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 |
|---|---|---|---|---|---|---|
Set.star_mem_center | Mathlib/Algebra/Star/Center.lean | theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where
comm | R : Type u_1
inst✝¹ : Mul R
inst✝ : StarMul R
a : R
ha : a ∈ center R
b c : R
⊢ star a * star (star c * star b) = star (star c * star b * a) | rw [← star_mul] | no goals | c15bad2514aa2cf1 |
Quiver.Push.lift_comp | Mathlib/Combinatorics/Quiver/Push.lean | theorem lift_comp : (of σ ⋙q lift σ φ τ h) = φ | case h_map.h
V : Type u_1
inst✝¹ : Quiver V
W : Type u_2
σ : V → W
W' : Type u_3
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
X Y : V
f : X ⟶ Y
⊢ HEq (⋯.mpr (φ.map f)) (⋯ ▸ ⋯ ▸ φ.map f) | apply (cast_heq _ _).trans | case h_map.h
V : Type u_1
inst✝¹ : Quiver V
W : Type u_2
σ : V → W
W' : Type u_3
inst✝ : Quiver W'
φ : V ⥤q W'
τ : W → W'
h : ∀ (x : V), φ.obj x = τ (σ x)
X Y : V
f : X ⟶ Y
⊢ HEq (φ.map f) (⋯ ▸ ⋯ ▸ φ.map f) | 2ad23c8af09b67fc |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.clear_insert_inductive_case | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem clear_insert_inductive_case {n : Nat} (f : DefaultFormula n) (f_assignments_size : f.assignments.size = n)
(units : Array (Literal (PosFin n))) (units_nodup : ∀ i : Fin units.size, ∀ j : Fin units.size, i ≠ j → units[i] ≠ units[j])
(idx : Fin units.size) (assignments : Array Assignment)
(ih : ClearI... | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
... | intro k_eq_idx | n : Nat
f : DefaultFormula n
f_assignments_size : f.assignments.size = n
units : Array (Literal (PosFin n))
units_nodup : ∀ (i j : Fin units.size), i ≠ j → units[i] ≠ units[j]
idx : Fin units.size
assignments : Array Assignment
hsize : assignments.size = n
hsize' : (clearUnit assignments units[idx]).size = n
i : Fin n
... | c90be518eb885cb1 |
dite_ne_right_iff | Mathlib/Logic/Basic.lean | theorem dite_ne_right_iff : (dite P A fun _ ↦ b) ≠ b ↔ ∃ h, A h ≠ b | α : Sort u_1
P : Prop
inst✝ : Decidable P
b : α
A : P → α
⊢ (dite P A fun x => b) ≠ b ↔ ∃ h, A h ≠ b | simp only [Ne, dite_eq_right_iff, not_forall] | no goals | f3fd6e49415d8434 |
Matroid.IsBasis.isBasis_isRestriction | Mathlib/Data/Matroid/Restrict.lean | theorem IsBasis.isBasis_isRestriction (hI : M.IsBasis I X) (hNM : N ≤r M) (hX : X ⊆ N.E) :
N.IsBasis I X | α : Type u_1
M : Matroid α
I X : Set α
N : Matroid α
hI : M.IsBasis I X
hNM : N ≤r M
hX : X ⊆ N.E
⊢ N.IsBasis I X | obtain ⟨R, hR, rfl⟩ := hNM | case intro.intro
α : Type u_1
M : Matroid α
I X : Set α
hI : M.IsBasis I X
R : Set α
hR : R ⊆ M.E
hX : X ⊆ (M ↾ R).E
⊢ (M ↾ R).IsBasis I X | 6c3f3b707679054a |
BoxIntegral.unitPartition.prepartition_tag | Mathlib/Analysis/BoxIntegral/UnitPartition.lean | theorem prepartition_tag {ν : ι → ℤ} {B : Box ι} (hν : ν ∈ admissibleIndex n B) :
(prepartition n B).tag (box n ν) = tag n ν | ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
⊢ (prepartition n B).tag (box n ν) = tag n ν | dsimp only [prepartition] | ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
⊢ (if hI : ∃ ν_1 ∈ admissibleIndex n B, box n ν = box n ν_1 then tag n hI.choose else ⋯.choose) = tag n ν | 50d6717226e9f93f |
SimpleGraph.Subgraph.IsMatching.even_card | Mathlib/Combinatorics/SimpleGraph/Matching.lean | theorem IsMatching.even_card [Fintype M.verts] (h : M.IsMatching) : Even M.verts.toFinset.card | V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype ↑M.verts
