name stringlengths 3 112 | file stringlengths 21 116 | statement stringlengths 17 8.64k | state stringlengths 7 205k | tactic stringlengths 3 4.55k | result stringlengths 7 205k | id stringlengths 16 16 |
|---|---|---|---|---|---|---|
Besicovitch.exist_finset_disjoint_balls_large_measure | Mathlib/MeasureTheory/Covering/Besicovitch.lean | theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
{τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
(rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
∃ t : Finset α, ↑t ⊆ s ∧ μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * ... | case inr.inr.intro.intro.intro.intro.intro.intro.intro.intro.refine_3.intro.intro.intro.intro
α : Type u_1
inst✝⁴ : MetricSpace α
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N... | exact hu i k'.2 l'.2 k'nel' | no goals | 689472630553a827 |
Nat.self_sub_mod | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem self_sub_mod (n k : Nat) [NeZero k] : (n - k) % n = n - k | case succ
k : Nat
inst✝ : NeZero k
n : Nat
⊢ n + 1 - k < n + 1 | cases k with
| zero => simp_all
| succ k => omega | no goals | b1b3cc122d8b73de |
Std.DHashMap.Internal.Raw₀.toListModel_insertListₘ | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/WF.lean | theorem toListModel_insertListₘ [BEq α] [Hashable α] [EquivBEq α][LawfulHashable α]
{m : Raw₀ α β} {l : List ((a : α) × β a)} (h : Raw.WFImp m.1) :
Perm (toListModel (insertListₘ m l).1.buckets)
(List.insertList (toListModel m.1.buckets) l) | case cons
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
hd : (a : α) × β a
tl : List ((a : α) × β a)
ih :
∀ {m : Raw₀ α β},
Raw.WFImp m.val → toListModel (m.insertListₘ tl).val.buckets ~ insertList (toListModel m.val.buckets) tl
m : Raw₀ α β
h : Raw.WFIm... | apply Perm.trans (ih (wfImp_insert h)) | case cons
α : Type u
β : α → Type v
inst✝³ : BEq α
inst✝² : Hashable α
inst✝¹ : EquivBEq α
inst✝ : LawfulHashable α
hd : (a : α) × β a
tl : List ((a : α) × β a)
ih :
∀ {m : Raw₀ α β},
Raw.WFImp m.val → toListModel (m.insertListₘ tl).val.buckets ~ insertList (toListModel m.val.buckets) tl
m : Raw₀ α β
h : Raw.WFIm... | ac07dfc59e878a3e |
GenContFract.sub_convs_eq | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | theorem sub_convs_eq {ifp : IntFractPair K}
(stream_nth_eq : IntFractPair.stream v n = some ifp) :
let g := of v
let B := (g.contsAux (n + 1)).b
let pB := (g.contsAux n).b
v - g.convs n = if ifp.fr = 0 then 0 else (-1) ^ n / (B * (ifp.fr⁻¹ * B + pB)) | case inl
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
g_finite_correctness : v = GenContFract.compExactValue pred_cont... | suffices v - g.convs n = 0 by simpa [ifp_fr_eq_zero] | case inl
K : Type u_1
v : K
n : ℕ
inst✝¹ : LinearOrderedField K
inst✝ : FloorRing K
ifp : IntFractPair K
stream_nth_eq : IntFractPair.stream v n = some ifp
g : GenContFract K := of v
conts : Pair K := g.contsAux (n + 1)
pred_conts : Pair K := g.contsAux n
g_finite_correctness : v = GenContFract.compExactValue pred_cont... | 703fcc23f4d0f739 |
Finset.Colex.erase_le_erase_min' | Mathlib/Combinatorics/Colex.lean | /-- If `s ≤ t` in colex and `#s ≤ #t`, then `s \ {a} ≤ t \ {min t}` for any `a ∈ s`. -/
lemma erase_le_erase_min' (hst : toColex s ≤ toColex t) (hcard : #s ≤ #t) (ha : a ∈ s) :
toColex (s.erase a) ≤
toColex (t.erase <| min' t <| card_pos.1 <| (card_pos.2 ⟨a, ha⟩).trans_le hcard) | case inr.intro.intro.intro.inl.inr
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : #s ≤ #t
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ... | by_contra! hab | case inr.intro.intro.intro.inl.inr
α : Type u_1
inst✝ : LinearOrder α
s t : Finset α
a : α
hcard : #s ≤ #t
ha : a ∈ s
ht : t.Nonempty
m : α := t.min' ht
h' : s ≠ t
w : α
hwt : w ∈ t
hws : w ∉ s
hw : ∀ ⦃a : α⦄, w < a → (a ∈ s ↔ a ∈ t)
hwa : w < a
hma : m < a
b : α
hbs : b ∈ { ofColex := { val := s.val.erase a, nodup := ... | 94805d4e030ffc34 |
Polynomial.abc | Mathlib/NumberTheory/FLT/MasonStothers.lean | theorem Polynomial.abc
{a b c : k[X]} (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0)
(hab : IsCoprime a b) (hsum : a + b + c = 0) :
( natDegree a + 1 ≤ (radical (a * b * c)).natDegree ∧
natDegree b + 1 ≤ (radical (a * b * c)).natDegree ∧
natDegree c + 1 ≤ (radical (a * b * c)).natDegree ) ∨
deriva... | k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hsum : a + b + c = 0
w : k[X] := a.wronskian b
wab : w = a.wronskian b
hbc : IsCoprime b c
hsum' : b + c + a = 0
hca : IsCoprime c a
wbc : w = b.wronskian c
wca : w = c.wronskian a
adr_dvd_w : divRadica... | rw [divRadical_mul (hca.symm.mul_left hbc), divRadical_mul hab] | k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hsum : a + b + c = 0
w : k[X] := a.wronskian b
wab : w = a.wronskian b
hbc : IsCoprime b c
hsum' : b + c + a = 0
hca : IsCoprime c a
wbc : w = b.wronskian c
wca : w = c.wronskian a
adr_dvd_w : divRadica... | 0effd224fcbff26d |
Real.exp_neg_one_gt_d9 | Mathlib/Data/Complex/ExponentialBounds.lean | theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) | ⊢ 0 < 0.36787944116 | norm_num | no goals | bcf9d28551578c4c |
List.getElem_splitWrtCompositionAux | Mathlib/Combinatorics/Enumerative/Composition.lean | theorem getElem_splitWrtCompositionAux (l : List α) (ns : List ℕ) {i : ℕ}
(hi : i < (l.splitWrtCompositionAux ns).length) :
(l.splitWrtCompositionAux ns)[i] =
(l.take (ns.take (i + 1)).sum).drop (ns.take i).sum | case nil
α : Type u_1
l : List α
i : ℕ
hi : i < (l.splitWrtCompositionAux []).length
⊢ (l.splitWrtCompositionAux [])[i] = drop (take i []).sum (take (take (i + 1) []).sum l) | cases hi | no goals | e1e405149acdbdb8 |
Rat.divInt_num_den | Mathlib/.lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | theorem divInt_num_den (z : d ≠ 0) (h : n /. d = ⟨n', d', z', c⟩) :
∃ m, m ≠ 0 ∧ n = n' * m ∧ d = d' * m | case intro.inl
n n' : Int
d' : Nat
z' : d' ≠ 0
c : n'.natAbs.Coprime d'
w✝ : Nat
z : ¬w✝ = 0
h : mkRat n w✝ = { num := n', den := d', den_nz := z', reduced := c }
⊢ ∃ m, ¬m = 0 ∧ n = n' * m ∧ ↑w✝ = ↑d' * m | have ⟨m, h₁, h₂⟩ := mkRat_num_den z h | case intro.inl
n n' : Int
d' : Nat
z' : d' ≠ 0
c : n'.natAbs.Coprime d'
w✝ : Nat
z : ¬w✝ = 0
h : mkRat n w✝ = { num := n', den := d', den_nz := z', reduced := c }
m : Nat
h₁ : m ≠ 0
h₂ : n = n' * ↑m ∧ w✝ = d' * m
⊢ ∃ m, ¬m = 0 ∧ n = n' * m ∧ ↑w✝ = ↑d' * m | 272a3f3229e80a30 |
PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective | Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean | lemma _root_.PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective
{Y : C} (r₀ r₀' : R₀.obj (Opposite.op Y))
(m₀ m₀' : M₀.obj (Opposite.op Y))
(hr₀ : α.app _ r₀ = α.app _ r₀')
(hm₀ : φ.app _ m₀ = φ.app _ m₀') :
φ.app _ (r₀ • m₀) = φ.app _ (r₀' • m₀') | case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ R : Cᵒᵖ ⥤ RingCat
α : R₀ ⟶ R
inst✝¹ : Presheaf.IsLocallyInjective J α
M₀ : PresheafOfModules R₀
A : Cᵒᵖ ⥤ AddCommGrp
φ : M₀.presheaf ⟶ A
inst✝ : Presheaf.IsLocallyInjective J φ
hA : Presheaf.IsSeparated J A
Y : C
r₀ r₀' : ↑(R₀.obj (Opposite.o... | intro Z g hg | case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ R : Cᵒᵖ ⥤ RingCat
α : R₀ ⟶ R
inst✝¹ : Presheaf.IsLocallyInjective J α
M₀ : PresheafOfModules R₀
A : Cᵒᵖ ⥤ AddCommGrp
φ : M₀.presheaf ⟶ A
inst✝ : Presheaf.IsLocallyInjective J φ
hA : Presheaf.IsSeparated J A
Y : C
r₀ r₀' : ↑(R₀.obj (Opposite.o... | cf5858c53eb20f7d |
OrthogonalFamily.range_linearIsometry | Mathlib/Analysis/InnerProductSpace/l2Space.lean | theorem range_linearIsometry [∀ i, CompleteSpace (G i)] :
LinearMap.range hV.linearIsometry.toLinearMap =
(⨆ i, LinearMap.range (V i).toLinearMap).topologicalClosure | case refine_2.h.intro
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFa... | use lp.single 2 i x | case h
