_im (hre : AEMeasurable (fun x => RCLike.re (f x)) μ)
(him : AEMeasurable (fun x => RCLike.im (f x)) μ) : AEMeasurable f μ := by
convert AEMeasurable.add (M := 𝕜) (RCLike.measurable_ofReal.comp_aemeasurable hre) |
convert AEMeasurable.add (M := 𝕜) (RCLike.measurable_ofReal.comp_aemeasurable hre)
((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I)
exact (RCLike.re_add_im _).symm
| true |
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.Algebra.GradedMulAction
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.Module.BigOperators
#align_import algebra.module.graded_module from "leanprover-community/mathlib"@"59cdeb0da2480abbc235b7e611ccd9a7e5603d7c"
section
open Dir... | Mathlib/Algebra/Module/GradedModule.lean | 99 | 102 | theorem smulAddMonoidHom_apply_of_of [DecidableEq ιA] [DecidableEq ιB] [GMonoid A] [Gmodule A M]
{i j} (x : A i) (y : M j) :
smulAddMonoidHom A M (DirectSum.of A i x) (of M j y) = of M (i +ᵥ j) (GSMul.smul x y) := by |
simp [smulAddMonoidHom]
| 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 | 125 | 125 | theorem symmDiff_bot : a ∆ ⊥ = a := by | rw [symmDiff, sdiff_bot, bot_sdiff, sup_bot_eq]
| 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 | 113 | 113 | theorem symmDiff_comm : a ∆ b = b ∆ a := by | simp only [symmDiff, sup_comm]
| true |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 95 | 99 | theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors] |
rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
exact le_of_dvd hm.bot_lt
| true |
import Mathlib.CategoryTheory.Action
import Mathlib.Combinatorics.Quiver.Arborescence
import Mathlib.Combinatorics.Quiver.ConnectedComponent
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
#align_import group_theory.nielsen_schreier from "leanprover-community/mathlib"@"1bda4fc53de6ade5ab9da36f2192e24e2084a2ce"
n... | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | 275 | 288 | theorem path_nonempty_of_hom {G} [Groupoid.{u, u} G] [IsFreeGroupoid G] {a b : G} :
Nonempty (a ⟶ b) → Nonempty (Path (symgen a) (symgen b)) := by
rintro ⟨p⟩ |
rintro ⟨p⟩
rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←
mul_inv_eq_one]
let X := FreeGroup (WeaklyConnectedComponent <| Generators G)
let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)
let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Type... | true |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 29 | 32 | theorem eval_one_cyclotomic_prime {R : Type*} [CommRing R] {p : ℕ} [hn : Fact p.Prime] :
eval 1 (cyclotomic p R) = p := by
simp only [cyclotomic_prime, eval_X, one_pow, Finset.sum_const, eval_pow, eval_finset_sum, |
simp only [cyclotomic_prime, eval_X, one_pow, Finset.sum_const, eval_pow, eval_finset_sum,
Finset.card_range, smul_one_eq_cast]
| true |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 78 | 81 | theorem hammingDist_triangle_right (x y z : ∀ i, β i) :
hammingDist x y ≤ hammingDist x z + hammingDist y z := by
rw [hammingDist_comm y] |
rw [hammingDist_comm y]
exact hammingDist_triangle _ _ _
| true |
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
#align_import analysis.mellin_transform from "leanprover-community/mathlib"@"917c3c072e487b3cccdbfeff17e75b40e45f66cb"
open MeasureTheory Set Filter A... | Mathlib/Analysis/MellinTransform.lean | 47 | 50 | theorem MellinConvergent.const_smul {f : ℝ → E} {s : ℂ} (hf : MellinConvergent f s) {𝕜 : Type*}
[NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [SMulCommClass ℂ 𝕜 E] (c : 𝕜) :
MellinConvergent (fun t => c • f t) s := by |
simpa only [MellinConvergent, smul_comm] using hf.smul c