h : M.IsMatching
⊢ Even M.verts.toFinset.card | classical
rw [isMatching_iff_forall_degree] at h
use M.coe.edgeFinset.card
rw [← two_mul, ← M.coe.sum_degrees_eq_twice_card_edges]
convert_to _ = Finset.sum Finset.univ fun v => SimpleGraph.degree (Subgraph.coe M) v using 3
simp [h, Finset.card_univ] | no goals | c50af873c5d088dd |
List.perm_dlookup | Mathlib/Data/List/Sigma.lean | theorem perm_dlookup (a : α) {l₁ l₂ : List (Sigma β)} (nd₁ : l₁.NodupKeys) (nd₂ : l₂.NodupKeys)
(p : l₁ ~ l₂) : dlookup a l₁ = dlookup a l₂ | α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
l₁ l₂ : List (Sigma β)
nd₁ : l₁.NodupKeys
nd₂ : l₂.NodupKeys
p : l₁ ~ l₂
⊢ dlookup a l₁ = dlookup a l₂ | ext b | case a
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
l₁ l₂ : List (Sigma β)
nd₁ : l₁.NodupKeys
nd₂ : l₂.NodupKeys
p : l₁ ~ l₂
b : β a
⊢ b ∈ dlookup a l₁ ↔ b ∈ dlookup a l₂ | 0a0c8107a3d4edc7 |
Set.Ioo_union_Ioo' | Mathlib/Order/Interval/Set/Basic.lean | theorem Ioo_union_Ioo' (h₁ : c < b) (h₂ : a < d) : Ioo a b ∪ Ioo c d = Ioo (min a c) (max b d) | case pos
α : Type u_1
inst✝ : LinearOrder α
a b c d : α
h₁ : c < b
h₂ : a < d
x : α
hc : ¬c < x
hd : x < d
hxb : x < b
⊢ a < x ∧ x < b ∨ c < x ∧ x < d ↔ (a < x ∨ c < x) ∧ (x < b ∨ x < d) | simp only [hxb, and_true, hc, false_and, or_false, true_or] | no goals | 925542e85b57dafe |
PiNat.mem_cylinder_iff_eq | Mathlib/Topology/MetricSpace/PiNat.lean | theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
y ∈ cylinder x n ↔ cylinder y n = cylinder x n | case mp.h₂
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder x n
i : ℕ
hi : i < n
⊢ z i = y i | rw [hy i hi] | case mp.h₂
E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hy : y ∈ cylinder x n
z : (n : ℕ) → E n
hz : z ∈ cylinder x n
i : ℕ
hi : i < n
⊢ z i = x i | 1650d6a7263b81d8 |
OreLocalization.smul'_char | Mathlib/GroupTheory/OreLocalization/Basic.lean | theorem smul'_char (r₁ : R) (r₂ : X) (s₁ s₂ : S) (u : S) (v : R) (huv : u * r₁ = v * s₂) :
OreLocalization.smul' r₁ s₁ r₂ s₂ = v • r₂ /ₒ (u * s₁) | R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type u_2
inst✝ : MulAction R X
r₁ : R
r₂ : X
s₁ s₂ u : ↥S
v : R
huv : ↑u * r₁ = v * ↑s₂
⊢ oreNum r₁ s₂ • r₂ /ₒ (oreDenom r₁ s₂ * s₁) = v • r₂ /ₒ (u * s₁) | have h₀ := ore_eq r₁ s₂ | R : Type u_1
inst✝² : Monoid R
S : Submonoid R
inst✝¹ : OreSet S
X : Type u_2
inst✝ : MulAction R X
r₁ : R
r₂ : X
s₁ s₂ u : ↥S
v : R
huv : ↑u * r₁ = v * ↑s₂
h₀ : ↑(oreDenom r₁ s₂) * r₁ = oreNum r₁ s₂ * ↑s₂
⊢ oreNum r₁ s₂ • r₂ /ₒ (oreDenom r₁ s₂ * s₁) = v • r₂ /ₒ (u * s₁) | 158523f845373f5e |
List.max?_le_iff | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/MinMax.lean | theorem max?_le_iff [Max α] [LE α]
(max_le_iff : ∀ a b c : α, max b c ≤ a ↔ b ≤ a ∧ c ≤ a) :
{xs : List α} → xs.max? = some a → ∀ {x}, a ≤ x ↔ ∀ b ∈ xs, b ≤ x
| nil => by simp
| cons x xs => by
rw [max?]; rintro ⟨⟩ y
induction xs generalizing x with
| nil => simp
| cons y xs ih => simp [ih, ... | α : Type u_1
a : α
inst✝¹ : Max α
inst✝ : LE α
max_le_iff : ∀ (a b c : α), max b c ≤ a ↔ b ≤ a ∧ c ≤ a
x : α
xs : List α
⊢ some (foldl max x xs) = some a → ∀ {x_1 : α}, a ≤ x_1 ↔ ∀ (b : α), b ∈ x :: xs → b ≤ x_1 | rintro ⟨⟩ y | case refl
α : Type u_1
inst✝¹ : Max α
inst✝ : LE α
max_le_iff : ∀ (a b c : α), max b c ≤ a ↔ b ≤ a ∧ c ≤ a
x : α
xs : List α
y : α
⊢ foldl max x xs ≤ y ↔ ∀ (b : α), b ∈ x :: xs → b ≤ y | a37680aba98f711a |
