Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
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import Mathlib.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 | 162 | 165 | theorem consCases_cons {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀ x, P (Fin.cons x₀ x))
(x₀ : α 0) (x : ∀ i : Fin n, α i.succ) : @consCases _ _ _ h (cons x₀ x) = h x₀ x := by |
rw [consCases, cast_eq]
congr
| false |
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import model_theory.graph from "leanprover-community/mathlib"@"e56b8fea84d60fe434632b9d3b829ee685fb0c8f"
set_option linter.uppercaseLean3 false
universe u v w w'
namespace FirstOrder
namespace Language
open FirstOr... | Mathlib/ModelTheory/Graph.lean | 75 | 79 | theorem Theory.simpleGraph_model_iff [Language.graph.Structure V] :
V ⊨ Theory.simpleGraph ↔
(Irreflexive fun x y : V => RelMap adj ![x, y]) ∧
Symmetric fun x y : V => RelMap adj ![x, y] := by |
simp [Theory.simpleGraph]
| false |
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.Ideal.Maps
#align_import data.polynomial.div from "leanprover-community/mathlib"@"e1e7190efdcefc925cb36f257a8362ef22944204"
noncomputable section
open Polynomial
... | Mathlib/Algebra/Polynomial/Div.lean | 186 | 192 | theorem zero_divByMonic (p : R[X]) : 0 /ₘ p = 0 := by |
classical
unfold divByMonic divModByMonicAux
dsimp
by_cases hp : Monic p
· rw [dif_pos hp, if_neg (mt And.right (not_not_intro rfl))]
· rw [dif_neg hp]
| false |
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open List Function
universe u v
variable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι} {f : ∀ i, α i}
namespace List
variable (α)
abbrev TProd (l : List ι) : Type v... | Mathlib/Data/Prod/TProd.lean | 99 | 103 | theorem elim_of_mem (hl : (i :: l).Nodup) (hj : j ∈ l) (v : TProd α (i :: l)) :
v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj := by |
apply elim_of_ne
rintro rfl
exact hl.not_mem hj
| false |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Polynomial
variable {R : Type*} [CommRing R] {n : ℕ}
theorem isRoot_... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 56 | 59 | theorem _root_.isRoot_of_unity_iff (h : 0 < n) (R : Type*) [CommRing R] [IsDomain R] {ζ : R} :
ζ ^ n = 1 ↔ ∃ i ∈ n.divisors, (cyclotomic i R).IsRoot ζ := by |
rw [← mem_nthRoots h, nthRoots, mem_roots <| X_pow_sub_C_ne_zero h _, C_1, ←
prod_cyclotomic_eq_X_pow_sub_one h, isRoot_prod]
| false |
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Order.Atoms.Finite
import Mathlib.Data.Set.Lattice
#align_import group_theory.sylow from "leanprove... | Mathlib/GroupTheory/Sylow.lean | 493 | 495 | theorem QuotientGroup.card_preimage_mk [Fintype G] (s : Subgroup G) (t : Set (G ⧸ s)) :
Fintype.card (QuotientGroup.mk ⁻¹' t) = Fintype.card s * Fintype.card t := by |
rw [← Fintype.card_prod, Fintype.card_congr (preimageMkEquivSubgroupProdSet _ _)]
| false |
import Mathlib.MeasureTheory.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
... | Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 97 | 100 | theorem isCaratheodory_inter (h₁ : IsCaratheodory m s₁) (h₂ : IsCaratheodory m s₂) :
IsCaratheodory m (s₁ ∩ s₂) := by |
rw [← isCaratheodory_compl_iff, Set.compl_inter]
exact isCaratheodory_union _ (isCaratheodory_compl _ h₁) (isCaratheodory_compl _ h₂)
| false |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 319 | 321 | theorem HasFDerivAt.smul_const (hc : HasFDerivAt c c' x) (f : F) :
HasFDerivAt (fun y => c y • f) (c'.smulRight f) x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasFDerivAt_const f x)
| false |
import Mathlib.Data.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
open Nat
variable {α β : Type*}
namespace List
variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
section Fix
#align list.prefix_append List.prefix_append
#align list.... | Mathlib/Data/List/Infix.lean | 70 | 70 | theorem prefix_concat (a : α) (l) : l <+: concat l a := by | simp
| false |
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 144 | 146 | theorem contractLeft_algebraMap_mul (r : R) (b : CliffordAlgebra Q) :
d⌋(algebraMap _ _ r * b) = algebraMap _ _ r * (d⌋b) := by |
rw [← Algebra.smul_def, map_smul, Algebra.smul_def]
| false |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 75 | 80 | theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by |
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
| false |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 103 | 105 | theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
∫ x, f x ∂μ = 0 := by |
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_right_eq_self]
| false |
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 | 110 | 112 | theorem bernoulli'_one : bernoulli' 1 = 1 / 2 := by
rw [bernoulli'_def] |
rw [bernoulli'_def]
norm_num
| true |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 254 | 255 | theorem coprodᵢ_eq_bot_iff [∀ i, Nonempty (α i)] : Filter.coprodᵢ f = ⊥ ↔ f = ⊥ := by |
simpa [funext_iff] using coprodᵢ_neBot_iff.not
| true |