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
ins... | e67b0f768537d5e1 |
List.take_append_eq_append_take | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Nat/TakeDrop.lean | theorem take_append_eq_append_take {l₁ l₂ : List α} {n : Nat} :
take n (l₁ ++ l₂) = take n l₁ ++ take (n - l₁.length) l₂ | case nil
α : Type u_1
l₂ : List α
n : Nat
⊢ take n ([] ++ l₂) = take n [] ++ take (n - [].length) l₂ | simp | no goals | f0800663d0e95bfd |
Holor.slice_sum | Mathlib/Data/Holor.lean | theorem slice_sum [AddCommMonoid α] {β : Type} (i : ℕ) (hid : i < d) (s : Finset β)
(f : β → Holor α (d :: ds)) : (∑ x ∈ s, slice (f x) i hid) = slice (∑ x ∈ s, f x) i hid | α : Type
d : ℕ
ds : List ℕ
inst✝ : AddCommMonoid α
β : Type
i : ℕ
hid : i < d
s : Finset β
f : β → Holor α (d :: ds)
this : DecidableEq β := Classical.decEq β
⊢ ∑ x ∈ s, (f x).slice i hid = (∑ x ∈ s, f x).slice i hid | refine Finset.induction_on s ?_ ?_ | case refine_1
α : Type
d : ℕ
ds : List ℕ
inst✝ : AddCommMonoid α
β : Type
i : ℕ
hid : i < d
s : Finset β
f : β → Holor α (d :: ds)
this : DecidableEq β := Classical.decEq β
⊢ ∑ x ∈ ∅, (f x).slice i hid = (∑ x ∈ ∅, f x).slice i hid
case refine_2
α : Type
d : ℕ
ds : List ℕ
inst✝ : AddCommMonoid α
β : Type
i : ℕ
hid : i ... | 5c8ff6afcaca4525 |
QuotientGroup.strictMono_comap_prod_image | Mathlib/GroupTheory/Coset/Basic.lean | theorem strictMono_comap_prod_image :
StrictMono fun t : Subgroup α ↦ (t.comap s.subtype, mk (s := s) '' t) | case intro.intro
α : Type u_1
inst✝ : Group α
s t₁ t₂ : Subgroup α
h : t₁ < t₂
x✝ : (fun t => (Subgroup.comap s.subtype t, mk '' ↑t)) t₂ ≤ (fun t => (Subgroup.comap s.subtype t, mk '' ↑t)) t₁
a : α
ha : a ∈ ↑t₂
le1 :
((fun t => (Subgroup.comap s.subtype t, mk '' ↑t)) t₂).1 ≤ ((fun t => (Subgroup.comap s.subtype t, mk... | convert ← t₁.mul_mem h' (@le1 ⟨_, QuotientGroup.eq.1 eq⟩ <| t₂.mul_mem (t₂.inv_mem <| h.1 h') ha) | case h.e'_1
α : Type u_1
inst✝ : Group α
s t₁ t₂ : Subgroup α
h : t₁ < t₂
x✝ : (fun t => (Subgroup.comap s.subtype t, mk '' ↑t)) t₂ ≤ (fun t => (Subgroup.comap s.subtype t, mk '' ↑t)) t₁
a : α
ha : a ∈ ↑t₂
le1 :
((fun t => (Subgroup.comap s.subtype t, mk '' ↑t)) t₂).1 ≤ ((fun t => (Subgroup.comap s.subtype t, mk '' ↑... | b09250fa61dca065 |
LieSubmodule.lcs_le_lcs_of_is_nilpotent_span_sup_eq_top | Mathlib/Algebra/Lie/Engel.lean | theorem lcs_le_lcs_of_is_nilpotent_span_sup_eq_top {n i j : ℕ}
(hxn : toEnd R L M x ^ n = 0) (hIM : lowerCentralSeries R L M i ≤ I.lcs M j) :
lowerCentralSeries R L M (i + n) ≤ I.lcs M (j + 1) | case succ.refine_2
R : Type u₁
L : Type u₂
M : Type u₄
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
I : LieIdeal R L
x : L
hxI : Submodule.span R {x} ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤
n i j ... | exact antitone_lowerCentralSeries R L M le_self_add | no goals | ff6796e644485966 |
Equiv.Perm.nodup_of_pairwise_disjoint | Mathlib/GroupTheory/Perm/Support.lean | theorem nodup_of_pairwise_disjoint {l : List (Perm α)} (h1 : (1 : Perm α) ∉ l)
(h2 : l.Pairwise Disjoint) : l.Nodup | α : Type u_1
l : List (Perm α)
h1 : 1 ∉ l
h2 : List.Pairwise Disjoint l
σ : Perm α
a✝ h_mem : σ ∈ l
h_disjoint : σ.Disjoint σ
this : σ = 1
⊢ False | rw [this] at h_mem | α : Type u_1
l : List (Perm α)
h1 : 1 ∉ l
h2 : List.Pairwise Disjoint l
σ : Perm α
a✝ : σ ∈ l
h_mem : 1 ∈ l
h_disjoint : σ.Disjoint σ
this : σ = 1
⊢ False | d28a7770c33ffe6c |
isRightRegular_of_mul_eq_one | Mathlib/Algebra/Regular/Basic.lean | theorem isRightRegular_of_mul_eq_one (h : a * b = 1) : IsRightRegular a :=
IsRightRegular.of_mul (a := b) (by rw [h]; exact isRegular_one.right)
| R : Type u_1
inst✝ : Monoid R
a b : R
h : a * b = 1
⊢ IsRightRegular (a * b) | rw [h] | R : Type u_1
inst✝ : Monoid R
a b : R
h : a * b = 1
⊢ IsRightRegular 1 | 8d95ccfb5145237a |
UniformSpace.Core.nhds_toTopologicalSpace | Mathlib/Topology/UniformSpace/Defs.lean | theorem UniformSpace.Core.nhds_toTopologicalSpace {α : Type u} (u : Core α) (x : α) :
@nhds α u.toTopologicalSpace x = comap (Prod.mk x) u.uniformity | case hpure
α : Type u
u : Core α
x : α
⊢ ∀ (a : α), ∀ i ∈ u.uniformity, a ∈ Prod.mk a ⁻¹' id i | exact fun a U hU ↦ u.refl hU rfl | no goals | 567766e05e617e3f |
Int.subNatNat_add | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean | theorem subNatNat_add (m n k : Nat) : subNatNat (m + n) k = m + subNatNat n k | case inl
m n k : Nat
h' : n < k
⊢ subNatNat (m + n) (k - n + n) = subNatNat m (k - n) | apply subNatNat_add_add | no goals | 9fc3dd7335f8add0 |
hasFDerivWithinAt_congr_set' | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | theorem hasFDerivWithinAt_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f f' t x :=
calc
HasFDerivWithinAt f f' s x ↔ HasFDerivWithinAt f f' (s \ {y}) x :=
(hasFDerivWithinAt_diff_singleton _).symm
_ ↔ HasFDerivWithinAt f f' (t \ {y}) x | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
s t : Set E
y : E
h : s =ᶠ[𝓝[{y}ᶜ] x] t
⊢ 𝓝[s \ {y}] x = 𝓝[t \ {y}] x | simpa only [set_eventuallyEq_iff_inf_principal, ← nhdsWithin_inter', diff_eq,
inter_comm] using h | no goals | f09aa9d80822de67 |
Unitization.inr_nonneg_iff | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean | @[simp, norm_cast]
lemma inr_nonneg_iff {a : A} : 0 ≤ (a : A⁺¹) ↔ 0 ≤ a | case neg.refine_2
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ¬IsSelfAdjoint a
h : 0 ≤ a
⊢ IsSelfAdjoint a | exact .of_nonneg h | no goals | d0ef500fa938a011 |
Pell.eq_of_xn_modEq' | Mathlib/NumberTheory/PellMatiyasevic.lean | theorem eq_of_xn_modEq' {i j n} (ipos : 0 < i) (hin : i ≤ n) (j4n : j ≤ 4 * n)
(h : xn a1 j ≡ xn a1 i [MOD xn a1 n]) : j = i ∨ j + i = 4 * n :=
have i2n : i ≤ 2 * n | a : ℕ
a1 : 1 < a
i j n : ℕ
ipos : 0 < i
hin : i ≤ n
j4n : j ≤ 4 * n
h : xn a1 j ≡ xn a1 i [MOD xn a1 n]
⊢ n ≤ 2 * n | rw [two_mul] | a : ℕ
a1 : 1 < a
i j n : ℕ
ipos : 0 < i
hin : i ≤ n
j4n : j ≤ 4 * n
h : xn a1 j ≡ xn a1 i [MOD xn a1 n]
⊢ n ≤ n + n | 98971c8e087e01f4 |
QPF.liftp_iff | Mathlib/Data/QPF/Univariate/Basic.lean | theorem liftp_iff {α : Type u} (p : α → Prop) (x : F α) :
Liftp p x ↔ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ i, p (f i) | case mp.intro
F : Type u → Type u
q : QPF F
α : Type u
p : α → Prop
x : F α
y : F (Subtype p)
hy : Subtype.val <$> y = x
⊢ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ (i : (P F).B a), p (f i) | rcases h : repr y with ⟨a, f⟩ | case mp.intro.mk
F : Type u → Type u
q : QPF F
α : Type u
p : α → Prop
x : F α
y : F (Subtype p)
hy : Subtype.val <$> y = x
a : (P F).A
f : (P F).B a → Subtype p
h : repr y = ⟨a, f⟩
⊢ ∃ a f, x = abs ⟨a, f⟩ ∧ ∀ (i : (P F).B a), p (f i) | 6ca161c11c1a2e96 |
CategoryTheory.Triangulated.TStructure.shift_ge | Mathlib/CategoryTheory/Triangulated/TStructure/Basic.lean | lemma shift_ge (a n n' : ℤ) (hn' : a + n = n') :
(t.ge n).shift a = t.ge n' | case h.a.mpr
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
⊢ t.ge n' X → (t.ge n).shift a X | intro hX | case h.a.mpr
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
a n n' : ℤ
hn' : a + n = n'
X : C
hX : t.ge n' X
⊢ (t.ge n).shift a X | ae81308233c6fce1 |
Ideal.comap_map_eq_self_iff_of_isPrime | Mathlib/RingTheory/Ideal/Maps.lean | /-- For a prime ideal `p` of `R`, `p` extended to `S` and
restricted back to `R` is `p` if and only if `p` is the restriction of a prime in `S`. -/
lemma comap_map_eq_self_iff_of_isPrime {S : Type*} [CommSemiring S] {f : R →+* S}
(p : Ideal R) [p.IsPrime] :
(p.map f).comap f = p ↔ (∃ (q : Ideal S), q.IsPrime ∧ ... | case refine_2.intro.intro
R : Type u
inst✝² : CommSemiring R
S : Type u_2
inst✝¹ : CommSemiring S
f : R →+* S
q : Ideal S
hq : q.IsPrime
inst✝ : (comap f q).IsPrime
⊢ comap f (map f (comap f q)) = comap f q | simp | no goals | 841a6f7eb86eb1ab |