Finset.centerMass_empty | Mathlib/Analysis/Convex/Combination.lean | theorem Finset.centerMass_empty : (∅ : Finset ι).centerMass w z = 0 | R : Type u_1
E : Type u_3
ι : Type u_5
inst✝² : LinearOrderedField R
inst✝¹ : AddCommGroup E
inst✝ : Module R E
w : ι → R
z : ι → E
⊢ ∅.centerMass w z = 0 | simp only [centerMass, sum_empty, smul_zero] | no goals | 4677fe032673d938 |
GromovHausdorff.totallyBounded | Mathlib/Topology/MetricSpace/GromovHausdorff.lean | theorem totallyBounded {t : Set GHSpace} {C : ℝ} {u : ℕ → ℝ} {K : ℕ → ℕ}
(ulim : Tendsto u atTop (𝓝 0)) (hdiam : ∀ p ∈ t, diam (univ : Set (GHSpace.Rep p)) ≤ C)
(hcov : ∀ p ∈ t, ∀ n : ℕ, ∃ s : Set (GHSpace.Rep p),
(#s) ≤ K n ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)) :
TotallyBounded t | case intro.refine_2.mk.mk
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p... | let Φ : s p → q.Rep := fun x => Ψ x | case intro.refine_2.mk.mk
t : Set GHSpace
C : ℝ
u : ℕ → ℝ
K : ℕ → ℕ
ulim : Tendsto u atTop (𝓝 0)
hdiam : ∀ p ∈ t, diam univ ≤ C
hcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)
δ : ℝ
δpos : δ > 0
ε : ℝ := 1 / 5 * δ
εpos : 0 < ε
n : ℕ
hn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε
u_le_ε : u n ≤ ε
s : (p... | c10025d34debaff9 |
Matroid.IsBasis.closure_eq_closure | Mathlib/Data/Matroid/Closure.lean | lemma IsBasis.closure_eq_closure (h : M.IsBasis I X) : M.closure I = M.closure X | α : Type u_2
M : Matroid α
X I : Set α
h : M.IsBasis I X
⊢ M.closure X ⊆ M.closure {x | M.IsBasis I (insert x I)} | exact M.closure_subset_closure fun e he ↦ (h.isBasis_subset (subset_insert _ _)
(insert_subset he h.subset)) | no goals | 7e99c226dd2c4c16 |
ENNReal.div_zero | Mathlib/Data/ENNReal/Inv.lean | theorem div_zero (h : a ≠ 0) : a / 0 = ∞ | a : ℝ≥0∞
h : a ≠ 0
⊢ a / 0 = ⊤ | simp [div_eq_mul_inv, h] | no goals | 6209f8f22d805448 |
MeasureTheory.lintegral_rpow_eq_lintegral_meas_lt_mul | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | theorem lintegral_rpow_eq_lintegral_meas_lt_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (t ^ (p - 1)) | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) =
ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1)) | apply congr_arg fun z => ENNReal.ofReal p * z | α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᶠ[ae μ] f
f_mble : AEMeasurable f μ
p : ℝ
p_pos : 0 < p
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) =
∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1)) | a9a2c0230dbc8681 |
Array.fst_unzip | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean | theorem fst_unzip (as : Array (α × β)) : (Array.unzip as).fst = as.map Prod.fst | case mk
α : Type u_1
β : Type u_2
as : List (α × β)
⊢ (foldl (fun x x_1 => (x.fst.push x_1.fst, x.snd.push x_1.snd)) (#[], #[]) { toList := as }).fst =
map Prod.fst { toList := as } | simp only [List.foldl_toArray'] | case mk
α : Type u_1
β : Type u_2
as : List (α × β)
⊢ (List.foldl (fun x x_1 => (x.fst.push x_1.fst, x.snd.push x_1.snd)) (#[], #[]) as).fst = map Prod.fst { toList := as } | 3740fb34d2c56eb2 |
RootPairing.exists_ge_zero_eq_rootForm | Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean | theorem exists_ge_zero_eq_rootForm [Fintype ι] (x : M) (hx : x ∈ span S (range P.root)) :
∃ s ≥ 0, algebraMap S R s = P.RootForm x x | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : AddCommGroup N
inst✝⁷ : Module R N
P : RootPairing ι R M N
S : Type u_5
inst✝⁶ : LinearOrderedCommRing S
inst✝⁵ : Algebra S R