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 38 | 74 | theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {P : Ideal R}
(hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P ^ 2)
(hxQ : ∀ Q : Ideal R, IsPrime Q → Q ≠ P → x ∉ Q) : P = Ideal.span {x} := by
letI := Classical.decEq (Ideal R) |
letI := Classical.decEq (Ideal R)
have hx0 : x ≠ 0 := by
rintro rfl
exact hxP2 (zero_mem _)
by_cases hP0 : P = ⊥
· subst hP0
-- Porting note: was `simpa using hxP2` but that hypothesis didn't even seem relevant in Lean 3
rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
have hspan0 : span ({x} ... | true |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 116 | 119 | theorem congrArg_mpr_hom_left {X Y Z : C} (p : X = Y) (q : Y ⟶ Z) :
(congrArg (fun W : C => W ⟶ Z) p).mpr q = eqToHom p ≫ q := by
cases p |
cases p
simp
| true |
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 | 189 | 197 | theorem continuous_transAssocReparamAux : Continuous transAssocReparamAux := by
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_) |
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_)
(continuous_if_le ?_ ?_
(Continuous.continuousOn ?_) (Continuous.continuousOn ?_) ?_).continuousOn
?_ <;>
[continuity; continuity; continuity; continuity; continuity; continuity; continuity; skip;
skip] <;>
· intro x hx
se... | true |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 132 | 136 | theorem to_quotient_square_range :
LinearMap.range I.cotangentToQuotientSquare = I.cotangentIdeal.restrictScalars R := by
trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent) |
trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)
· rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]
· rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype]; ext; rfl
| true |
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 32 | 34 | theorem coprime_multiset_prod_right_iff {k : ℕ} {m : Multiset ℕ} :
Coprime k m.prod ↔ ∀ n ∈ m, Coprime k n := by |
induction m using Quotient.inductionOn; simpa using coprime_list_prod_right_iff
| true |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 174 | 174 | theorem nonpos_iff_ne_one {a : SignType} : a ≤ 0 ↔ a ≠ 1 := by | cases a <;> decide
| true |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 111 | 111 | theorem sub : mapFun f (x - y) = mapFun f x - mapFun f y := by | map_fun_tac
| true |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 316 | 321 | theorem isZero_zero [HasZeroObject V] : IsZero (zero : HomologicalComplex V c) := by
refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩ |
refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩
all_goals
ext
dsimp [zero]
apply Subsingleton.elim
| true |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.egorov from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open MeasureTheory NNReal ENNReal Topology
namespace MeasureTheory
open Set Filt... | Mathlib/MeasureTheory/Function/Egorov.lean | 59 | 70 | theorem measure_inter_notConvergentSeq_eq_zero [SemilatticeSup ι] [Nonempty ι]
(hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
μ (s ∩ ⋂ j, notConvergentSeq f g n j) = 0 := by
simp_rw [Metric.tendsto_atTop, ae_iff] at hfg |
simp_rw [Metric.tendsto_atTop, ae_iff] at hfg
rw [← nonpos_iff_eq_zero, ← hfg]
refine measure_mono fun x => ?_
simp only [Set.mem_inter_iff, Set.mem_iInter, ge_iff_le, mem_notConvergentSeq_iff]
push_neg
rintro ⟨hmem, hx⟩
refine ⟨hmem, 1 / (n + 1 : ℝ), Nat.one_div_pos_of_nat, fun N => ?_⟩
obtain ⟨n, hn₁... | true |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 69 | 70 | theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
¬A i j = 1 ↔ A i j = 0 := by | obtain h | h := h.zero_or_one i j <;> simp [h]
| true |
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.Topology.StoneCech
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.CategoryTheory.Elementwise
#align_import topol... | Mathlib/Topology/Category/CompHaus/Basic.lean | 123 | 135 | theorem isIso_of_bijective {X Y : CompHaus.{u}} (f : X ⟶ Y) (bij : Function.Bijective f) :
IsIso f := by
let E := Equiv.ofBijective _ bij |
let E := Equiv.ofBijective _ bij
have hE : Continuous E.symm := by
rw [continuous_iff_isClosed]
intro S hS
rw [← E.image_eq_preimage]
exact isClosedMap f S hS
refine ⟨⟨⟨E.symm, hE⟩, ?_, ?_⟩⟩
· ext x
apply E.symm_apply_apply
· ext x
apply E.apply_symm_apply
| true |
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 60 | 61 | theorem destutter'_singleton : [b].destutter' R a = if R a b then [a, b] else [a] := by |
split_ifs with h <;> simp! [h]
| true |
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 107 | 109 | theorem comap_prod (f : α → β × γ) (b : Filter β) (c : Filter γ) :
comap f (b ×ˢ c) = comap (Prod.fst ∘ f) b ⊓ comap (Prod.snd ∘ f) c := by |
erw [comap_inf, Filter.comap_comap, Filter.comap_comap]
| true |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Topology.Category.TopCat.Limits.Basic
