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 | num_lines int64 1 150 | complexity_score float64 2.72 139,370,958,066,637,970,000,000,000,000,000,000,000,000,000,000,000,000,000B | diff_level int64 0 2 | file_diff_level float64 0 2 | theorem_same_file int64 1 32 | rank_file int64 0 2.51k |
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import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 95 | 103 | theorem hasFDerivAt_polarCoord_symm (p : ℝ × ℝ) :
HasFDerivAt polarCoord.symm
(LinearMap.toContinuousLinearMap (Matrix.toLin (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)
!![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])) p := by |
rw [Matrix.toLin_finTwoProd_toContinuousLinearMap]
convert HasFDerivAt.prod (𝕜 := ℝ)
(hasFDerivAt_fst.mul ((hasDerivAt_cos p.2).comp_hasFDerivAt p hasFDerivAt_snd))
(hasFDerivAt_fst.mul ((hasDerivAt_sin p.2).comp_hasFDerivAt p hasFDerivAt_snd)) using 2 <;>
simp [smul_smul, add_comm, neg_mul, smul_neg, n... | 5 | 148.413159 | 2 | 2 | 2 | 2,067 |
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 110 | 123 | theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by |
have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
intro x hx
simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
Classical.not_not] at hx
exact hx.2
have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
a... | 13 | 442,413.392009 | 2 | 2 | 2 | 2,067 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
#align_import ring_theory.graded_algebra.radical from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a"
open GradedRing DirectSum SetLike Finset
variable {ι σ A : Type*}
variable [CommRing A]
variable [LinearOrderedCancelAddCommMono... | Mathlib/RingTheory/GradedAlgebra/Radical.lean | 47 | 136 | theorem Ideal.IsHomogeneous.isPrime_of_homogeneous_mem_or_mem {I : Ideal A} (hI : I.IsHomogeneous 𝒜)
(I_ne_top : I ≠ ⊤)
(homogeneous_mem_or_mem :
∀ {x y : A}, Homogeneous 𝒜 x → Homogeneous 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :
Ideal.IsPrime I :=
⟨I_ne_top, by
intro x y hxy
by_contra! rid
... |
intro x hx
rw [filter_nonempty_iff]
contrapose! hx
simp_rw [proj_apply] at hx
rw [← sum_support_decompose 𝒜 x]
exact Ideal.sum_mem _ hx
set max₁ := set₁.max' (nonempty x rid₁)
set max₂ := set₂.max' (nonempty y rid₂)
have mem_max₁ : max₁ ∈ set₁ := max'_... | 62 | 843,835,666,874,145,400,000,000,000 | 2 | 2 | 1 | 2,068 |
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.FunctionalCalculus
import Mathlib.Topology.UniformSpace.CompactConvergence
local notation "σₙ" => quasispectrum
open... | Mathlib/Topology/ContinuousFunction/NonUnitalFunctionalCalculus.lean | 147 | 167 | theorem cfcₙHom_comp [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(σₙ R a, R)₀)
(f' : C(σₙ R a, σₙ R (cfcₙHom ha f))₀)
(hff' : ∀ x, f x = f' x) (g : C(σₙ R (cfcₙHom ha f), R)₀) :
cfcₙHom ha (g.comp f') = cfcₙHom (cfcₙHom_predicate ha f) g := by |
let ψ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] C(σₙ R a, R)₀ :=
{ toFun := (ContinuousMapZero.comp · f')
map_smul' := fun _ _ ↦ rfl
map_add' := fun _ _ ↦ rfl
map_mul' := fun _ _ ↦ rfl
map_zero' := rfl
map_star' := fun _ ↦ rfl }
let φ : C(σₙ R (cfcₙHom ha f), R)₀ →⋆ₙₐ[R] A := (cfcₙHom ... | 17 | 24,154,952.753575 | 2 | 2 | 1 | 2,069 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import algebra.char_p.local_ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/Algebra/CharP/LocalRing.lean | 25 | 67 | theorem charP_zero_or_prime_power (R : Type*) [CommRing R] [LocalRing R] (q : ℕ)
[char_R_q : CharP R q] : q = 0 ∨ IsPrimePow q := by |
-- Assume `q := char(R)` is not zero.
apply or_iff_not_imp_left.2
intro q_pos
let K := LocalRing.ResidueField R
haveI RM_char := ringChar.charP K
let r := ringChar K
let n := q.factorization r
-- `r := char(R/m)` is either prime or zero:
cases' CharP.char_is_prime_or_zero K r with r_prime r_zero
· ... | 41 | 639,843,493,530,055,000 | 2 | 2 | 1 | 2,070 |
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,285 | 1,309 | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m ∈ filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m ∈ filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m ∈ range (j - n), (1 / ((m + n).fac... |
refine sum_nbij' (· - n) (· + n) ?_ ?_ ?_ ?_ ?_ <;>
simp (config := { contextual := true }) [lt_tsub_iff_right, tsub_add_cancel_of_le]
_ ≤ ∑ m ∈ range (j - n), ((n.factorial : α) * (n.succ : α) ^ m)⁻¹ := by
simp_rw [one_div]
gcongr
rw [← Nat.cast_pow, ← Nat.cast_mul, Nat.cast_le... | 19 | 178,482,300.963187 | 2 | 2 | 1 | 2,071 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 37 | 50 | theorem isConj_of_support_equiv
(f : { x // x ∈ (σ.support : Set α) } ≃ { x // x ∈ (τ.support : Set α) })
(hf : ∀ (x : α) (hx : x ∈ (σ.support : Set α)),
(f ⟨σ x, apply_mem_support.2 hx⟩ : α) = τ ↑(f ⟨x, hx⟩)) :
IsConj σ τ := by |
refine isConj_iff.2 ⟨Equiv.extendSubtype f, ?_⟩
rw [mul_inv_eq_iff_eq_mul]
ext x
simp only [Perm.mul_apply]
by_cases hx : x ∈ σ.support
· rw [Equiv.extendSubtype_apply_of_mem, Equiv.extendSubtype_apply_of_mem]
· exact hf x (Finset.mem_coe.2 hx)
· rwa [Classical.not_not.1 ((not_congr mem_support).1 (E... | 9 | 8,103.083928 | 2 | 2 | 5 | 2,072 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 57 | 65 | theorem perm_inv_on_of_perm_on_finset {s : Finset α} {f : Perm α} (h : ∀ x ∈ s, f x ∈ s) {y : α}
(hy : y ∈ s) : f⁻¹ y ∈ s := by |
have h0 : ∀ y ∈ s, ∃ (x : _) (hx : x ∈ s), y = (fun i (_ : i ∈ s) => f i) x hx :=
Finset.surj_on_of_inj_on_of_card_le (fun x hx => (fun i _ => f i) x hx) (fun a ha => h a ha)
(fun a₁ a₂ ha₁ ha₂ heq => (Equiv.apply_eq_iff_eq f).mp heq) rfl.ge
obtain ⟨y2, hy2, heq⟩ := h0 y hy
convert hy2
rw [heq]
sim... | 7 | 1,096.633158 | 2 | 2 | 5 | 2,072 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 68 | 75 | theorem perm_inv_mapsTo_of_mapsTo (f : Perm α) {s : Set α} [Finite s] (h : Set.MapsTo f s s) :
Set.MapsTo (f⁻¹ : _) s s := by |
cases nonempty_fintype s
exact fun x hx =>
Set.mem_toFinset.mp <|
perm_inv_on_of_perm_on_finset
(fun a ha => Set.mem_toFinset.mpr (h (Set.mem_toFinset.mp ha)))
(Set.mem_toFinset.mpr hx)
| 6 | 403.428793 | 2 | 2 | 5 | 2,072 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 111 | 129 | theorem perm_mapsTo_inl_iff_mapsTo_inr {m n : Type*} [Finite m] [Finite n] (σ : Perm (Sum m n)) :
Set.MapsTo σ (Set.range Sum.inl) (Set.range Sum.inl) ↔
Set.MapsTo σ (Set.range Sum.inr) (Set.range Sum.inr) := by |
constructor <;>
( intro h
classical
rw [← perm_inv_mapsTo_iff_mapsTo] at h
intro x
cases' hx : σ x with l r)
· rintro ⟨a, rfl⟩
obtain ⟨y, hy⟩ := h ⟨l, rfl⟩
rw [← hx, σ.inv_apply_self] at hy
exact absurd hy Sum.inl_ne_inr
· rintro _; exact ⟨r, rfl⟩
· rintro _; exact... | 16 | 8,886,110.520508 | 2 | 2 | 5 | 2,072 |
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.sign from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u v
o... | Mathlib/GroupTheory/Perm/Finite.lean | 132 | 163 | theorem mem_sumCongrHom_range_of_perm_mapsTo_inl {m n : Type*} [Finite m] [Finite n]
{σ : Perm (Sum m n)} (h : Set.MapsTo σ (Set.range Sum.inl) (Set.range Sum.inl)) :
σ ∈ (sumCongrHom m n).range := by |
classical
have h1 : ∀ x : Sum m n, (∃ a : m, Sum.inl a = x) → ∃ a : m, Sum.inl a = σ x := by
rintro x ⟨a, ha⟩
apply h
rw [← ha]
exact ⟨a, rfl⟩
have h3 : ∀ x : Sum m n, (∃ b : n, Sum.inr b = x) → ∃ b : n, Sum.inr b = σ x := by
rintro x ⟨b, hb⟩
apply (perm_mapsTo_inl_iff_map... | 29 | 3,931,334,297,144.042 | 2 | 2 | 5 | 2,072 |
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Closeds
open Function Set Filter TopologicalSpace
open scoped Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace Y]
| Mathlib/Topology/ClopenBox.lean | 36 | 44 | theorem TopologicalSpace.Clopens.exists_prod_subset (W : Clopens (X × Y)) {a : X × Y} (h : a ∈ W) :
∃ U : Clopens X, a.1 ∈ U ∧ ∃ V : Clopens Y, a.2 ∈ V ∧ U ×ˢ V ≤ W := by |
have hp : Continuous (fun y : Y ↦ (a.1, y)) := Continuous.Prod.mk _
let V : Set Y := {y | (a.1, y) ∈ W}
have hV : IsCompact V := (W.2.1.preimage hp).isCompact
let U : Set X := {x | MapsTo (Prod.mk x) V W}
have hUV : U ×ˢ V ⊆ W := fun ⟨_, _⟩ hw ↦ hw.1 hw.2
exact ⟨⟨U, (ContinuousMap.isClopen_setOf_mapsTo hV ... | 7 | 1,096.633158 | 2 | 2 | 2 | 2,073 |
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Closeds
open Function Set Filter TopologicalSpace
open scoped Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace Y]
theorem TopologicalSpace.Clopens.exists_prod_subset (W : Clopens (X × Y)) {a : X × Y} (h : a ∈ W... | Mathlib/Topology/ClopenBox.lean | 50 | 61 | theorem TopologicalSpace.Clopens.exists_finset_eq_sup_prod (W : Clopens (X × Y)) :