inst✝⁴ : FaithfulSMul S R
inst✝³ : Module S M
inst✝² : IsScalarTowe... | refine ⟨(P.posRootForm S).posForm ⟨x, hx⟩ ⟨x, hx⟩, IsSumSq.nonneg ?_, by simp [posRootForm]⟩ | ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝¹¹ : CommRing R
inst✝¹⁰ : AddCommGroup M
inst✝⁹ : Module R M
inst✝⁸ : AddCommGroup N
inst✝⁷ : Module R N
P : RootPairing ι R M N
S : Type u_5
inst✝⁶ : LinearOrderedCommRing S
inst✝⁵ : Algebra S R
inst✝⁴ : FaithfulSMul S R
inst✝³ : Module S M
inst✝² : IsScalarTowe... | cf1b0b2424f372a7 |
OrderIso.preimage_Ioc | Mathlib/Order/Interval/Set/OrderIso.lean | theorem preimage_Ioc (e : α ≃o β) (a b : β) : e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) | α : Type u_1
β : Type u_2
inst✝¹ : Preorder α
inst✝ : Preorder β
e : α ≃o β
a b : β
⊢ ⇑e ⁻¹' Ioc a b = Ioc (e.symm a) (e.symm b) | simp [← Ioi_inter_Iic] | no goals | 21a1b5a3fc34c6f4 |
CategoryTheory.Equivalence.funInvIdAssoc_hom_app | Mathlib/CategoryTheory/Equivalence.lean | theorem funInvIdAssoc_hom_app (e : C ≌ D) (F : C ⥤ E) (X : C) :
(funInvIdAssoc e F).hom.app X = F.map (e.unitInv.app X) | C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
e : C ≌ D
F : C ⥤ E
X : C
⊢ (e.funInvIdAssoc F).hom.app X = F.map (e.unitInv.app X) | dsimp [funInvIdAssoc] | C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
e : C ≌ D
F : C ⥤ E
X : C
⊢ 𝟙 (F.obj (e.inverse.obj (e.functor.obj X))) ≫ F.map (e.unitIso.inv.app X) ≫ 𝟙 (F.obj X) = F.map (e.unitInv.app X) | a493c6cd0239349e |
DirichletCharacter.differentiable_completedLFunction | Mathlib/NumberTheory/LSeries/DirichletContinuation.lean | /-- The completed L-function of a non-trivial Dirichlet character is differentiable everywhere. -/
lemma differentiable_completedLFunction {χ : DirichletCharacter ℂ N} (hχ : χ ≠ 1) :
Differentiable ℂ (completedLFunction χ) | N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ ≠ 1
⊢ Differentiable ℂ (completedLFunction χ) | refine fun s ↦ differentiableAt_completedLFunction _ _ (Or.inr ?_) (Or.inr hχ) | N : ℕ
inst✝ : NeZero N
χ : DirichletCharacter ℂ N
hχ : χ ≠ 1
s : ℂ
⊢ N ≠ 1 | 6832c38e9a2c9f2e |
FirstOrder.Language.BoundedFormula.realize_toFormula | Mathlib/ModelTheory/Semantics.lean | theorem realize_toFormula (φ : L.BoundedFormula α n) (v : α ⊕ (Fin n) → M) :
φ.toFormula.Realize v ↔ φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr) | case all
L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
n n✝ : ℕ
f✝ : L.BoundedFormula α (n✝ + 1)
ih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), f✝.toFormula.Realize v ↔ f✝.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)
v : α ⊕ Fin n✝ → M
a : M
h :
f✝.toFormula.Realize (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔ f✝.Realiz... | rw [← h, realize_relabel, Formula.Realize, iff_iff_eq] | case all
L : Language
M : Type w
inst✝ : L.Structure M
α : Type u'
n n✝ : ℕ
f✝ : L.BoundedFormula α (n✝ + 1)
ih3 : ∀ (v : α ⊕ Fin (n✝ + 1) → M), f✝.toFormula.Realize v ↔ f✝.Realize (v ∘ Sum.inl) (v ∘ Sum.inr)
v : α ⊕ Fin n✝ → M
a : M
h :
f✝.toFormula.Realize (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a)) ↔ f✝.Realiz... | 9745031cf13e24da |
Zsqrtd.norm_eq_zero | Mathlib/NumberTheory/Zsqrtd/Basic.lean | theorem norm_eq_zero {d : ℤ} (h_nonsquare : ∀ n : ℤ, d ≠ n * n) (a : ℤ√d) : norm a = 0 ↔ a = 0 | d : ℤ
h_nonsquare : ∀ (n : ℤ), d ≠ n * n
a : ℤ√d
ha : a.re * a.re = d * a.im * a.im
h : d < 0