#align_import topology.category.Top.limits.konig from "leanprover-community/mathlib"@"dbdf71cee7bb20367cb7e37279c08b0c218cf967"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 ... | Mathlib/Topology/Category/TopCat/Limits/Konig.lean | 130 | 146 | theorem nonempty_limitCone_of_compact_t2_cofiltered_system (F : J ⥤ TopCat.{max v u})
[IsCofilteredOrEmpty J]
[∀ j : J, Nonempty (F.obj j)] [∀ j : J, CompactSpace (F.obj j)] [∀ j : J, T2Space (F.obj j)] :
Nonempty (TopCat.limitCone F).pt := by
classical |
classical
obtain ⟨u, hu⟩ :=
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed (fun G => partialSections F _)
(partialSections.directed F) (fun G => partialSections.nonempty F _)
(fun G => IsClosed.isCompact (partialSections.closed F _)) fun G =>
partialSections.closed F _
us... | true |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 111 | 114 | theorem encard_union_eq (h : Disjoint s t) : (s ∪ t).encard = s.encard + t.encard := by
classical |
classical
have e := (Equiv.Set.union (by rwa [subset_empty_iff, ← disjoint_iff_inter_eq_empty])).symm
simp [encard, ← PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv]
| true |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Restrict
import Mathlib.Tactic.FailIfNoProgress
#align_import analysis.normed_space.affine_isomet... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 86 | 88 | theorem toAffineMap_injective : Injective (toAffineMap : (P →ᵃⁱ[𝕜] P₂) → P →ᵃ[𝕜] P₂) := by
rintro ⟨f, _⟩ ⟨g, _⟩ rfl |
rintro ⟨f, _⟩ ⟨g, _⟩ rfl
rfl
| true |
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 | 180 | 183 | theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i := by |
rw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant] <;> assumption
| true |
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 53 | 54 | theorem toPGame_leftMoves (o : Ordinal) : o.toPGame.LeftMoves = o.out.α := by |
rw [toPGame, LeftMoves]
| true |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 52 | 60 | theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
(h : x ∈ derivedSeries R L 1) : χ x = 0 := by
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero, ← |
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_zero, ←
LieSubmodule.mem_coeSubmodule, LieSubmodule.lieIdeal_oper_eq_linear_span] at h
refine Submodule.span_induction h ?_ ?_ ?_ ?_
· rintro y ⟨⟨z, hz⟩, ⟨⟨w, hw⟩, rfl⟩⟩; apply lieCharacter_apply_lie
· exact χ.map_zero
· intro y z hy ... | true |
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 | 44 | 47 | theorem inv_gold : φ⁻¹ = -ψ := by
have : 1 + √5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num)) |
have : 1 + √5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
| true |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 148 | 151 | theorem A_mono (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : A f L r ε ⊆ A f L r δ := by
rintro x ⟨r', r'r, hr'⟩ |
rintro x ⟨r', r'r, hr'⟩
refine ⟨r', r'r, fun y hy z hz => (hr' y hy z hz).trans_le (mul_le_mul_of_nonneg_right h ?_)⟩
linarith [mem_ball.1 hy, r'r.2, @dist_nonneg _ _ y x]
| true |
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Localization.FractionRing
#align_import algebra.char_p.algebra from "leanprover-community/mathlib"@"96782a2d6dcded92116d8ac9ae48efb41d46a27c"
theorem charP_of_injective_ringHom {R A : Type*} [NonAssocSemiring R] [NonAs... | Mathlib/Algebra/CharP/Algebra.lean | 121 | 123 | theorem Algebra.ringChar_eq : ringChar K = ringChar L := by
rw [ringChar.eq_iff, Algebra.charP_iff K L] |
rw [ringChar.eq_iff, Algebra.charP_iff K L]
apply ringChar.charP
| true |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatorics.hales_jewett from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open scoped Classical
universe u v
namespace ... | Mathlib/Combinatorics/HalesJewett.lean | 197 | 200 | theorem horizontal_apply {α ι ι'} (l : Line α ι) (v : ι' → α) (x : α) :
l.horizontal v x = Sum.elim (l x) v := by
funext i |
funext i
cases i <;> rfl
| true |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 116 | 120 | theorem fold_union_inter [DecidableEq α] {s₁ s₂ : Finset α} {b₁ b₂ : β} :
((s₁ ∪ s₂).fold op b₁ f * (s₁ ∩ s₂).fold op b₂ f) = s₁.fold op b₂ f * s₂.fold op b₁ f := by
unfold fold |
unfold fold
rw [← fold_add op, ← Multiset.map_add, union_val, inter_val, union_add_inter, Multiset.map_add,
hc.comm, fold_add]
| true |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.adjunction.reflective from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
universe v₁ v₂ v₃ u₁ u₂ u₃
noncomputable s... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 87 | 89 | theorem Functor.essImage.unit_isIso [Reflective i] {A : C} (h : A ∈ i.essImage) :
IsIso ((reflectorAdjunction i).unit.app A) := by |
rwa [isIso_unit_app_iff_mem_essImage]
| true |
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.GroupTheory.GroupAction.Ring
suppress_compilation
open scoped TensorProduct
variable {R ι A B : Type*}
variable [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι]
variable [CommRing R] [R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 133 | 135 | theorem auxEquiv_one : auxEquiv R 𝒜 ℬ 1 = 1 := by
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one, |