| true |
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 | 75 | 112 | theorem condexp_indicator (hf_int : Integrable f μ) (hs : MeasurableSet[m] s) :
μ[s.indicator f|m] =ᵐ[μ] s.indicator (μ[f|m]) := by
by_cases hm : m ≤ m0 |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm, Set.indicator_zero']; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm, Set.indicator_zero']; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
-- use `have` to perform what should be the first calc step becau... | true |
import Mathlib.Algebra.Module.Card
import Mathlib.SetTheory.Cardinal.CountableCover
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Topology.MetricSpace.Perfect
universe u v
open Filter Pointwise Set Function Cardinal
open scoped Cardinal Topology
theorem c... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 110 | 115 | theorem cardinal_eq_of_isOpen
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [AddCommGroup E] [Module 𝕜 E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul 𝕜 E] {s : Set E}
(hs : IsOpen s) (h's : s.Nonempty) : #s = #E := by
rcases h's with ⟨x, hx⟩ |
rcases h's with ⟨x, hx⟩
exact cardinal_eq_of_mem_nhds 𝕜 (hs.mem_nhds hx)
| true |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 165 | 167 | theorem vsub_vadd_eq_vsub_sub (p₁ p₂ : P) (g : G) : p₁ -ᵥ (g +ᵥ p₂) = p₁ -ᵥ p₂ - g := by
rw [← add_right_inj (p₂ -ᵥ p₁ : G), vsub_add_vsub_cancel, ← neg_vsub_eq_vsub_rev, vadd_vsub, ← |
rw [← add_right_inj (p₂ -ᵥ p₁ : G), vsub_add_vsub_cancel, ← neg_vsub_eq_vsub_rev, vadd_vsub, ←
add_sub_assoc, ← neg_vsub_eq_vsub_rev, neg_add_self, zero_sub]
| true |
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 101 | 101 | theorem singleton_mem_nhds_of_units (γ : Γ₀ˣ) : ({↑γ} : Set Γ₀) ∈ 𝓝 (γ : Γ₀) := by | simp
| true |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
variable {R M N N' : Type*}
variable [CommRing R] [AddCommGroup M] [AddCo... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 96 | 99 | theorem liftAlternating_algebraMap (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (r : R) :
liftAlternating (R := R) (M := M) (N := N) f (algebraMap _ (ExteriorAlgebra R M) r) =
r • f 0 0 := by |
rw [Algebra.algebraMap_eq_smul_one, map_smul, liftAlternating_one]
| true |
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 | 97 | 101 | theorem LocallyIntegrable.integrable_of_isBigO_atBot [IsMeasurablyGenerated (atBot (α := α))]
[OrderTop α] (hf : LocallyIntegrable f μ) (ho : f =O[atBot] g)
(hg : IntegrableAtFilter g atBot μ) : Integrable f μ := by
refine integrable_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩ |
refine integrable_iff_integrableAtFilter_atBot.mpr ⟨ho.integrableAtFilter ?_ hg, hf⟩
exact hf.aestronglyMeasurable.stronglyMeasurableAtFilter
| true |
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import linear_algebra.clifford_algebra.grading from "leanprover-community/mathlib"@"34020e531ebc4e8aac6d449d9eecbcd1508ea8d0"
namespace CliffordAlgebra
variable {R M : Type*} [Co... | Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean | 58 | 65 | theorem evenOdd_mul_le (i j : ZMod 2) : evenOdd Q i * evenOdd Q j ≤ evenOdd Q (i + j) := by
simp_rw [evenOdd, Submodule.iSup_eq_span, Submodule.span_mul_span] |
simp_rw [evenOdd, Submodule.iSup_eq_span, Submodule.span_mul_span]
apply Submodule.span_mono
simp_rw [Set.iUnion_mul, Set.mul_iUnion, Set.iUnion_subset_iff, Set.mul_subset_iff]
rintro ⟨xi, rfl⟩ ⟨yi, rfl⟩ x hx y hy
refine Set.mem_iUnion.mpr ⟨⟨xi + yi, Nat.cast_add _ _⟩, ?_⟩