∃ (I : Finset (Clopens X × Clopens Y)), W = I.sup fun i ↦ i.1 ×ˢ i.2 := by |
choose! U hxU V hxV hUV using fun x ↦ W.exists_prod_subset (a := x)
rcases W.2.1.isCompact.elim_nhds_subcover (fun x ↦ U x ×ˢ V x) (fun x hx ↦
(U x ×ˢ V x).2.isOpen.mem_nhds ⟨hxU x hx, hxV x hx⟩) with ⟨I, hIW, hWI⟩
classical
use I.image fun x ↦ (U x, V x)
rw [Finset.sup_image]
refine le_antisymm (fun x... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,073 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 67 | 98 | theorem listDecode_encode_list (l : List (L.Term α)) :
listDecode (l.bind listEncode) = l.map Option.some := by |
suffices h : ∀ (t : L.Term α) (l : List (Sum α (Σi, L.Functions i))),
listDecode (t.listEncode ++ l) = some t::listDecode l by
induction' l with t l lih
· rfl
· rw [cons_bind, h t (l.bind listEncode), lih, List.map]
intro t
induction' t with a n f ts ih <;> intro l
· rw [listEncode, singleton... | 30 | 10,686,474,581,524.463 | 2 | 2 | 3 | 2,074 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 122 | 151 | theorem card_sigma : #(Σn, L.Term (Sum α (Fin n))) = max ℵ₀ #(Sum α (Σi, L.Functions i)) := by |
refine le_antisymm ?_ ?_
· rw [mk_sigma]
refine (sum_le_iSup_lift _).trans ?_
rw [mk_nat, lift_aleph0, mul_eq_max_of_aleph0_le_left le_rfl, max_le_iff,
ciSup_le_iff' (bddAbove_range _)]
· refine ⟨le_max_left _ _, fun i => card_le.trans ?_⟩
refine max_le (le_max_left _ _) ?_
rw [← add_... | 29 | 3,931,334,297,144.042 | 2 | 2 | 3 | 2,074 |
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model_theory.encoding from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u v w u' v'
namespace FirstOrder
namespace... | Mathlib/ModelTheory/Encoding.lean | 235 | 287 | theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
(listDecode (l.bind fun φ => φ.2.listEncode)).1 = l.headI := by |
suffices h : ∀ (φ : Σn, L.BoundedFormula α n) (l),
(listDecode (listEncode φ.2 ++ l)).1 = φ ∧ (listDecode (listEncode φ.2 ++ l)).2.1 = l by
induction' l with φ l _
· rw [List.nil_bind]
simp [listDecode]
· rw [cons_bind, (h φ _).1, headI_cons]
rintro ⟨n, φ⟩
induction' φ with _ _ _ _ φ_n φ_... | 51 | 14,093,490,824,269,389,000,000 | 2 | 2 | 3 | 2,074 |
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq from "leanprover-community/mathlib"@"9116dd6709f303dcf781632e15fdef382b0fc579"
open Finset IsAbsoluteValue
namespace IsCauSeq
variable {α β : Type*} [LinearOrderedField... | Mathlib/Algebra/Order/CauSeq/BigOperators.lean | 57 | 141 | theorem _root_.cauchy_product (ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n))
(hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ, ∀ j ≥ i,
abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k -
∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε := by |
let ⟨P, hP⟩ := ha.bounded
let ⟨Q, hQ⟩ := hb.bounded
have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
let ⟨N, hN⟩ := hb.cauchy₂ hPε0
have hQε0 : 0 < ε / (4 * Q) :=
div_pos ε0 (mul_pos (show (0 : α) < 4 by norm... | 79 | 20,382,810,665,126,688,000,000,000,000,000,000 | 2 | 2 | 1 | 2,075 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 45 | 51 | theorem dense_irrational : Dense { x : ℝ | Irrational x } := by |
refine Real.isTopologicalBasis_Ioo_rat.dense_iff.2 ?_
simp only [gt_iff_lt, Rat.cast_lt, not_lt, ge_iff_le, Rat.cast_le, mem_iUnion, mem_singleton_iff,
exists_prop, forall_exists_index, and_imp]
rintro _ a b hlt rfl _
rw [inter_comm]
exact exists_irrational_btwn (Rat.cast_lt.2 hlt)
| 6 | 403.428793 | 2 | 2 | 2 | 2,076 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 78 | 89 | theorem eventually_forall_le_dist_cast_div (hx : Irrational x) (n : ℕ) :
∀ᶠ ε : ℝ in 𝓝 0, ∀ m : ℤ, ε ≤ dist x (m / n) := by |
have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.closedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with... | 10 | 22,026.465795 | 2 | 2 | 2 | 2,076 |
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 41 | 70 | theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) {r : ℝ}
(hr : r < 1) : ∃ x₀ : E, x₀ ∉ F ∧ ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖ := by |
classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1... | 28 | 1,446,257,064,291.475 | 2 | 2 | 3 | 2,077 |
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 83 | 105 | theorem riesz_lemma_of_norm_lt {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) {F : Subspace 𝕜 E}
(hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) :
∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ := by |
have Rpos : 0 < R := (norm_nonneg _).trans_lt hR
have : ‖c‖ / R < 1 := by
rw [div_lt_iff Rpos]
simpa using hR
rcases riesz_lemma hFc hF this with ⟨x, xF, hx⟩
have x0 : x ≠ 0 := fun H => by simp [H] at xF
obtain ⟨d, d0, dxlt, ledx, -⟩ :
∃ d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < R ∧ R / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ R... | 20 | 485,165,195.40979 | 2 | 2 | 3 | 2,077 |
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 108 | 114 | theorem Metric.closedBall_infDist_compl_subset_closure {x : F} {s : Set F} (hx : x ∈ s) :
closedBall x (infDist x sᶜ) ⊆ closure s := by |
rcases eq_or_ne (infDist x sᶜ) 0 with h₀ | h₀
· rw [h₀, closedBall_zero']
exact closure_mono (singleton_subset_iff.2 hx)
· rw [← closure_ball x h₀]
exact closure_mono ball_infDist_compl_subset
| 5 | 148.413159 | 2 | 2 | 3 | 2,077 |
import Mathlib.RingTheory.WittVector.StructurePolynomial
#align_import ring_theory.witt_vector.defs from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a"
noncomputable section
structure WittVector (p : ℕ) (R : Type*) where mk' ::
coeff : ℕ → R
#align witt_vector WittVector
-- Port... | Mathlib/RingTheory/WittVector/Defs.lean | 74 | 78 | theorem ext {x y : 𝕎 R} (h : ∀ n, x.coeff n = y.coeff n) : x = y := by |
cases x
cases y
simp only at h
simp [Function.funext_iff, h]
| 4 | 54.59815 | 2 | 2 | 1 | 2,078 |
import Batteries.Data.Nat.Gcd
import Batteries.Data.Int.DivMod
import Batteries.Lean.Float
-- `Rat` is not tagged with the `ext` attribute, since this is more often than not undesirable
structure Rat where
mk' ::
num : Int
den : Nat := 1
den_nz : den ≠ 0 := by decide
reduced : num.natAbs.C... | .lake/packages/batteries/Batteries/Data/Rat/Basic.lean | 60 | 66 | theorem Rat.normalize.reduced {num : Int} {den g : Nat} (den_nz : den ≠ 0)
(e : g = num.natAbs.gcd den) : (num.div g).natAbs.Coprime (den / g) :=
have : Int.natAbs (num.div ↑g) = num.natAbs / g := by |
match num, num.eq_nat_or_neg with
| _, ⟨_, .inl rfl⟩ => rfl
| _, ⟨_, .inr rfl⟩ => rw [Int.neg_div, Int.natAbs_neg, Int.natAbs_neg]; rfl
this ▸ e ▸ Nat.coprime_div_gcd_div_gcd (Nat.gcd_pos_of_pos_right _ (Nat.pos_of_ne_zero den_nz))
| 4 | 54.59815 | 2 | 2 | 1 | 2,079 |
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {α X : Type*} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
| Mathlib/Dynamics/Ergodic/Function.lean | 27 | 35 | theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
{s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
(h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
(hg_eq : g ∘ f =ᵐ[μ] g) :
∃ c, g =ᵐ[μ] const α c := by |
refine exists_eventuallyEq_const_of_eventually_mem_of_forall_separating MeasurableSet hs ?_
refine fun U hU ↦ h.ae_mem_or_ae_nmem₀ (s := g ⁻¹' U) (hgm hU) ?_b
refine (hg_eq.mono fun x hx ↦ ?_).set_eq
rw [← preimage_comp, mem_preimage, mem_preimage, hx]
| 4 | 54.59815 | 2 | 2 | 2 | 2,080 |
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {α X : Type*} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace... | Mathlib/Dynamics/Ergodic/Function.lean | 77 | 82 | theorem ae_eq_const_of_ae_eq_comp_ae {g : α → X} (h : QuasiErgodic f μ)
(hgm : AEStronglyMeasurable g μ) (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c := by |
borelize X
rcases hgm.isSeparable_ae_range with ⟨t, ht, hgt⟩
haveI := ht.secondCountableTopology
exact h.ae_eq_const_of_ae_eq_comp_of_ae_range₀ hgt hgm.aemeasurable.nullMeasurable hg_eq
| 4 | 54.59815 | 2 | 2 | 2 | 2,080 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.Topology.Algebra.Module.Cardinality
open Convex Set Metric
section TopologicalVectorSpace
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E]
| Mathlib/Analysis/NormedSpace/Connected.lean | 34 | 103 | theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
(h : 1 < Module.rank ℝ E) {s : Set E} (hs : s.Countable) :
IsPathConnected sᶜ := by |
have : Nontrivial E := (rank_pos_iff_nontrivial (R := ℝ)).1 (zero_lt_one.trans h)
-- the set `sᶜ` is dense, therefore nonempty. Pick `a ∈ sᶜ`. We have to show that any
-- `b ∈ sᶜ` can be joined to `a`.