⊢ d * (a.im * a.im) ≤ 0 | exact mul_nonpos_of_nonpos_of_nonneg h.le (mul_self_nonneg _) | no goals | 1741f43679b93652 |
Std.DHashMap.Internal.List.mem_eraseKey_of_key_beq_eq_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem mem_eraseKey_of_key_beq_eq_false [BEq α] {a : α}
{l : List ((a : α) × β a)} (p : (a : α) × β a) (hne : (p.1 == a) = false) :
p ∈ eraseKey a l ↔ p ∈ l | case isFalse
α : Type u
β : α → Type v
inst✝ : BEq α
a : α
p : (a : α) × β a
hne : (p.fst == a) = false
head✝ : (a : α) × β a
tail✝ : List ((a : α) × β a)
ih : p ∈ eraseKey a tail✝ ↔ p ∈ tail✝
h✝ : ¬(head✝.fst == a) = true
⊢ p ∈ ⟨head✝.fst, head✝.snd⟩ :: eraseKey a tail✝ ↔ p = head✝ ∨ p ∈ tail✝ | next h =>
simp only [List.mem_cons, ih] | no goals | 997dbbc69c0b9544 |
CompHaus.effectiveEpiFamily_tfae | Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean | theorem effectiveEpiFamily_tfae
{α : Type} [Finite α] {B : CompHaus.{u}}
(X : α → CompHaus.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] | α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b) → Epi (Sigma.desc π)
x✝ : Epi (Sigma.desc π)
e : F... | obtain ⟨t, rfl⟩ := e b | case intro
α : Type
inst✝ : Finite α
B : CompHaus
X : α → CompHaus
π : (a : α) → X a ⟶ B
tfae_2_to_1 : Epi (Sigma.desc π) → EffectiveEpiFamily X π
tfae_1_to_2 : EffectiveEpiFamily X π → Epi (Sigma.desc π)
tfae_3_to_2 : (∀ (b : ↑B.toTop), ∃ a x, (ConcreteCategory.hom (π a)) x = b) → Epi (Sigma.desc π)
x✝ : Epi (Sigma.de... | 0aeead2cda12faf0 |
exists_associated_pow_of_mul_eq_pow | Mathlib/Algebra/GCDMonoid/Basic.lean | theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : IsUnit (gcd a b))
{k : ℕ} (h : a * b = c ^ k) : ∃ d : α, Associated (d ^ k) a | case h
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₁ * d₂
⊢ Associated (d₁ ^ k) a | obtain ⟨h0₁, ⟨a', ha'⟩⟩ := pow_dvd_of_mul_eq_pow ha hab h hc hd₁ | case h.intro.intro
α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : GCDMonoid α
a b c : α
hab : IsUnit (gcd a b)
k : ℕ
h : a * b = c ^ k
h✝ : Nontrivial α
ha : ¬a = 0
hb : ¬b = 0
hk : k > 0
hc✝ : c ∣ a * b
d₁ d₂ : α
hd₁ : d₁ ∣ a
hd₂ : d₂ ∣ b
hc : c = d₁ * d₂
h0₁ : d₁ ^ k ≠ 0
a' : α
ha' : a = d₁ ^ k * a'
⊢ Assoc... | 4887d5e0abe68a27 |
Polynomial.degree_X_pow | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | theorem degree_X_pow : degree ((X : R[X]) ^ n) = n | R : Type u
inst✝¹ : Semiring R
inst✝ : Nontrivial R
n : ℕ
⊢ (X ^ n).degree = ↑n | rw [X_pow_eq_monomial, degree_monomial _ (one_ne_zero' R)] | no goals | 3ebf6c97e2df715a |
FirstOrder.Language.PartialEquiv.cod_le_cod | Mathlib/ModelTheory/PartialEquiv.lean | theorem cod_le_cod {f g : M ≃ₚ[L] N} : f ≤ g → f.cod ≤ g.cod | case intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f g : M ≃ₚ[L] N
w✝ : f.dom ≤ g.dom
eq_fun : g.cod.subtype.comp (g.toEquiv.toEmbedding.comp (inclusion w✝)) = f.cod.subtype.comp f.toEquiv.toEmbedding
n : N
hn : n ∈ f.cod
m : ↥f.dom := f.toEquiv.symm ⟨n, hn⟩
⊢ n ∈ g.cod | have : ((subtype _).comp f.toEquiv.toEmbedding) m = n := by simp only [m, Embedding.comp_apply,
Equiv.coe_toEmbedding, Equiv.apply_symm_apply, coe_subtype] | case intro
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
f g : M ≃ₚ[L] N
w✝ : f.dom ≤ g.dom
eq_fun : g.cod.subtype.comp (g.toEquiv.toEmbedding.comp (inclusion w✝)) = f.cod.subtype.comp f.toEquiv.toEmbedding
n : N
hn : n ∈ f.cod
m : ↥f.dom := f.toEquiv.symm ⟨n, hn⟩
this : (f.cod.subtyp... | 058229ae64672064 |
