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import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 184 | 184 | theorem refl_trans (α : X ≅ Y) : Iso.refl X ≪≫ α = α := by | ext; apply Category.id_comp
|
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 132 | 145 | theorem IsGδ.dense_iUnion_interior_of_closed [Countable ι] {s : Set X} (hs : IsGδ s) (hd : Dense s)
{f : ι → Set X} (hc : ∀ i, IsClosed (f i)) (hU : s ⊆ ⋃ i, f i) :
Dense (⋃ i, interior (f i)) := by |
let g i := (frontier (f i))ᶜ
have hgo : ∀ i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl
have hgd : Dense (⋂ i, g i) := by
refine dense_iInter_of_isOpen hgo fun i x => ?_
rw [closure_compl, interior_frontier (hc _)]
exact id
refine (hd.inter_of_Gδ hs (.iInter_of_isOpen fun i => (hgo i))... |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstances
import Mathlib.Data.Set.Subsingleton
#align_import algebra.module.submodule.lattice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
universe v
variable {R S M : Ty... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 277 | 280 | theorem mem_sup_right {S T : Submodule R M} : ∀ {x : M}, x ∈ T → x ∈ S ⊔ T := by |
have : T ≤ S ⊔ T := le_sup_right
rw [LE.le] at this
exact this
|
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
#align_import category_theory.sites.surjective from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u w v' u' w'
open ... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 119 | 124 | theorem isLocallySurjective_of_surjective {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G)
(H : ∀ U, Function.Surjective (f.app U)) : IsLocallySurjective J f where
imageSieve_mem {U} s := by |
obtain ⟨t, rfl⟩ := H _ s
rw [imageSieve_app]
exact J.top_mem _
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 141 | 150 | theorem cons_self_tail : cons (q 0) (tail q) = q := by |
ext j
by_cases h : j = 0
· rw [h]
simp
· let j' := pred j h
have : j'.succ = j := succ_pred j h
rw [← this]
unfold tail
rw [cons_succ]
|
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 67 | 75 | theorem germ_injective_of_isIntegral [IsIntegral X] {U : Opens X.carrier} (x : U) :
Function.Injective (X.presheaf.germ x) := by |
rw [injective_iff_map_eq_zero]
intro y hy
rw [← (X.presheaf.germ x).map_zero] at hy
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ x.prop x.prop _ _ hy
cases Subsingleton.elim iU iV
haveI : Nonempty W := ⟨⟨_, hW⟩⟩
exact map_injective_of_isIntegral X iU e
|
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {𝕜 E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 106 | 111 | theorem continuousWithinAt_dslope_of_ne (h : b ≠ a) :
ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b := by |
refine ⟨ContinuousWithinAt.of_dslope, fun hc => ?_⟩
simp only [dslope, continuousWithinAt_update_of_ne h]
exact ((continuousWithinAt_id.sub continuousWithinAt_const).inv₀ (sub_ne_zero.2 h)).smul
(hc.sub continuousWithinAt_const)
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Preimage
variable {f : α → β} {g : β → γ... | Mathlib/Data/Set/Image.lean | 186 | 193 | theorem preimage_subtype_coe_eq_compl {s u v : Set α} (hsuv : s ⊆ u ∪ v)
(H : s ∩ (u ∩ v) = ∅) : ((↑) : s → α) ⁻¹' u = ((↑) ⁻¹' v)ᶜ := by |
ext ⟨x, x_in_s⟩
constructor
· intro x_in_u x_in_v
exact eq_empty_iff_forall_not_mem.mp H x ⟨x_in_s, ⟨x_in_u, x_in_v⟩⟩
· intro hx
exact Or.elim (hsuv x_in_s) id fun hx' => hx.elim hx'
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 671 | 678 | theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}
(h : (o.oangle x y).sign = 0) :
x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
rw [o.angle_eq_abs_oangle_toReal hx hy]
rw [Real.Angle.sign_eq_zero_iff] at h
rcases h with (h | h) <;> simp [h, Real.pi_pos.le]
|
import Mathlib.Algebra.Category.MonCat.Limits
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
import Mathlib.CategoryTheory.Limits.TypesFiltered
#align_import algebra.category.Mon.filtered_colimits from "leanprover-community/mathlib"@"70fd9563a21e7b96... | Mathlib/Algebra/Category/MonCat/FilteredColimits.lean | 190 | 202 | theorem colimit_mul_mk_eq (x y : Σ j, F.obj j) (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k) :
M.mk.{v, u} F x * M.mk F y = M.mk F ⟨k, F.map f x.2 * F.map g y.2⟩ := by |
cases' x with j₁ x; cases' y with j₂ y
obtain ⟨s, α, β, h₁, h₂⟩ := IsFiltered.bowtie (IsFiltered.leftToMax j₁ j₂) f
(IsFiltered.rightToMax j₁ j₂) g
refine M.mk_eq F _ _ ?_
use s, α, β
dsimp
simp_rw [MonoidHom.map_mul]
-- Porting note: Lean cannot seem to use lemmas from concrete categories directly
... |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 124 | 130 | theorem Sized.induction {t} (hl : @Sized α t) {C : Ordnode α → Prop} (H0 : C nil)
(H1 : ∀ l x r, C l → C r → C (.node' l x r)) : C t := by |
induction t with
| nil => exact H0
| node _ _ _ _ t_ih_l t_ih_r =>
rw [hl.eq_node']
exact H1 _ _ _ (t_ih_l hl.2.1) (t_ih_r hl.2.2)
|
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 219 | 221 | theorem transpose_transpose (μ : YoungDiagram) : μ.transpose.transpose = μ := by |
ext x
simp
|
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.WithBotTop
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Function Set NNReal