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def]
| true |
import Mathlib.RingTheory.Finiteness
import Mathlib.Logic.Equiv.TransferInstance
universe u v w
open Function
variable (R : Type u) [Semiring R]
@[mk_iff]
class OrzechProperty : Prop where
injective_of_surjective_of_submodule' : ∀ {M : Type u} [AddCommMonoid M] [Module R M]
[Module.Finite R M] {N : Submod... | Mathlib/RingTheory/OrzechProperty.lean | 69 | 82 | theorem injective_of_surjective_of_injective
{N : Type w} [AddCommMonoid N] [Module R N]
(i f : N →ₗ[R] M) (hi : Injective i) (hf : Surjective f) : Injective f := by
obtain ⟨n, g, hg⟩ := Module.Finite.exists_fin' R M |
obtain ⟨n, g, hg⟩ := Module.Finite.exists_fin' R M
haveI := small_of_surjective hg
letI := Equiv.addCommMonoid (equivShrink M).symm
letI := Equiv.module R (equivShrink M).symm
let j : Shrink.{u} M ≃ₗ[R] M := Equiv.linearEquiv R (equivShrink M).symm
haveI := Module.Finite.equiv j.symm
let i' := j.symm.toL... | true |
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 102 | 105 | theorem toDualRight_symm_comp_toDualLeft :
p.toDualRight.symm.dualMap ∘ₗ (p.toDualLeft : M →ₗ[R] Dual R N) = Dual.eval R M := by
ext1 x |
ext1 x
exact p.toDualRight_symm_toDualLeft x
| true |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 141 | 146 | theorem monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
HasDerivAt (fun y => (x - y) ^ (n + 1)) (-(n + 1) * (x - t) ^ n) t := by
simp_rw [sub_eq_neg_add] |
simp_rw [sub_eq_neg_add]
rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc]
convert HasDerivAt.pow (n + 1) ((hasDerivAt_id t).neg.add_const x)
simp only [Nat.cast_add, Nat.cast_one]
| true |
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.quotient from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
def HomRel (C) [Quiver C] :=
∀ ⦃X Y : C⦄, (X ⟶ Y) → (X ⟶ Y) → Prop
#align hom_rel HomRel
-- Porting Note: `deriving I... | Mathlib/CategoryTheory/Quotient.lean | 65 | 66 | theorem CompClosure.of {a b : C} (m₁ m₂ : a ⟶ b) (h : r m₁ m₂) : CompClosure r m₁ m₂ := by |
simpa using CompClosure.intro (𝟙 _) m₁ m₂ (𝟙 _) h
| true |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 63 | 67 | theorem IsMatching.toEdge_eq_of_adj {M : Subgraph G} (h : M.IsMatching) {v w : V} (hv : v ∈ M.verts)
(hvw : M.Adj v w) : h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩ := by
simp only [IsMatching.toEdge, Subtype.mk_eq_mk] |
simp only [IsMatching.toEdge, Subtype.mk_eq_mk]
congr
exact ((h (M.edge_vert hvw)).choose_spec.2 w hvw).symm
| true |
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 | 80 | 87 | theorem integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
∫ x : ℝ in (0)..1, bernoulliFun k x = 0 := by
rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x) |
rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)
((Polynomial.continuous _).intervalIntegrable _ _)]
rw [bernoulliFun_eval_one]
split_ifs with h
· exfalso; exact hk (Nat.succ_inj'.mp h)
· simp
| true |
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 65 | 69 | theorem discr_prime_pow_eq_unit_mul_pow' [IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) :
∃ (u : ℤˣ) (n : ℕ), discr ℚ (hζ.subOnePowerBasis ℚ).basis = u * p ^ n := by
rw [hζ.discr_zeta_eq_discr_zeta_sub_one.symm] |
rw [hζ.discr_zeta_eq_discr_zeta_sub_one.symm]
exact discr_prime_pow_eq_unit_mul_pow hζ (cyclotomic.irreducible_rat (p ^ k).pos)
| true |
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 144 | 146 | theorem card_of_subsingleton (a : α) [Subsingleton α] : Nat.card α = 1 := by
letI := Fintype.ofSubsingleton a |
letI := Fintype.ofSubsingleton a
rw [card_eq_fintype_card, Fintype.card_ofSubsingleton a]
| true |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 204 | 208 | theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
(h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p s := by
refine ⟨fun x hx => ?_, h.fderivWithin, h.cont⟩ |
refine ⟨fun x hx => ?_, h.fderivWithin, h.cont⟩
rw [h₁ x hx]
exact h.zero_eq x hx
| true |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 44 | 45 | theorem tsub_le_tsub_iff_right (h : c ≤ b) : a - c ≤ b - c ↔ a ≤ b := by |
rw [tsub_le_iff_right, tsub_add_cancel_of_le h]
| true |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 110 | 113 | theorem WithTop.coe_iInf [Nonempty ι] [InfSet α] {f : ι → α} (hf : BddBelow (range f)) :
↑(⨅ i, f i) = (⨅ i, f i : WithTop α) := by
rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp] |
rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp]
rfl
| true |
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 138 | 138 | theorem one_le_cast : 1 ≤ (n : α) ↔ 1 ≤ n := by | rw [← cast_one, cast_le]
| true |
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 110 | 134 | theorem remainder_lt' (n : ℕ) {m : ℝ} (m1 : 1 < m) :
remainder m n < (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) :=
-- two useful inequalities
have m0 : 0 < m := zero_lt_one.trans m1
have mi : 1 / m < 1 := (div_lt_one m0).mpr m1
-- to show the strict inequality between these series, we prove that:
