Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
namespace MulAction
universe u v
variable {α : Type v}
variable {G : Type u} [Group G] [MulAction G α]
variable {M : Type u} [Monoid M] [MulAction M α]
@[to_additive "If the action is periodic, t... | Mathlib/GroupTheory/GroupAction/Period.lean | 117 | 120 | theorem period_bounded_of_exponent_pos (exp_pos : 0 < Monoid.exponent M) (m : M) :
BddAbove (Set.range (fun a : α => period m a)) := by |
use Monoid.exponent M
simpa [upperBounds] using period_le_exponent exp_pos _
| 0.40625 |
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 Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 249 | 251 | theorem image_congr {f g : α → β} {s : Set α} (h : ∀ a ∈ s, f a = g a) : f '' s = g '' s := by |
ext x
exact exists_congr fun a ↦ and_congr_right fun ha ↦ by rw [h a ha]
| 0.40625 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 60 | 62 | theorem count_eq_card_fintype (n : ℕ) : count p n = Fintype.card { k : ℕ // k < n ∧ p k } := by |
rw [count_eq_card_filter_range, ← Fintype.card_ofFinset, ← CountSet.fintype]
rfl
| 0.40625 |
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
noncomputable section
universe u
variable {K : Type u}
namespace RatFunc
section IntDegree
open Polynomial
variable [Field... | Mathlib/FieldTheory/RatFunc/Degree.lean | 44 | 45 | theorem intDegree_zero : intDegree (0 : RatFunc K) = 0 := by |
rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]
| 0.40625 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 67 | 74 | theorem affineSegment_same (x : P) : affineSegment R x x = {x} := by |
-- Porting note: added as this doesn't do anything in `simp_rw` any more
rw [affineSegment]
-- Note: when adding "simp made no progress" in lean4#2336,
-- had to change `lineMap_same` to `lineMap_same _`. Not sure why?
-- Porting note: added `_ _` and `Function.const`
simp_rw [lineMap_same _, AffineMap.coe... | 0.40625 |
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.split from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open Filter
open Function Set Filter
namespace BoxIntegral
variable {ι M : Type*} {... | Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 130 | 136 | theorem splitUpper_def [DecidableEq ι] {i x} (h : x ∈ Ioo (I.lower i) (I.upper i))
(h' : ∀ j, update I.lower i x j < I.upper j :=
(forall_update_iff I.lower fun j y => y < I.upper j).2
⟨h.2, fun j _ => I.lower_lt_upper _⟩) :
I.splitUpper i x = (⟨update I.lower i x, I.upper, h'⟩ : Box ι) := by |
simp (config := { unfoldPartialApp := true }) only [splitUpper, mk'_eq_coe, max_eq_left h.1.le,
update, and_self]
| 0.40625 |
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable {R : Type*} [CommRing R]
namespace Ideal
open Submodule
variable (R) in
def isPrincipalSubmonoid : Submonoid (Ideal R) where
carrier := ... | Mathlib/RingTheory/Ideal/IsPrincipal.lean | 67 | 72 | theorem associatesEquivIsPrincipal_mul (x y : Associates R) :
(associatesEquivIsPrincipal R (x * y) : Ideal R) =
(associatesEquivIsPrincipal R x) * (associatesEquivIsPrincipal R y) := by |
rw [← Associates.quot_out x, ← Associates.quot_out y]
simp_rw [Associates.mk_mul_mk, ← Associates.quotient_mk_eq_mk, associatesEquivIsPrincipal_apply,
span_singleton_mul_span_singleton]
| 0.40625 |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 256 | 259 | theorem finsuppTensorFinsupp_single (i : ι) (m : M) (k : κ) (n : N) :
finsuppTensorFinsupp R S M N ι κ (Finsupp.single i m ⊗ₜ Finsupp.single k n) =
Finsupp.single (i, k) (m ⊗ₜ n) := by |
simp [finsuppTensorFinsupp]
| 0.40625 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 107 | 110 | theorem relabel_id (t : L.Term α) : t.relabel id = t := by |
induction' t with _ _ _ _ ih
· rfl
· simp [ih]
| 0.40625 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 161 | 169 | theorem StrongMono.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) [h : StrongMono f] : StrongMono g :=
{ mono := by |
rw [Arrow.iso_w' e]
haveI := mono_comp f e.hom.right
apply mono_comp
rlp := fun {X Y} z => by
intro
apply HasLiftingProperty.of_arrow_iso_right z e }