variable {α : Typ... | Mathlib/Data/ENNReal/Basic.lean | 297 | 298 | theorem toReal_eq_zero_iff (x : ℝ≥0∞) : x.toReal = 0 ↔ x = 0 ∨ x = ∞ := by |
simp [ENNReal.toReal, toNNReal_eq_zero_iff]
|
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 | 624 | 626 | theorem frobenius_norm_col (v : n → α) : ‖col v‖ = ‖(WithLp.equiv 2 _).symm v‖ := by |
simp_rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]
simp only [col_apply, Real.rpow_two, WithLp.equiv_symm_pi_apply]
|
import Mathlib.Topology.Category.LightProfinite.Limits
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
universe u
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
namespace LightProfinite
noncomputable
def EffectiveEpi.struct {B X : LightProfinite.{u}} (... | Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean | 54 | 58 | theorem effectiveEpi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
EffectiveEpi f ↔ Function.Surjective f := by |
refine ⟨fun h ↦ ?_, fun h ↦ ⟨⟨EffectiveEpi.struct f h⟩⟩⟩
rw [← epi_iff_surjective]
infer_instance
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 814 | 816 | theorem append_get_eq [Append α] (a b : Part α) (hab : Dom (a ++ b)) : (a ++ b).get hab =
a.get (left_dom_of_append_dom hab) ++ b.get (right_dom_of_append_dom hab) := by |
simp [append_def]; aesop
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 301 | 304 | theorem HasStrictDerivAt.const_mul (c : 𝔸) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => c * d y) (c * d') x := by |
convert (hasStrictDerivAt_const _ _).mul hd using 1
rw [zero_mul, zero_add]
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
#align_import ring_theory.adjoin.basic fr... | Mathlib/RingTheory/Adjoin/Basic.lean | 134 | 141 | theorem adjoin_induction'' {x : A} (hx : x ∈ adjoin R s)
{p : (x : A) → x ∈ adjoin R s → Prop} (mem : ∀ x (h : x ∈ s), p x (subset_adjoin h))
(algebraMap : ∀ (r : R), p (algebraMap R A r) (algebraMap_mem _ r))
(add : ∀ x hx y hy, p x hx → p y hy → p (x + y) (add_mem hx hy))
(mul : ∀ x hx y hy, p x hx → ... |
refine adjoin_induction' mem algebraMap ?_ ?_ ⟨x, hx⟩ (p := fun x : adjoin R s ↦ p x.1 x.2)
exacts [fun x y ↦ add x.1 x.2 y.1 y.2, fun x y ↦ mul x.1 x.2 y.1 y.2]
|
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 66 | 70 | theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab]
|
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 | 96 | 106 | theorem mk_one_pow_eq_mk_choose_add :
(mk 1 : S⟦X⟧) ^ (d + 1) = (mk fun n => Nat.choose (d + n) d : S⟦X⟧) := by |
induction d with
| zero => ext; simp
| succ d hd =>
ext n
rw [pow_add, hd, pow_one, mul_comm, coeff_mul]
simp_rw [coeff_mk, Pi.one_apply, one_mul]
norm_cast
rw [Finset.sum_antidiagonal_choose_add, ← Nat.choose_succ_succ, Nat.succ_eq_add_one,
add_right_comm]
|
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 342 | 344 | theorem w_compact_support {D : ℝ} (Dpos : 0 < D) : HasCompactSupport (w D : E → ℝ) := by |
rw [hasCompactSupport_def, w_support E Dpos, closure_ball (0 : E) Dpos.ne']
exact isCompact_closedBall _ _
|
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Products.Bifunctor
#align_import category_theory.limits.fubini from "leanprover-community/mathlib"@"59382264386afdbaf1727e617f5fdda511992eb9"
uni... | Mathlib/CategoryTheory/Limits/Fubini.lean | 346 | 351 | theorem colimitUncurryIsoColimitCompColim_ι_ι_inv {j} {k} :
colimit.ι (F.obj j) k ≫ colimit.ι (F ⋙ colim) j ≫ (colimitUncurryIsoColimitCompColim F).inv =
colimit.ι (uncurry.obj F) (j, k) := by |
dsimp [colimitUncurryIsoColimitCompColim, IsColimit.coconePointUniqueUpToIso,
IsColimit.uniqueUpToIso]
simp
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 112 | 134 | theorem nat_is_prime_iff : Prime (n : Cardinal) ↔ n.Prime := by |
simp only [Prime, Nat.prime_iff]
refine and_congr (by simp) (and_congr ?_ ⟨fun h b c hbc => ?_, fun h b c hbc => ?_⟩)
· simp only [isUnit_iff, Nat.isUnit_iff]
exact mod_cast Iff.rfl
· exact mod_cast h b c (mod_cast hbc)
cases' lt_or_le (b * c) ℵ₀ with h' h'
· rcases mul_lt_aleph0_iff.mp h' with (rfl | ... |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 179 | 183 | theorem join_drop_length_sub_one {L : List (List α)} (h : L ≠ []) :
(L.drop (L.length - 1)).join = L.getLast h := by |
induction L using List.reverseRecOn
· cases h rfl
· simp
|
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual
#align_import analysis.calculus.parametric_integral from "leanprover-community/mathlib"@"8f9fea08977f7e4... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 291 | 309 | theorem hasDerivAt_integral_of_dominated_loc_of_deriv_le (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
{F' : 𝕜 → α → E} (hF'_meas : AEStronglyMeasurable (F' x₀) μ)
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F' x a‖ ≤ bound a) (bound_integrable : Integrabl... |
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' x₀ a) x₀ :=
h_diff.mono fun a ha ↦ ha x₀ x₀_in
have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 ↦ F x a) (ball x₀ ε) := by
apply (h_diff.and h_bound).mono
rintro a ⟨ha_deriv, ha_bound⟩
... |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 1,002 | 1,006 | theorem integrable_withDensity_iff {f : α → ℝ≥0∞} (hf : Measurable f) (hflt : ∀ᵐ x ∂μ, f x < ∞)