PiTensorProduct.mul_comm | Mathlib/RingTheory/PiTensorProduct.lean | protected lemma mul_comm (x y : ⨂[R] i, A i) : mul x y = mul y x | ι : Type u_1
R : Type u_3
A : ι → Type u_4
inst✝² : CommSemiring R
inst✝¹ : (i : ι) → CommSemiring (A i)
inst✝ : (i : ι) → Algebra R (A i)
x y : ⨂[R] (i : ι), A i
⊢ mul = mul.flip | ext x y | case H.H.H.H
ι : Type u_1
R : Type u_3
A : ι → Type u_4
inst✝² : CommSemiring R
inst✝¹ : (i : ι) → CommSemiring (A i)
inst✝ : (i : ι) → Algebra R (A i)
x✝ y✝ : ⨂[R] (i : ι), A i
x y : (i : ι) → A i
⊢ (((mul.compMultilinearMap (tprod R)) x).compMultilinearMap (tprod R)) y =
(((mul.flip.compMultilinearMap (tprod R)) ... | ad1d587b41a97abc |
PresentedGroup.generated_by | Mathlib/GroupTheory/PresentedGroup.lean | theorem generated_by (rels : Set (FreeGroup α)) (H : Subgroup (PresentedGroup rels))
(h : ∀ j : α, PresentedGroup.of j ∈ H) (x : PresentedGroup rels) : x ∈ H | case H.Cm
α : Type u_1
rels : Set (FreeGroup α)
H : Subgroup (PresentedGroup rels)
h : ∀ (j : α), of j ∈ H
x✝ y✝ : FreeGroup α
a✝¹ : (mk rels) x✝ ∈ H
a✝ : (mk rels) y✝ ∈ H
⊢ (mk rels) (x✝ * y✝) ∈ H | rename_i h1 h2 | case H.Cm
α : Type u_1
rels : Set (FreeGroup α)
H : Subgroup (PresentedGroup rels)
h : ∀ (j : α), of j ∈ H
x✝ y✝ : FreeGroup α
h1 : (mk rels) x✝ ∈ H
h2 : (mk rels) y✝ ∈ H
⊢ (mk rels) (x✝ * y✝) ∈ H | 36af71a0cd1686c5 |
AList.lookupFinsupp_support | Mathlib/Data/Finsupp/AList.lean | theorem lookupFinsupp_support [DecidableEq α] [DecidableEq M] (l : AList fun _x : α => M) :
l.lookupFinsupp.support = (l.1.filter fun x => Sigma.snd x ≠ 0).keys.toFinset | α : Type u_1
M : Type u_2
inst✝² : Zero M
inst✝¹ : DecidableEq α
inst✝ : DecidableEq M
l : AList fun _x => M
⊢ (filter (fun x => decide (x.snd ≠ 0)) l.entries).keys.toFinset =
(filter (fun x => decide (x.snd ≠ 0)) l.entries).keys.toFinset | congr! | no goals | f877487f8fda3140 |
ContinuousLinearMap.inner_map_map_iff_adjoint_comp_self | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | theorem inner_map_map_iff_adjoint_comp_self (u : H →L[𝕜] K) :
(∀ x y : H, ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜) ↔ adjoint u ∘L u = 1 | case refine_1
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
u : H →L[𝕜] K
h : ∀ (x y : H), ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜
x : H
⊢ ((adjoin... | refine ext_inner_right 𝕜 fun y ↦ ?_ | case refine_1
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
u : H →L[𝕜] K
h : ∀ (x y : H), ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜
x y : H
⊢ ⟪((adj... | 76c8f2fe76b6a005 |
ValuationRing.le_total | Mathlib/RingTheory/Valuation/ValuationRing.lean | theorem le_total (a b : ValueGroup A K) : a ≤ b ∨ b ≤ a | case mk.mk.intro.intro.intro.intro.intro.intro.intro.inl.h
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a b : ValueGroup A K
xa ya : A
hya : ya ∈ nonZeroDivisors A
xb yb : A
hyb : yb ∈ nonZeroDivisors A
this✝ : (a... | use c | case h
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a b : ValueGroup A K
xa ya : A
hya : ya ∈ nonZeroDivisors A
xb yb : A
hyb : yb ∈ nonZeroDivisors A
this✝ : (algebraMap A K) ya ≠ 0
this : (algebraMap A K) yb ≠ 0... | 43f1ff2b88bd99b5 |
IncidenceAlgebra.mu_toDual | Mathlib/Combinatorics/Enumerative/IncidenceAlgebra.lean | @[simp]
lemma mu_toDual (a b : α) : mu 𝕜 (toDual a) (toDual b) = mu 𝕜 b a | 𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯... | symm | 𝕜 : Type u_2
α : Type u_5
inst✝³ : Ring 𝕜
inst✝² : PartialOrder α