| 0.40625 |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 76 | 78 | theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by |
ext i
simp [mem_image, ht.exists_mem]
| 0.40625 |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 104 | 106 | theorem IsUpperSet.div_right (hs : IsUpperSet s) : IsUpperSet (s / t) := by |
rw [div_eq_mul_inv]
exact hs.mul_right
| 0.40625 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace LocalizedModule
universe u v
variable {R : Type u} [CommSemiring R] (S : Submonoid R)
variab... | Mathlib/Algebra/Module/LocalizedModule.lean | 120 | 121 | theorem liftOn_mk {α : Type*} {f : M × S → α} (wd : ∀ (p p' : M × S), p ≈ p' → f p = f p')
(m : M) (s : S) : liftOn (mk m s) f wd = f ⟨m, s⟩ := by | convert Quotient.liftOn_mk f wd ⟨m, s⟩
| 0.40625 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 110 | 113 | theorem ascPochhammer_eval_zero {n : ℕ} : (ascPochhammer S n).eval 0 = if n = 0 then 1 else 0 := by |
cases n
· simp
· simp [X_mul, Nat.succ_ne_zero, ascPochhammer_succ_left]
| 0.40625 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 313 | 314 | theorem map_smul₂ (f : E →L[𝕜] F →L[𝕜] G) (c : 𝕜) (x : E) (y : F) : f (c • x) y = c • f x y := by |
rw [f.map_smul, smul_apply]
| 0.40625 |
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 104 | 105 | theorem dist_vadd_cancel_right (v₁ v₂ : V) (x : P) : dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂ := by |
rw [dist_eq_norm_vsub V, dist_eq_norm, vadd_vsub_vadd_cancel_right]
| 0.40625 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 76 | 76 | theorem not_eq_true_eq_eq_false (a : Bool) : (not a = true) = (a = false) := by | cases a <;> simp
| 0.40625 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 32 | 35 | theorem comm {a b c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c := by |
delta Fermat42
rw [add_comm]
tauto
| 0.40625 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 126 | 127 | theorem taylor_eval_sub {R} [CommRing R] (r : R) (f : R[X]) (s : R) :
(taylor r f).eval (s - r) = f.eval s := by | rw [taylor_eval, sub_add_cancel]
| 0.40625 |
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Vector
variable {α β : Type*} {n : ℕ} (a a' : α)
@[simp]
| Mathlib/Data/Vector/Mem.lean | 26 | 28 | theorem get_mem (i : Fin n) (v : Vector α n) : v.get i ∈ v.toList := by |
rw [get_eq_get]
exact List.get_mem _ _ _
| 0.40625 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 77 | 83 | theorem symmetry (X Y : C) :
(Limits.BinaryFan.braiding (ℬ X Y).isLimit (ℬ Y X).isLimit).hom ≫
(Limits.BinaryFan.braiding (ℬ Y X).isLimit (ℬ X Y).isLimit).hom =
𝟙 (tensorObj ℬ X Y) := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext;
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
| 0.40625 |
import Mathlib.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.antilipschitz from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
... | Mathlib/Topology/MetricSpace/Antilipschitz.lean | 110 | 113 | theorem mul_le_edist (hf : AntilipschitzWith K f) (x y : α) :
(K : ℝ≥0∞)⁻¹ * edist x y ≤ edist (f x) (f y) := by |
rw [mul_comm, ← div_eq_mul_inv]
exact ENNReal.div_le_of_le_mul' (hf x y)
| 0.40625 |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 100 | 103 | theorem map_zero (hf : IsRingHom f) : f 0 = 0 :=
calc
f 0 = f (0 + 0) - f 0 := by | rw [hf.map_add]; simp
_ = 0 := by simp
| 0.40625 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
assert_not_exists MonoidWithZero
open Prod Decidable Function
namespace Nat
-- Porting note: no pp_nodot
--@[pp_nodot]
def pair (a b : ... | Mathlib/Data/Nat/Pairing.lean | 104 | 106 | theorem unpair_zero : unpair 0 = 0 := by |
rw [unpair]
simp
| 0.40625 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 62 | 64 | theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support ≤ p.support := by |
intro
simpa using left_ne_zero_of_mul
| 0.40625 |
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian
import Mathlib.RingTheory.ChainOfDivisors
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Operations