{g : α → ℝ} : Integrable g (μ.withDensity f) ↔ Integrable (fun x => g x * (f x).toReal) μ := by |
have : (fun x => g x * (f x).toReal) = fun x => (f x).toReal • g x := by simp [mul_comm]
rw [this]
exact integrable_withDensity_iff_integrable_smul' hf hflt
|
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Equiv
#align_import topology.uniform_space.abstract_completion from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
attribute [local instance] Classical.propDecidable
open F... | Mathlib/Topology/UniformSpace/AbstractCompletion.lean | 143 | 149 | theorem uniformContinuous_extend : UniformContinuous (pkg.extend f) := by |
by_cases hf : UniformContinuous f
· rw [pkg.extend_def hf]
exact uniformContinuous_uniformly_extend pkg.uniformInducing pkg.dense hf
· change UniformContinuous (ite _ _ _)
rw [if_neg hf]
exact uniformContinuous_of_const fun a b => by congr 1
|
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section TopologicalGroup
variable [CommGroup α] [TopologicalSpace α] [TopologicalGroup α]
variable {f g : β → α} {a a₁... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 75 | 81 | theorem HasProd.update (hf : HasProd f a₁) (b : β) [DecidableEq β] (a : α) :
HasProd (update f b a) (a / f b * a₁) := by |
convert (hasProd_ite_eq b (a / f b)).mul hf with b'
by_cases h : b' = b
· rw [h, update_same]
simp [eq_self_iff_true, if_true, sub_add_cancel]
· simp only [h, update_noteq, if_false, Ne, one_mul, not_false_iff]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,421 | 1,424 | theorem set_lintegral_map [MeasurableSpace β] {f : β → ℝ≥0∞} {g : α → β} {s : Set β}
(hs : MeasurableSet s) (hf : Measurable f) (hg : Measurable g) :
∫⁻ y in s, f y ∂map g μ = ∫⁻ x in g ⁻¹' s, f (g x) ∂μ := by |
rw [restrict_map hg hs, lintegral_map hf hg]
|
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 144 | 145 | theorem transReflReparamAux_zero : transReflReparamAux 0 = 0 := by |
set_option tactic.skipAssignedInstances false in norm_num [transReflReparamAux]
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.fixed_point from "leanprover-community/mathlib"@"0dd4319a17376eda5763cd0a7e0d35bbaaa50e83"
noncomputable section
universe u v
open Function Order
namespace Ordinal
section
variable {ι ... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 497 | 499 | theorem fp_unbounded (H : IsNormal f) : (Function.fixedPoints f).Unbounded (· < ·) := by |
convert fp_family_unbounded fun _ : Unit => H
exact (Set.iInter_const _).symm
|
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 95 | 97 | theorem χ₄_nat_one_mod_four {n : ℕ} (hn : n % 4 = 1) : χ₄ n = 1 := by |
rw [χ₄_nat_mod_four, hn]
rfl
|
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open sco... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 372 | 414 | theorem finite_of_discr_bdd_of_isComplex :
{K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
haveI : NumberField K := @NumberField.mk _ _ inferInstance K.prop
{w : InfinitePlace K | IsComplex w}.Nonempty ∧ |discr K| ≤ N }.Finite := by |
-- The bound on the degree of the generating polynomials
let D := rankOfDiscrBdd N
-- The bound on the Minkowski bound
let B := boundOfDiscBdd N
-- The bound on the coefficients of the generating polynomials
let C := Nat.ceil ((max (sqrt (1 + B ^ 2)) 1) ^ D * Nat.choose D (D / 2))
refine finite_of_finite... |
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.WithBotTop
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Function Set NNReal
variable {α : Typ... | Mathlib/Data/ENNReal/Basic.lean | 601 | 603 | theorem toReal_le_coe_of_le_coe {a : ℝ≥0∞} {b : ℝ≥0} (h : a ≤ b) : a.toReal ≤ b := by |
lift a to ℝ≥0 using ne_top_of_le_ne_top coe_ne_top h
simpa using h
|
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 175 | 176 | theorem tail_think (s : Computation α) : tail (think s) = s := by |
cases' s with f al; apply Subtype.eq; dsimp [tail, think]
|
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.dist from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
namespace Nat
def dist (n m : ℕ) :=
n - m + (m - n)
#align nat.dist Nat.dist
-- Should be aligned to `Nat.dist.eq_def`, but that is generated on demand and isn't pr... | Mathlib/Data/Nat/Dist.lean | 60 | 60 | theorem dist_tri_left' (n m : ℕ) : n ≤ dist n m + m := by | rw [dist_comm]; apply dist_tri_left
|
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 74 | 77 | theorem antideriv_bernoulliFun (k : ℕ) (x : ℝ) :
HasDerivAt (fun x => bernoulliFun (k + 1) x / (k + 1)) (bernoulliFun k x) x := by |
convert (hasDerivAt_bernoulliFun (k + 1) x).div_const _ using 1
field_simp [Nat.cast_add_one_ne_zero k]
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 235 | 238 | theorem formPerm_apply_nthLe (xs : List α) (h : Nodup xs) (n : ℕ) (hn : n < xs.length) :
formPerm xs (xs.nthLe n hn) =
xs.nthLe ((n + 1) % xs.length) (Nat.mod_lt _ (n.zero_le.trans_lt hn)) := by |
apply formPerm_apply_get _ h
|
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 63 | 70 | theorem mk_mem_enumFrom_iff_le_and_get?_sub {n i : ℕ} {x : α} {l : List α} :
(i, x) ∈ enumFrom n l ↔ n ≤ i ∧ l.get? (i - n) = x := by |
if h : n ≤ i then
rcases Nat.exists_eq_add_of_le h with ⟨i, rfl⟩