CategoryTheory.hasInitial_of_isCoseparating | Mathlib/CategoryTheory/Generator/Basic.lean | theorem hasInitial_of_isCoseparating [LocallySmall.{w} C] [WellPowered.{w} C]
[HasLimitsOfSize.{w, w} C] {𝒢 : Set C} [Small.{w} 𝒢]
(h𝒢 : IsCoseparating 𝒢) : HasInitial C | C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} C
inst✝² : WellPowered.{w, v₁, u₁} C
inst✝¹ : HasLimitsOfSize.{w, w, v₁, u₁} C
𝒢 : Set C
inst✝ : Small.{w, u₁} ↑𝒢
h𝒢 : IsCoseparating 𝒢
this✝³ : HasFiniteLimits C
this✝² : HasProductsOfShape (↑𝒢) C
this✝¹ : ∀ (A : C), HasProductsOfShape ((G... | exact hasInitial_of_unique ((⊥ : Subobject (piObj (Subtype.val : 𝒢 → C))) : C) | no goals | 5635e0b68399a531 |
ENNReal.rpow_sum_le_const_mul_sum_rpow | Mathlib/Analysis/MeanInequalities.lean | theorem rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i ∈ s, f i) ^ p ≤ (card s : ℝ≥0∞) ^ (p - 1) * ∑ i ∈ s, f i ^ p | case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p | have hpq : p.IsConjExponent q := .conjExponent hp | case inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
q : ℝ := p / (p - 1)
hpq : p.IsConjExponent q
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p | 3feff6e3e07bbcb1 |
CategoryTheory.Grothendieck.map_id_eq | Mathlib/CategoryTheory/Grothendieck.lean | theorem map_id_eq : map (𝟙 F) = 𝟙 (Cat.of <| Grothendieck <| F) | case h_map
C : Type u
inst✝ : Category.{v, u} C
F : C ⥤ Cat
X Y : Grothendieck F
f : X ⟶ Y
⊢ { base := f.base, fiber := f.fiber } = f | rfl | no goals | 5867a0755a02dad4 |
MeasureTheory.NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const | Mathlib/MeasureTheory/Measure/Restrict.lean | lemma NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
{β : Type*} [MeasurableSpace β] {b : β} {f : α → β} {s : Set α}
(f_mble : NullMeasurable f (μ.restrict s)) (hs : f =ᵐ[Measure.restrict μ sᶜ] (fun _ ↦ b))
{t : Set β} (t_mble : MeasurableSet t) (ht : b ∉ t) :
μ (f ⁻¹'... | α : Type u_2
m0 : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : MeasurableSpace β
b : β
f : α → β
s : Set α
f_mble : NullMeasurable f (μ.restrict s)
hs : (μ.restrict sᶜ) {a | ¬f a = b} = 0
t : Set β
t_mble : MeasurableSet t
ht : b ∉ t
⊢ NullMeasurableSet {a | ¬f a = b} (μ.restrict sᶜ) | exact NullMeasurableSet.of_null hs | no goals | 38ba84a38a03d7e4 |
CategoryTheory.InitiallySmall.exists_small_weakly_initial_set | Mathlib/CategoryTheory/Limits/FinallySmall.lean | theorem InitiallySmall.exists_small_weakly_initial_set [InitiallySmall.{w} J] :
∃ (s : Set J) (_ : Small.{w} s), ∀ i, ∃ j ∈ s, Nonempty (j ⟶ i) | J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : InitiallySmall J
i : J
⊢ ∃ j ∈ Set.range (fromInitialModel J).obj, Nonempty (j ⟶ i) | obtain ⟨f⟩ : Nonempty (CostructuredArrow (fromInitialModel J) i) := IsConnected.is_nonempty | case intro
J : Type u
inst✝¹ : Category.{v, u} J
inst✝ : InitiallySmall J
i : J
f : CostructuredArrow (fromInitialModel J) i
⊢ ∃ j ∈ Set.range (fromInitialModel J).obj, Nonempty (j ⟶ i) | a939e0d8e64f8cce |
List.length_tail_add_one | Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean | theorem length_tail_add_one (l : List α) (h : 0 < length l) : (length (tail l)) + 1 = length l | α : Type u_1
l : List α
h : 0 < l.length
⊢ l.tail.length + 1 = l.length | simp [Nat.sub_add_cancel h] | no goals | fd0fc35c1c65a92b |
Nat.logC_zero | Mathlib/Data/Nat/Log.lean | private lemma logC_zero {b : ℕ} :
logC b 0 = 0 | case inl
b : ℕ
hb : b ≤ 1
⊢ logC b 0 = 0
case inr
b : ℕ
hb : 1 < b
⊢ logC b 0 = 0 | case inl => exact logC_of_left_le_one hb | case inr
b : ℕ
hb : 1 < b
⊢ logC b 0 = 0 | 06c838807c040f81 |
LSeries_eq_zero_iff | Mathlib/NumberTheory/LSeries/Injectivity.lean | /-- Assuming `f 0 = 0`, the `LSeries` of `f` is zero if and only if either `f = 0` or the
L-series converges nowhere. -/
lemma LSeries_eq_zero_iff {f : ℕ → ℂ} (hf : f 0 = 0) :
LSeries f = 0 ↔ f = 0 ∨ abscissaOfAbsConv f = ⊤ | case a.h.zero
f : ℕ → ℂ
hf : f 0 = 0
h : ¬abscissaOfAbsConv f = ⊤
H : LSeries f = 0
H' : ∀ (n : ℕ), n ≠ 0 → f n = 0
⊢ f 0 = 0 0 | simp [hf] | no goals | 8588821b561f6ae9 |
LieModule.exists_forall_lie_eq_smul_of_isSolvable_of_finite | Mathlib/Algebra/Lie/LieTheorem.lean | private lemma exists_forall_lie_eq_smul_of_isSolvable_of_finite
(L : Type*) [LieRing L] [LieAlgebra k L] [LieRingModule L V] [LieModule k L V]
[IsSolvable L] [LieModule.IsTriangularizable k L V] [Module.Finite k L] :
∃ χ : Module.Dual k L, Nontrivial (weightSpace V χ) | case inr.intro.intro.intro
k : Type u_1
inst✝¹² : Field k
V : Type u_3
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module k V
inst✝⁹ : CharZero k
inst✝⁸ : Module.Finite k V
inst✝⁷ : Nontrivial V
L : Type u_4
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra k L
inst✝⁴ : LieRingModule L V
inst✝³ : LieModule k L V
inst✝² : IsSolvable L
inst... | obtain ⟨χ', _⟩ := exists_forall_lie_eq_smul_of_isSolvable_of_finite A | case inr.intro.intro.intro.intro
k : Type u_1
inst✝¹² : Field k
V : Type u_3
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module k V
inst✝⁹ : CharZero k
inst✝⁸ : Module.Finite k V
inst✝⁷ : Nontrivial V
L : Type u_4
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra k L
inst✝⁴ : LieRingModule L V
inst✝³ : LieModule k L V
inst✝² : IsSolvable ... | d7b003a641e390c7 |
HasFDerivAt.mul | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | theorem HasFDerivAt.mul (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x) :
HasFDerivAt (fun y => c y * d y) (c x • d' + d x • c') x | case h.e'_12.h.e'_6
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivAt c c' x
hd : HasFDerivAt d d' x
⊢ d x • c' = c'.smul... | ext z | case h.e'_12.h.e'_6.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
𝔸' : Type u_6
inst✝¹ : NormedCommRing 𝔸'
inst✝ : NormedAlgebra 𝕜 𝔸'
c d : E → 𝔸'
c' d' : E →L[𝕜] 𝔸'
hc : HasFDerivAt c c' x
hd : HasFDerivAt d d' x
z : E
⊢ (d x • c')... | cc5229819c36d1e9 |
WeierstrassCurve.Projective.Point.toAffine_of_equiv | Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean | lemma toAffine_of_equiv {P Q : Fin 3 → F} (h : P ≈ Q) : toAffine W P = toAffine W Q | case intro
F : Type u
inst✝ : Field F
W : Projective F
Q : Fin 3 → F
u : Fˣ
⊢ toAffine W ((fun m => m • Q) u) = toAffine W Q | exact toAffine_smul Q u.isUnit | no goals | 0e8c9fd95689da61 |
Ideal.span_pair_add_mul_left | Mathlib/RingTheory/Ideal/Span.lean | theorem span_pair_add_mul_left {R : Type u} [CommRing R] {x y : R} (z : R) :
(span {x + y * z, y} : Ideal R) = span {x, y} | R : Type u
inst✝ : CommRing R
x y z x✝¹ : R
x✝ : ∃ a b, a * (x + y * z) + b * y = x✝¹
a b : R
h : a * (x + y * z) + b * y = x✝¹
⊢ a * x + (b + a * z) * y = x✝¹ | rw [← h] | R : Type u
inst✝ : CommRing R
x y z x✝¹ : R
x✝ : ∃ a b, a * (x + y * z) + b * y = x✝¹
a b : R
h : a * (x + y * z) + b * y = x✝¹
⊢ a * x + (b + a * z) * y = a * (x + y * z) + b * y | f2a26d862bf33a33 |
Real.cos_eq_one_iff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | theorem cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * π) = x :=
⟨fun h =>
let ⟨n, hn⟩ := sin_eq_zero_iff.1 (sin_eq_zero_iff_cos_eq.2 (Or.inl h))
⟨n / 2,
(Int.emod_two_eq_zero_or_one n).elim
(fun hn0 => by
rwa [← mul_assoc, ← @Int.cast_two ℝ, ← Int.cast_mul,
I... | x : ℝ
h : cos x = 1
n : ℤ
hn : ↑n * π = x
hn1 : n % 2 = 1
⊢ ↑(n / 2) * (2 * π) = x | rw [← Int.emod_add_ediv n 2, hn1, Int.cast_add, Int.cast_one, add_mul, one_mul, add_comm,
mul_comm (2 : ℤ), Int.cast_mul, mul_assoc, Int.cast_two] at hn | x : ℝ
h : cos x = 1
n : ℤ
hn : ↑(n / 2) * (2 * π) + π = x
hn1 : n % 2 = 1
⊢ ↑(n / 2) * (2 * π) = x | 2ee05dd5f96f3a7c |
BoxIntegral.Prepartition.isPartition_single_iff | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | theorem isPartition_single_iff (h : J ≤ I) : IsPartition (single I J h) ↔ J = I | ι : Type u_1
I J : Box ι
h : J ≤ I