#align_impo... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 136 | 137 | theorem map_inv (I : FractionalIdeal R₁⁰ K) (h : K ≃ₐ[R₁] K') :
I⁻¹.map (h : K →ₐ[R₁] K') = (I.map h)⁻¹ := by | rw [inv_eq, map_div, map_one, inv_eq]
| 0.40625 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 53 | 60 | theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
{a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) := by |
intro p hp
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')
· exact ⟨a, left_mem_Icc.mpr hab, rfl⟩
· exact ⟨b, right_mem_Icc.mpr hab, rfl⟩
· have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
exact image_subset f Ioo_subset_Icc_self this
| 0.40625 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 159 | 165 | theorem IsDiag.fromBlocks [Zero α] {A : Matrix m m α} {D : Matrix n n α} (ha : A.IsDiag)
(hd : D.IsDiag) : (A.fromBlocks 0 0 D).IsDiag := by |
rintro (i | i) (j | j) hij
· exact ha (ne_of_apply_ne _ hij)
· rfl
· rfl
· exact hd (ne_of_apply_ne _ hij)
| 0.40625 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 218 | 219 | theorem units_apply_inv_apply (f : CircleDeg1Liftˣ) (x : ℝ) :
f ((f⁻¹ : CircleDeg1Liftˣ) x) = x := by | simp only [← mul_apply, f.mul_inv, coe_one, id]
| 0.40625 |
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α : Type*} {N : α → Type*}
namespace DFinsupp
variable [DecidableEq α]
section NHasZero
variable [∀ a, DecidableEq (N a)] [∀ a, Zero (N a)] (f g : Π₀... | Mathlib/Data/DFinsupp/NeLocus.lean | 72 | 74 | theorem neLocus_zero_right : f.neLocus 0 = f.support := by |
ext
rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]
| 0.40625 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 107 | 108 | theorem SameCycle.conj : SameCycle f x y → SameCycle (g * f * g⁻¹) (g x) (g y) := by |
simp [sameCycle_conj]
| 0.40625 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s... | Mathlib/Data/List/InsertNth.lean | 116 | 119 | theorem insertNth_length_self (l : List α) (x : α) : insertNth l.length x l = l ++ [x] := by |
induction' l with hd tl IH
· simp
· simpa using IH
| 0.40625 |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 153 | 154 | theorem comp.mk_get (x : comp P Q α) : comp.mk (comp.get x) = x := by |
rfl
| 0.40625 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.PowerBasis
#align_import field_theory.separable from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
... | Mathlib/FieldTheory/Separable.lean | 66 | 67 | theorem separable_of_subsingleton [Subsingleton R] (f : R[X]) : f.Separable := by |
simp [Separable, IsCoprime, eq_iff_true_of_subsingleton]
| 0.40625 |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 102 | 104 | theorem IsEquivalent.refl : u ~[l] u := by |
rw [IsEquivalent, sub_self]
exact isLittleO_zero _ _
| 0.40625 |
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a fu... | Mathlib/MeasureTheory/Group/LIntegral.lean | 34 | 37 | theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (g * x) ∂μ) = ∫⁻ x, f x ∂μ := by |
convert (lintegral_map_equiv f <| MeasurableEquiv.mulLeft g).symm
simp [map_mul_left_eq_self μ g]
| 0.40625 |
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 | 170 | 172 | theorem symmDiff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b := by |
rw [symmDiff_sdiff]
simp [symmDiff]
| 0.40625 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 123 | 123 | theorem norm_star_mul_self' {x : E} : ‖x⋆ * x‖ = ‖x⋆‖ * ‖x‖ := by | rw [norm_star_mul_self, norm_star]
| 0.40625 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 126 | 129 | theorem polar_comp {F : Type*} [CommRing S] [FunLike F R S] [AddMonoidHomClass F R S]
(f : M → R) (g : F) (x y : M) :
polar (g ∘ f) x y = g (polar f x y) := by |
simp only [polar, Pi.smul_apply, Function.comp_apply, map_sub]
| 0.40625 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 | theorem gcd_zero_left (a : R) : gcd 0 a = a := by |
rw [gcd]
exact if_pos rfl
| 0.40625 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 106 | 107 | theorem asAlgebraHom_single (g : G) (r : k) : asAlgebraHom ρ (Finsupp.single g r) = r • ρ g := by |