inst✝¹ : LocallyFiniteOrder α
inst✝ : DecidableEq α
a b : α
this✝ : DecidableRel fun x1 x2 => x1 ≤ x2 := Classical.decRel fun x1 x2 => x1 ≤ x2
mud : IncidenceAlgebra 𝕜 αᵒᵈ := { toFun := fun a b => (mu 𝕜) (ofDual b) (ofDual a), eq_zero_of_not_le' := ⋯... | 95f4eed14862eb7a |
Submodule.mem_sSup_iff_exists_finset | Mathlib/LinearAlgebra/Finsupp/Span.lean | theorem Submodule.mem_sSup_iff_exists_finset {S : Set (Submodule R M)} {m : M} :
m ∈ sSup S ↔ ∃ s : Finset (Submodule R M), ↑s ⊆ S ∧ m ∈ ⨆ i ∈ s, i | case h.e'_3.h.e'_4.h.pq.a.a
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
S : Set (Submodule R M)
m : M
x✝¹ : ∃ s, ↑s ⊆ S ∧ m ∈ ⨆ i ∈ s, i
s : Finset (Submodule R M)
hsS : ↑s ⊆ S
hs : m ∈ ⨆ i ∈ s, i
x✝ : Submodule R M
⊢ x✝ ∈ s ↔ x✝ ∈ S ∧ x✝ ∈ s | aesop | no goals | 4a53431e2aac6cd1 |
ContDiffWithinAt.contDiffOn' | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffWithinAt.contDiffOn' (hm : m ≤ n) (h' : m = ∞ → n = ω)
(h : ContDiffWithinAt 𝕜 n f s x) :
∃ u, IsOpen u ∧ x ∈ u ∧ ContDiffOn 𝕜 m f (insert x s ∩ u) | case inl.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
m : WithTop ℕ∞
hm : m ≤ ω
h' : m = ∞ → ω = ω
t : Set E
ht : ... | rw [inter_comm] at hut | case inl.intro.intro.intro.intro.intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
m : WithTop ℕ∞
hm : m ≤ ω
h' : m = ∞ → ω = ω
t : Set E
ht : ... | 252a1c681d4d0950 |
Ring.choose_eq_nat_choose | Mathlib/RingTheory/Binomial.lean | theorem choose_eq_nat_choose [NatPowAssoc R] (n k : ℕ) : choose (n : R) k = Nat.choose n k | case zero
R : Type u_1
inst✝³ : NonAssocRing R
inst✝² : Pow R ℕ
inst✝¹ : BinomialRing R
inst✝ : NatPowAssoc R
k : ℕ
⊢ choose (↑0) k = ↑(Nat.choose 0 k) | cases k with
| zero => rw [choose_zero_right, Nat.choose_zero_right, Nat.cast_one]
| succ k => rw [Nat.cast_zero, choose_zero_succ, Nat.choose_zero_succ, Nat.cast_zero] | no goals | 1e26b94f806077b3 |
hasFDerivAt_exp_of_mem_ball | Mathlib/Analysis/SpecialFunctions/Exponential.lean | theorem hasFDerivAt_exp_of_mem_ball [CharZero 𝕂] {x : 𝔸}
(hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
HasFDerivAt (exp 𝕂) (exp 𝕂 x • (1 : 𝔸 →L[𝕂] 𝔸)) x | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
⊢ (fun h => exp 𝕂 x * (exp 𝕂 (0 + h) - exp 𝕂 0 - (Continuou... | have : ∀ᶠ h in 𝓝 (0 : 𝔸), h ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius :=
EMetric.ball_mem_nhds _ hpos | 𝕂 : Type u_1
𝔸 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕂
inst✝³ : NormedCommRing 𝔸
inst✝² : NormedAlgebra 𝕂 𝔸
inst✝¹ : CompleteSpace 𝔸
inst✝ : CharZero 𝕂
x : 𝔸
hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius
hpos : 0 < (expSeries 𝕂 𝔸).radius
this : ∀ᶠ (h : 𝔸) in 𝓝 0, h ∈ EMetric.ball 0 (expSeries 𝕂 �... | b912975acf3383c5 |
Filter.HasBasis.liminf_eq_ite | Mathlib/Order/LiminfLimsup.lean | theorem HasBasis.liminf_eq_ite {v : Filter ι} {p : ι' → Prop} {s : ι' → Set ι}
[Countable (Subtype p)] [Nonempty (Subtype p)] (hv : v.HasBasis p s) (f : ι → α) :
liminf f v = if ∃ (j : Subtype p), s j = ∅ then sSup univ else
if ∀ (j : Subtype p), ¬BddBelow (range (fun (i : s j) ↦ f i)) then sSup ∅
e... | case neg.H
α : Type u_1
ι : Type u_4
ι' : Type u_5
inst✝² : ConditionallyCompleteLinearOrder α
v : Filter ι
p : ι' → Prop
s : ι' → Set ι
inst✝¹ : Countable (Subtype p)
inst✝ : Nonempty (Subtype p)
hv : v.HasBasis p s
f : ι → α
H : ¬∃ j, s ↑j = ∅
H' : ¬∀ (j : Subtype p), ¬BddBelow (range fun i => f ↑i)
⊢ ∃ j, BddBelow (... | push_neg at H' | case neg.H