simp only [Subtype.coe_mk, Nat.ca... | true |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 116 | 118 | theorem nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
‖A.map f‖₊ = ‖A‖₊ := by |
simp only [nnnorm_def, Pi.nnnorm_def, Matrix.map_apply, hf]
| true |
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.Algebra.Exact
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.Derivation
#align_import ring_theory.kaehler from "leanprover-community/mathli... | Mathlib/RingTheory/Kaehler.lean | 78 | 99 | theorem Derivation.tensorProductTo_mul (D : Derivation R S M) (x y : S ⊗[R] S) :
D.tensorProductTo (x * y) =
TensorProduct.lmul' (S := S) R x • D.tensorProductTo y +
TensorProduct.lmul' (S := S) R y • D.tensorProductTo x := by
refine TensorProduct.induction_on x ?_ ?_ ?_ |
refine TensorProduct.induction_on x ?_ ?_ ?_
· rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
· intro x₁ y₁ h₁ h₂
rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm]
intro x₁ x₂
refine TensorProduct.induction_on y ?_ ?_ ?_
· rw [mul_zero, map_... | true |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 312 | 316 | theorem symmetric (h : Symmetric r) : Symmetric (ReflTransGen r) := by
intro x y h |
intro x y h
induction' h with z w _ b c
· rfl
· apply Relation.ReflTransGen.head (h b) c
| true |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 133 | 137 | theorem midpoint_vsub (p₁ p₂ p : P) :
midpoint R p₁ p₂ -ᵥ p = (⅟ 2 : R) • (p₁ -ᵥ p) + (⅟ 2 : R) • (p₂ -ᵥ p) := by
rw [← vsub_sub_vsub_cancel_right p₁ p p₂, smul_sub, sub_eq_add_neg, ← smul_neg, |
rw [← vsub_sub_vsub_cancel_right p₁ p p₂, smul_sub, sub_eq_add_neg, ← smul_neg,
neg_vsub_eq_vsub_rev, add_assoc, invOf_two_smul_add_invOf_two_smul, ← vadd_vsub_assoc,
midpoint_comm, midpoint, lineMap_apply]
| true |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 115 | 117 | theorem coe_bind (l : List α) (f : α → List β) : (@bind α β l fun a => f a) = l.bind f := by
rw [List.bind, ← coe_join, List.map_map] |
rw [List.bind, ← coe_join, List.map_map]
rfl
| true |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Multiset.Powerset
#align_import data.finset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Finset
open Function Multiset
variable {α : Type*} {s t : Finset α}
section powersetCard
variable {n} {s t : Fi... | Mathlib/Data/Finset/Powerset.lean | 220 | 225 | theorem powersetCard_zero (s : Finset α) : s.powersetCard 0 = {∅} := by
ext; rw [mem_powersetCard, mem_singleton, card_eq_zero] |
ext; rw [mem_powersetCard, mem_singleton, card_eq_zero]
refine
⟨fun h => h.2, fun h => by
rw [h]
exact ⟨empty_subset s, rfl⟩⟩
| true |
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 125 | 132 | theorem locallyConnectedSpace_of_connected_bases {ι : Type*} (b : α → ι → Set α) (p : α → ι → Prop)
(hbasis : ∀ x, (𝓝 x).HasBasis (p x) (b x))
(hconnected : ∀ x i, p x i → IsPreconnected (b x i)) : LocallyConnectedSpace α := by
rw [locallyConnectedSpace_iff_connected_basis] |
rw [locallyConnectedSpace_iff_connected_basis]
exact fun x =>
(hbasis x).to_hasBasis
(fun i hi => ⟨b x i, ⟨(hbasis x).mem_of_mem hi, hconnected x i hi⟩, subset_rfl⟩) fun s hs =>
⟨(hbasis x).index s hs.1, ⟨(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1⟩⟩
| true |
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.stirling from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Topology Real Nat Asymptotics
open Finset Filter Nat Real
namespace... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 77 | 93 | theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
HasSum (fun k : ℕ => (1 : ℝ) / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ ↑(k + 1))