⊢ (single I J h).IsPartition ↔ J = I | simp [isPartition_iff_iUnion_eq] | no goals | d18f561a20463158 |
RelSeries.smash_succ_castAdd | Mathlib/Order/RelSeries.lean | lemma smash_succ_castAdd {p q : RelSeries r} (h : p.last = q.head)
(i : Fin p.length) : p.smash q h (i.castAdd q.length).succ = p i.succ | case neg
α : Type u_1
r : Rel α α
p q : RelSeries r
h : p.last = q.head
i : Fin p.length
H : ¬↑i + 1 < p.length
⊢ q.toFun ⟨↑(Fin.castAdd q.length i).succ - p.length, ⋯⟩ = p.toFun i.succ | convert h.symm | case h.e'_2
α : Type u_1
r : Rel α α
p q : RelSeries r
h : p.last = q.head
i : Fin p.length
H : ¬↑i + 1 < p.length
⊢ q.toFun ⟨↑(Fin.castAdd q.length i).succ - p.length, ⋯⟩ = q.head
case h.e'_3
α : Type u_1
r : Rel α α
p q : RelSeries r
h : p.last = q.head
i : Fin p.length
H : ¬↑i + 1 < p.length
⊢ p.toFun i.succ = p.la... | fd7866c3f8830725 |
NormedField.discreteTopology_of_bddAbove_range_norm | Mathlib/Analysis/Normed/Field/Lemmas.lean | lemma discreteTopology_of_bddAbove_range_norm {𝕜 : Type*} [NormedField 𝕜]
(h : BddAbove (Set.range fun k : 𝕜 ↦ ‖k‖)) :
DiscreteTopology 𝕜 | 𝕜 : Type u_4
inst✝ : NormedField 𝕜
h : BddAbove (Set.range fun k => ‖k‖)
⊢ ¬Nonempty { h' // NontriviallyNormedField.toNormedField = inst✝ } | rintro ⟨_, rfl⟩ | case intro.mk
𝕜 : Type u_4
val✝ : NontriviallyNormedField 𝕜
h : BddAbove (Set.range fun k => ‖k‖)
⊢ False | 79eb47b5eea4349e |
Filter.tendsto_of_subseq_tendsto | Mathlib/Order/Filter/AtTopBot/CountablyGenerated.lean | theorem tendsto_of_subseq_tendsto {ι : Type*} {x : ι → α} {f : Filter α} {l : Filter ι}
[l.IsCountablyGenerated]
(hxy : ∀ ns : ℕ → ι, Tendsto ns atTop l →
∃ ms : ℕ → ℕ, Tendsto (fun n => x (ns <| ms n)) atTop f) :
Tendsto x l f | α : Type u_1
ι : Type u_3
x : ι → α
f : Filter α
l : Filter ι
inst✝ : l.IsCountablyGenerated
hxy : ¬Tendsto x l f
⊢ ∃ s ∈ f, ∃ᶠ (n : ι) in l, x n ∉ s | rwa [not_tendsto_iff_exists_frequently_nmem] at hxy | no goals | c6c67ce922c3a724 |
Finset.card_mul_cast_addConst | Mathlib/Combinatorics/Additive/DoublingConst.lean | lemma card_mul_cast_addConst (A B : Finset G') : (#A * σ[A, B] : 𝕜) = #(A + B) | G' : Type u_2
inst✝³ : AddGroup G'
inst✝² : DecidableEq G'
𝕜 : Type u_3
inst✝¹ : Semifield 𝕜
inst✝ : CharZero 𝕜
A B : Finset G'
⊢ ↑(#A) * ↑σ[A, B] = ↑(#(A + B)) | norm_cast | G' : Type u_2
inst✝³ : AddGroup G'
inst✝² : DecidableEq G'
𝕜 : Type u_3
inst✝¹ : Semifield 𝕜
inst✝ : CharZero 𝕜
A B : Finset G'
⊢ ↑(#A) * σ[A, B] = ↑(#(A + B)) | 7f9b1c678f114375 |
DenomsClearable.add | Mathlib/Algebra/Polynomial/DenomsClearable.lean | theorem DenomsClearable.add {N : ℕ} {f g : R[X]} :
DenomsClearable a b N f i → DenomsClearable a b N g i → DenomsClearable a b N (f + g) i :=
fun ⟨Df, bf, bfu, Hf⟩ ⟨Dg, bg, bgu, Hg⟩ =>
⟨Df + Dg, bf, bfu, by
rw [RingHom.map_add, Polynomial.map_add, eval_add, mul_add, Hf, Hg]
congr
refine @inv_unique ... | case e_a.e_a.e_a.e_a
R : Type u_1
K : Type u_2
inst✝¹ : Semiring R
inst✝ : CommSemiring K
i : R →+* K
a b : R
N : ℕ
f g : R[X]
x✝¹ : DenomsClearable a b N f i
x✝ : DenomsClearable a b N g i
Df : R
bf : K
bfu : bf * i b = 1
Hf : i Df = i b ^ N * eval (i a * bf) (Polynomial.map i f)
Dg : R
bg : K
bgu : bg * i b = 1
Hg : ... | refine @inv_unique K _ (i b) bg bf ?_ ?_ <;> rwa [mul_comm] | no goals | 3fdc87808c3ef765 |
Batteries.UnionFind.rankD_lt_rankMax | Mathlib/.lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | theorem rankD_lt_rankMax (self : UnionFind) (i : Nat) :
rankD self.arr i < self.rankMax | self : UnionFind
i : Nat
⊢ rankD self.arr i < self.rankMax | simp [rankD] | self : UnionFind
i : Nat
⊢ (if h : i < self.arr.size then self.arr[i].rank else 0) < self.rankMax | f96352b23db23583 |
Finsupp.weight_sub_single_add | Mathlib/Data/Finsupp/Weight.lean | lemma weight_sub_single_add {f : σ →₀ ℕ} {i : σ} (hi : f i ≠ 0) :
(f - single i 1).weight w + w i = f.weight w | σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ (weight w) (f - single i 1) + w i = (f - single i 1 + single i 1).sum fun i c => c • w i | rw [sum_add_index', sum_single_index, one_smul, weight_apply] | σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ 0 • w i = 0
case h_zero
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ
hi : f i ≠ 0
⊢ ∀ (a : σ), 0 • w a = 0
case h_add
σ : Type u_1
M : Type u_2
w : σ → M
inst✝ : AddCommMonoid M
f : σ →₀ ℕ
i : σ... | 830d7279c7065ac5 |
MvPolynomial.induction_on_monomial | Mathlib/Algebra/MvPolynomial/Basic.lean | theorem induction_on_monomial {M : MvPolynomial σ R → Prop} (h_C : ∀ a, M (C a))
(h_X : ∀ p n, M p → M (p * X n)) : ∀ s a, M (monomial s a) | R : Type u
σ : Type u_1
inst✝ : CommSemiring R
M : MvPolynomial σ R → Prop
h_C : ∀ (a : R), M (C a)
h_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)
s : σ →₀ ℕ
a : R
⊢ M ((monomial s) a) | apply @Finsupp.induction σ ℕ _ _ s | case h0
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
M : MvPolynomial σ R → Prop
h_C : ∀ (a : R), M (C a)
h_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * X n)
s : σ →₀ ℕ
a : R
⊢ M ((monomial 0) a)
case ha
R : Type u
σ : Type u_1
inst✝ : CommSemiring R
M : MvPolynomial σ R → Prop
h_C : ∀ (a : R), M (C a)
h_X : ∀... | f250a78fcc947e26 |
LinearMap.map_coprod_prod | Mathlib/LinearAlgebra/Prod.lean | theorem map_coprod_prod (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) (p : Submodule R M)
(q : Submodule R M₂) : map (coprod f g) (p.prod q) = map f p ⊔ map g q | case refine_3
R : Type u
M : Type v
M₂ : Type w
M₃ : Type y
inst✝⁶ : Semiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid M₂
inst✝³ : AddCommMonoid M₃
inst✝² : Module R M
inst✝¹ : Module R M₂
inst✝ : Module R M₃
f : M →ₗ[R] M₃
g : M₂ →ₗ[R] M₃
p : Submodule R M
q : Submodule R M₂
⊢ q ≤ comap g (map (f.coprod g) (p... | exact fun x hx => ⟨(0, x), by simp [hx]⟩ | no goals | cc3d0c0d4b68950f |
MeasureTheory.Measure.mutuallySingular_singularPart | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | theorem mutuallySingular_singularPart (μ ν : Measure α) : μ.singularPart ν ⟂ₘ ν | α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
⊢ μ.singularPart ν ⟂ₘ ν | by_cases h : HaveLebesgueDecomposition μ ν | case pos
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν
case neg
α : Type u_1
m : MeasurableSpace α
μ ν : Measure α
h : ¬μ.HaveLebesgueDecomposition ν
⊢ μ.singularPart ν ⟂ₘ ν | 630b067128bbb175 |
Nat.chineseRemainderOfList_modEq_unique | Mathlib/Data/Nat/ChineseRemainder.lean | theorem chineseRemainderOfList_modEq_unique (l : List ι)
(co : l.Pairwise (Coprime on s)) {z} (hz : ∀ i ∈ l, z ≡ a i [MOD s i]) :
z ≡ chineseRemainderOfList a s l co [MOD (l.map s).prod] | case nil
ι : Type u_1
a s : ι → ℕ
z : ℕ
co : List.Pairwise (Coprime on s) []
hz : ∀ i ∈ [], z ≡ a i [MOD s i]
⊢ z ≡ ↑(chineseRemainderOfList a s [] co) [MOD (List.map s []).prod] | simp [modEq_one] | no goals | 9a8d4e3838c69fb4 |
Nat.Linear.Poly.denote_eq_cancelAux | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Linear.lean | theorem Poly.denote_eq_cancelAux (ctx : Context) (fuel : Nat) (m₁ m₂ r₁ r₂ : Poly)
(h : denote_eq ctx (r₁.reverse ++ m₁, r₂.reverse ++ m₂)) : denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂) | case succ.h_3
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
r₁ r₂ m₁✝¹ m₂✝¹ : Poly
k₁✝ : Nat
v₁✝ : Var
m₁✝ : List (Nat × Var)
k₂✝ : Nat
v₂✝ : Var
m₂✝ : List (Nat × Var)
h : denote_eq ctx (List.reverse... | rename_i k₁ v₁ m₁ k₂ v₂ m₂ | case succ.h_3
ctx : Context
fuel : Nat
ih :
∀ (m₁ m₂ r₁ r₂ : Poly),
denote_eq ctx (List.reverse r₁ ++ m₁, List.reverse r₂ ++ m₂) → denote_eq ctx (cancelAux fuel m₁ m₂ r₁ r₂)
r₁ r₂ m₁✝ m₂✝ : Poly
k₁ : Nat
v₁ : Var
m₁ : List (Nat × Var)
k₂ : Nat
v₂ : Var
m₂ : List (Nat × Var)
h : denote_eq ctx (List.reverse r₁ ++ (... | 9bfcd466b3bdd638 |