simp only [asAlgebraHom_def, MonoidAlgebra.lift_single]
| 0.40625 |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 69 | 71 | theorem of_mul_t (b : B) :
(of (b : G) : HNNExtension G A B φ) * t = t * of (φ.symm b : G) := by |
rw [t_mul_of]; simp
| 0.40625 |
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 100 | 104 | theorem singleton_unop_op (x : αᵒᵖ) : ({unop x} : Set α).op = {x} := by |
ext
constructor
· apply unop_injective
· apply op_injective
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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]
exact WithTop.coe_eq_coe
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import Mathlib.Order.Filter.AtTopBot
#align_import order.filter.indicator_function from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α β M E : Type*}
open Set Filter
@[to_additive]
| Mathlib/Order/Filter/IndicatorFunction.lean | 63 | 66 | theorem Monotone.mulIndicator_eventuallyEq_iUnion {ι} [Preorder ι] [One β] (s : ι → Set α)
(hs : Monotone s) (f : α → β) (a : α) :
(fun i => mulIndicator (s i) f a) =ᶠ[atTop] fun _ ↦ mulIndicator (⋃ i, s i) f a := by |
classical exact hs.piecewise_eventually_eq_iUnion f 1 a
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import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 64 | 65 | theorem withDensityᵥ_zero : μ.withDensityᵥ (0 : α → E) = 0 := by |
ext1 s hs; erw [withDensityᵥ_apply (integrable_zero α E μ) hs]; simp
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import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.trigonometric.inverse_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classic... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 66 | 71 | theorem hasDerivWithinAt_arcsin_Ici {x : ℝ} (h : x ≠ -1) :
HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Ici x) x := by |
rcases eq_or_ne x 1 with (rfl | h')
· convert (hasDerivWithinAt_const (1 : ℝ) _ (π / 2)).congr _ _ <;>
simp (config := { contextual := true }) [arcsin_of_one_le]
· exact (hasDerivAt_arcsin h h').hasDerivWithinAt
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import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 160 | 162 | theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
sSup (s / t) = sSup s / sInf t := by |
rw [div_eq_mul_inv, csSup_mul hs₀ hs₁ ht₀.inv ht₁.inv, csSup_inv ht₀ ht₁, div_eq_mul_inv]
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import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 131 | 134 | theorem δ_comp_δ_self {n} {i : Fin (n + 2)} :
X.δ (Fin.castSucc i) ≫ X.δ i = X.δ i.succ ≫ X.δ i := by |
dsimp [δ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_δ_self]
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import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 164 | 164 | theorem kernelSubobjectMap_id : kernelSubobjectMap (𝟙 (Arrow.mk f)) = 𝟙 _ := by | aesop_cat
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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 | 159 | 165 | theorem sup_mem_closed_subalgebra (A : Subalgebra ℝ C(X, ℝ)) (h : IsClosed (A : Set C(X, ℝ)))
(f g : A) : (f : C(X, ℝ)) ⊔ (g : C(X, ℝ)) ∈ A := by |
convert sup_mem_subalgebra_closure A f g
apply SetLike.ext'
symm
erw [closure_eq_iff_isClosed]
exact h
| 0.375 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | Mathlib/Order/Interval/Set/OrderIso.lean | 68 | 69 | theorem image_Iic (e : α ≃o β) (a : α) : e '' Iic a = Iic (e a) := by |
rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]
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import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 98 | 100 | theorem convexJoin_iUnion_right (s : Set E) (t : ι → Set E) :
convexJoin 𝕜 s (⋃ i, t i) = ⋃ i, convexJoin 𝕜 s (t i) := by |
simp_rw [convexJoin_comm s, convexJoin_iUnion_left]
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import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Ring.Action.Basic
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import algebra.polynomial.group_ring_action from "leanprover-community/mathlib"@"afad8e438d03f... | Mathlib/Algebra/Polynomial/GroupRingAction.lean | 76 | 78 | theorem smul_eval [MulSemiringAction G S] (g : G) (f : S[X]) (x : S) :