(log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
let f (k : ℕ) := (1 : ℝ) / (2 * k + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ k |
let f (k : ℕ) := (1 : ℝ) / (2 * k + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ k
change HasSum (fun k => f (k + 1)) _
rw [hasSum_nat_add_iff]
convert (hasSum_log_one_add_inv m.cast_add_one_pos).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
· ext k
dsimp only [f]
rw [← pow_mul, pow_add]
push_cast
field... | true |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
variable {R M N N' : Type*}
variable [CommRing R] [AddCommGroup M] [AddCo... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 125 | 135 | theorem liftAlternating_comp (g : N →ₗ[R] N') (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) :
(liftAlternating (R := R) (M := M) (N := N') fun i => g.compAlternatingMap (f i)) =
g ∘ₗ liftAlternating (R := R) (M := M) (N := N) f := by
ext v |
ext v
rw [LinearMap.comp_apply]
induction' v using CliffordAlgebra.left_induction with r x y hx hy x m hx generalizing f
· rw [liftAlternating_algebraMap, liftAlternating_algebraMap, map_smul,
LinearMap.compAlternatingMap_apply]
· rw [map_add, map_add, map_add, hx, hy]
· rw [liftAlternating_ι_mul, li... | true |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 96 | 99 | theorem toPEquiv_mul_matrix [Fintype m] [DecidableEq m] [Semiring α] (f : m ≃ m)
(M : Matrix m n α) : f.toPEquiv.toMatrix * M = M.submatrix f id := by
ext i j |
ext i j
rw [mul_matrix_apply, Equiv.toPEquiv_apply, submatrix_apply, id]
| true |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 91 | 98 | theorem length_inv (w : W) : ℓ (w⁻¹) = ℓ w := by
apply Nat.le_antisymm |
apply Nat.le_antisymm
· rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
have := cs.length_wordProd_le (List.reverse ω)
rwa [wordProd_reverse, length_reverse, hω] at this
· rcases cs.exists_reduced_word w⁻¹ with ⟨ω, hω, h'ω⟩
have := cs.length_wordProd_le (List.reverse ω)
rwa [wordProd_reverse, l... | true |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 109 | 119 | theorem _root_.Function.Injective.wOppSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WOppSide (f x) (f y) ↔ s.WOppSide x y := by
refine ⟨fun h => ?_, fun h => h.map _⟩ |
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
| true |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe u₀ u₁ u₂ v₀ v₁ v₂
open Function
class Bifunctor (F : Type u₀ → Type u₁ → Type u₂) where
bimap : ∀ {α α' β β'}, (α → α') → (β → β'... | Mathlib/Control/Bifunctor.lean | 86 | 87 | theorem comp_fst {α₀ α₁ α₂ β} (f : α₀ → α₁) (f' : α₁ → α₂) (x : F α₀ β) :
fst f' (fst f x) = fst (f' ∘ f) x := by | simp [fst, bimap_bimap]
| true |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 48 | 50 | theorem comap_id_apply (x : σ → R) : comap (AlgHom.id R (MvPolynomial σ R)) x = x := by
funext i |
funext i
simp only [comap, AlgHom.id_apply, id, aeval_X]
| true |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 65 | 66 | theorem normSq_eq_norm_mul_self (a : ℍ) : normSq a = ‖a‖ * ‖a‖ := by |
rw [← inner_self, real_inner_self_eq_norm_mul_norm]
| true |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 63 | 64 | theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by |
rw [← exp_mul_I, abs_mul_exp_arg_mul_I]
| true |
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 113 | 116 | theorem N₁Γ₀_inv_app_f_f (K : ChainComplex C ℕ) (n : ℕ) :
(N₁Γ₀.inv.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom.f.f n := by
rw [N₁Γ₀_inv_app] |
rw [N₁Γ₀_inv_app]