simp [mk_add_mem_enumFrom_iff_get?, Nat.add_sub_cancel_left]
else
have : ∀ k, n + k ≠ i := by rintro k rfl; simp at h
simp [h, mem_iff_get?, this]
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876... | Mathlib/Analysis/Convex/Topology.lean | 109 | 112 | theorem closure_openSegment (x y : E) : closure (openSegment 𝕜 x y) = [x -[𝕜] y] := by |
rw [segment_eq_image, openSegment_eq_image, ← closure_Ioo (zero_ne_one' 𝕜)]
exact (image_closure_of_isCompact (isBounded_Ioo _ _).isCompact_closure <|
Continuous.continuousOn <| by continuity).symm
|
import Mathlib.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift
import Mathlib.Data.Option.Basic
#align_import order.with_bot from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
variabl... | Mathlib/Order/WithBot.lean | 135 | 136 | theorem unbot'_eq_iff {d y : α} {x : WithBot α} : unbot' d x = y ↔ x = y ∨ x = ⊥ ∧ y = d := by |
induction x <;> simp [@eq_comm _ d]
|
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 | 253 | 253 | theorem Prime.factorization_self {p : ℕ} (hp : Prime p) : p.factorization p = 1 := by | simp [hp]
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,305 | 1,305 | theorem cos_add_pi_div_two (x : ℂ) : cos (x + π / 2) = -sin x := by | simp [cos_add]
|
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 | 127 | 142 | theorem ergodic_zsmul_add (x : AddCircle T) {n : ℤ} (h : 1 < |n|) : Ergodic fun y => n • y + x := by |
set f : AddCircle T → AddCircle T := fun y => n • y + x
let e : AddCircle T ≃ᵐ AddCircle T := MeasurableEquiv.addLeft (DivisibleBy.div x <| n - 1)
have he : MeasurePreserving e volume volume :=
measurePreserving_add_left volume (DivisibleBy.div x <| n - 1)
suffices e ∘ f ∘ e.symm = fun y => n • y by
rw... |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
#align_import ring_theory.polynomial.tower from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Polynomial
variable (R A B : Type*)
namespace Polynomial
section CommSemiring
variable [CommSemiring ... | Mathlib/RingTheory/Polynomial/Tower.lean | 54 | 56 | theorem aeval_algebraMap_apply (x : A) (p : R[X]) :
aeval (algebraMap A B x) p = algebraMap A B (aeval x p) := by |
rw [aeval_def, aeval_def, hom_eval₂, ← IsScalarTower.algebraMap_eq]
|
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.calculus.inverse from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Function Set Filter Metric
open scoped Topolo... | Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean | 283 | 291 | theorem open_image (hf : ApproximatesLinearOn f f' s c) (f'symm : f'.NonlinearRightInverse)
(hs : IsOpen s) (hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹) : IsOpen (f '' s) := by |
cases' hc with hE hc
· exact isOpen_discrete _
simp only [isOpen_iff_mem_nhds, nhds_basis_closedBall.mem_iff, forall_mem_image] at hs ⊢
intro x hx
rcases hs x hx with ⟨ε, ε0, hε⟩
refine ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, ?_⟩
exact (hf.surjOn_closedBall_of_nonlinearRightInverse f'symm ... |
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 321 | 324 | theorem oangle_smul_left_self_of_nonneg (x : V) {r : ℝ} (hr : 0 ≤ r) : o.oangle (r • x) x = 0 := by |
rcases hr.lt_or_eq with (h | h)
· simp [h]
· simp [h.symm]
|
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 359 | 360 | theorem associator_inv_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
f ◁ η ▷ h ≫ (α_ f g' h).inv = (α_ f g h).inv ≫ (f ◁ η) ▷ h := by | simp
|
import Mathlib.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
#align_import algebraic_topology.dold_kan.faces from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Limits CategoryTheory.Category
CategoryTheory.Preadditive CategoryTheor... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 172 | 219 | theorem induction {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ) :
HigherFacesVanish (q + 1) (φ ≫ (𝟙 _ + Hσ q).f (n + 1)) := by |
intro j hj₁
dsimp
simp only [comp_add, add_comp, comp_id]
-- when n < q, the result follows immediately from the assumption
by_cases hqn : n < q
· rw [v.comp_Hσ_eq_zero hqn, zero_comp, add_zero, v j (by omega)]
-- we now assume that n≥q, and write n=a+q
cases' Nat.le.dest (not_lt.mp hqn) with a ha
rw... |
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 446 | 453 | theorem le_ofWithBot {boxes : Finset (WithBot (Box ι))}
{le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I}
{pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint}
(H : ∀ J ∈ π, ∃ J' ∈ boxes, ↑J ≤ J') : π ≤ ofWithBot boxes le_of_mem pairwise_disjoint := by |
intro J hJ
rcases H J hJ with ⟨J', J'mem, hle⟩
lift J' to Box ι using ne_bot_of_le_ne_bot WithBot.coe_ne_bot hle
exact ⟨J', mem_ofWithBot.2 J'mem, WithBot.coe_le_coe.1 hle⟩
|
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 127 | 132 | theorem norm_le_half_period {x : AddCircle p} (hp : p ≠ 0) : ‖x‖ ≤ |p| / 2 := by |
obtain ⟨x⟩ := x
change ‖(x : AddCircle p)‖ ≤ |p| / 2
rw [norm_eq, ← mul_le_mul_left (abs_pos.mpr (inv_ne_zero hp)), ← abs_mul, mul_sub, mul_left_comm,
← mul_div_assoc, ← abs_mul, inv_mul_cancel hp, mul_one, abs_one]
exact abs_sub_round (p⁻¹ * x)
|
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 92 | 93 | theorem slope_comm (f : k → PE) (a b : k) : slope f a b = slope f b a := by |
rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 93 | 94 | theorem image_coe_Ici : (some : α → WithTop α) '' Ici a = Ico (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ici, image_preimage_eq_inter_range, range_coe, Ici_inter_Iio]
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 282 | 300 | theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
[IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
gauge (a • s) = a⁻¹ • gauge s := by |
obtain rfl | ha' := ha.eq_or_lt
· rw [inv_zero, zero_smul, gauge_of_subset_zero (zero_smul_set_subset _)]
ext x
rw [gauge_def', Pi.smul_apply, gauge_def', ← Real.sInf_smul_of_nonneg (inv_nonneg.2 ha)]
congr 1
ext r
simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
constructor
· rintro ⟨hr, y, hy, h⟩
si... |
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 | 148 | 161 | theorem normed_le_div_measure_closedBall_rOut [IsAddHaarMeasure μ] (K : ℝ) (h : f.rOut ≤ K * f.rIn)
(x : E) :
f.normed μ x ≤ K ^ finrank ℝ E / (μ (closedBall c f.rOut)).toReal := by |
have K_pos : 0 < K := by
simpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)
have : f x / ∫ y, f y ∂μ ≤ 1 / ∫ y, f y ∂μ := by
gcongr
· exact f.integral_pos.le
· exact f.le_one
apply this.trans
rw [div_le_div_iff f.integral_pos, one_mul, ← div_le_iff' (pow_pos K_... |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Algebra.Ring.NegOnePow
namespace Matrix
variable {R : Type*} [CommRing R]
| Mathlib/LinearAlgebra/Matrix/Determinant/Misc.lean | 21 | 47 | theorem submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_det {n : ℕ}
(M : Matrix (Fin (n + 1)) (Fin n) R) (hv : ∑ j, M j = 0) (j₁ j₂ : Fin (n + 1)) :
(M.submatrix (Fin.succAbove j₁) id).det =
Int.negOnePow (j₁ - j₂) • (M.submatrix (Fin.succAbove j₂) id).det := by |
suffices ∀ j, (M.submatrix (Fin.succAbove j) id).det =
Int.negOnePow j • (M.submatrix (Fin.succAbove 0) id).det by
rw [this j₁, this j₂, smul_smul, ← Int.negOnePow_add, sub_add_cancel]
intro j
induction j using Fin.induction with
| zero => rw [Fin.val_zero, Nat.cast_zero, Int.negOnePow_zero, one_smul... |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 243 | 251 | theorem dist_log_im_le (z w : ℍ) : dist (log z.im) (log w.im) ≤ dist z w :=
calc
dist (log z.im) (log w.im) = dist (mk ⟨0, z.im⟩ z.im_pos) (mk ⟨0, w.im⟩ w.im_pos) :=
Eq.symm <| dist_of_re_eq rfl
_ ≤ dist z w := by |
simp_rw [dist_eq]
dsimp only [coe_mk, mk_im]
gcongr
simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
|
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 | 184 | 185 | theorem Ioo_subset_Ioo (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ := by |
simpa [← coe_subset] using Set.Ioo_subset_Ioo ha hb
|
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 162 | 168 | theorem aestronglyMeasurable_snd [TopologicalSpace γ] [MetrizableSpace γ] [BorelSpace γ]
(h : IdentDistrib f g μ ν) (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable g ν := by |
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨h.aemeasurable_snd, ?_⟩
rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with ⟨t, t_sep, ht⟩
refine ⟨closure t, t_sep.closure, ?_⟩
apply h.ae_mem_snd isClosed_closure.measurableSet
filter_upwards [ht] with x hx using subset_closure hx
|
import Mathlib.Algebra.Order.GroupWithZero.Synonym
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Algebra.Ring.Hom.Defs
#align_import algebra.order.ring.with_top from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
variable {α : Type... | Mathlib/Algebra/Order/Ring/WithTop.lean | 89 | 91 | theorem mul_lt_top' [LT α] {a b : WithTop α} (ha : a < ⊤) (hb : b < ⊤) : a * b < ⊤ := by |
rw [WithTop.lt_top_iff_ne_top] at *
simp only [Ne, mul_eq_top_iff, *, and_false, false_and, or_self, not_false_eq_true]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 85 | 85 | theorem gcd_self_add_left (m n : ℕ) : gcd (m + n) m = gcd n m := by | rw [add_comm, gcd_add_self_left]
|
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
noncomputable section
v... | Mathlib/RingTheory/HahnSeries/Addition.lean | 81 | 89 | theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R}
(hxy : x + y ≠ 0) : min x.order y.order ≤ (x + y).order := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
rw [order_of_ne hx, order_of_ne hy, order_of_ne hxy]
apply le_of_eq_of_le _ (Set.IsWF.min_le_min_of_subset (support_add_subset (x := x) (y := y)))
· simp
· simp [hy]
· exact (Set.IsWF.min_union _ _ _ _).symm
|
import Mathlib.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_dimensional from "leanprover-community/mathlib"@"67e606eaea14c7854bdc556bd53d98aefdf76ec0"
noncomputable section
open Affine
section AffineSpace... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 602 | 610 | theorem Collinear.collinear_insert_iff_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
(hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₃ : p₂ ≠ p₃) :
Collinear k (insert p₁ s) ↔ Collinear k ({p₁, p₂, p₃} : Set P) := by |
have hv : vectorSpan k (insert p₁ s) = vectorSpan k ({p₁, p₂, p₃} : Set P) := by
-- Porting note: Original proof used `conv_lhs` and `conv_rhs`, but these tactics timed out.