(g • f).eval x = g • f.eval (g⁻¹ • x) := by |
rw [← smul_eval_smul, smul_inv_smul]
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import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
#align_import number_theory.padics.padic_integers from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open Padic Metric LocalRing
noncomputable section
open scoped Classical
def Pad... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 145 | 145 | theorem coe_eq_zero (z : ℤ_[p]) : (z : ℚ_[p]) = 0 ↔ z = 0 := by | rw [← coe_zero, Subtype.coe_inj]
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import Mathlib.Data.Option.Basic
import Mathlib.Data.Set.Basic
#align_import data.pequiv from "leanprover-community/mathlib"@"7c3269ca3fa4c0c19e4d127cd7151edbdbf99ed4"
universe u v w x
structure PEquiv (α : Type u) (β : Type v) where
toFun : α → Option β
invFun : β → Option α
inv : ∀ (a : α) (b :... | Mathlib/Data/PEquiv.lean | 174 | 175 | theorem trans_refl (f : α ≃. β) : f.trans (PEquiv.refl β) = f := by |
ext; dsimp [PEquiv.trans]; simp
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import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 155 | 156 | theorem sub (hf : IsBoundedLinearMap 𝕜 f) (hg : IsBoundedLinearMap 𝕜 g) :
IsBoundedLinearMap 𝕜 fun e => f e - g e := by | simpa [sub_eq_add_neg] using add hf (neg hg)
| 0.375 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 270 | 271 | theorem le_pre : μ ≤ pre m r ↔ ∀ s : Set X, diam s ≤ r → μ s ≤ m s := by |
simp only [pre, le_boundedBy, extend, le_iInf_iff]
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import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 31 | 34 | theorem continuantsAux_stable_step_of_terminated (terminated_at_n : g.TerminatedAt n) :
g.continuantsAux (n + 2) = g.continuantsAux (n + 1) := by |
rw [terminatedAt_iff_s_none] at terminated_at_n
simp only [continuantsAux, Nat.add_eq, Nat.add_zero, terminated_at_n]
| 0.375 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.lemmas from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
open Function Nat
namespace Int
variable {a b : ℤ} {n : ℕ}
| Mathlib/Data/Int/Order/Lemmas.lean | 28 | 30 | theorem natAbs_eq_iff_mul_self_eq {a b : ℤ} : a.natAbs = b.natAbs ↔ a * a = b * b := by |
rw [← abs_eq_iff_mul_self_eq, abs_eq_natAbs, abs_eq_natAbs]
exact Int.natCast_inj.symm
| 0.375 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import data.nat.choose.bounds from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
open Nat
variable {α : Type*} [LinearOrderedSemif... | Mathlib/Data/Nat/Choose/Bounds.lean | 41 | 46 | theorem pow_le_choose (r n : ℕ) : ((n + 1 - r : ℕ) ^ r : α) / r ! ≤ n.choose r := by |
rw [div_le_iff']
· norm_cast
rw [← Nat.descFactorial_eq_factorial_mul_choose]
exact n.pow_sub_le_descFactorial r
exact mod_cast r.factorial_pos
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import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 128 | 130 | theorem Periodic.const_inv_smul₀ [AddCommMonoid α] [DivisionSemiring γ] [Module γ α]
(h : Periodic f c) (a : γ) : Periodic (fun x => f (a⁻¹ • x)) (a • c) := by |
simpa only [inv_inv] using h.const_smul₀ a⁻¹
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import Mathlib.Logic.Function.Conjugate
#align_import logic.function.iterate from "leanprover-community/mathlib"@"792a2a264169d64986541c6f8f7e3bbb6acb6295"
universe u v
variable {α : Type u} {β : Type v}
def Nat.iterate {α : Sort u} (op : α → α) : ℕ → α → α
| 0, a => a
| succ k, a => iterate op k (op a)
#a... | Mathlib/Logic/Function/Iterate.lean | 80 | 82 | theorem iterate_add_apply (m n : ℕ) (x : α) : f^[m + n] x = f^[m] (f^[n] x) := by |
rw [iterate_add f m n]
rfl
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import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 167 | 171 | theorem not_linearIndependent_iff :
¬LinearIndependent R v ↔
∃ s : Finset ι, ∃ g : ι → R, ∑ i ∈ s, g i • v i = 0 ∧ ∃ i ∈ s, g i ≠ 0 := by |