calc
(∑' i, ... |
simp only [pow_add, one_div, mul_inv, inv_pow]
-- factor the constant `(1 / m ^ (n + 1)!)` out of the series
_ = (∑' i, (1 / m) ^ i) * (1 / m ^ (n + 1)!) := tsum_mul_right
-- the series is the geometric series
_ = (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) := by rw [tsum_geometric_of_lt_one (by positivit... | true |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 118 | 120 | theorem sum_tensor {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(∑ j ∈ s, g j) ⊗ f = ∑ j ∈ s, g j ⊗ f := by |
simp only [tensorHom_def, sum_whiskerRight, Preadditive.sum_comp]
| true |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.basis.bilinear from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
namespace LinearMap
variable {ι₁ ι₂ : Type*}
variable {R R₂ S S₂ M N P Rₗ : Type*}
variable {Mₗ Nₗ Pₗ : Type*}
--... | Mathlib/LinearAlgebra/Basis/Bilinear.lean | 44 | 49 | theorem sum_repr_mul_repr_mulₛₗ {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x y) :
((b₁.repr x).sum fun i xi => (b₂.repr y).sum fun j yj => ρ₁₂ xi • σ₁₂ yj • B (b₁ i) (b₂ j)) =
B x y := by
conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y] |
conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y]
simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂,
LinearMap.map_smulₛₗ]
| true |
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 103 | 106 | theorem CancelMonoid.toLeftCancelMonoid_injective {M : Type u} :
Function.Injective (@CancelMonoid.toLeftCancelMonoid M) := by
rintro ⟨⟩ ⟨⟩ h |
rintro ⟨⟩ ⟨⟩ h
congr
| true |
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.Analytic.Basic
#align_import measure_theory.integral.circle_integral from "leanprover-communit... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 130 | 131 | theorem circleMap_not_mem_ball (c : ℂ) (R : ℝ) (θ : ℝ) : circleMap c R θ ∉ ball c R := by |
simp [dist_eq, le_abs_self]
| true |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 124 | 128 | theorem log_conj_eq_ite (x : ℂ) : log (conj x) = if x.arg = π then log x else conj (log x) := by
simp_rw [log, abs_conj, arg_conj, map_add, map_mul, conj_ofReal] |
simp_rw [log, abs_conj, arg_conj, map_add, map_mul, conj_ofReal]
split_ifs with hx
· rw [hx]
simp_rw [ofReal_neg, conj_I, mul_neg, neg_mul]
| true |
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 | 114 | 120 | theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) := by
rintro ⟨⟨ha, hb, heq⟩, h2⟩ |
rintro ⟨⟨ha, hb, heq⟩, h2⟩
constructor
· apply And.intro ha (And.intro hb _)
rw [heq]
exact (neg_sq c).symm
rwa [Int.natAbs_neg c]
| true |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
open MeasureTheory Measure FiniteDimensional
variable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
[NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCo... | Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean | 101 | 151 | theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
{f f' : E → F} {g g' : E → G} {v : E} {B : F →L[ℝ] G →L[ℝ] W}
(hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ) (hfg' : Integrable (fun x ↦ B (f x) (g' x)) μ)
(hfg : Integrable (fun x ↦ B (f x) (g x)) μ)
(hf : ∀ x, HasLineDerivAt ℝ f ... |
by_cases hW : CompleteSpace W; swap
· simp [integral, hW]
rcases eq_or_ne v 0 with rfl|hv
· have Hf' x : f' x = 0 := by
simpa [(hasLineDerivAt_zero (f := f) (x := x)).lineDeriv] using (hf x).lineDeriv.symm
have Hg' x : g' x = 0 := by
simpa [(hasLineDerivAt_zero (f := g) (x := x)).lineDeriv] usi... | true |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 88 | 93 | theorem of_comp_eq_base (i : ι) : (of i).comp (φ i) = (base φ) := by
ext x |
ext x
apply (Con.eq _).2
refine ConGen.Rel.of _ _ ?_
simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range]
exact ⟨_, _, rfl, rfl⟩
| true |
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
open Finset
section birkhoffAverage
variable (R : Type*) {α M : Type*} [DivisionSemiring R] [AddCommMonoid M] [Module R M]
def birkhoffAverage (f : α → α) (g : α → M) (n : ℕ) (x : α) : M := (n : R)⁻¹ • birkhoffSum f g n x
theorem bir... | Mathlib/Dynamics/BirkhoffSum/Average.lean | 68 | 70 | theorem birkhoffAverage_congr_ring' (S : Type*) [DivisionSemiring S] [Module S M] :
birkhoffAverage (α := α) (M := M) R = birkhoffAverage S := by |
ext; apply birkhoffAverage_congr_ring
| true |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 124 | 127 | theorem verschiebung_frobenius [CharP R p] (x : 𝕎 R) : verschiebung (frobenius x) = x * p := by
ext ⟨i⟩ |
ext ⟨i⟩
· rw [mul_charP_coeff_zero, verschiebung_coeff_zero]
· rw [mul_charP_coeff_succ, verschiebung_coeff_succ, coeff_frobenius_charP]
| true |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 126 | 126 | theorem pow (n : ℕ) : mapFun f (x ^ n) = mapFun f x ^ n := by | map_fun_tac
| true |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 114 | 123 | theorem truncation_eq_of_nonneg {f : α → ℝ} {A : ℝ} (h : ∀ x, 0 ≤ f x) :
truncation f A = indicator (Set.Ioc 0 A) id ∘ f := by
ext x |
ext x
rcases (h x).lt_or_eq with (hx | hx)