obtain ⟨a, ha⟩ : sᶜ.Nonempty := (hs.dense_compl ℝ).nonempty
refine ⟨a, ha, ?_⟩
intro b hb
rcases eq_or_... | 67 | 125,236,317,084,221,370,000,000,000,000 | 2 | 2 | 1 | 2,081 |
import Mathlib.CategoryTheory.Limits.ColimitLimit
import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.ConcreteCatego... | Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean | 72 | 142 | theorem colimitLimitToLimitColimit_injective :
Function.Injective (colimitLimitToLimitColimit F) := by |
classical
cases nonempty_fintype J
-- Suppose we have two terms `x y` in the colimit (over `K`) of the limits (over `J`),
-- and that these have the same image under `colimitLimitToLimitColimit F`.
intro x y h
-- These elements of the colimit have representatives somewhere:
obtain ⟨kx, x, rfl... | 69 | 925,378,172,558,778,900,000,000,000,000 | 2 | 2 | 1 | 2,082 |
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.AlgebraicGeometry.AffineScheme
#align_import algebraic_geometry.limits from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
suppress_compilation
set_option linter.uppercaseLean3 false
universe u
open CategoryTheory CategoryTheor... | Mathlib/AlgebraicGeometry/Limits.lean | 133 | 139 | theorem bot_isAffineOpen (X : Scheme) : IsAffineOpen (⊥ : Opens X.carrier) := by |
convert rangeIsAffineOpenOfOpenImmersion (initial.to X)
ext
-- Porting note: added this `erw` to turn LHS to `False`
erw [Set.mem_empty_iff_false]
rw [false_iff_iff]
exact fun x => isEmptyElim (show (⊥_ Scheme).carrier from x.choose)
| 6 | 403.428793 | 2 | 2 | 1 | 2,083 |
import Mathlib.Algebra.Group.Nat
set_option autoImplicit true
open Lean hiding Literal HashMap
open Batteries
namespace Sat
inductive Literal
| pos : Nat → Literal
| neg : Nat → Literal
def Literal.ofInt (i : Int) : Literal :=
if i < 0 then Literal.neg (-i-1).toNat else Literal.pos (i-1).toNat
def Lit... | Mathlib/Tactic/Sat/FromLRAT.lean | 156 | 166 | theorem Valuation.mk_implies {as ps} (as₁) : as = List.reverseAux as₁ ps →
(Valuation.mk as).implies p ps as₁.length → p := by |
induction ps generalizing as₁ with
| nil => exact fun _ ↦ id
| cons a as ih =>
refine fun e H ↦ @ih (a::as₁) e (H ?_)
subst e; clear ih H
suffices ∀ n n', n' = List.length as₁ + n →
∀ bs, mk (as₁.reverseAux bs) n' ↔ mk bs n from this 0 _ rfl (a::as)
induction as₁ with simp
| cons b as₁ ... | 9 | 8,103.083928 | 2 | 2 | 2 | 2,084 |
import Mathlib.Algebra.Group.Nat
set_option autoImplicit true
open Lean hiding Literal HashMap
open Batteries
namespace Sat
inductive Literal
| pos : Nat → Literal
| neg : Nat → Literal
def Literal.ofInt (i : Int) : Literal :=
if i < 0 then Literal.neg (-i-1).toNat else Literal.pos (i-1).toNat
def Lit... | Mathlib/Tactic/Sat/FromLRAT.lean | 180 | 185 | theorem Fmla.reify_or (h₁ : Fmla.reify v f₁ a) (h₂ : Fmla.reify v f₂ b) :
Fmla.reify v (f₁.and f₂) (a ∨ b) := by |
refine ⟨fun H ↦ by_contra fun hn ↦ H ⟨fun c h ↦ by_contra fun hn' ↦ ?_⟩⟩
rcases List.mem_append.1 h with h | h
· exact hn <| Or.inl <| h₁.1 fun Hc ↦ hn' <| Hc.1 _ h
· exact hn <| Or.inr <| h₂.1 fun Hc ↦ hn' <| Hc.1 _ h
| 4 | 54.59815 | 2 | 2 | 2 | 2,084 |
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 123 | 135 | theorem monomial_smul_single (i : ℕ) (r : R) (j : ℕ) (m : M) :
monomial i r • single R j m = single R (i + j) (r • m) := by |
simp only [LinearMap.mul_apply, Polynomial.aeval_monomial, LinearMap.pow_apply,
Module.algebraMap_end_apply, smul_def]
induction i generalizing r j m with
| zero =>
rw [Function.iterate_zero, zero_add]
exact Finsupp.smul_single r j m
| succ n hn =>
rw [Function.iterate_succ, Function.comp_apply... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,085 |
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 139 | 153 | theorem monomial_smul_apply (i : ℕ) (r : R) (g : PolynomialModule R M) (n : ℕ) :
(monomial i r • g) n = ite (i ≤ n) (r • g (n - i)) 0 := by |
induction' g using PolynomialModule.induction_linear with p q hp hq
· simp only [smul_zero, zero_apply, ite_self]
· simp only [smul_add, add_apply, hp, hq]
split_ifs
exacts [rfl, zero_add 0]
· rw [monomial_smul_single, single_apply, single_apply, smul_ite, smul_zero, ← ite_and]
congr
rw [eq_iff... | 13 | 442,413.392009 | 2 | 2 | 3 | 2,085 |
import Mathlib.Algebra.Polynomial.Module.AEval
#align_import data.polynomial.module from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
universe u v
open Polynomial BigOperators
@[nolint unusedArguments]
def PolynomialModule (R M : Type*) [CommRing R] [AddCommGroup M] [Module R M] := ℕ ... | Mathlib/Algebra/Polynomial/Module/Basic.lean | 157 | 169 | theorem smul_single_apply (i : ℕ) (f : R[X]) (m : M) (n : ℕ) :
(f • single R i m) n = ite (i ≤ n) (f.coeff (n - i) • m) 0 := by |
induction' f using Polynomial.induction_on' with p q hp hq
· rw [add_smul, Finsupp.add_apply, hp, hq, coeff_add, add_smul]
split_ifs
exacts [rfl, zero_add 0]
· rw [monomial_smul_single, single_apply, coeff_monomial, ite_smul, zero_smul]
by_cases h : i ≤ n
· simp_rw [eq_tsub_iff_add_eq_of_le h, if... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,085 |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 148 | 155 | theorem range_inl : range (inl R M M₂) = ker (snd R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.fst, Prod.ext rfl h.symm⟩
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,086 |
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
import Mathlib.Order.PartialSups
#align_import linear_algebra.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
universe u v w x y z u' v' w' y'
variable {R : Type u} {K : Ty... | Mathlib/LinearAlgebra/Prod.lean | 162 | 169 | theorem range_inr : range (inr R M M₂) = ker (fst R M M₂) := by |
ext x
simp only [mem_ker, mem_range]
constructor
· rintro ⟨y, rfl⟩
rfl
· intro h
exact ⟨x.snd, Prod.ext h.symm rfl⟩
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,086 |
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Top... | Mathlib/Probability/Process/Stopping.lean | 72 | 82 | theorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[f i] {ω | τ ω < i} := by |
by_cases hi_min : IsMin i
· suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)
ext1 ω
simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
rw [isMin_iff_forall_not_lt] at hi_min
exact hi_min (τ ω)
have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i) := by... | 9 | 8,103.083928 | 2 | 2 | 1 | 2,087 |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 31 | 41 | theorem unitary.spectrum_subset_circle (u : unitary E) :
spectrum 𝕜 (u : E) ⊆ Metric.sphere 0 1 := by |
nontriviality E
refine fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm ?_ ?_)
· simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk
· rw [← unitary.val_toUnits_apply u] at hk
have hnk := ne_zero_of_mem_of_unit hk
rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_in... | 9 | 8,103.083928 | 2 | 2 | 3 | 2,088 |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 60 | 69 | theorem IsSelfAdjoint.spectralRadius_eq_nnnorm {a : A} (ha : IsSelfAdjoint a) :
spectralRadius ℂ a = ‖a‖₊ := by |
have hconst : Tendsto (fun _n : ℕ => (‖a‖₊ : ℝ≥0∞)) atTop _ := tendsto_const_nhds
refine tendsto_nhds_unique ?_ hconst
convert
(spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp
(Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two) using 1
refine funext fun n => ?_