apply id_comp
| true |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 44 | 49 | theorem ComponentCompl.supp_injective :
Function.Injective (ComponentCompl.supp : G.ComponentCompl K → Set V) := by
refine ConnectedComponent.ind₂ ?_ |
refine ConnectedComponent.ind₂ ?_
rintro ⟨v, hv⟩ ⟨w, hw⟩ h
simp only [Set.ext_iff, ConnectedComponent.eq, Set.mem_setOf_eq, ComponentCompl.supp] at h ⊢
exact ((h v).mp ⟨hv, Reachable.refl _⟩).choose_spec
| true |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 88 | 89 | theorem add_self_eq_zero {a : R} : a + a = 0 ↔ a = 0 := by |
simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or_iff]
| true |
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Set.Finite
import Mathlib.Order.Category.FinPartOrd
import Mathlib.Order.Category.LinOrd
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
import Mathlib.Data.Set.Subsingleton
#align_import order.category.No... | Mathlib/Order/Category/NonemptyFinLinOrd.lean | 150 | 163 | theorem mono_iff_injective {A B : NonemptyFinLinOrd.{u}} (f : A ⟶ B) :
Mono f ↔ Function.Injective f := by
refine ⟨?_, ConcreteCategory.mono_of_injective f⟩ |
refine ⟨?_, ConcreteCategory.mono_of_injective f⟩
intro
intro a₁ a₂ h
let X := NonemptyFinLinOrd.of (ULift (Fin 1))
let g₁ : X ⟶ A := ⟨fun _ => a₁, fun _ _ _ => by rfl⟩
let g₂ : X ⟶ A := ⟨fun _ => a₂, fun _ _ _ => by rfl⟩
change g₁ (ULift.up (0 : Fin 1)) = g₂ (ULift.up (0 : Fin 1))
have eq : g₁ ≫ f = g... | true |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
variable (R A B : Type*) {σ : Type*}
namespace MvPolynomial
section CommSemiring
variable [CommSemiring R] ... | Mathlib/RingTheory/MvPolynomial/Tower.lean | 48 | 53 | theorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :
aeval (algebraMap A B ∘ x) p = algebraMap A B (MvPolynomial.aeval x p) := by
rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ← |
rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←
IsScalarTower.algebraMap_eq]
-- Porting note: added
simp only [Function.comp]
| true |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 82 | 88 | theorem revAt_add {N O n o : ℕ} (hn : n ≤ N) (ho : o ≤ O) :
revAt (N + O) (n + o) = revAt N n + revAt O o := by
rcases Nat.le.dest hn with ⟨n', rfl⟩ |
rcases Nat.le.dest hn with ⟨n', rfl⟩
rcases Nat.le.dest ho with ⟨o', rfl⟩
repeat' rw [revAt_le (le_add_right rfl.le)]
rw [add_assoc, add_left_comm n' o, ← add_assoc, revAt_le (le_add_right rfl.le)]
repeat' rw [add_tsub_cancel_left]
| true |
import Mathlib.Data.Real.Basic
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Algebra.Order.EuclideanAbsoluteValue
#align_import number_theory.class_number.admissible_absolute_value from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
local infixl:50 " ≺ " => EuclideanDomain.r
na... | Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean | 117 | 123 | theorem exists_approx {ι : Type*} [Fintype ι] {ε : ℝ} (hε : 0 < ε) {b : R} (hb : b ≠ 0)
(h : abv.IsAdmissible) (A : Fin (h.card ε ^ Fintype.card ι).succ → ι → R) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ k, (abv (A i₁ k % b - A i₀ k % b) : ℝ) < abv b • ε := by
let e := Fintype.equivFin ι |
let e := Fintype.equivFin ι
obtain ⟨i₀, i₁, ne, h⟩ := h.exists_approx_aux (Fintype.card ι) hε hb fun x y ↦ A x (e.symm y)
refine ⟨i₀, i₁, ne, fun k ↦ ?_⟩
convert h (e k) <;> simp only [e.symm_apply_apply]
| true |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 99 | 102 | theorem hasFDerivAt_of_restrictScalars {g' : E →L[𝕜] F} (h : HasFDerivAt f g' x)
(H : f'.restrictScalars 𝕜 = g') : HasFDerivAt f f' x := by