· simp only [truncation, indicator, hx, Set.mem_Ioc, id, Function.comp_apply, true_and_iff]
by_cases h'x : f x ≤ A
· have : -A < f x := by linarith [h x]
simp only [this, true_and_iff]
· simp only [h'x, and_false_iff]
· simp only [truncation, indicat... | true |
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 | 61 | 66 | theorem mem_memberSubfamily : s ∈ 𝒜.memberSubfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := by
simp_rw [memberSubfamily, mem_image, mem_filter] |
simp_rw [memberSubfamily, mem_image, mem_filter]
refine ⟨?_, fun h => ⟨insert a s, ⟨h.1, by simp⟩, erase_insert h.2⟩⟩
rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩
rw [insert_erase hs2]
exact ⟨hs1, not_mem_erase _ _⟩
| true |
import Mathlib.Algebra.Module.Defs
import Mathlib.SetTheory.Cardinal.Basic
open Function
universe u v
namespace Cardinal
| Mathlib/Algebra/Module/Card.lean | 24 | 29 | theorem mk_le_of_module (R : Type u) (E : Type v)
[AddCommGroup E] [Ring R] [Module R E] [Nontrivial E] [NoZeroSMulDivisors R E] :
Cardinal.lift.{v} (#R) ≤ Cardinal.lift.{u} (#E) := by
obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0 |
obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0
have : Injective (fun k ↦ k • x) := smul_left_injective R hx
exact lift_mk_le_lift_mk_of_injective this
| true |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 119 | 121 | theorem mul_charP_coeff_succ [CharP R p] (x : 𝕎 R) (i : ℕ) :
(x * p).coeff (i + 1) = x.coeff i ^ p := by |
rw [← frobenius_verschiebung, coeff_frobenius_charP, verschiebung_coeff_succ]
| true |
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 | 56 | 58 | theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl, |
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl,
subset_compl_iff_disjoint_left]
| true |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 68 | 80 | theorem isAcyclic_iff_forall_adj_isBridge :
G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w) := by
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem] |
simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]
constructor
· intro ha v w hvw
apply And.intro hvw
intro u p hp
cases ha p hp
· rintro hb v (_ | ⟨ha, p⟩) hp
· exact hp.not_of_nil
· apply (hb ha).2 _ hp
rw [Walk.edges_cons]
apply List.mem_cons_self
| true |
import Mathlib.Data.ENNReal.Real
#align_import data.real.conjugate_exponents from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
noncomputable section
open scoped ENNReal
namespace Real
@[mk_iff]
structure IsConjExponent (p q : ℝ) : Prop where
one_lt : 1 < p
inv_add_inv_conj : p⁻... | Mathlib/Data/Real/ConjExponents.lean | 101 | 102 | theorem mul_eq_add : p * q = p + q := by |
simpa only [sub_mul, sub_eq_iff_eq_add, one_mul] using h.sub_one_mul_conj
| true |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 76 | 82 | theorem isOpen_iff : IsOpen s ↔ ∀ o ∈ s, IsLimit o → ∃ a < o, Set.Ioo a o ⊆ s := by
refine isOpen_iff_mem_nhds.trans <| forall₂_congr fun o ho => ?_ |
refine isOpen_iff_mem_nhds.trans <| forall₂_congr fun o ho => ?_
by_cases ho' : IsLimit o
· simp only [(nhdsBasis_Ioc ho'.1).mem_iff, ho', true_implies]
refine exists_congr fun a => and_congr_right fun ha => ?_
simp only [← Set.Ioo_insert_right ha, Set.insert_subset_iff, ho, true_and]
· simp [nhds_eq_p... | true |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 119 | 123 | theorem sign_inv (r : ℝ) : sign r⁻¹ = sign r := by
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ) |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)]
· rw [sign_zero, inv_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)]
| true |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 168 | 197 | theorem cantorFunction_injective (h1 : 0 < c) (h2 : c < 1 / 2) :
Function.Injective (cantorFunction c) := by
intro f g hfg |
intro f g hfg
classical
by_contra h
revert hfg
have : ∃ n, f n ≠ g n := by
rw [← not_forall]
intro h'
apply h
ext
apply h'
let n := Nat.find this
have hn : ∀ k : ℕ, k < n → f k = g k := by
intro k hk
apply of_not_not
exact Nat.find_min this hk
... | true |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 368 | 371 | theorem HasFDerivAt.comp_hasDerivWithinAt_of_eq (hl : HasFDerivAt l l' y)
(hf : HasDerivWithinAt f f' s x) (hy : y = f x) :
HasDerivWithinAt (l ∘ f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf
| true |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 94 | 102 | theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b := by
have h0 : 0 < a ^ (0 : Ordinal) := by simp only [opow_zero, zero_lt_one] |
have h0 : 0 < a ^ (0 : Ordinal) := by simp only [opow_zero, zero_lt_one]
induction b using limitRecOn with
| H₁ => exact h0
| H₂ b IH =>
rw [opow_succ]
exact mul_pos IH a0
| H₃ b l _ =>
exact (lt_opow_of_limit (Ordinal.pos_iff_ne_zero.1 a0) l).2 ⟨0, l.pos, h0⟩
| true |
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 | 239 | 241 | theorem inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by
rw [← inner_conj_symm, inner_smul_left]; |
rw [← inner_conj_symm, inner_smul_left];