rw [Function... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,088 |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 72 | 86 | theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
spectralRadius ℂ a = ‖a‖₊ := by |
refine (ENNReal.pow_strictMono two_ne_zero).injective ?_
have heq :
(fun n : ℕ => (‖(a⋆ * a) ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) =
(fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ) := by
funext n
rw [Function.comp_apply, ← rpow_natCast, ← rpow_mul, mul_comm, rpow_mul, rpow_natCast, ←
... | 13 | 442,413.392009 | 2 | 2 | 3 | 2,088 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.IndicatorConstPointwise
#align_import measure_theory.constructions.borel_space.metrizable from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Filter MeasureT... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 31 | 47 | theorem measurable_of_tendsto_metrizable' {ι} {f : ι → α → β} {g : α → β} (u : Filter ι) [NeBot u]
[IsCountablyGenerated u] (hf : ∀ i, Measurable (f i)) (lim : Tendsto f u (𝓝 g)) :
Measurable g := by |
letI : PseudoMetricSpace β := pseudoMetrizableSpacePseudoMetric β
apply measurable_of_isClosed'
intro s h1s h2s h3s
have : Measurable fun x => infNndist (g x) s := by
suffices Tendsto (fun i x => infNndist (f i x) s) u (𝓝 fun x => infNndist (g x) s) from
NNReal.measurable_of_tendsto' u (fun i => (hf... | 14 | 1,202,604.284165 | 2 | 2 | 3 | 2,089 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.IndicatorConstPointwise
#align_import measure_theory.constructions.borel_space.metrizable from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Filter MeasureT... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 57 | 78 | theorem aemeasurable_of_tendsto_metrizable_ae {ι} {μ : Measure α} {f : ι → α → β} {g : α → β}
(u : Filter ι) [hu : NeBot u] [IsCountablyGenerated u] (hf : ∀ n, AEMeasurable (f n) μ)
(h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) : AEMeasurable g μ := by |
rcases u.exists_seq_tendsto with ⟨v, hv⟩
have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n)
set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
have hp : ∀ᵐ x ∂μ, p x fun n => f (v n) x := by
filter_upwards [h_tendsto] with x hx using hx.comp hv
set aeSeqLim := f... | 19 | 178,482,300.963187 | 2 | 2 | 3 | 2,089 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.IndicatorConstPointwise
#align_import measure_theory.constructions.borel_space.metrizable from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Filter MeasureT... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 87 | 101 | theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
{μ : Measure α} {g : α → β}
(hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
AEMeasurable g μ := by |
obtain ⟨u, -, u_pos, u_lim⟩ :
∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
exists_seq_strictAnti_tendsto (0 : ℝ)
choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u ... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,089 |
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
#align_import category_theory.sites.closed from "leanprover-community/mathlib"@"4cfc30e317caad46858393f1a7a33f609296cc30"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable (J₁ J₂ : GrothendieckTopol... | Mathlib/CategoryTheory/Sites/Closed.lean | 124 | 132 | theorem pullback_close {X Y : C} (f : Y ⟶ X) (S : Sieve X) :
J₁.close (S.pullback f) = (J₁.close S).pullback f := by |
apply le_antisymm
· refine J₁.le_close_of_isClosed (Sieve.pullback_monotone _ (J₁.le_close S)) ?_
apply J₁.isClosed_pullback _ _ (J₁.close_isClosed _)
· intro Z g hg
change _ ∈ J₁ _
rw [← Sieve.pullback_comp]
apply hg
| 7 | 1,096.633158 | 2 | 2 | 2 | 2,090 |
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.Order.Closure
#align_import category_theory.sites.closed from "leanprover-community/mathlib"@"4cfc30e317caad46858393f1a7a33f609296cc30"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable (J₁ J₂ : GrothendieckTopol... | Mathlib/CategoryTheory/Sites/Closed.lean | 149 | 159 | theorem close_eq_top_iff_mem {X : C} (S : Sieve X) : J₁.close S = ⊤ ↔ S ∈ J₁ X := by |
constructor
· intro h
apply J₁.transitive (J₁.top_mem X)
intro Y f hf
change J₁.close S f
rwa [h]
· intro hS
rw [eq_top_iff]
intro Y f _
apply J₁.pullback_stable _ hS
| 10 | 22,026.465795 | 2 | 2 | 2 | 2,090 |
import Mathlib.Analysis.NormedSpace.Star.ContinuousFunctionalCalculus.Restrict
import Mathlib.Analysis.NormedSpace.Star.ContinuousFunctionalCalculus
import Mathlib.Analysis.NormedSpace.Star.Spectrum
import Mathlib.Analysis.NormedSpace.Star.Unitization
import Mathlib.Topology.ContinuousFunction.UniqueCFC
noncomputab... | Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus/Instances.lean | 120 | 136 | theorem RCLike.nonUnitalContinuousFunctionalCalculus :
NonUnitalContinuousFunctionalCalculus 𝕜 (p : A → Prop) where
exists_cfc_of_predicate a ha := by |
let ψ : C(σₙ 𝕜 a, 𝕜)₀ →⋆ₙₐ[𝕜] A := comp (inrRangeEquiv 𝕜 A).symm <|
codRestrict (cfcₙAux hp₁ a ha) _ (cfcₙAux_mem_range_inr hp₁ a ha)
have coe_ψ (f : C(σₙ 𝕜 a, 𝕜)₀) : ψ f = cfcₙAux hp₁ a ha f :=
congr_arg Subtype.val <| (inrRangeEquiv 𝕜 A).apply_symm_apply
⟨cfcₙAux hp₁ a ha f, cfcₙAu... | 14 | 1,202,604.284165 | 2 | 2 | 1 | 2,091 |
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import Mathlib.Topology.ShrinkingLemma
#align_import geometry.manifold.partition_of_unity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 157 | 162 | theorem exists_pos_of_mem {x} (hx : x ∈ s) : ∃ i, 0 < f i x := by |
by_contra! h
have H : ∀ i, f i x = 0 := fun i ↦ le_antisymm (h i) (f.nonneg i x)
have := f.sum_eq_one hx
simp_rw [H] at this
simpa
| 5 | 148.413159 | 2 | 2 | 1 | 2,092 |
import Mathlib.FieldTheory.Finite.Basic
#align_import field_theory.chevalley_warning from "leanprover-community/mathlib"@"e001509c11c4d0f549d91d89da95b4a0b43c714f"
universe u v
section FiniteField
open MvPolynomial
open Function hiding eval
open Finset FiniteField
variable {K σ ι : Type*} [Fintype K] [Field ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 53 | 97 | theorem MvPolynomial.sum_eval_eq_zero (f : MvPolynomial σ K)
(h : f.totalDegree < (q - 1) * Fintype.card σ) : ∑ x, eval x f = 0 := by |
haveI : DecidableEq K := Classical.decEq K
calc
∑ x, eval x f = ∑ x : σ → K, ∑ d ∈ f.support, f.coeff d * ∏ i, x i ^ d i := by
simp only [eval_eq']
_ = ∑ d ∈ f.support, ∑ x : σ → K, f.coeff d * ∏ i, x i ^ d i := sum_comm
_ = 0 := sum_eq_zero ?_
intro d hd
obtain ⟨i, hi⟩ : ∃ i, d i < q - 1 := ... | 43 | 4,727,839,468,229,346,000 | 2 | 2 | 2 | 2,093 |
import Mathlib.FieldTheory.Finite.Basic
#align_import field_theory.chevalley_warning from "leanprover-community/mathlib"@"e001509c11c4d0f549d91d89da95b4a0b43c714f"
universe u v
section FiniteField
open MvPolynomial
open Function hiding eval
open Finset FiniteField
variable {K σ ι : Type*} [Fintype K] [Field ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 107 | 160 | theorem char_dvd_card_solutions_of_sum_lt {s : Finset ι} {f : ι → MvPolynomial σ K}
(h : (∑ i ∈ s, (f i).totalDegree) < Fintype.card σ) :
p ∣ Fintype.card { x : σ → K // ∀ i ∈ s, eval x (f i) = 0 } := by |
have hq : 0 < q - 1 := by rw [← Fintype.card_units, Fintype.card_pos_iff]; exact ⟨1⟩
let S : Finset (σ → K) := { x ∈ univ | ∀ i ∈ s, eval x (f i) = 0 }.toFinset
have hS : ∀ x : σ → K, x ∈ S ↔ ∀ i : ι, i ∈ s → eval x (f i) = 0 := by
intro x
simp only [S, Set.toFinset_setOf, mem_univ, true_and, mem_filter]... | 51 | 14,093,490,824,269,389,000,000 | 2 | 2 | 2 | 2,093 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.conformal_linear_map from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {E F : Type*}