rw [← H] at h |
rw [← H] at h
exact .of_isLittleO h.1
| true |
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
#align_import linear_algebra.direct_sum.tensor_product from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
suppress_compilation
universe u v₁ v₂ w₁ w₁' w₂ w₂'
section Ring
namespace TensorProduct
... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 150 | 153 | theorem directSum_lof_tmul_lof (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) :
TensorProduct.directSum R S M₁ M₂ (DirectSum.lof S ι₁ M₁ i₁ m₁ ⊗ₜ DirectSum.lof R ι₂ M₂ i₂ m₂) =
DirectSum.lof S (ι₁ × ι₂) (fun i => M₁ i.1 ⊗[R] M₂ i.2) (i₁, i₂) (m₁ ⊗ₜ m₂) := by |
simp [TensorProduct.directSum]
| true |
import Mathlib.Analysis.BoxIntegral.Partition.Additive
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import analysis.box_integral.partition.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set
noncomputable section
open scoped ENNReal Classical BoxIntegral... | Mathlib/Analysis/BoxIntegral/Partition/Measure.lean | 74 | 76 | theorem coe_ae_eq_Icc : (I : Set (ι → ℝ)) =ᵐ[volume] Box.Icc I := by
rw [coe_eq_pi] |
rw [coe_eq_pi]
exact Measure.univ_pi_Ioc_ae_eq_Icc
| true |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 57 | 59 | theorem lineMap_strict_mono_left (ha : a < a') (hr : r < 1) : lineMap a b r < lineMap a' b r := by
simp only [lineMap_apply_module] |
simp only [lineMap_apply_module]
exact add_lt_add_right (smul_lt_smul_of_pos_left ha (sub_pos.2 hr)) _
| true |
import Mathlib.Algebra.Category.GroupCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Group.images from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open CategoryTheory
open CategoryTheory.Limits
universe u
namespace AddCommGroupCat
set... | Mathlib/Algebra/Category/GroupCat/Images.lean | 87 | 91 | theorem image.lift_fac (F' : MonoFactorisation f) : image.lift F' ≫ F'.m = image.ι f := by
ext x |
ext x
change (F'.e ≫ F'.m) _ = _
rw [F'.fac, (Classical.indefiniteDescription _ x.2).2]
rfl
| true |
import Mathlib.LinearAlgebra.Ray
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.ray from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Real
variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*}
[NormedAddCommGroup F] [NormedSp... | Mathlib/Analysis/NormedSpace/Ray.lean | 59 | 65 | theorem norm_injOn_ray_left (hx : x ≠ 0) : { y | SameRay ℝ x y }.InjOn norm := by
rintro y hy z hz h |
rintro y hy z hz h
rcases hy.exists_nonneg_left hx with ⟨r, hr, rfl⟩
rcases hz.exists_nonneg_left hx with ⟨s, hs, rfl⟩
rw [norm_smul, norm_smul, mul_left_inj' (norm_ne_zero_iff.2 hx), norm_of_nonneg hr,
norm_of_nonneg hs] at h
rw [h]
| true |
import Mathlib.Order.Interval.Set.OrdConnectedComponent
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.t5 from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter Set Function OrderDual Topology Interval
variable {X : Type*} [LinearOrder X] [Topological... | Mathlib/Topology/Order/T5.lean | 27 | 30 | theorem ordConnectedComponent_mem_nhds : ordConnectedComponent s a ∈ 𝓝 a ↔ s ∈ 𝓝 a := by
refine ⟨fun h => mem_of_superset h ordConnectedComponent_subset, fun h => ?_⟩ |
refine ⟨fun h => mem_of_superset h ordConnectedComponent_subset, fun h => ?_⟩
rcases exists_Icc_mem_subset_of_mem_nhds h with ⟨b, c, ha, ha', hs⟩
exact mem_of_superset ha' (subset_ordConnectedComponent ha hs)
| true |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import Mathlib.RingTheory.TensorProduct.Basic
variable {R A V : Type*}
variable [CommRing R] [CommRing A] [AddCommGroup V]