rw [linearIndependent_iff']
simp only [exists_prop, not_forall]
| 0.375 |
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Data.Real.ConjExponents
#align_import analysis.mean_inequalities from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
universe u... | Mathlib/Analysis/MeanInequalities.lean | 169 | 177 | theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i ∈ s, w i = 1)
(hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i ∈ s, w i * z i = x :=
calc
∑ i ∈ s, w i * z i = ∑ i ∈ s, w i * x := by |
refine sum_congr rfl fun i hi => ?_
rcases eq_or_ne (w i) 0 with hwi | hwi
· rw [hwi, zero_mul, zero_mul]
· rw [hx i hi hwi]
_ = x := by rw [← sum_mul, hw', one_mul]
| 0.375 |
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 | 131 | 131 | theorem self_mem_cylinder (x : ∀ n, E n) (n : ℕ) : x ∈ cylinder x n := by | simp
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import Mathlib.Analysis.MeanInequalities
import Mathlib.Data.Fintype.Order
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.Analysis.NormedSpace.WithLp
#align_import analysis.normed_space.pi_Lp from "leanprover-community/mathlib"@"9d013ad8430ddddd350cff5c3db830278ded3c79"
set_option linter.uppercaseLean3 f... | Mathlib/Analysis/NormedSpace/PiLp.lean | 247 | 249 | theorem dist_eq_iSup (f g : PiLp ∞ α) : dist f g = ⨆ i, dist (f i) (g i) := by |
dsimp [dist]
exact if_neg ENNReal.top_ne_zero
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import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 384 | 384 | theorem graph_id : graph id = @Eq α := by | simp (config := { unfoldPartialApp := true }) [graph]
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import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 556 | 558 | theorem pullback_id (x : Subobject X) : (pullback (𝟙 X)).obj x = x := by |
induction' x using Quotient.inductionOn' with f
exact Quotient.sound ⟨MonoOver.pullbackId.app f⟩
| 0.375 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 67 | 68 | theorem fold_cons'_right (b a : α) (s : Multiset α) : (a ::ₘ s).fold op b = s.fold op (b * a) := by |
rw [fold_eq_foldl, foldl_cons, ← fold_eq_foldl]
| 0.375 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 40 | 45 | theorem measurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun x ↦ lineDeriv 𝕜 f x v) := by |
borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact (measurable_deriv_with_param hg).comp measurable_prod_mk_right
| 0.375 |
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.monoidal.functor from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
open CategoryTheory
universe v₁ v₂ v₃ u... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 113 | 116 | theorem LaxMonoidalFunctor.μ_natural (F : LaxMonoidalFunctor C D) {X Y X' Y' : C}
(f : X ⟶ Y) (g : X' ⟶ Y') :
(F.map f ⊗ F.map g) ≫ F.μ Y Y' = F.μ X X' ≫ F.map (f ⊗ g) := by |
simp [tensorHom_def]
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import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 490 | 493 | theorem toSheafify_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
η ≫ J.toSheafify _ = J.toSheafify _ ≫ J.sheafifyMap η := by |
dsimp [sheafifyMap, sheafify, toSheafify]
simp
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import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 96 | 97 | theorem toOuterMeasure_map_apply : (p.map f).toOuterMeasure s = p.toOuterMeasure (f ⁻¹' s) := by |
simp [map, Set.indicator, toOuterMeasure_apply p (f ⁻¹' s)]
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import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 162 | 163 | theorem preimage_const_add_Ioo : (fun x => a + x) ⁻¹' Ioo b c = Ioo (b - a) (c - a) := by |
simp [← Ioi_inter_Iio]
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import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 119 | 123 | theorem Finset.untrop_sum' [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → Tropical R) :
untrop (∑ i ∈ s, f i) = s.inf (untrop ∘ f) := by |
convert Multiset.untrop_sum (s.val.map f)