variable [NormedAddCommGroup E] [NormedAddCom... | Mathlib/Analysis/InnerProductSpace/ConformalLinearMap.lean | 29 | 43 | theorem isConformalMap_iff (f : E →L[ℝ] F) :
IsConformalMap f ↔ ∃ c : ℝ, 0 < c ∧ ∀ u v : E, ⟪f u, f v⟫ = c * ⟪u, v⟫ := by |
constructor
· rintro ⟨c₁, hc₁, li, rfl⟩
refine ⟨c₁ * c₁, mul_self_pos.2 hc₁, fun u v => ?_⟩
simp only [real_inner_smul_left, real_inner_smul_right, mul_assoc, coe_smul',
coe_toContinuousLinearMap, Pi.smul_apply, inner_map_map]
· rintro ⟨c₁, hc₁, huv⟩
obtain ⟨c, hc, rfl⟩ : ∃ c : ℝ, 0 < c ∧ c₁ = ... | 13 | 442,413.392009 | 2 | 2 | 1 | 2,094 |
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.Algebra.Category.GroupCat.EpiMono
#align_import category_theory.preadditive.yoneda.projective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9ed226956e26dee00975946"
universe v u
open... | Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean | 31 | 39 | theorem projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj (P : C) :
Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms := by |
rw [projective_iff_preservesEpimorphisms_coyoneda_obj]
refine ⟨fun h : (preadditiveCoyoneda.obj (op P) ⋙
forget AddCommGroupCat).PreservesEpimorphisms => ?_, ?_⟩
· exact Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveCoyoneda.obj (op P))
(forget _)
· intro
exact (inferInst... | 7 | 1,096.633158 | 2 | 2 | 2 | 2,095 |
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.Algebra.Category.GroupCat.EpiMono
#align_import category_theory.preadditive.yoneda.projective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9ed226956e26dee00975946"
universe v u
open... | Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean | 42 | 50 | theorem projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj' (P : C) :
Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms := by |
rw [projective_iff_preservesEpimorphisms_coyoneda_obj]
refine ⟨fun h : (preadditiveCoyoneda.obj (op P) ⋙
forget AddCommGroupCat).PreservesEpimorphisms => ?_, ?_⟩
· exact Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveCoyoneda.obj (op P))
(forget _)
· intro
exact (inferInst... | 7 | 1,096.633158 | 2 | 2 | 2 | 2,095 |
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
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
| 11 | 59,874.141715 | 2 | 2 | 1 | 2,096 |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.Data.Set.Pairwise.Lattice
#align_import measure_theory.covering.vitali from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
variable {α ι : Type*}
open Set Metri... | Mathlib/MeasureTheory/Covering/Vitali.lean | 58 | 153 | theorem exists_disjoint_subfamily_covering_enlargment (B : ι → Set α) (t : Set ι) (δ : ι → ℝ)
(τ : ℝ) (hτ : 1 < τ) (δnonneg : ∀ a ∈ t, 0 ≤ δ a) (R : ℝ) (δle : ∀ a ∈ t, δ a ≤ R)
(hne : ∀ a ∈ t, (B a).Nonempty) :
∃ u ⊆ t,
u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ δ a ≤ τ * δ b := b... |
/- The proof could be formulated as a transfinite induction. First pick an element of `t` with `δ`
as large as possible (up to a factor of `τ`). Then among the remaining elements not intersecting
the already chosen one, pick another element with large `δ`. Go on forever (transfinitely) until
there is nothing l... | 90 | 1,220,403,294,317,840,800,000,000,000,000,000,000,000 | 2 | 2 | 1 | 2,097 |
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 67 | 75 | theorem germ_injective_of_isIntegral [IsIntegral X] {U : Opens X.carrier} (x : U) :
Function.Injective (X.presheaf.germ x) := by |
rw [injective_iff_map_eq_zero]
intro y hy
rw [← (X.presheaf.germ x).map_zero] at hy
obtain ⟨W, hW, iU, iV, e⟩ := X.presheaf.germ_eq _ x.prop x.prop _ _ hy
cases Subsingleton.elim iU iV
haveI : Nonempty W := ⟨⟨_, hW⟩⟩
exact map_injective_of_isIntegral X iU e
| 7 | 1,096.633158 | 2 | 2 | 3 | 2,098 |
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 83 | 93 | theorem genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[hX : IrreducibleSpace X.carrier] [IrreducibleSpace Y.carrier] :
f.1.base (genericPoint X.carrier : _) = (genericPoint Y.carrier : _) := by |
apply ((genericPoint_spec Y).eq _).symm
convert (genericPoint_spec X.carrier).image (show Continuous f.1.base by continuity)
symm
rw [eq_top_iff, Set.top_eq_univ, Set.top_eq_univ]
convert subset_closure_inter_of_isPreirreducible_of_isOpen _ H.base_open.isOpen_range _
· rw [Set.univ_inter, Set.image_univ]
... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,098 |
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 115 | 121 | theorem genericPoint_eq_bot_of_affine (R : CommRingCat) [IsDomain R] :
genericPoint (Scheme.Spec.obj <| op R).carrier = (⟨0, Ideal.bot_prime⟩ : PrimeSpectrum R) := by |
apply (genericPoint_spec (Scheme.Spec.obj <| op R).carrier).eq
rw [isGenericPoint_def]
rw [← PrimeSpectrum.zeroLocus_vanishingIdeal_eq_closure, PrimeSpectrum.vanishingIdeal_singleton]
rw [Set.top_eq_univ, ← PrimeSpectrum.zeroLocus_singleton_zero]
simp_rw [Submodule.zero_eq_bot, Submodule.bot_coe]
| 5 | 148.413159 | 2 | 2 | 3 | 2,098 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 40 | 80 | theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t).toSubmodule.FG) :
(adjoin R (s ∪ t)).toSubmodule.FG := by |
rcases fg_def.1 h1 with ⟨p, hp, hp'⟩
rcases fg_def.1 h2 with ⟨q, hq, hq'⟩
refine fg_def.2 ⟨p * q, hp.mul hq, le_antisymm ?_ ?_⟩
· rw [span_le, Set.mul_subset_iff]
intro x hx y hy
change x * y ∈ adjoin R (s ∪ t)
refine Subalgebra.mul_mem _ ?_ ?_
· have : x ∈ Subalgebra.toSubmodule (adjoin R s) :... | 39 | 86,593,400,423,993,740 | 2 | 2 | 3 | 2,099 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 129 | 137 | theorem FG.prod {S : Subalgebra R A} {T : Subalgebra R B} (hS : S.FG) (hT : T.FG) :
(S.prod T).FG := by |
obtain ⟨s, hs⟩ := fg_def.1 hS
obtain ⟨t, ht⟩ := fg_def.1 hT
rw [← hs.2, ← ht.2]
exact fg_def.2 ⟨LinearMap.inl R A B '' (s ∪ {1}) ∪ LinearMap.inr R A B '' (t ∪ {1}),
Set.Finite.union (Set.Finite.image _ (Set.Finite.union hs.1 (Set.finite_singleton _)))
(Set.Finite.image _ (Set.Finite.union ht.1 (Set.f... | 7 | 1,096.633158 | 2 | 2 | 3 | 2,099 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 170 | 179 | theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A → Prop) (base : P ⊥)
(ih : ∀ (S : Subalgebra R A) (x : A), P S → P (Algebra.adjoin R (insert x S)))
(S : Subalgebra R A) : P S := by |
classical
obtain ⟨t, rfl⟩ := S.fg_of_noetherian
refine Finset.induction_on t ?_ ?_
· simpa using base
intro x t _ h
rw [Finset.coe_insert]
simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
| 7 | 1,096.633158 | 2 | 2 | 3 | 2,099 |
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Topology.Category.TopCat.Limits.Products
universe w w' v u
open CategoryTheory Opposit... | Mathlib/Topology/Category/TopCat/Yoneda.lean | 48 | 58 | theorem piComparison_fac {α : Type} (X : α → TopCat) :
piComparison (yonedaPresheaf'.{w, w'} Y) (fun x ↦ op (X x)) =
(yonedaPresheaf' Y).map ((opCoproductIsoProduct X).inv ≫ (TopCat.sigmaIsoSigma X).inv.op) ≫
(equivEquivIso (sigmaEquiv Y (fun x ↦ (X x).1))).inv ≫ (Types.productIso _).inv := by |
rw [← Category.assoc, Iso.eq_comp_inv]
ext
simp only [yonedaPresheaf', unop_op, piComparison, types_comp_apply,
Types.productIso_hom_comp_eval_apply, Types.pi_lift_π_apply, comp_apply, TopCat.coe_of,