simp only [conj_conj, inner_conj_symm, RingHom.map_mul]
| true |
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.FunctorCategory
import Mathlib.CategoryTheory.Limits.Shapes.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
#align_import category_theory.abelian.functor_category from "leanprover-community/mathlib"@"8a... | Mathlib/CategoryTheory/Abelian/FunctorCategory.lean | 79 | 83 | theorem coimageImageComparison_app' :
(coimageImageComparison α).app X =
(coimageObjIso α X).hom ≫ coimageImageComparison (α.app X) ≫ (imageObjIso α X).inv := by
simp only [coimageImageComparison_app, Iso.hom_inv_id_assoc, Iso.hom_inv_id, Category.assoc, |
simp only [coimageImageComparison_app, Iso.hom_inv_id_assoc, Iso.hom_inv_id, Category.assoc,
Category.comp_id]
| true |
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 | 132 | 134 | theorem projIcc_val (x : Icc a b) : projIcc a b h x = x := by
cases x |
cases x
apply projIcc_of_mem
| true |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 359 | 362 | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t := by
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffOn_iff, contDiffOn_pi] |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffOn_iff, contDiffOn_pi]
rfl
| true |
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import algebra.bounds from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29... | Mathlib/Algebra/Bounds.lean | 175 | 176 | theorem ciSup_div (hf : BddAbove (range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by |
simp only [div_eq_mul_inv, ciSup_mul hf]
| true |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by
rw [← LinearMap.det_toMat... | Mathlib/Data/Complex/Determinant.lean | 31 | 33 | theorem linearEquiv_det_conjAe : LinearEquiv.det conjAe.toLinearEquiv = -1 := by
rw [← Units.eq_iff, LinearEquiv.coe_det, AlgEquiv.toLinearEquiv_toLinearMap, det_conjAe, |
rw [← Units.eq_iff, LinearEquiv.coe_det, AlgEquiv.toLinearEquiv_toLinearMap, det_conjAe,
Units.coe_neg_one]
| true |
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 | 154 | 156 | theorem δ_comp_σ_self {n} {i : Fin (n + 1)} : X.σ i ≫ X.δ (Fin.castSucc i) = 𝟙 _ := by
dsimp [δ, σ] |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_self, op_id, X.map_id]
| true |
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTheory.GroupAction.SubMulAction
open scoped BigOperators Pointwise
namespace MulAction
section SMul
variable (G : Type*) {X : Type*} [SMul G X]
-- Change termin... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 102 | 103 | theorem isBlock_empty : IsBlock G (⊥ : Set X) := by |
simp [IsBlock.def, Set.bot_eq_empty, Set.smul_set_empty]
| true |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 115 | 117 | theorem vanishingIdeal_singleton (x : ProjectiveSpectrum 𝒜) :
vanishingIdeal ({x} : Set (ProjectiveSpectrum 𝒜)) = x.asHomogeneousIdeal := by |
simp [vanishingIdeal]
| true |
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Category.ModuleCat.Biproducts
import Mathlib.RingTheory.SimpleModule
#align_import algebra.module.pid from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198b... | Mathlib/Algebra/Module/PID.lean | 75 | 84 | theorem Submodule.isInternal_prime_power_torsion_of_pid [Module.Finite R M]
(hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (⊤ : Submodule R M).annihilator).toFinset =>
torsionBy R M
(IsPrincipal.generator (p : Ideal R) ^
(factors (⊤ : Submodule R M).annihilator).coun... |
convert isInternal_prime_power_torsion hM
ext p : 1
rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ← Ideal.span_singleton_pow,
Ideal.span_singleton_generator]
| true |
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.MvPolynomial.Symmetric
#align_import ring_theory.polynomial.vieta from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open Polynomial
namespace Multiset
open Polynomial
section Semiring
variable {R : Type*} [CommSemi... | Mathlib/RingTheory/Polynomial/Vieta.lean | 75 | 77 | theorem prod_X_add_C_coeff' {σ} (s : Multiset σ) (r : σ → R) {k : ℕ} (h : k ≤ Multiset.card s) :
(s.map fun i => X + C (r i)).prod.coeff k = (s.map r).esymm (Multiset.card s - k) := by |
erw [← map_map (fun r => X + C r) r, prod_X_add_C_coeff] <;> rw [s.card_map r]; assumption
| true |
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 | 309 | 315 | theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞)
(hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r := by
refine le_antisymm (le_pre.2 fun s hs => ?_) (le_trim _) |
refine le_antisymm (le_pre.2 fun s hs => ?_) (le_trim _)
rw [trim_eq_iInf]
refine iInf_le_of_le (closure s) <| iInf_le_of_le subset_closure <|
iInf_le_of_le measurableSet_closure ((pre_le ?_).trans_eq (hcl _))
rwa [diam_closure]
| true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 595 | 596 | theorem iterate_prod_map (f : α → α) (g : β → β) (n : ℕ) :
(Prod.map f g)^[n] = Prod.map (f^[n]) (g^[n]) := by | induction n <;> simp [*, Prod.map_comp_map]
| true |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 55 | 57 | theorem natAbs_le_iff_sq_le {a b : ℤ} : a.natAbs ≤ b.natAbs ↔ a ^ 2 ≤ b ^ 2 := by