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan]
symm
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, h.... |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 573 | 592 | theorem isNormal_map (hφ : Function.Injective φ.obj) (hφ' : im φ hφ = ⊤) (Sn : S.IsNormal) :
(map φ hφ S).IsNormal :=
{ wide := fun d => by
obtain ⟨c, rfl⟩ := obj_surjective_of_im_eq_top φ hφ hφ' d
change Map.Arrows φ hφ S _ _ (𝟙 _); rw [← Functor.map_id]
constructor; exact Sn.wide c
conj :... | rw [hφ']; apply mem_top_objs
rw [mem_im_objs_iff] at this
obtain ⟨c', rfl⟩ := this
have : g ∈ (im φ hφ).arrows (φ.obj c) (φ.obj c') := by rw [hφ']; trivial
rw [mem_im_iff] at this
obtain ⟨b, b', f, hb, hb', _, hf⟩ := this; cases hφ hb; cases hφ hb'
change Map.Arrows φ hφ S (φ.obj c'... |
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Logic.Equiv.Embedding
#align_import data.fintype.card_embedding from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
local notation "|" x "|" => Finset.card x
local notation "‖" x "‖" => Fintype.card x
open Function
open Nat
nam... | Mathlib/Data/Fintype/CardEmbedding.lean | 36 | 50 | theorem card_embedding_eq {α β : Type*} [Fintype α] [Fintype β] [emb : Fintype (α ↪ β)] :
‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖ := by |
rw [Subsingleton.elim emb Embedding.fintype]
refine Fintype.induction_empty_option (P := fun t ↦ ‖t ↪ β‖ = ‖β‖.descFactorial ‖t‖)
(fun α₁ α₂ h₂ e ih ↦ ?_) (?_) (fun γ h ih ↦ ?_) α <;> dsimp only <;> clear! α
· letI := Fintype.ofEquiv _ e.symm
rw [← card_congr (Equiv.embeddingCongr e (Equiv.refl β)), ... |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
noncomputable section
open scoped Classical
open Topology Filter
open TopologicalSpace Set Metric Filter Function
attribute [local simp... | Mathlib/Topology/MetricSpace/PiNat.lean | 119 | 119 | theorem cylinder_zero (x : ∀ n, E n) : cylinder x 0 = univ := by | simp [cylinder_eq_pi]
|
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 358 | 360 | theorem ball_sub_ball (hε : 0 < ε) (hδ : 0 < δ) (a b : E) :
ball a ε - ball b δ = ball (a - b) (ε + δ) := by |
simp_rw [sub_eq_add_neg, neg_ball, ball_add_ball hε hδ]
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 620 | 644 | theorem tfae_modEq :
TFAE
[a ≡ b [PMOD p], ∀ z : ℤ, b - z • p ∉ Set.Ioo a (a + p), toIcoMod hp a b ≠ toIocMod hp a b,
toIcoMod hp a b + p = toIocMod hp a b] := by |
rw [modEq_iff_toIcoMod_eq_left hp]
tfae_have 3 → 2
· rw [← not_exists, not_imp_not]
exact fun ⟨i, hi⟩ =>
((toIcoMod_eq_iff hp).2 ⟨Set.Ioo_subset_Ico_self hi, i, (sub_add_cancel b _).symm⟩).trans
((toIocMod_eq_iff hp).2 ⟨Set.Ioo_subset_Ioc_self hi, i, (sub_add_cancel b _).symm⟩).symm
tfae_have... |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 305 | 306 | theorem inner_sub_sub_self (x y : F) : ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫ := by |
simp only [inner_sub_left, inner_sub_right]; ring
|
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Data.ZMod.Quotient
#align_import measure_theory.group.add_circle from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter MeasureTheory MeasureTheory.Measure Metric
open scoped MeasureTheory Pointwise Top... | Mathlib/MeasureTheory/Group/AddCircle.lean | 34 | 48 | theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =ᵐ[volume] ball x ε := by |
rcases le_or_lt ε 0 with hε | hε
· rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closedBall,
min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ENNReal.ofReal_eq_zero]
exact mul_nonpos_of_nonneg_of_nonpos zero_le_two hε
· suffices volume (closedBall x ε) ≤ volume (ball x ε) by
exact (ae_eq_of_subse... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 181 | 184 | theorem coeff_monomial (m n : ℕ) (a : R) : coeff R m (monomial R n a) = if m = n then a else 0 :=
calc
coeff R m (monomial R n a) = _ := MvPowerSeries.coeff_monomial _ _ _
_ = if m = n then a else 0 := by | simp only [Finsupp.unique_single_eq_iff]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 385 | 388 | theorem subset_generated (c d : C) : X c d ⊆ (generated X).arrows c d := by |
dsimp only [generated, sInf]
simp only [subset_iInter₂_iff]
exact fun S hS f fS => hS _ _ fS
|
import Batteries.Data.RBMap.Basic
import Batteries.Tactic.SeqFocus
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] All
theorem All.trivial (H : ∀ {x : α}, p x) : ∀ {t : RBNode α}, t.All p
| nil => _root_.trivial
| node .. => ⟨H, All.trivial H, All.trivial H⟩
theorem All_and {t : RBNode α}... | .lake/packages/batteries/Batteries/Data/RBMap/WF.lean | 51 | 52 | theorem reverse_eq_iff {t t' : RBNode α} : t.reverse = t' ↔ t = t'.reverse := by |