unop_comp, Quiver.Hom.unop_op, sigmaEquiv, equivEquivIso_hom, Equiv.toIso_inv,
Equiv.coe_fn_symm_mk, ... | 7 | 1,096.633158 | 2 | 2 | 1 | 2,100 |
import Mathlib.Data.List.Basic
import Mathlib.Order.MinMax
import Mathlib.Order.WithBot
#align_import data.list.min_max from "leanprover-community/mathlib"@"6d0adfa76594f304b4650d098273d4366edeb61b"
namespace List
variable {α β : Type*}
section ArgAux
variable (r : α → α → Prop) [DecidableRel r] {l : List α} {o... | Mathlib/Data/List/MinMax.lean | 69 | 86 | theorem not_of_mem_foldl_argAux (hr₀ : Irreflexive r) (hr₁ : Transitive r) :
∀ {a m : α} {o : Option α}, a ∈ l → m ∈ foldl (argAux r) o l → ¬r a m := by |
induction' l using List.reverseRecOn with tl a ih
· simp
intro b m o hb ho
rw [foldl_append, foldl_cons, foldl_nil, argAux] at ho
cases' hf : foldl (argAux r) o tl with c
· rw [hf] at ho
rw [foldl_argAux_eq_none] at hf
simp_all [hf.1, hf.2, hr₀ _]
rw [hf, Option.mem_def] at ho
dsimp only at ho
... | 16 | 8,886,110.520508 | 2 | 2 | 1 | 2,101 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.Cyclotomic.Rat
section case1
open ZMod
private lemma cube_of_castHom_ne_zero {n : ZMod 9} :
castHom (show 3 ∣ 9 by norm_num) (ZMod 3) n ≠ 0 → n ^ 3 = 1 ∨ n ^ 3 = 8 := by
revert n; decide
private lemma cube_of_n... | Mathlib/NumberTheory/FLT/Three.lean | 36 | 44 | theorem fermatLastTheoremThree_case_1 {a b c : ℤ} (hdvd : ¬ 3 ∣ a * b * c) :
a ^ 3 + b ^ 3 ≠ c ^ 3 := by |
simp_rw [Int.prime_three.dvd_mul, not_or] at hdvd
apply mt (congrArg (Int.cast : ℤ → ZMod 9))
simp_rw [Int.cast_add, Int.cast_pow]
rcases cube_of_not_dvd hdvd.1.1 with ha | ha <;>
rcases cube_of_not_dvd hdvd.1.2 with hb | hb <;>
rcases cube_of_not_dvd hdvd.2 with hc | hc <;>
rw [ha, hb, hc] <;> decide
| 7 | 1,096.633158 | 2 | 2 | 1 | 2,102 |
import Mathlib.ModelTheory.ElementarySubstructures
#align_import model_theory.skolem from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
universe u v w w'
namespace FirstOrder
namespace Language
open Structure Cardinal
open Cardinal
variable (L : Language.{u, v}) {M : Type w} [None... | Mathlib/ModelTheory/Skolem.lean | 50 | 62 | theorem card_functions_sum_skolem₁ :
#(Σ n, (L.sum L.skolem₁).Functions n) = #(Σ n, L.BoundedFormula Empty (n + 1)) := by |
simp only [card_functions_sum, skolem₁_Functions, mk_sigma, sum_add_distrib']
conv_lhs => enter [2, 1, i]; rw [lift_id'.{u, v}]
rw [add_comm, add_eq_max, max_eq_left]
· refine sum_le_sum _ _ fun n => ?_
rw [← lift_le.{_, max u v}, lift_lift, lift_mk_le.{v}]
refine ⟨⟨fun f => (func f default).bdEqual (f... | 11 | 59,874.141715 | 2 | 2 | 3 | 2,103 |
import Mathlib.ModelTheory.ElementarySubstructures
#align_import model_theory.skolem from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
universe u v w w'
namespace FirstOrder
namespace Language
open Structure Cardinal
open Cardinal
variable (L : Language.{u, v}) {M : Type w} [None... | Mathlib/ModelTheory/Skolem.lean | 65 | 73 | theorem card_functions_sum_skolem₁_le : #(Σ n, (L.sum L.skolem₁).Functions n) ≤ max ℵ₀ L.card := by |
rw [card_functions_sum_skolem₁]
trans #(Σ n, L.BoundedFormula Empty n)
· exact
⟨⟨Sigma.map Nat.succ fun _ => id,
Nat.succ_injective.sigma_map fun _ => Function.injective_id⟩⟩
· refine _root_.trans BoundedFormula.card_le (lift_le.{max u v}.1 ?_)
simp only [mk_empty, lift_zero, lift_uzero, ze... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,103 |
import Mathlib.ModelTheory.ElementarySubstructures
#align_import model_theory.skolem from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
universe u v w w'
namespace FirstOrder
namespace Language
open Structure Cardinal
open Cardinal
variable (L : Language.{u, v}) {M : Type w} [None... | Mathlib/ModelTheory/Skolem.lean | 86 | 95 | theorem skolem₁_reduct_isElementary (S : (L.sum L.skolem₁).Substructure M) :
(LHom.sumInl.substructureReduct S).IsElementary := by |
apply (LHom.sumInl.substructureReduct S).isElementary_of_exists
intro n φ x a h
let φ' : (L.sum L.skolem₁).Functions n := LHom.sumInr.onFunction φ
exact
⟨⟨funMap φ' ((↑) ∘ x), S.fun_mem (LHom.sumInr.onFunction φ) ((↑) ∘ x) (by
exact fun i => (x i).2)⟩,
by exact Classical.epsilon_spec (p := fun ... | 8 | 2,980.957987 | 2 | 2 | 3 | 2,103 |
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Functor.EpiMono
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.concrete_category.basic from "leanprover-community/mathlib"@"311ef8c4b4ae2804ea76b8a611bc5ea1d9c16872"
universe w w' v v' v'' u u' u''
namespa... | Mathlib/CategoryTheory/ConcreteCategory/Basic.lean | 106 | 110 | theorem ConcreteCategory.hom_ext {X Y : C} (f g : X ⟶ Y) (w : ∀ x : X, f x = g x) : f = g := by |
apply (forget C).map_injective
dsimp [forget]
funext x
exact w x
| 4 | 54.59815 | 2 | 2 | 1 | 2,104 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
| Mathlib/Analysis/Convex/StoneSeparation.lean | 30 | 77 | theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ segment 𝕜 x y)
(hu : u ∈ segment 𝕜 x p) (hv : v ∈ segment 𝕜 y q) :
¬Disjoint (segment 𝕜 u v) (convexHull 𝕜 {p, q, z}) := by |
rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain r... | 45 | 34,934,271,057,485,095,000 | 2 | 2 | 2 | 2,105 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Join
#align_import analysis.convex.stone_separation from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
variable {𝕜 E ι : Type*} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E}
th... | Mathlib/Analysis/Convex/StoneSeparation.lean | 81 | 109 | theorem exists_convex_convex_compl_subset (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hst : Disjoint s t) :
∃ C : Set E, Convex 𝕜 C ∧ Convex 𝕜 Cᶜ ∧ s ⊆ C ∧ t ⊆ Cᶜ := by |
let S : Set (Set E) := { C | Convex 𝕜 C ∧ Disjoint C t }
obtain ⟨C, hC, hsC, hCmax⟩ :=
zorn_subset_nonempty S
(fun c hcS hc ⟨_, _⟩ =>
⟨⋃₀ c,
⟨hc.directedOn.convex_sUnion fun s hs => (hcS hs).1,
disjoint_sUnion_left.2 fun c hc => (hcS hc).2⟩,
fun s => subset_sUnion... | 27 | 532,048,240,601.79865 | 2 | 2 | 2 | 2,105 |
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
open Set Filter Uniformity Topology Function UniformConvergence
variable {ι X Y α β : Type*} [TopologicalSpace X] [UniformSpace α] [UniformSpace β]
variable {F : ι ... | Mathlib/Topology/UniformSpace/Ascoli.lean | 85 | 125 | theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
(UniformFun.uniformSpace X α).comap F =
(Pi.uniformSpace _).comap F := by |
-- The `≤` inequality is trivial
refine le_antisymm (UniformSpace.comap_mono UniformFun.uniformContinuous_toFun) ?_
-- A bit of rewriting to get a nice intermediate statement.
change comap _ _ ≤ comap _ _
simp_rw [Pi.uniformity, Filter.comap_iInf, comap_comap, Function.comp]
refine ((UniformFun.hasBasis_un... | 38 | 31,855,931,757,113,756 | 2 | 2 | 2 | 2,106 |
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
open Set Filter Uniformity Topology Function UniformConvergence
variable {ι X Y α β : Type*} [TopologicalSpace X] [UniformSpace α] [UniformSpace β]
variable {F : ι ... | Mathlib/Topology/UniformSpace/Ascoli.lean | 163 | 199 | theorem Equicontinuous.tendsto_uniformFun_iff_pi [CompactSpace X]
(F_eqcont : Equicontinuous F) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformFun.ofFun ∘ F) ℱ (𝓝 <| UniformFun.ofFun f) ↔
Tendsto F ℱ (𝓝 f) := by |
-- Assume `ℱ` is non trivial.
rcases ℱ.eq_or_neBot with rfl | ℱ_ne
· simp
constructor <;> intro H
-- The forward direction is always true, the interesting part is the converse.