SzemerediRegularity.card_aux₂ | Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean | theorem card_aux₂ (hP : P.IsEquipartition) (hu : u ∈ P.parts) (hucard : #u ≠ m * 4 ^ #P.parts + a) :
(4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u | α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
hP : P.IsEquipartition
hu : u ∈ P.parts
hucard : #u ≠ Fintype.card α / #P.parts
this : m * 4 ^ #P.parts ≤ Fintype.card α / #P.parts
⊢ (4 ^ #P.parts - (a + 1)) * m + (a + 1) * (m + 1) = #u | rw [(hP.card_parts_eq_average hu).resolve_left hucard, mul_add, mul_one, ← add_assoc, ← add_mul,
Nat.sub_add_cancel a_add_one_le_four_pow_parts_card, ← add_assoc, mul_comm,
Nat.add_sub_of_le this, card_univ] | no goals | c434f1e0b19cc540 |
MeasureTheory.Measure.hausdorffMeasure_le_one_of_subsingleton | Mathlib/MeasureTheory/Measure/Hausdorff.lean | theorem hausdorffMeasure_le_one_of_subsingleton {s : Set X} (hs : s.Subsingleton) {d : ℝ}
(hd : 0 ≤ d) : μH[d] s ≤ 1 | case inr.intro.inl
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
s : Set X
hs : s.Subsingleton
x : X
hx : x ∈ s
hd : 0 ≤ 0
⊢ μH[0] {x} ≤ 1 | simp only [le_refl, hausdorffMeasure_zero_singleton] | no goals | ecec90d28f4c1dc1 |
IsLocalization.comap_map_of_isPrime_disjoint | Mathlib/RingTheory/Localization/Ideal.lean | theorem comap_map_of_isPrime_disjoint (I : Ideal R) (hI : I.IsPrime) (hM : Disjoint (M : Set R) I) :
Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) = I | R : Type u_1
inst✝³ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝² : CommSemiring S
inst✝¹ : Algebra R S
inst✝ : IsLocalization M S
I : Ideal R
hI : I.IsPrime
hM : Disjoint ↑M ↑I
a : R
ha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)
b : ↥I
s : ↥M
h : (algebraMap R S) a * (algebraMap R S) ↑(b,... | simpa only [← map_mul, mul_comm] using h | no goals | 62d3fb0e13b1d4c6 |
CochainComplex.HomComplex.Cocycle.coe_zero | Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | @[simp]
lemma coe_zero : (↑(0 : Cocycle F G n) : Cochain F G n) = 0 | C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n : ℤ
⊢ ↑0 = 0 | rfl | no goals | 28eefd843031088c |
LinearMap.BilinForm.isCompl_orthogonal_iff_disjoint | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | lemma isCompl_orthogonal_iff_disjoint (hB₀ : B.IsRefl) :
IsCompl W (B.orthogonal W) ↔ Disjoint W (B.orthogonal W) | V : Type u_5
K : Type u_6
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
B : BilinForm K V
W : Submodule K V
hB₀ : B.IsRefl
h : Disjoint W (B.orthogonal W)
⊢ Codisjoint W (B.orthogonal W) | rw [codisjoint_iff] | V : Type u_5
K : Type u_6
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
B : BilinForm K V
W : Submodule K V
hB₀ : B.IsRefl
h : Disjoint W (B.orthogonal W)
⊢ W ⊔ B.orthogonal W = ⊤ | 226f3a2eeabd2066 |
Equiv.Perm.sign_prodCongrLeft | Mathlib/GroupTheory/Perm/Sign.lean | theorem sign_prodCongrLeft (σ : α → Perm β) : sign (prodCongrLeft σ) = ∏ k, sign (σ k) | α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
σ : α → Perm β
⊢ sign (prodCongrLeft σ) = ∏ k : α, sign (σ k) | refine (sign_eq_sign_of_equiv _ _ (prodComm β α) ?_).trans (sign_prodCongrRight σ) | α : Type u
inst✝³ : DecidableEq α
β : Type v
inst✝² : Fintype α
inst✝¹ : DecidableEq β
inst✝ : Fintype β
σ : α → Perm β
⊢ ∀ (x : β × α), (prodComm β α) ((prodCongrLeft σ) x) = (prodCongrRight σ) ((prodComm β α) x) | f81489a19834a7a5 |
IsCompact.union | Mathlib/Topology/Compactness/Compact.lean | theorem IsCompact.union (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s ∪ t) | X : Type u
inst✝ : TopologicalSpace X
s t : Set X
hs : IsCompact s
ht : IsCompact t
b : Bool
⊢ IsCompact (bif b then s else t) | cases b <;> assumption | no goals | e71b84057fcf3b7c |
EuclideanGeometry.reflection_symm | Mathlib/Geometry/Euclidean/Basic.lean | theorem reflection_symm (s : AffineSubspace ℝ P) [Nonempty s]
[HasOrthogonalProjection s.direction] : (reflection s).symm = reflection s | case h
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥s
inst✝ : HasOrthogonalProjection s.direction
x✝ : P
⊢ (reflection s).symm x✝ = (reflection s) x✝ | rw [← (reflection s).injective.eq_iff] | case h
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥s
inst✝ : HasOrthogonalProjection s.direction
x✝ : P
⊢ (reflection s) ((reflection s).symm x✝) = (reflection s) ((reflection s) x✝) | cde77f2d1ec536bc |
String.Pos.zero_addString_byteIdx | Mathlib/.lake/packages/lean4/src/lean/Init/Data/String/Basic.lean | theorem zero_addString_byteIdx (s : String) : ((0 : Pos) + s).byteIdx = s.utf8ByteSize | s : String
⊢ (0 + s).byteIdx = s.utf8ByteSize | simp only [addString_byteIdx, byteIdx_zero, Nat.zero_add] | no goals | d2d85cde96a3afa1 |
Ordinal.blsub_le_iff | Mathlib/SetTheory/Ordinal/Arithmetic.lean | theorem blsub_le_iff {o : Ordinal.{u}} {f : ∀ a < o, Ordinal.{max u v}} {a} :
blsub.{_, v} o f ≤ a ↔ ∀ i h, f i h < a | o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
a : Ordinal.{max u v}
⊢ o.blsub f ≤ a ↔ ∀ (i : Ordinal.{u}) (h : i < o), f i h < a | convert bsup_le_iff.{_, v} (f := fun a ha => succ (f a ha)) (a := a) using 2 | case h.e'_2.h.a
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
a : Ordinal.{max u v}
a✝ : Ordinal.{u}
⊢ (∀ (h : a✝ < o), f a✝ h < a) ↔ ∀ (h : a✝ < o), succ (f a✝ h) ≤ a | 70b2c6d9d2a2607d |
PowerSeries.trunc_trunc_mul_trunc | Mathlib/RingTheory/PowerSeries/Trunc.lean | theorem trunc_trunc_mul_trunc {n} (f g : R⟦X⟧) :
trunc n (trunc n f * trunc n g : R⟦X⟧) = trunc n (f * g) | R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f g : R⟦X⟧
⊢ trunc n (↑(trunc n f) * ↑(trunc n g)) = trunc n (f * g) | rw [trunc_trunc_mul, trunc_mul_trunc] | no goals | 790281afe74d45d7 |
ADEInequality.Admissible.one_lt_sumInv | Mathlib/NumberTheory/ADEInequality.lean | theorem Admissible.one_lt_sumInv {pqr : Multiset ℕ+} : Admissible pqr → 1 < sumInv pqr | case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < (↑↑2)⁻¹ + (↑↑3)⁻¹ + (↑↑5)⁻¹ | norm_num | no goals | 441519598e95d509 |
SimpleGraph.chromaticNumber_pos | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | theorem chromaticNumber_pos [Nonempty V] {n : ℕ} (hc : G.Colorable n) : 0 < G.chromaticNumber | V : Type u
G : SimpleGraph V
inst✝ : Nonempty V
n : ℕ
hc : G.Colorable n
⊢ 0 < G.chromaticNumber | rw [hc.chromaticNumber_eq_sInf, Nat.cast_pos] | V : Type u
G : SimpleGraph V
inst✝ : Nonempty V
n : ℕ
hc : G.Colorable n
⊢ 0 < sInf {n' | G.Colorable n'} | c828f021de3822a4 |
Substring.Valid.prev | Mathlib/.lake/packages/batteries/Batteries/Data/String/Lemmas.lean | theorem prev : ∀ {s}, Valid s → s.toString.1 = m₁ ++ c :: m₂ →
s.prev ⟨utf8Len m₁ + c.utf8Size⟩ = ⟨utf8Len m₁⟩
| _, h, e => by
let ⟨l, m, r, h⟩ := h.validFor
simp only [h.toString] at e; subst e; simp [h.prev]
| m₁ : List Char
c : Char
m₂ : List Char
x✝ : Substring
h : x✝.Valid
e : x✝.toString.data = m₁ ++ c :: m₂
⊢ x✝.prev { byteIdx := utf8Len m₁ + c.utf8Size } = { byteIdx := utf8Len m₁ } | let ⟨l, m, r, h⟩ := h.validFor | m₁ : List Char
c : Char
m₂ : List Char
x✝ : Substring
h✝ : x✝.Valid
e : x✝.toString.data = m₁ ++ c :: m₂
l m r : List Char
h : ValidFor l m r x✝
⊢ x✝.prev { byteIdx := utf8Len m₁ + c.utf8Size } = { byteIdx := utf8Len m₁ } | 4517c7cc3f576567 |
cauchySeq_finset_iff_prod_vanishing | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | theorem cauchySeq_finset_iff_prod_vanishing :
(CauchySeq fun s : Finset β ↦ ∏ b ∈ s, f b) ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e | case mp
α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformGroup α
f : β → α
h : ∀ s ∈ 𝓝 1, ∃ a, ∀ b ≥ a, (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s
e : Set α
he : e ∈ 𝓝 1
⊢ ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e | obtain ⟨⟨s₁, s₂⟩, h⟩ := h e he | case mp.intro.mk
α : Type u_1
β : Type u_2
inst✝² : CommGroup α