constructor <;> rintro rfl <;> simp
|
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 | 87 | 88 | theorem gold_pow_sub_gold_pow (n : ℕ) : φ ^ (n + 2) - φ ^ (n + 1) = φ ^ n := by |
rw [goldenRatio]; ring_nf; norm_num; ring
|
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 251 | 252 | theorem finrank_matrix (m n : Type*) [Fintype m] [Fintype n] :
finrank R (Matrix m n R) = card m * card n := by | simp [finrank]
|
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 | 95 | 100 | theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
(Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom := by |
change (N₁Γ₀.app K).inv = _
simp only [N₁Γ₀_app]
rfl
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 390 | 391 | theorem casesOn'_none_coe (f : Option α → β) (o : Option α) :
casesOn' o (f none) (f ∘ (fun a ↦ ↑a)) = f o := by | cases o <;> rfl
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 91 | 95 | theorem bernoulli'_spec' (n : ℕ) :
(∑ k ∈ antidiagonal n, ((k.1 + k.2).choose k.2 : ℚ) / (k.2 + 1) * bernoulli' k.1) = 1 := by |
refine ((sum_antidiagonal_eq_sum_range_succ_mk _ n).trans ?_).trans (bernoulli'_spec n)
refine sum_congr rfl fun x hx => ?_
simp only [add_tsub_cancel_of_le, mem_range_succ_iff.mp hx, cast_sub]
|
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 | 217 | 220 | theorem natTrailingDegree_le_of_trailingDegree_le {n : ℕ} {hp : p ≠ 0}
(H : (n : ℕ∞) ≤ trailingDegree p) : n ≤ natTrailingDegree p := by |
rw [trailingDegree_eq_natTrailingDegree hp] at H
exact WithTop.coe_le_coe.mp H
|
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 143 | 145 | theorem zip_of_prod {l : List α} {l' : List β} {lp : List (α × β)} (hl : lp.map Prod.fst = l)
(hr : lp.map Prod.snd = l') : lp = l.zip l' := by |
rw [← hl, ← hr, ← zip_unzip lp, ← unzip_left, ← unzip_right, zip_unzip, zip_unzip]
|
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 74 | 78 | theorem memberSubfamily_inter (a : α) (𝒜 ℬ : Finset (Finset α)) :
(𝒜 ∩ ℬ).memberSubfamily a = 𝒜.memberSubfamily a ∩ ℬ.memberSubfamily a := by |
unfold memberSubfamily
rw [filter_inter_distrib, image_inter_of_injOn _ _ ((erase_injOn' _).mono _)]
simp
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanprover-community/mathlib"@"a6ece35404f60597c651689c1b46ead86de5ac1b"
open Function
variable {R A : Type*}
def IsSelfAdjoint [Star R] (x : R) : Prop :=
... | Mathlib/Algebra/Star/SelfAdjoint.lean | 92 | 93 | theorem mul_star_self [Mul R] [StarMul R] (x : R) : IsSelfAdjoint (x * star x) := by |
simpa only [star_star] using star_mul_self (star x)
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 146 | 165 | theorem condexpIndL1Fin_disjoint_union (hs : MeasurableSet s) (ht : MeasurableSet t) (hμs : μ s ≠ ∞)
(hμt : μ t ≠ ∞) (hst : s ∩ t = ∅) (x : G) :
condexpIndL1Fin hm (hs.union ht) ((measure_union_le s t).trans_lt
(lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne x =
condexpIndL1Fin hm hs h... |
ext1
have hμst :=
((measure_union_le s t).trans_lt (lt_top_iff_ne_top.mpr (ENNReal.add_ne_top.mpr ⟨hμs, hμt⟩))).ne
refine (condexpIndL1Fin_ae_eq_condexpIndSMul hm (hs.union ht) hμst x).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_add _ _).symm
have hs_eq := condexpIndL1Fin_ae_eq_condexpIndSMul hm hs h... |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 355 | 356 | theorem _root_.Wbtw.angle₃₁₂_eq_zero_of_ne {p₁ p₂ p₃ : P} (h : Wbtw ℝ p₁ p₂ p₃) (hp₂p₁ : p₂ ≠ p₁) :
∠ p₃ p₁ p₂ = 0 := by | rw [← h.angle₂₁₃_eq_zero_of_ne hp₂p₁, angle_comm]
|
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.Algebra.Star
import Mathlib.Topology.Instances.Int
import Mathlib.Topology.Order.Bornology
#align_import topology.instances.real fro... | Mathlib/Topology/Instances/Real.lean | 77 | 78 | theorem Real.cobounded_eq : cobounded ℝ = atBot ⊔ atTop := by |
simp only [← comap_dist_right_atTop (0 : ℝ), Real.dist_eq, sub_zero, comap_abs_atTop]
|
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 | 236 | 238 | theorem linearIndependent_le_span_finset {ι : Type*} (v : ι → M) (i : LinearIndependent R v)
(w : Finset M) (s : span R (w : Set M) = ⊤) : #ι ≤ w.card := by |
simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
|
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 87 | 89 | theorem add (hxy₁ : x₁ ≡ y₁ [SMOD U]) (hxy₂ : x₂ ≡ y₂ [SMOD U]) : x₁ + x₂ ≡ y₁ + y₂ [SMOD U] := by |
rw [SModEq.def] at hxy₁ hxy₂ ⊢
simp_rw [Quotient.mk_add, hxy₁, hxy₂]
|
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 206 | 209 | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by |
rw [upperCrossingTime_succ]
exact le_hitting lowerCrossingTime_le ω