· exact UniformFun.uniformContinuous_toFun.continuous.tendsto _|>.comp H
-- To prove it, assume that `F` tends to `f` *pointwise... | 33 | 214,643,579,785,916.06 | 2 | 2 | 2 | 2,106 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.rational_root from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
open scoped Polynomial
section ScaleRoots
var... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 39 | 44 | theorem scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : A[X]} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) : aeval (algebraMap A S r) (scaleRoots p s) = 0 := by |
convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr
-- Porting note: added
funext
rw [aeval_def, mk'_spec' _ r s]
| 4 | 54.59815 | 2 | 2 | 2 | 2,107 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.rational_root from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
open scoped Polynomial
section ScaleRoots
var... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 49 | 54 | theorem num_isRoot_scaleRoots_of_aeval_eq_zero [UniqueFactorizationMonoid A] {p : A[X]} {x : K}
(hr : aeval x p = 0) : IsRoot (scaleRoots p (den A x)) (num A x) := by |
apply isRoot_of_eval₂_map_eq_zero (IsFractionRing.injective A K)
refine scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero ?_
rw [mk'_num_den]
exact hr
| 4 | 54.59815 | 2 | 2 | 2 | 2,107 |
import Mathlib.Order.Zorn
import Mathlib.Order.Atoms
#align_import order.zorn_atoms from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
open Set
| Mathlib/Order/ZornAtoms.lean | 24 | 36 | theorem IsCoatomic.of_isChain_bounded {α : Type*} [PartialOrder α] [OrderTop α]
(h :
∀ c : Set α,
IsChain (· ≤ ·) c → c.Nonempty → ⊤ ∉ c → ∃ x ≠ ⊤, x ∈ upperBounds c) :
IsCoatomic α := by |
refine ⟨fun x => le_top.eq_or_lt.imp_right fun hx => ?_⟩
have : ∃ y ∈ Ico x ⊤, x ≤ y ∧ ∀ z ∈ Ico x ⊤, y ≤ z → z = y := by
refine zorn_nonempty_partialOrder₀ (Ico x ⊤) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)
rcases h c hc ⟨y, hy⟩ fun h => (hxc h).2.ne rfl with ⟨z, hz, hcz⟩
exact ⟨z, ⟨le_trans (h... | 8 | 2,980.957987 | 2 | 2 | 1 | 2,108 |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Star.Pointwise
import Mathlib.Algebra.Group.Centralizer
variable {R : Type*} [Mul R] [StarMul R] {a : R} {s : Set R}
| Mathlib/Algebra/Star/Center.lean | 14 | 34 | theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where
comm := by | simpa only [star_mul, star_star] using fun g =>
congr_arg star (((Set.mem_center_iff R).mp ha).comm <| star g).symm
left_assoc b c := calc
star a * (b * c) = star a * (star (star b) * star (star c)) := by rw [star_star, star_star]
_ = star a * star (star c * star b) := by rw [star_mul]
_ = star ((sta... | 20 | 485,165,195.40979 | 2 | 2 | 1 | 2,109 |
import Mathlib.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.UpperLower
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Order.UpperLower
import Mathlib.Topology.MetricSpace.Sequences
#align_import analysis.no... | Mathlib/Analysis/Normed/Order/UpperLower.lean | 94 | 109 | theorem IsUpperSet.mem_interior_of_forall_lt (hs : IsUpperSet s) (hx : x ∈ closure s)
(h : ∀ i, x i < y i) : y ∈ interior s := by |
cases nonempty_fintype ι
obtain ⟨ε, hε, hxy⟩ := Pi.exists_forall_pos_add_lt h
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε
rw [dist_pi_lt_iff hε] at hxz
have hyz : ∀ i, z i < y i := by
refine fun i => (hxy _).trans_le' (sub_le_iff_le_add'.1 <| (le_abs_self _).trans ?_)
rw [← Real.norm_eq_a... | 14 | 1,202,604.284165 | 2 | 2 | 2 | 2,110 |
import Mathlib.Algebra.Order.Field.Pi
import Mathlib.Algebra.Order.UpperLower
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Order.UpperLower
import Mathlib.Topology.MetricSpace.Sequences
#align_import analysis.no... | Mathlib/Analysis/Normed/Order/UpperLower.lean | 112 | 128 | theorem IsLowerSet.mem_interior_of_forall_lt (hs : IsLowerSet s) (hx : x ∈ closure s)
(h : ∀ i, y i < x i) : y ∈ interior s := by |
cases nonempty_fintype ι
obtain ⟨ε, hε, hxy⟩ := Pi.exists_forall_pos_add_lt h
obtain ⟨z, hz, hxz⟩ := Metric.mem_closure_iff.1 hx _ hε
rw [dist_pi_lt_iff hε] at hxz
have hyz : ∀ i, y i < z i := by
refine fun i =>
(lt_sub_iff_add_lt.2 <| hxy _).trans_le (sub_le_comm.1 <| (le_abs_self _).trans ?_)
... | 15 | 3,269,017.372472 | 2 | 2 | 2 | 2,110 |
import Mathlib.CategoryTheory.Idempotents.Karoubi
#align_import category_theory.idempotents.functor_extension from "leanprover-community/mathlib"@"5f68029a863bdf76029fa0f7a519e6163c14152e"
namespace CategoryTheory
namespace Idempotents
open Category Karoubi
variable {C D E : Type*} [Category C] [Category D] [Ca... | Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean | 35 | 40 | theorem natTrans_eq {F G : Karoubi C ⥤ D} (φ : F ⟶ G) (P : Karoubi C) :
φ.app P = F.map (decompId_i P) ≫ φ.app P.X ≫ G.map (decompId_p P) := by |
rw [← φ.naturality, ← assoc, ← F.map_comp]
conv_lhs => rw [← id_comp (φ.app P), ← F.map_id]
congr
apply decompId
| 4 | 54.59815 | 2 | 2 | 1 | 2,111 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 43 | 48 | theorem mul_average_left (g : G) : ↑(Finsupp.single g 1) * average k G = average k G := by |
simp only [mul_one, Finset.mul_sum, Algebra.mul_smul_comm, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
show ⅟ (Fintype.card G : k) • ∑ x : G, f (g * x) = ⅟ (Fintype.card G : k) • ∑ x : G, f x
rw [Function.B... | 5 | 148.413159 | 2 | 2 | 3 | 2,112 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 54 | 59 | theorem mul_average_right (g : G) : average k G * ↑(Finsupp.single g 1) = average k G := by |
simp only [mul_one, Finset.sum_mul, Algebra.smul_mul_assoc, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
show ⅟ (Fintype.card G : k) • ∑ x : G, f (x * g) = ⅟ (Fintype.card G : k) • ∑ x : G, f x
rw [Function.... | 5 | 148.413159 | 2 | 2 | 3 | 2,112 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace Representation
namespace linHom... | Mathlib/RepresentationTheory/Invariants.lean | 133 | 139 | theorem mem_invariants_iff_comm {X Y : Rep k G} (f : X.V →ₗ[k] Y.V) (g : G) :
(linHom X.ρ Y.ρ) g f = f ↔ f.comp (X.ρ g) = (Y.ρ g).comp f := by |
dsimp
erw [← ρAut_apply_inv]
rw [← LinearMap.comp_assoc, ← ModuleCat.comp_def, ← ModuleCat.comp_def, Iso.inv_comp_eq,
ρAut_apply_hom]
exact comm
| 5 | 148.413159 | 2 | 2 | 3 | 2,112 |
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 89 | 109 | theorem μ_natural {X Y X' Y' : Type u} (f : X ⟶ Y) (g : X' ⟶ Y') :
((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom = (μ R X X').hom ≫ (free R).map (f ⊗ g) := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro x'
apply LinearMap.ext_ring
apply Finsupp.ext
intro ⟨y, y'⟩
-- Porting note (#10934): used to be dsimp [μ]
change (finsuppTensorFinsupp' R Y Y')
... | 18 | 65,659,969.137331 | 2 | 2 | 3 | 2,113 |
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 112 | 129 | theorem left_unitality (X : Type u) :
(λ_ ((free R).obj X)).hom =
(ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [ε, μ]
let q : X →₀ R := ((λ_ (of R (X →₀ R))).hom) (1 ⊗ₜ[R] Finsupp.single x 1)
cha... | 15 | 3,269,017.372472 | 2 | 2 | 3 | 2,113 |
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.CategoryTheory.Linear.LinearFunctor
#align_import algebra.category.Module.adjunctions from "leanpr... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 132 | 149 | theorem right_unitality (X : Type u) :
(ρ_ ((free R).obj X)).hom =
(𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [ε, μ]
let q : X →₀ R := ((ρ_ (of R (X →₀ R))).hom) (Finsupp.single x 1 ⊗ₜ[R] 1)
cha... | 15 | 3,269,017.372472 | 2 | 2 | 3 | 2,113 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Algebra.Group.UniqueProds
#align_import algebra.monoid_algebra.no_zero_divisors from "leanprover-community/mathlib"@"3e067975886cf5801e597925328c335609511b1a"
open Finsupp
variable {R A : Type*} [Semiring R]
namespace MonoidAlgebra
| Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean | 68 | 79 | theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
(h : UniqueMul f.support g.support a0 b0) :
(f * g) (a0 * b0) = f a0 * g b0 := by |
classical
simp_rw [mul_apply, sum, ← Finset.sum_product']
refine (Finset.sum_eq_single (a0, b0) ?_ ?_).trans (if_pos rfl) <;> simp_rw [Finset.mem_product]
· refine fun ab hab hne => if_neg (fun he => hne <| Prod.ext ?_ ?_)
exacts [(h hab.1 hab.2 he).1, (h hab.1 hab.2 he).2]
· refine fun hnmem => ite_eq_r... | 9 | 8,103.083928 | 2 | 2 | 1 | 2,114 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 59 | 69 | theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
{ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
(hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine b.ext_elem fun i ↦ ?_
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
exact
(Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
⟨x, hx, rfl⟩
| 8 | 2,980.957987 | 2 | 2 | 3 | 2,115 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section Ring
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 72 | 81 | theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
(hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
[(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ := by |
rw [Submodule.eq_bot_iff]
intro x hx
refine (mk_eq_zero _ _).mp (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i ↦ ?_)
rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
rw [LinearEquiv.map_zero, Finsupp.zero_apply]
refine (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)... | 7 | 1,096.633158 | 2 | 2 | 3 | 2,115 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
universe u v
section IsDomain
variable {ι : Type*} {R : Type*} [CommRing R] [IsDoma... | Mathlib/LinearAlgebra/FreeModule/PID.lean | 93 | 98 | theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
x ∣ generator I ↔ I = Ideal.span {x} := by |
conv_rhs => rw [← span_singleton_generator I]
rw [Ideal.submodule_span_eq, Ideal.span_singleton_eq_span_singleton, ← dvd_dvd_iff_associated,
← mem_iff_generator_dvd]
exact ⟨fun h ↦ ⟨hx, h⟩, fun h ↦ h.2⟩
| 4 | 54.59815 | 2 | 2 | 3 | 2,115 |
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.Topology.Algebra.Polynomial
#align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open Polyn... | Mathlib/Analysis/Complex/Polynomial.lean | 34 | 45 | theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z := by |
by_contra! hf'
/- Since `f` has no roots, `f⁻¹` is differentiable. And since `f` is a polynomial, it tends to
infinity at infinity, thus `f⁻¹` tends to zero at infinity. By Liouville's theorem, `f⁻¹ = 0`. -/
have (z : ℂ) : (f.eval z)⁻¹ = 0 :=
(f.differentiable.inv hf').apply_eq_of_tendsto_cocompact z <|
... | 11 | 59,874.141715 | 2 | 2 | 1 | 2,116 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 106 | 137 | theorem norm_max_aux₁ [CompleteSpace F] {f : ℂ → F} {z w : ℂ}
(hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ := by |
-- Consider a circle of radius `r = dist w z`.