inst✝¹ : UniformSpace α
inst✝ : UniformGroup α
f : β → α
h✝ : ∀ s ∈ 𝓝 1, ∃ a, ∀ b ≥ a, (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s
e : Set α
he : e ∈ 𝓝 1
s₁ s₂ : Finset β
h : ∀ b ≥ (s₁, s₂), (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ e
⊢ ∃ s, ∀ (t : Finset β), Disjoi... | b5f6a526c2a1495a |
Polynomial.Monic.mul_natDegree_lt_iff | Mathlib/Algebra/Polynomial/Monic.lean | theorem Monic.mul_natDegree_lt_iff (h : Monic p) {q : R[X]} :
(p * q).natDegree < p.natDegree ↔ p ≠ 1 ∧ q = 0 | case pos
R : Type u
inst✝ : Semiring R
p : R[X]
h : p.Monic
q : R[X]
hq : q = 0
⊢ (p * q).natDegree < p.natDegree ↔ p ≠ 1 ∧ q = 0 | suffices 0 < p.natDegree ↔ p.natDegree ≠ 0 by simpa [hq, ← h.natDegree_eq_zero_iff_eq_one] | case pos
R : Type u
inst✝ : Semiring R
p : R[X]
h : p.Monic
q : R[X]
hq : q = 0
⊢ 0 < p.natDegree ↔ p.natDegree ≠ 0 | 2aa58058563eb3f3 |
CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le | Mathlib/Order/CompactlyGenerated/Basic.lean | theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
IsCompactElement k ↔
∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x | case mp.intro
α : Type u_2
inst✝ : CompleteLattice α
k : α
hk : IsCompactElement k
s : Set α
hne : s.Nonempty
hdir : DirectedOn (fun x1 x2 => x1 ≤ x2) s
hsup : k ≤ sSup s
t : Finset α
ht : ↑t ⊆ s ∧ k ≤ t.sup id
⊢ ∃ x ∈ s, k ≤ x | have t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y := fun x hxt => ⟨x, ht.left hxt, le_rfl⟩ | case mp.intro
α : Type u_2
inst✝ : CompleteLattice α
k : α
hk : IsCompactElement k
s : Set α
hne : s.Nonempty
hdir : DirectedOn (fun x1 x2 => x1 ≤ x2) s
hsup : k ≤ sSup s
t : Finset α
ht : ↑t ⊆ s ∧ k ≤ t.sup id
t_below_s : ∀ x ∈ t, ∃ y ∈ s, x ≤ y
⊢ ∃ x ∈ s, k ≤ x | 445bf3d7fb93c3e6 |
MonoidHom.isOpenQuotientMap_of_isQuotientMap | Mathlib/Topology/Algebra/Group/Basic.lean | /-- Let `A` and `B` be topological groups, and let `φ : A → B` be a continuous surjective group
homomorphism. Assume furthermore that `φ` is a quotient map (i.e., `V ⊆ B`
is open iff `φ⁻¹ V` is open). Then `φ` is an open quotient map, and in particular an open map. -/
@[to_additive "Let `A` and `B` be topological addit... | A : Type u_1
inst✝⁶ : Group A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : IsTopologicalGroup A
B : Type u_2
inst✝³ : Group B
inst✝² : TopologicalSpace B
F : Type u_3
inst✝¹ : FunLike F A B
inst✝ : MonoidHomClass F A B
φ : F
hφ : IsQuotientMap ⇑φ
⊢ IsOpenMap ⇑φ | intro U hU | A : Type u_1
inst✝⁶ : Group A
inst✝⁵ : TopologicalSpace A
inst✝⁴ : IsTopologicalGroup A
B : Type u_2
inst✝³ : Group B
inst✝² : TopologicalSpace B
F : Type u_3
inst✝¹ : FunLike F A B
inst✝ : MonoidHomClass F A B
φ : F
hφ : IsQuotientMap ⇑φ
U : Set A
hU : IsOpen U
⊢ IsOpen (⇑φ '' U) | 46fa017a40a72a1b |
MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | theorem MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient [LocallyCompactSpace G]
[QuotientMeasureEqMeasurePreimage ν μ] [i : HasFundamentalDomain Γ.op G ν]
[IsFiniteMeasure μ] : IsHaarMeasure μ | case intro
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν :... | obtain ⟨s, fund_dom_s⟩ := i | case intro.mk.intro
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countab... | d4b9c01a9d05b936 |
AkraBazziRecurrence.smoothingFn_mul_asympBound_isBigO_T | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | /-- The main proof of the lower bound part of the Akra-Bazzi theorem. The factor
`1 + ε n` does not change the asymptotic order, but is needed for the induction step to go
through. -/
lemma smoothingFn_mul_asympBound_isBigO_T :
(fun (n : ℕ) => (1 + ε n) * asympBound g a b n) =O[atTop] T | α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_ge_Rn₀ : R.n₀ ≤ n₀... | obtain ⟨m, hm_mem, hm⟩ :=
Finset.exists_mem_eq_inf' h_base_nonempty (fun n => T n / ((1 + ε n) * asympBound g a b n)) | case intro.intro
α : Type u_1
inst✝¹ : Fintype α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
inst✝ : Nonempty α
R : AkraBazziRecurrence T g a b r
b' : ℝ := b (min_bi b) / 2
hb_pos : 0 < b'
c₁ : ℝ
hc₁ : c₁ > 0
h_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g ↑n
n₀ : ℕ
n₀_g... | 82bb51116c3f899e |
Std.Tactic.BVDecide.LRAT.Internal.addRupCaseSound | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/LRATCheckerSound.lean | theorem addRupCaseSound [DecidableEq α] [Clause α β] [Entails α σ] [Formula α β σ] (f : σ)
(f_readyForRupAdd : ReadyForRupAdd f)
(f_readyForRatAdd : ReadyForRatAdd f) (c : β) (f' : σ) (rupHints : Array Nat)
(heq : performRupAdd f c rupHints = (f', true))
(restPrf : List (Action β α)) (restPrfWellFormed ... | α : Type u_1
β : Type u_2
σ : Type u_3
inst✝³ : DecidableEq α
inst✝² : Clause α β
inst✝¹ : Entails α σ
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : ReadyForRupAdd f
f_readyForRatAdd : ReadyForRatAdd f
c : β
f' : σ
rupHints : Array Nat
heq : performRupAdd f c rupHints = (f', true)
restPrf : List (Action β α)
restPrfWe... | have f'_def := rupAdd_result f c rupHints f' f_readyForRupAdd heq | α : Type u_1
β : Type u_2
σ : Type u_3
inst✝³ : DecidableEq α
inst✝² : Clause α β
inst✝¹ : Entails α σ
inst✝ : Formula α β σ
f : σ
f_readyForRupAdd : ReadyForRupAdd f
f_readyForRatAdd : ReadyForRatAdd f
c : β
f' : σ
rupHints : Array Nat
heq : performRupAdd f c rupHints = (f', true)
restPrf : List (Action β α)
restPrfWe... | f4a1ecc7f037720c |
isLindelof_of_countable_subcover | Mathlib/Topology/Compactness/Lindelof.lean | theorem isLindelof_of_countable_subcover
(h : ∀ {ι : Type u} (U : ι → Set X), (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) →
∃ t : Set ι, t.Countable ∧ s ⊆ ⋃ i ∈ t, U i) :
IsLindelof s := fun f hf hfs ↦ by
contrapose! h
simp only [ClusterPt, not_neBot, ← disjoint_iff, SetCoe.forall',
(nhds_basis_opens _).di... | X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : CountableInterFilter f
fsub : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
t : Set ↑s
ht : t.Countable
h : s ⊆ ⋃ i ∈ t, U i
uinf : ⋃ i ∈ t, U i ∈ f.sets
⊢ False | have uninf : ⋂ i ∈ t, (U i)ᶜ ∈ f := (countable_bInter_mem ht).mpr (fun _ _ ↦ hUf _) | X : Type u
inst✝ : TopologicalSpace X
s : Set X
f : Filter X
hf : f.NeBot
hfs : CountableInterFilter f
fsub : f ≤ 𝓟 s
U : ↑s → Set X
hU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen (U x)
hUf : ∀ (x : ↑s), (U x)ᶜ ∈ f
t : Set ↑s
ht : t.Countable
h : s ⊆ ⋃ i ∈ t, U i
uinf : ⋃ i ∈ t, U i ∈ f.sets
uninf : ⋂ i ∈ t, (U i)ᶜ ∈ f
⊢ False | 55d21432f93098bb |
Multiset.Nodup.le_nsmul_iff_le | Mathlib/Algebra/Order/Group/Multiset.lean | lemma Nodup.le_nsmul_iff_le {s t : Multiset α} {n : ℕ} (h : s.Nodup) (hn : n ≠ 0) :
s ≤ n • t ↔ s ≤ t | α : Type u_1
s t : Multiset α
n : ℕ
h : s.Nodup
hn : n ≠ 0
⊢ s ≤ n • t ↔ s ≤ t | classical simp [← h.le_dedup_iff_le, Iff.comm, ← h.le_dedup_iff_le, hn] | no goals | b2f659a6a35e4502 |
FirstOrder.Language.BoundedFormula.castLE_castLE | Mathlib/ModelTheory/Syntax.lean | theorem castLE_castLE {k m n} (km : k ≤ m) (mn : m ≤ n) (φ : L.BoundedFormula α k) :
(φ.castLE km).castLE mn = φ.castLE (km.trans mn) | L : Language
α : Type u'
k : ℕ
φ : L.BoundedFormula α k
⊢ ∀ {m n : ℕ} (km : k ≤ m) (mn : m ≤ n), castLE mn (castLE km φ) = castLE ⋯ φ | induction φ with
| falsum => intros; rfl
| equal => simp
| rel =>
intros
simp only [castLE, eq_self_iff_true, heq_iff_eq]
rw [← Function.comp_assoc, Term.relabel_comp_relabel]
simp
| imp _ _ ih1 ih2 => simp [ih1, ih2]
| all _ ih3 => intros; simp only [castLE, ih3] | no goals | ed6234c9d544357e |
tendsto_div_of_monotone_of_exists_subseq_tendsto_div | Mathlib/Analysis/SpecificLimits/FloorPow.lean | theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
(hmono : Monotone u)
(hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) | u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) *... | gcongr | no goals | 2aab76659c6dcba8 |