simp only [Multiset.map_map, Function.comp_apply]
rfl
| 0.375 |
import Mathlib.Analysis.Convex.StrictConvexBetween
import Mathlib.Geometry.Euclidean.Basic
#align_import geometry.euclidean.sphere.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type... | Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 141 | 143 | theorem dist_center_eq_dist_center_of_mem_sphere' {p₁ p₂ : P} {s : Sphere P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) : dist s.center p₁ = dist s.center p₂ := by |
rw [mem_sphere'.1 hp₁, mem_sphere'.1 hp₂]
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import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
universe u v w w₁ w₂
variable {R : Type u} {L : Type v}
variable [CommRing R] [LieRing L] [LieAlgebra R L] (H : Lie... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 65 | 69 | theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
H.toLieSubmodule.ucs k = H.toLieSubmodule := by |
induction' k with k ih
· simp
· simp [ih]
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import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 99 | 99 | theorem projIci_eq_self : projIci a x = ⟨a, le_rfl⟩ ↔ x ≤ a := by | simp [projIci, Subtype.ext_iff]
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import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 244 | 245 | theorem mul_boole {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) :
(a * if P then 1 else 0) = if P then a else 0 := by | simp
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import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 110 | 112 | theorem nhdsSet_eq_principal_iff : 𝓝ˢ s = 𝓟 s ↔ IsOpen s := by |
rw [← principal_le_nhdsSet.le_iff_eq, le_principal_iff, mem_nhdsSet_iff_forall,
isOpen_iff_mem_nhds]
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import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
noncomputable section
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w}
open Function Set
open scoped Topology ... | Mathlib/Topology/MetricSpace/Isometry.lean | 155 | 157 | theorem ediam_range (hf : Isometry f) : EMetric.diam (range f) = EMetric.diam (univ : Set α) := by |
rw [← image_univ]
exact hf.ediam_image univ
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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 | 71 | 73 | theorem exists_reduced_word (w : W) : ∃ ω, ω.length = ℓ w ∧ w = π ω := by |
have := Nat.find_spec (cs.exists_word_with_prod w)
tauto
| 0.375 |
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 | 56 | 58 | theorem image.fac : factorThruImage f ≫ image.ι f = f := by |
ext
rfl
| 0.375 |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 213 | 215 | theorem gcd_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).gcd = GCDMonoid.gcd s₁.gcd s₂.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add]
simp
| 0.375 |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 124 | 127 | theorem comp (hf : IsRingHom f) {γ} [Ring γ] {g : β → γ} (hg : IsRingHom g) : IsRingHom (g ∘ f) :=
{ map_add := fun x y => by simp only [Function.comp_apply, map_add hf, map_add hg]
map_mul := fun x y => by simp only [Function.comp_apply, map_mul hf, map_mul hg]
map_one := by | simp only [Function.comp_apply, map_one hf, map_one hg] }
| 0.375 |
import Mathlib.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
suppress_compilation
universe uR uM₁ uM₂ uM₃ uM₄
variable {R : Type uR} {M₁ : Type uM₁} {M₂ : Type uM₂} {M₃ : Type uM₃} {M₄ : Type uM₄}
open scoped TensorProduct
namespace QuadraticForm
variable [Co... | Mathlib/LinearAlgebra/QuadraticForm/TensorProduct/Isometries.lean | 153 | 159 | theorem comp_tensorRId_eq (Q₁ : QuadraticForm R M₁) :
Q₁.comp (TensorProduct.rid R M₁) = Q₁.tmul (sq (R := R)) := by |
refine (QuadraticForm.associated_rightInverse R).injective ?_
ext m₁ m₁'
dsimp [-associated_apply]
simp only [associated_tmul, QuadraticForm.associated_comp]
simp [-associated_apply, one_mul]
| 0.375 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 160 | 167 | theorem card_le_multiplicity {s : Finset E} (hs : ∀ c ∈ s, ‖c‖ ≤ 2)