rw [sq, sq] |
rw [sq, sq]
exact natAbs_le_iff_mul_self_le
| true |
import Mathlib.CategoryTheory.Category.Basic
import Mathlib.CategoryTheory.Functor.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Tactic.NthRewrite
import Mathlib.CategoryTheory.PathCategory
import Mathlib.CategoryTheory.Quotient
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import category_theory... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 93 | 117 | theorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :
Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =
Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (𝟙 X) := by
apply Quot.EqvGen_sound |
apply Quot.EqvGen_sound
induction' p with a b q f ih
· apply EqvGen.refl
· simp only [Quiver.Path.reverse]
fapply EqvGen.trans
-- Porting note: `Quiver.Path.*` and `Quiver.Hom.*` notation not working
· exact q ≫ Quiver.Path.reverse q
· apply EqvGen.symm
apply EqvGen.rel
have : Quoti... | true |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 161 | 168 | theorem vars_C_mul (a : A) (ha : a ≠ 0) (φ : MvPolynomial σ A) :
(C a * φ : MvPolynomial σ A).vars = φ.vars := by
ext1 i |
ext1 i
simp only [mem_vars, exists_prop, mem_support_iff]
apply exists_congr
intro d
apply and_congr _ Iff.rfl
rw [coeff_C_mul, mul_ne_zero_iff, eq_true ha, true_and_iff]
| true |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import analysis.normed_space.complemented from "leanprover-community/mathlib"@"3397560e65278e5f31acefcdea63138bd53d1cd4"
variable {𝕜 E F G : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedS... | Mathlib/Analysis/NormedSpace/Complemented.lean | 139 | 143 | theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace 𝕜]
[FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented := by
obtain ⟨q, hq⟩ : ∃ q, IsCompl p q := p.exists_isCompl |
obtain ⟨q, hq⟩ : ∃ q, IsCompl p q := p.exists_isCompl
haveI : FiniteDimensional 𝕜 q := (p.quotientEquivOfIsCompl q hq).finiteDimensional
exact .of_isCompl_isClosed hq hp q.closed_of_finiteDimensional
| true |
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 | 150 | 156 | theorem goldConj_irrational : Irrational ψ := by
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num) |
have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
have := this.rat_sub 1
have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
convert this
norm_num
field_simp
| true |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 146 | 163 | theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra R A}
{φ ψ : S.topologicalClosure →⋆ₐ[R] B} (hφ : Continuous φ) (hψ : Continuous ψ)
(h :
φ.comp (inclusion (le_topologicalClosure S)) = ψ.comp (inclusion (le_topologicalClosure S))) :
φ = ψ := by
rw [DFunLike.ext'_iff] |
rw [DFunLike.ext'_iff]
have : Dense (Set.range <| inclusion (le_topologicalClosure S)) := by
refine embedding_subtype_val.toInducing.dense_iff.2 fun x => ?_
convert show ↑x ∈ closure (S : Set A) from x.prop
rw [← Set.range_comp]
exact
Set.ext fun y =>
⟨by
rintro ⟨y, rfl⟩
... | true |
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 102 | 105 | theorem _root_.IsClosed.measure_eq_univ_iff_eq [OpensMeasurableSpace X] [IsFiniteMeasure μ]
(hF : IsClosed F) :
μ F = μ univ ↔ F = univ := by |
rw [← ae_eq_univ_iff_measure_eq hF.measurableSet.nullMeasurableSet, hF.ae_eq_univ_iff_eq]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 129 | 136 | theorem trailingDegree_mul : (p * q).trailingDegree = p.trailingDegree + q.trailingDegree := by
by_cases hp : p = 0 |
by_cases hp : p = 0
· rw [hp, zero_mul, trailingDegree_zero, top_add]
by_cases hq : q = 0
· rw [hq, mul_zero, trailingDegree_zero, add_top]
· rw [trailingDegree_eq_natTrailingDegree hp, trailingDegree_eq_natTrailingDegree hq,
trailingDegree_eq_natTrailingDegree (mul_ne_zero hp hq), natTrailingDegree_mul ... | true |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 143 | 146 | theorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by
rw [prod_univ_castSucc, prod_univ_four] |
rw [prod_univ_castSucc, prod_univ_four]
rfl
| true |
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open Set
open Computability
-- "ε_NFA"
set_option linter.uppercaseLean3 false
universe u v
structure εNFA (α : Type u) (σ : Type v) where
step : σ → Opt... | Mathlib/Computability/EpsilonNFA.lean | 87 | 88 | theorem stepSet_empty (a : α) : M.stepSet ∅ a = ∅ := by |
simp_rw [stepSet, mem_empty_iff_false, iUnion_false, iUnion_empty]
| true |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section Group
variable [Group α] (e : α) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 150 | 153 | theorem mulETransformRight.fst_mul_snd_subset :
(mulETransformRight e x).1 * (mulETransformRight e x).2 ⊆ x.1 * x.2 := by
refine union_mul_inter_subset_union.trans (union_subset Subset.rfl ?_) |
refine union_mul_inter_subset_union.trans (union_subset Subset.rfl ?_)
rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]
| true |
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