List.Perm.drop_inter | Mathlib/Data/List/Perm/Lattice.lean | theorem Perm.drop_inter {xs ys : List α} (n : ℕ) (h : xs ~ ys) (h' : ys.Nodup) :
xs.drop n ~ ys.inter (xs.drop n) | case pos
α : Type u_1
inst✝ : DecidableEq α
xs ys : List α
n : ℕ
h : xs ~ ys
h' : ys.Nodup
h'' : n ≤ xs.length
n' : ℕ := xs.length - n
h₀ : n = xs.length - n'
h₁ : drop n xs = (take n' xs.reverse).reverse
⊢ drop n xs ~ ys.inter (drop n xs) | rw [h₁] | case pos
α : Type u_1
inst✝ : DecidableEq α
xs ys : List α
n : ℕ
h : xs ~ ys
h' : ys.Nodup
h'' : n ≤ xs.length
n' : ℕ := xs.length - n
h₀ : n = xs.length - n'
h₁ : drop n xs = (take n' xs.reverse).reverse
⊢ (take n' xs.reverse).reverse ~ ys.inter (take n' xs.reverse).reverse | d21abc3a6eb3c832 |
Transcendental.of_aeval | Mathlib/RingTheory/Algebraic/Basic.lean | theorem Transcendental.of_aeval {r : A} {f : R[X]}
(H : Transcendental R (Polynomial.aeval r f)) : Transcendental R f | R : Type u
A : Type v
inst✝² : CommRing R
inst✝¹ : Ring A
inst✝ : Algebra R A
r : A
f : R[X]
H : ∀ (p : R[X]), (Polynomial.aeval ((Polynomial.aeval r) f)) p = 0 → p = 0
p : R[X]
hp : (Polynomial.aeval f) p = 0
⊢ (Polynomial.aeval ((Polynomial.aeval r) f)) p = 0 | rw [← aeval_comp, comp_eq_aeval, hp, map_zero] | no goals | f95d2ba8e4b6f764 |
mellin_inversion | Mathlib/Analysis/MellinInversion.lean | theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ)
(hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) :
mellinInv σ (mellin f) x = f x | E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
σ : ℝ
f : ℝ → E
x : ℝ
hx : 0 < x
g : ℝ → E := fun u => rexp (-σ * u) • f (rexp (-u))
hf : Integrable g volume
hFf : Integrable (𝓕 g) volume
hfx : ContinuousAt g (-Real.log x)
⊢ ↑x ^ (-↑σ) • 𝓕⁻ (fun y => 𝓕 g ((↑σ + 2 * ↑π * ↑y... | simp [mul_div_cancel_left₀ _ (show 2 * π ≠ 0 by norm_num; exact pi_ne_zero)] | no goals | eb76f337444b2548 |
Cardinal.mk_image2_le | Mathlib/SetTheory/Cardinal/Basic.lean | lemma mk_image2_le {α β γ : Type u} {f : α → β → γ} {s : Set α} {t : Set β} :
#(image2 f s t) ≤ #s * #t | α β γ : Type u
f : α → β → γ
s : Set α
t : Set β
⊢ #↑(uncurry f '' s ×ˢ t) ≤ #↑(s ×ˢ t) | exact mk_image_le | no goals | 087c072c686cf709 |
ContinuousLinearMap.add_compLp | Mathlib/MeasureTheory/Function/LpSpace/Basic.lean | theorem add_compLp (L L' : E →L[𝕜] F) (f : Lp E p μ) :
(L + L').compLp f = L.compLp f + L'.compLp f | case h
α : Type u_1
E : Type u_4
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
𝕜 : Type u_7
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
L L' : E →L[𝕜] F
f : ↥(Lp E p μ)
⊢ ↑↑((L + L').compLp f) =ᶠ[ae μ] ... | refine (coeFn_compLp' (L + L') f).trans ?_ | case h
α : Type u_1
E : Type u_4
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
𝕜 : Type u_7
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
L L' : E →L[𝕜] F
f : ↥(Lp E p μ)
⊢ (fun a => (L + L') (↑↑f a)) =ᶠ[... | 85de0124603f69e0 |
WittVector.wittOne_zero_eq_one | Mathlib/RingTheory/WittVector/Defs.lean | theorem wittOne_zero_eq_one : wittOne p 0 = 1 | p : ℕ
hp : Fact (Nat.Prime p)
⊢ wittOne p 0 = 1 | apply MvPolynomial.map_injective (Int.castRingHom ℚ) Int.cast_injective | case a
p : ℕ
hp : Fact (Nat.Prime p)
⊢ (map (Int.castRingHom ℚ)) (wittOne p 0) = (map (Int.castRingHom ℚ)) 1 | ec1ec3508cea391b |
NumberField.house.asiegel_ne_0 | Mathlib/NumberTheory/NumberField/House.lean | theorem asiegel_ne_0 : asiegel K a ≠ 0 | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
⊢ ¬(fun k l => ((NumberField.house.newBasis K).repr (a k.1 l.1 * (NumberField.house.newBasis K) l.2)) k.2) = 0 | rw [funext_iff] | K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
α : Type u_2
β : Type u_3
a : Matrix α β (𝓞 K)
ha : a ≠ 0
⊢ ¬∀ (x : α × (K →+* ℂ)),
(fun l => ((NumberField.house.newBasis K).repr (a x.1 l.1 * (NumberField.house.newBasis K) l.2)) x.2) = 0 x | 0ad6e35554ac03cb |
MeasureTheory.addContent_eq_add_disjointOfDiffUnion_of_subset | Mathlib/MeasureTheory/Measure/AddContent.lean | lemma addContent_eq_add_disjointOfDiffUnion_of_subset (hC : IsSetSemiring C)
(hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s)
(h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
m s = ∑ i ∈ I, m i + ∑ i ∈ hC.disjointOfDiffUnion hs hI, m i | case h_ss
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
m : AddContent C
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
h_dis : (↑I).PairwiseDisjoint id
⊢ ↑(I ∪ hC.disjointOfDiffUnion hs hI) ⊆ C | rw [coe_union] | case h_ss
α : Type u_1
C : Set (Set α)
s : Set α
I : Finset (Set α)
m : AddContent C
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
h_dis : (↑I).PairwiseDisjoint id
⊢ ↑I ∪ ↑(hC.disjointOfDiffUnion hs hI) ⊆ C | eb6d1b6afb03f034 |
PFunctor.M.ichildren_mk | Mathlib/Data/PFunctor/Univariate/M.lean | theorem ichildren_mk [DecidableEq F.A] [Inhabited (M F)] (x : F (M F)) (i : F.Idx) :
ichildren i (M.mk x) = x.iget i | F : PFunctor.{u}
inst✝¹ : DecidableEq F.A
inst✝ : Inhabited F.M
x : ↑F F.M
i : F.Idx
⊢ ichildren i (M.mk x) = x.iget i | dsimp only [ichildren, PFunctor.Obj.iget] | F : PFunctor.{u}
inst✝¹ : DecidableEq F.A
inst✝ : Inhabited F.M
x : ↑F F.M
i : F.Idx
⊢ (if H' : i.fst = (M.mk x).head then (M.mk x).children (cast ⋯ i.snd) else default) =
if h : i.fst = x.fst then x.snd (cast ⋯ i.snd) else default | b52a11709dd8661a |
aemeasurable_indicator_const_iff | Mathlib/MeasureTheory/Measure/AEMeasurable.lean | /-- A characterization of the a.e.-measurability of the indicator function which takes a constant
value `b` on a set `A` and `0` elsewhere. -/
lemma aemeasurable_indicator_const_iff {s} [MeasurableSingletonClass β] (b : β) [NeZero b] :
AEMeasurable (s.indicator (fun _ ↦ b)) μ ↔ NullMeasurableSet s μ | case h.e'_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ s = (s.indicator fun x => b) ⁻¹' {0}ᶜ | rw [indicator_const_preimage_eq_union s {0}ᶜ b] | case h.e'_3
α : Type u_2
β : Type u_3
m0 : MeasurableSpace α
inst✝³ : MeasurableSpace β
μ : Measure α
inst✝² : Zero β
s : Set α
inst✝¹ : MeasurableSingletonClass β
b : β
inst✝ : NeZero b
h : AEMeasurable (s.indicator fun x => b) μ
⊢ s = (if b ∈ {0}ᶜ then s else ∅) ∪ if 0 ∈ {0}ᶜ then sᶜ else ∅ | 88ff3f717a1cbdcb |
Submodule.prod_le_iff | Mathlib/LinearAlgebra/Prod.lean | theorem prod_le_iff {p₁ : Submodule R M} {p₂ : Submodule R M₂} {q : Submodule R (M × M₂)} :
p₁.prod p₂ ≤ q ↔ map (LinearMap.inl R M M₂) p₁ ≤ q ∧ map (LinearMap.inr R M M₂) p₂ ≤ q | R : Type u
M : Type v
M₂ : Type w
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p₁ : Submodule R M
p₂ : Submodule R M₂
q : Submodule R (M × M₂)
hH : map (inl R M M₂) p₁ ≤ q
hK : map (inr R M M₂) p₂ ≤ q
x1 : M
x2 : M₂
h1 : (x1, x2).1 ∈ ↑p₁
h2 : (x1, x2).2 ... | apply hH | case a
R : Type u
M : Type v
M₂ : Type w
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p₁ : Submodule R M
p₂ : Submodule R M₂
q : Submodule R (M × M₂)
hH : map (inl R M M₂) p₁ ≤ q
hK : map (inr R M M₂) p₂ ≤ q
x1 : M
x2 : M₂
h1 : (x1, x2).1 ∈ ↑p₁
h2 : (x1,... | f6d4330f7f3f2538 |
Subfield.sInf_toSubring | Mathlib/Algebra/Field/Subfield/Basic.lean | theorem sInf_toSubring (s : Set (Subfield K)) :
(sInf s).toSubring = ⨅ t ∈ s, Subfield.toSubring t | K : Type u
inst✝ : DivisionRing K
s : Set (Subfield K)
⊢ (sInf s).toSubring = ⨅ t ∈ s, t.toSubring | ext x | case h
K : Type u
inst✝ : DivisionRing K
s : Set (Subfield K)
x : K
⊢ x ∈ (sInf s).toSubring ↔ x ∈ ⨅ t ∈ s, t.toSubring | 337447e91d0fdca6 |
ProbabilityTheory.lintegral_paretoPDF_eq_one | Mathlib/Probability/Distributions/Pareto.lean | /-- The pdf of the Pareto distribution integrates to `1`. -/
@[simp]
lemma lintegral_paretoPDF_eq_one (ht : 0 < t) (hr : 0 < r) :
∫⁻ x, paretoPDF t r x = 1 | case hf
t r : ℝ
ht : 0 < t
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0
rightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))
⊢ 0 ≤ᶠ[ae (ℙ.restrict (Ici t))] fun x => r * t ^ r * x ^ (-(r + 1)) | rw [EventuallyLE, ae_restrict_iff' measurableSet_Ici] | case hf
t r : ℝ
ht : 0 < t
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio t, paretoPDF t r x = 0
rightSide : ∫⁻ (x : ℝ) in Ici t, paretoPDF t r x = ∫⁻ (x : ℝ) in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1)))
⊢ ∀ᵐ (x : ℝ), x ∈ Ici t → 0 x ≤ r * t ^ r * x ^ (-(r + 1)) | e4acb7fa15218c7c |
ZLattice.covolume_comap | Mathlib/Algebra/Module/ZLattice/Covolume.lean | theorem covolume_comap {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
[MeasurableSpace F] [BorelSpace F] (ν : Measure F | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
L : Submodule ℤ E
inst✝⁸ : DiscreteTopology ↥L
inst✝⁷ : IsZLattice ℝ L
μ : autoParam (Measure E) _auto✝
inst✝⁶ : Measure.IsAddHaarMeasure μ
F : Type u_2
inst✝⁵ : Normed... | rw [covolume_eq_measure_fundamentalDomain _ _ (isAddFundamentalDomain (Free.chooseBasis ℤ L) μ),
covolume_eq_measure_fundamentalDomain _ _ ((isAddFundamentalDomain
((Free.chooseBasis ℤ L).ofZLatticeComap ℝ L e.toLinearEquiv) ν)), ← he.measure_preimage
(fundamentalDomain_measurableSet _).nullMeasurableSet, ← e.ima... | E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
L : Submodule ℤ E
inst✝⁸ : DiscreteTopology ↥L
inst✝⁷ : IsZLattice ℝ L
μ : autoParam (Measure E) _auto✝
inst✝⁶ : Measure.IsAddHaarMeasure μ
F : Type u_2
inst✝⁵ : Normed... | d24de32206919eed |
isClosedMap_smul_left | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | theorem isClosedMap_smul_left [T2Space E] (c : E) : IsClosedMap fun x : 𝕜 => x • c | case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module 𝕜 E
inst✝¹ : ContinuousSMul 𝕜 E
inst✝ : T2Space E
c : E
hc : c = 0
⊢ IsClosedMap fun x => 0 | exact isClosedMap_const | no goals | be83c856771190c7 |
Nat.digits_two_eq_bits | Mathlib/Data/Nat/Digits.lean | theorem digits_two_eq_bits (n : ℕ) : digits 2 n = n.bits.map fun b => cond b 1 0 | case f.false
n : ℕ
h : n ≠ 0
ih : digits 2 n = List.map (fun b => bif b then 1 else 0) n.bits
⊢ 0 < bit false n | simpa [Nat.bit, pos_iff_ne_zero] | no goals | 208c3d814ef4bafc |
WittVector.mul_pow_charP_coeff_succ | Mathlib/RingTheory/WittVector/Identities.lean | theorem mul_pow_charP_coeff_succ [CharP R p] (x : 𝕎 R) {m n : ℕ} :
(x * p ^ n).coeff (m + n) = x.coeff m ^ (p ^ n) | case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
x : 𝕎 R
n : ℕ
ih : ∀ {m : ℕ}, (x * ↑p ^ n).coeff (m + n) = x.coeff m ^ p ^ n
m : ℕ
⊢ (x * ↑p ^ n).coeff (m + n) ^ p = (x.coeff m ^ p ^ n) ^ p | congr | case succ.e_a
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CharP R p
x : 𝕎 R
n : ℕ
ih : ∀ {m : ℕ}, (x * ↑p ^ n).coeff (m + n) = x.coeff m ^ p ^ n
m : ℕ
⊢ (x * ↑p ^ n).coeff (m + n) = x.coeff m ^ p ^ n | 04bf0eae3f8313bd |
Equiv.Perm.exists_with_cycleType_iff | Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.lean | theorem Equiv.Perm.exists_with_cycleType_iff {m : Multiset ℕ} :
(∃ g : Equiv.Perm α, g.cycleType = m) ↔
(m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) | case h.h1.intro.intro.hn
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
m : Multiset ℕ
hc : m.sum ≤ Fintype.card α
h2c : ∀ a ∈ m, 2 ≤ a
hc' : m.toList.sum ≤ Fintype.card α
p : List (List α)
hp_length : List.map List.length p = m.toList
hp_nodup : ∀ s ∈ p, s.Nodup
hp_disj : List.Pairwise List.Disjoint p
hp2 : ∀ x... | exact hp2 x hx | no goals | 71fc8c11578523b5 |
IsLocalization.localization_localization_isLocalization_of_has_all_units | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | theorem localization_localization_isLocalization_of_has_all_units [IsLocalization N T]
(H : ∀ x : S, IsUnit x → x ∈ N) : IsLocalization (N.comap (algebraMap R S)) T | case h.e'_3.e_S
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S),... | rw [sup_eq_left] | case h.e'_3.e_S
R : Type u_1
inst✝⁸ : CommSemiring R
M : Submonoid R
S : Type u_2
inst✝⁷ : CommSemiring S
inst✝⁶ : Algebra R S
N : Submonoid S
T : Type u_3
inst✝⁵ : CommSemiring T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsLocalization M S
inst✝ : IsLocalization N T
H : ∀ (x : S),... | eed8005ab7c6cf25 |
ProbabilityTheory.Kernel.IsCondKernel.isProbabilityMeasure_ae | Mathlib/Probability/Kernel/Disintegration/Basic.lean | /-- A conditional kernel is almost everywhere a probability measure. -/
lemma IsCondKernel.isProbabilityMeasure_ae [IsFiniteKernel κ.fst] [κ.IsCondKernel κCond] (a : α) :
∀ᵐ b ∂(κ.fst a), IsProbabilityMeasure (κCond (a, b)) | case pos.left
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableS... | by_contra h_ne_zero | case pos.left
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond : Kernel (α × β) Ω
inst✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableS... | d615b2a173689899 |
CategoryTheory.LocalizerMorphism.isRightDerivabilityStructure_iff | Mathlib/CategoryTheory/Localization/DerivabilityStructure/Basic.lean | lemma isRightDerivabilityStructure_iff [Φ.HasRightResolutions] (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F) :
Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom | C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ :... | let e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂ | C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{u_4, u_1} D₁
inst✝³ : Category.{u_3, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ :... | 97c55d8f5d3aa5e9 |
DomMulAct.stabilizer_card' | Mathlib/GroupTheory/Perm/DomMulAct.lean | theorem stabilizer_card':
Fintype.card {g : Perm α // f ∘ g = f} =
∏ i ∈ Finset.univ.image f, (Fintype.card ({a // f a = i}))! | case h.h.f
α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
φ : α → { x // x ∈ Finset.image f Finset.univ } := Set.codRestrict f ↑(Finset.image f Finset.univ) ⋯
this : ∀ (g : Perm α), f ∘ ⇑g = f ↔ φ ∘ ⇑g = φ
i : { x // x ∈ Finset.image f Finset.univ }
ha✝ : i ∈ Finset.... | apply Equiv.subtypeEquiv (Equiv.refl α) | case h.h.f
α : Type u_1
ι : Type u_2
f : α → ι
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableEq ι
φ : α → { x // x ∈ Finset.image f Finset.univ } := Set.codRestrict f ↑(Finset.image f Finset.univ) ⋯
this : ∀ (g : Perm α), f ∘ ⇑g = f ↔ φ ∘ ⇑g = φ
i : { x // x ∈ Finset.image f Finset.univ }
ha✝ : i ∈ Finset.... | eb29591c1a59346a |
bernsteinPolynomial.linearIndependent_aux | Mathlib/RingTheory/Polynomial/Bernstein.lean | theorem linearIndependent_aux (n k : ℕ) (h : k ≤ n + 1) :
LinearIndependent ℚ fun ν : Fin k => bernsteinPolynomial ℚ n ν | n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)
⊢ eval 1 ((⇑derivative)^[n - k] p) = 0 | refine span_induction ?_ ?_ ?_ ?_ m | case refine_1
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)
⊢ ∀ x ∈ Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1, eval 1 ((⇑derivative)^[n - k] x) = 0
case refine_2
n k : ℕ
h : k ≤ n
p : ℚ[X]
m : p ∈ span ℚ (Set.range fun k_1 => bernsteinPolynomial ℚ n ↑k_1)
⊢ eval ... | 6da9df7f52f7e5d5 |
MeasureTheory.MemLp.mono_exponent | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | theorem MemLp.mono_exponent {p q : ℝ≥0∞} [IsFiniteMeasure μ] {f : α → E} (hfq : MemLp f q μ)
(hpq : p ≤ q) : MemLp f p μ | case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ eLpNormEssSup f μ * μ Set... | refine ENNReal.mul_lt_top hfq_lt_top ?_ | case pos
α : Type u_1
E : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup E
μ : Measure α
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
f : α → E
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNormEssSup f μ < ⊤
hp0 : p ≠ 0
hp_top : ¬p = ⊤
hp_pos : 0 < p.toReal
hq_top : q = ⊤
⊢ μ Set.univ ^ (1 / p.toRea... | 20f1c91936cbd48e |
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