set r : ℝ := dist w z
have hw : w ∈ closedBall z r := mem_closedBall.2 le_rfl
-- Assume the converse. Since `‖f w‖ ≤ ‖f z‖`, we have `‖f w‖ < ‖f z‖`.
refine (isMaxOn_iff.1 hz _ hw).antisymm (not_lt.1 ?_)
rintro hw_lt : ‖f w‖ < ‖f z‖
have hr : 0 < r := dist_p... | 29 | 3,931,334,297,144.042 | 2 | 2 | 4 | 2,117 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 144 | 151 | theorem norm_max_aux₂ {f : ℂ → F} {z w : ℂ} (hd : DiffContOnCl ℂ f (ball z (dist w z)))
(hz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z) : ‖f w‖ = ‖f z‖ := by |
set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
have he : ∀ x, ‖e x‖ = ‖x‖ := UniformSpace.Completion.norm_coe
replace hz : IsMaxOn (norm ∘ e ∘ f) (closedBall z (dist w z)) z := by
simpa only [IsMaxOn, (· ∘ ·), he] using hz
simpa only [he, (· ∘ ·)]
using norm_max_aux₁ (e.differentiable.comp_diff... | 6 | 403.428793 | 2 | 2 | 4 | 2,117 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 159 | 164 | theorem norm_max_aux₃ {f : ℂ → F} {z w : ℂ} {r : ℝ} (hr : dist w z = r)
(hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) : ‖f w‖ = ‖f z‖ := by |
subst r
rcases eq_or_ne w z with (rfl | hne); · rfl
rw [← dist_ne_zero] at hne
exact norm_max_aux₂ hd (closure_ball z hne ▸ hz.closure hd.continuousOn.norm)
| 4 | 54.59815 | 2 | 2 | 4 | 2,117 |
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Topology.Order.ExtrClosure
#align_import analysis.complex.abs_max from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpa... | Mathlib/Analysis/Complex/AbsMax.lean | 181 | 196 | theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
(hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by |
intro w hw
rw [mem_closedBall, dist_comm] at hw
rcases eq_or_ne z w with (rfl | hne); · rfl
set e := (lineMap z w : ℂ → E)
have hde : Differentiable ℂ e := (differentiable_id.smul_const (w - z)).add_const z
suffices ‖(f ∘ e) (1 : ℂ)‖ = ‖(f ∘ e) (0 : ℂ)‖ by simpa [e]
have hr : dist (1 : ℂ) 0 = 1 := by sim... | 13 | 442,413.392009 | 2 | 2 | 4 | 2,117 |
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import algebra.monoid_algebra.ideal from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k A G : Type*}
| Mathlib/Algebra/MonoidAlgebra/Ideal.lean | 23 | 58 | theorem MonoidAlgebra.mem_ideal_span_of_image [Monoid G] [Semiring k] {s : Set G}
{x : MonoidAlgebra k G} :
x ∈ Ideal.span (MonoidAlgebra.of k G '' s) ↔ ∀ m ∈ x.support, ∃ m' ∈ s, ∃ d, m = d * m' := by |
let RHS : Ideal (MonoidAlgebra k G) :=
{ carrier := { p | ∀ m : G, m ∈ p.support → ∃ m' ∈ s, ∃ d, m = d * m' }
add_mem' := fun {x y} hx hy m hm => by
classical exact (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
zero_mem' := fun m hm => by cases hm
smul_mem' := fun x... | 33 | 214,643,579,785,916.06 | 2 | 2 | 1 | 2,118 |
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 | 30 | 49 | theorem deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x := by |
cases' h₁.lt_or_lt with h₁ h₁
· have : 1 - x ^ 2 < 0 := by nlinarith [h₁]
rw [sqrt_eq_zero'.2 this.le, div_zero]
have : arcsin =ᶠ[𝓝 x] fun _ => -(π / 2) :=
(gt_mem_nhds h₁).mono fun y hy => arcsin_of_le_neg_one hy.le
exact ⟨(hasStrictDerivAt_const _ _).congr_of_eventuallyEq this.symm,
cont... | 18 | 65,659,969.137331 | 2 | 2 | 6 | 2,119 |
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
| 4 | 54.59815 | 2 | 2 | 6 | 2,119 |
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 | 74 | 79 | theorem hasDerivWithinAt_arcsin_Iic {x : ℝ} (h : x ≠ 1) :
HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Iic x) x := by |
rcases em (x = -1) with (rfl | h')
· convert (hasDerivWithinAt_const (-1 : ℝ) _ (-(π / 2))).congr _ _ <;>
simp (config := { contextual := true }) [arcsin_of_le_neg_one]
· exact (hasDerivAt_arcsin h' h).hasDerivWithinAt
| 4 | 54.59815 | 2 | 2 | 6 | 2,119 |
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 | 82 | 90 | theorem differentiableWithinAt_arcsin_Ici {x : ℝ} :
DifferentiableWithinAt ℝ arcsin (Ici x) x ↔ x ≠ -1 := by |
refine ⟨?_, fun h => (hasDerivWithinAt_arcsin_Ici h).differentiableWithinAt⟩
rintro h rfl
have : sin ∘ arcsin =ᶠ[𝓝[≥] (-1 : ℝ)] id := by
filter_upwards [Icc_mem_nhdsWithin_Ici ⟨le_rfl, neg_lt_self (zero_lt_one' ℝ)⟩] with x using
sin_arcsin'
have := h.hasDerivWithinAt.sin.congr_of_eventuallyEq this.s... | 7 | 1,096.633158 | 2 | 2 | 6 | 2,119 |
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 | 93 | 98 | theorem differentiableWithinAt_arcsin_Iic {x : ℝ} :
DifferentiableWithinAt ℝ arcsin (Iic x) x ↔ x ≠ 1 := by |
refine ⟨fun h => ?_, fun h => (hasDerivWithinAt_arcsin_Iic h).differentiableWithinAt⟩
rw [← neg_neg x, ← image_neg_Ici] at h
have := (h.comp (-x) differentiableWithinAt_id.neg (mapsTo_image _ _)).neg
simpa [(· ∘ ·), differentiableWithinAt_arcsin_Ici] using this
| 4 | 54.59815 | 2 | 2 | 6 | 2,119 |
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 | 108 | 114 | theorem deriv_arcsin : deriv arcsin = fun x => 1 / √(1 - x ^ 2) := by |
funext x
by_cases h : x ≠ -1 ∧ x ≠ 1
· exact (hasDerivAt_arcsin h.1 h.2).deriv
· rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_arcsin.1 h)]
simp only [not_and_or, Ne, Classical.not_not] at h
rcases h with (rfl | rfl) <;> simp
| 6 | 403.428793 | 2 | 2 | 6 | 2,119 |
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