(h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖) : s.card ≤ multiplicity E := by |
apply le_csSup
· refine ⟨5 ^ finrank ℝ E, ?_⟩
rintro _ ⟨s, ⟨rfl, h⟩⟩
exact Besicovitch.card_le_of_separated s h.1 h.2
· simp only [mem_setOf_eq, Ne]
exact ⟨s, rfl, hs, h's⟩
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import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 131 | 136 | theorem _root_.HasCompactSupport.convolution_integrand_bound_right_of_subset
(hcg : HasCompactSupport g) (hg : Continuous g)
{x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t := by |
refine convolution_integrand_bound_right_of_le_of_subset _ (fun i => ?_) hx hu
exact le_ciSup (hg.norm.bddAbove_range_of_hasCompactSupport hcg.norm) _
| 0.375 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {α : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace α} (μ : @Measure α m0) (hm : m ≤ m0) : @Measure α m :=
@OuterMeasure.toMeasure α m μ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 124 | 128 | theorem sigmaFinite_trim_bot_iff : SigmaFinite (μ.trim bot_le) ↔ IsFiniteMeasure μ := by |
rw [sigmaFinite_bot_iff]
refine ⟨fun h => ⟨?_⟩, fun h => ⟨?_⟩⟩ <;> have h_univ := h.measure_univ_lt_top
· rwa [trim_measurableSet_eq bot_le MeasurableSet.univ] at h_univ
· rwa [trim_measurableSet_eq bot_le MeasurableSet.univ]
| 0.375 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 411 | 413 | theorem mul_inv_lt_inv_mul_iff : a * b⁻¹ < d⁻¹ * c ↔ d * a < c * b := by |
rw [← mul_lt_mul_iff_left d, ← mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,
inv_mul_cancel_right]
| 0.375 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 103 | 105 | theorem det_invOf [Invertible A] [Invertible A.det] : (⅟ A).det = ⅟ A.det := by |
letI := detInvertibleOfInvertible A
convert (rfl : _ = ⅟ A.det)
| 0.375 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 110 | 111 | theorem disjoint_inv_right_iff : Disjoint f g⁻¹ ↔ Disjoint f g := by |
rw [disjoint_comm, disjoint_inv_left_iff, disjoint_comm]
| 0.375 |
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open ... | Mathlib/Analysis/NormedSpace/Dual.lean | 82 | 84 | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by |
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
| 0.375 |
import Mathlib.Analysis.SpecialFunctions.Complex.Log
#align_import analysis.special_functions.pow.complex from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open scoped Classical
open Real Topology Filter ComplexConjugate Finset Set
namespace Complex
noncomputable def cpow (x y : ℂ) ... | Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 107 | 108 | theorem cpow_sub {x : ℂ} (y z : ℂ) (hx : x ≠ 0) : x ^ (y - z) = x ^ y / x ^ z := by |
rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv]
| 0.375 |
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]
| 0.375 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 114 | 117 | theorem norm_id [Nontrivial E] : ‖id 𝕜 E‖ = 1 := by |
refine norm_id_of_nontrivial_seminorm ?_
obtain ⟨x, hx⟩ := exists_ne (0 : E)
exact ⟨x, ne_of_gt (norm_pos_iff.2 hx)⟩
| 0.375 |
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
universe ... | Mathlib/AlgebraicGeometry/Morphisms/Separated.lean | 57 | 60 | theorem respectsIso : MorphismProperty.RespectsIso @IsSeparated := by |
rw [isSeparated_eq_diagonal_isClosedImmersion]
apply MorphismProperty.RespectsIso.diagonal
exact IsClosedImmersion.respectsIso
| 0.375 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 87 | 90 | theorem log_of_left_le_one {b : ℕ} (hb : b ≤ 1) (r : R) : log b r = 0 := by |
rcases le_total 1 r with h | h
· rw [log_of_one_le_right _ h, Nat.log_of_left_le_one hb, Int.ofNat_zero]
· rw [log_of_right_le_one _ h, Nat.clog_of_left_le_one hb, Int.ofNat_zero, neg_zero]
| 0.375 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 121 | 124 | theorem fib_add_two_strictMono : StrictMono fun n => fib (n + 2) := by |
refine strictMono_nat_of_lt_succ fun n => ?_
rw [add_right_comm]
exact fib_lt_fib_succ (self_le_add_left _ _)
| 0.375 |
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