variable [Algebra R A] [Mod... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 124 | 137 | theorem toBaseChange_comp_reverseOp (Q : QuadraticForm R V) :
(toBaseChange A Q).op.comp reverseOp =
((Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)).toAlgHom.comp <|
(Algebra.TensorProduct.map
(AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))).comp
(toBaseChange A... |
ext v
show op (toBaseChange A Q (reverse (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))) =
Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)
(Algebra.TensorProduct.map (AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))
(toBaseChange A Q (ι (Q.baseChange A) (1 ⊗ₜ[R] v))))
rw [toBaseChange_ι, re... | true |
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def ImpartialAux : PGame → Prop
| G => (G ≈ -G) ∧ (∀ i... | Mathlib/SetTheory/Game/Impartial.lean | 137 | 142 | theorem equiv_or_fuzzy_zero : (G ≈ 0) ∨ G ‖ 0 := by
rcases lt_or_equiv_or_gt_or_fuzzy G 0 with (h | h | h | h) |
rcases lt_or_equiv_or_gt_or_fuzzy G 0 with (h | h | h | h)
· exact ((nonneg G) h).elim
· exact Or.inl h
· exact ((nonpos G) h).elim
· exact Or.inr h
| true |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 82 | 86 | theorem natAbs_coe_sub_coe_le_of_le {a b n : ℕ} (a_le_n : a ≤ n) (b_le_n : b ≤ n) :
natAbs (a - b : ℤ) ≤ n := by
rw [← Nat.cast_le (α := ℤ), natCast_natAbs] |
rw [← Nat.cast_le (α := ℤ), natCast_natAbs]
exact abs_sub_le_of_nonneg_of_le (ofNat_nonneg a) (ofNat_le.mpr a_le_n)
(ofNat_nonneg b) (ofNat_le.mpr b_le_n)
| true |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Function
variable {ι α β : Type*} {l : Filter α}
namespace Filter
def cofinite : Filter α :=
comk Set.Finite finite_e... | Mathlib/Order/Filter/Cofinite.lean | 101 | 104 | theorem le_cofinite_iff_compl_singleton_mem : l ≤ cofinite ↔ ∀ x, {x}ᶜ ∈ l := by
refine ⟨fun h x => h (finite_singleton x).compl_mem_cofinite, fun h s (hs : sᶜ.Finite) => ?_⟩ |
refine ⟨fun h x => h (finite_singleton x).compl_mem_cofinite, fun h s (hs : sᶜ.Finite) => ?_⟩
rw [← compl_compl s, ← biUnion_of_singleton sᶜ, compl_iUnion₂, Filter.biInter_mem hs]
exact fun x _ => h x
| true |
import Mathlib.Data.Finset.Lattice
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.partial_sups from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {α : Type*}
section SemilatticeSup
var... | Mathlib/Order/PartialSups.lean | 97 | 101 | theorem Monotone.partialSups_eq {f : ℕ → α} (hf : Monotone f) : (partialSups f : ℕ → α) = f := by
ext n |
ext n
induction' n with n ih
· rfl
· rw [partialSups_succ, ih, sup_eq_right.2 (hf (Nat.le_succ _))]
| true |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 146 | 147 | theorem degree_eq_iff_natDegree_eq {p : R[X]} {n : ℕ} (hp : p ≠ 0) :
p.degree = n ↔ p.natDegree = n := by | rw [degree_eq_natDegree hp]; exact WithBot.coe_eq_coe
| true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 78 | 79 | theorem map_add_nat' [AddMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (x : G) (n : ℕ) : f (x + n) = f x + n • b := by | simp [← map_add_nsmul]
| true |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 92 | 97 | theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩ |
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
| true |
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 129 | 129 | theorem measure_union_null (hs : μ s = 0) (ht : μ t = 0) : μ (s ∪ t) = 0 := by | simp [*]
| true |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import data.pnat.factors from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