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
#align_import algebra.star.self_adjoint from "leanprover-community/mathlib"@"a6ece35404f60597c651689c1b46ead86de5ac1b"
open Function
variable {R A : Type*}
def IsSelfAdjoint [Star R] (x : R) : Prop :=
... | Mathlib/Algebra/Star/SelfAdjoint.lean | 137 | 138 | theorem neg {x : R} (hx : IsSelfAdjoint x) : IsSelfAdjoint (-x) := by |
simp only [isSelfAdjoint_iff, star_neg, hx.star_eq]
|
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a22dacca4c20b63d"
noncomputable section
open scoped Manifold
-- See note [Design choices about smooth algebraic structures]
class LieAddGroup {𝕜 : Type*... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 186 | 188 | theorem ContMDiffOn.div {f g : M → G} {s : Set M} (hf : ContMDiffOn I' I n f s)
(hg : ContMDiffOn I' I n g s) : ContMDiffOn I' I n (fun x => f x / g x) s := by |
simp_rw [div_eq_mul_inv]; exact hf.mul hg.inv
|
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 | 124 | 134 | theorem inf_mem_subalgebra_closure (A : Subalgebra ℝ C(X, ℝ)) (f g : A) :
(f : C(X, ℝ)) ⊓ (g : C(X, ℝ)) ∈ A.topologicalClosure := by |
rw [inf_eq_half_smul_add_sub_abs_sub' ℝ]
refine
A.topologicalClosure.smul_mem
(A.topologicalClosure.sub_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.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, S... | Mathlib/Order/BooleanAlgebra.lean | 469 | 470 | theorem inf_sdiff_distrib_right (a b c : α) : a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c) := by |
simp_rw [inf_comm _ c, inf_sdiff_distrib_left]
|
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.Data.List.Chain
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theor... | Mathlib/GroupTheory/CoprodI.lean | 588 | 592 | theorem cons_eq_smul {i} {m : M i} {ls h1 h2} :
cons m ls h1 h2 = of m • ls := by |
rw [of_smul_def, equivPair_eq_of_fstIdx_ne _]
· simp [cons, rcons, h2]
· exact h1
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : ∅ ∈... | Mathlib/Topology/Basic.lean | 812 | 812 | theorem le_nhds_iff {f} : f ≤ 𝓝 x ↔ ∀ s : Set X, x ∈ s → IsOpen s → s ∈ f := by | simp [nhds_def]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 622 | 631 | theorem inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) :
⟪x, y⟫ = ‖x‖ * ‖y‖ * Real.Angle.cos (o.oangle x y) := by |
by_cases hx : x = 0; · simp [hx]
by_cases hy : y = 0; · simp [hy]
have : ‖x‖ ≠ 0 := by simpa using hx
have : ‖y‖ ≠ 0 := by simpa using hy
rw [oangle, Real.Angle.cos_coe, Complex.cos_arg, o.abs_kahler]
· simp only [kahler_apply_apply, real_smul, add_re, ofReal_re, mul_re, I_re, ofReal_im]
field_simp
·... |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 56 | 59 | theorem restrict_toOuterMeasure_eq_toOuterMeasure_restrict (h : MeasurableSet s) :
(μ.restrict s).toOuterMeasure = OuterMeasure.restrict s μ.toOuterMeasure := by |
simp_rw [restrict, restrictₗ, liftLinear, LinearMap.coe_mk, AddHom.coe_mk,
toMeasure_toOuterMeasure, OuterMeasure.restrict_trim h, μ.trimmed]
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 313 | 324 | theorem sum_lt {α : Type*} {F : α → ℚ} {t : ℚ} {s : Finset α} :
s.Nonempty → (∀ i ∈ s, padicNorm p (F i) < t) → padicNorm p (∑ i ∈ s, F i) < t := by |
classical
refine s.induction_on (by rintro ⟨-, ⟨⟩⟩) ?_
rintro a S haS IH - ht
by_cases hs : S.Nonempty
· rw [Finset.sum_insert haS]
exact
lt_of_le_of_lt padicNorm.nonarchimedean
(max_lt (ht a (Finset.mem_insert_self a S))
(IH hs fun b hb ↦ ht b (Finset.mem_insert_o... |
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {α X : Type*} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
| Mathlib/Dynamics/Ergodic/Function.lean | 27 | 35 | theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
{s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
(h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
(hg_eq : g ∘ f =ᵐ[μ] g) :
∃ c, g =ᵐ[μ] const α c := by |
refine exists_eventuallyEq_const_of_eventually_mem_of_forall_separating MeasurableSet hs ?_
refine fun U hU ↦ h.ae_mem_or_ae_nmem₀ (s := g ⁻¹' U) (hgm hU) ?_b
refine (hg_eq.mono fun x hx ↦ ?_).set_eq
rw [← preimage_comp, mem_preimage, mem_preimage, hx]
|
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 | 503 | 506 | theorem Wbtw.trans_left_ne [NoZeroSMulDivisors R V] {w x y z : P} (h₁ : Wbtw R w y z)
(h₂ : Wbtw R w x y) (h : y ≠ z) : x ≠ z := by |
rintro rfl
exact h (h₁.swap_right_iff.1 h₂)
|
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 295 | 296 | theorem inv_hom_id_assoc (f : X ⟶ Y) [I : IsIso f] {Z} (g : Y ⟶ Z) : inv f ≫ f ≫ g = g := by |
simp [← Category.assoc]
|
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