-- Porting note: `deriving` contained Inhabited, Canonic... | Mathlib/Data/PNat/Factors.lean | 121 | 123 | theorem coePNat_prime (v : PrimeMultiset) (p : ℕ+) (h : p ∈ (v : Multiset ℕ+)) : p.Prime := by
rcases Multiset.mem_map.mp h with ⟨⟨_, hp'⟩, ⟨_, h_eq⟩⟩ |
rcases Multiset.mem_map.mp h with ⟨⟨_, hp'⟩, ⟨_, h_eq⟩⟩
exact h_eq ▸ hp'
| true |
import Mathlib.RepresentationTheory.FdRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
#align_import representation_theory.character from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
noncomputable section
universe u
open CategoryTheory LinearMap ... | Mathlib/RepresentationTheory/Character.lean | 54 | 55 | theorem char_mul_comm (V : FdRep k G) (g : G) (h : G) :
V.character (h * g) = V.character (g * h) := by | simp only [trace_mul_comm, character, map_mul]
| true |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 78 | 80 | theorem mem_graph_iff {c : α × M} {f : α →₀ M} : c ∈ f.graph ↔ f c.1 = c.2 ∧ c.2 ≠ 0 := by
cases c |
cases c
exact mk_mem_graph_iff
| true |
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 83 | 87 | theorem comap_eq_id_of_eq_id (f : MvPolynomial σ R →ₐ[R] MvPolynomial σ R) (hf : ∀ φ, f φ = φ)
(x : σ → R) : comap f x = x := by
convert comap_id_apply x |
convert comap_id_apply x
ext1 φ
simp [hf, AlgHom.id_apply]
| true |
import Mathlib.MeasureTheory.Covering.DensityTheorem
#align_import measure_theory.covering.liminf_limsup from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
open Set Filter Metric MeasureTheory TopologicalSpace
open scoped NNReal ENNReal Topology
variable {α : Type*} [MetricSpace α] [... | Mathlib/MeasureTheory/Covering/LiminfLimsup.lean | 41 | 150 | theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
(hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
{M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
(blimsup (fun i => cthickening (r₁ i) (s i)) atTop... |
/- Sketch of proof:
Assume that `p` is identically true for simplicity. Let `Y₁ i = cthickening (r₁ i) (s i)`, define
`Y₂` similarly except using `r₂`, and let `(Z i) = ⋃_{j ≥ i} (Y₂ j)`. Our goal is equivalent to
showing that `μ ((limsup Y₁) \ (Z i)) = 0` for all `i`.
Assume for contradiction that `μ ((li... | true |
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function Set
variable {α : Type*} (r r₁ r₂ : α → α → Prop) (s... | Mathlib/Order/Minimal.lean | 113 | 115 | theorem mem_minimals_iff_forall_lt_not_mem' (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt y x → y ∉ s := by |
simp [minimals, right_iff_left_not_left_of r rlt, not_imp_not, imp.swap (a := _ ∈ _)]
| true |
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Finsupp.Fin
import Mathlib.Data.Finsupp.Indicator
#align_import algebra.bi... | Mathlib/Algebra/BigOperators/Finsupp.lean | 124 | 127 | theorem sum_ite_self_eq_aux [DecidableEq α] {N : Type*} [AddCommMonoid N] (f : α →₀ N) (a : α) :
(if a ∈ f.support then f a else 0) = f a := by
simp only [mem_support_iff, ne_eq, ite_eq_left_iff, not_not] |
simp only [mem_support_iff, ne_eq, ite_eq_left_iff, not_not]
exact fun h ↦ h.symm
| true |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 85 | 89 | theorem bottom_row_coprime {R : Type*} [CommRing R] (g : SL(2, R)) :
IsCoprime ((↑g : Matrix (Fin 2) (Fin 2) R) 1 0) ((↑g : Matrix (Fin 2) (Fin 2) R) 1 1) := by
use -(↑g : Matrix (Fin 2) (Fin 2) R) 0 1, (↑g : Matrix (Fin 2) (Fin 2) R) 0 0 |
use -(↑g : Matrix (Fin 2) (Fin 2) R) 0 1, (↑g : Matrix (Fin 2) (Fin 2) R) 0 0
rw [add_comm, neg_mul, ← sub_eq_add_neg, ← det_fin_two]
exact g.det_coe
| true |
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