α : Type u_1
ι : Type u_4
ι' : Type u_5
inst✝² : ConditionallyCompleteLinearOrder α
v : Filter ι
p : ι' → Prop
s : ι' → Set ι
inst✝¹ : Countable (Subtype p)
inst✝ : Nonempty (Subtype p)
hv : v.HasBasis p s
f : ι → α
H : ¬∃ j, s ↑j = ∅
H' : ∃ j, BddBelow (range fun i => f ↑i)
⊢ ∃ j, BddBelow (range fun i => f... | f25bd96869eaf9b2 |
MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | theorem map_linearMap_addHaar_eq_smul_addHaar {f : E →ₗ[ℝ] E} (hf : LinearMap.det f ≠ 0) :
Measure.map f μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ | case h
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝¹ : FiniteDimensional ℝ (ι → ℝ)
this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)
e :... | simp only [id, Function.comp_apply, LinearEquiv.symm_apply_apply] | no goals | 99679000a5dbc25a |
List.last_eq_of_concat_eq | Mathlib/.lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean | theorem last_eq_of_concat_eq {a b : α} {l₁ l₂ : List α} : concat l₁ a = concat l₂ b → a = b | α : Type u_1
a b : α
l₁ l₂ : List α
⊢ l₁.concat a = l₂.concat b → a = b | simp only [concat_eq_append] | α : Type u_1
a b : α
l₁ l₂ : List α
⊢ l₁ ++ [a] = l₂ ++ [b] → a = b | 478662c2630897dd |
Nat.max_sq_add_min_le_pair | Mathlib/Data/Nat/Pairing.lean | theorem max_sq_add_min_le_pair (m n : ℕ) : max m n ^ 2 + min m n ≤ pair m n | case inl
m n : ℕ
h : m < n
⊢ (m ⊔ n) ^ 2 + m ⊓ n ≤ if m < n then n * n + m else m * m + m + n | rw [if_pos h, max_eq_right h.le, min_eq_left h.le, Nat.pow_two] | no goals | bf37f67bbb78318d |
IsIntegrallyClosed.eq_map_mul_C_of_dvd | Mathlib/RingTheory/Polynomial/GaussLemma.lean | theorem IsIntegrallyClosed.eq_map_mul_C_of_dvd [IsIntegrallyClosed R] {f : R[X]} (hf : f.Monic)
{g : K[X]} (hg : g ∣ f.map (algebraMap R K)) :
∃ g' : R[X], g'.map (algebraMap R K) * (C <| leadingCoeff g) = g | R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
f : R[X]
hf : f.Monic
g : K[X]
hg : g ∣ map (algebraMap R K) f
g_ne_0 : g ≠ 0
g_mul_dvd : g * C g.leadingCoeff⁻¹ ∣ map (algebraMap R K) f
algeq : ↥(integralClosure R K) ≃ₐ[R] R :=... | have H :=
(mem_lifts _).1
(integralClosure.mem_lifts_of_monic_of_dvd_map K hf (monic_mul_leadingCoeff_inv g_ne_0)
g_mul_dvd) | R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
f : R[X]
hf : f.Monic
g : K[X]
hg : g ∣ map (algebraMap R K) f
g_ne_0 : g ≠ 0
g_mul_dvd : g * C g.leadingCoeff⁻¹ ∣ map (algebraMap R K) f
algeq : ↥(integralClosure R K) ≃ₐ[R] R :=... | 918c3567824cf666 |
MeasureTheory.JordanDecomposition.toSignedMeasure_injective | Mathlib/MeasureTheory/Decomposition/Jordan.lean | theorem toSignedMeasure_injective : Injective <| @JordanDecomposition.toSignedMeasure α _ | α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasure ≤[T] ... | rw [measure_union,
show j₂.posPart (i ∩ T) = 0 from
nonpos_iff_eq_zero.1 (hT₄ ▸ measure_mono Set.inter_subset_right),
zero_add] | case hd
α : Type u_1
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : j₁.toSignedMeasure ≤[S] 0
hS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : j₁.toSignedMeasu... | dcb73e0ca837cd73 |
cfcₙHom_comp | Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean | theorem cfcₙHom_comp [UniqueHom 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 | case refine_2
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower ... | simp only [φ, ψ, NonUnitalStarAlgHom.comp_apply, NonUnitalStarAlgHom.coe_mk',
NonUnitalAlgHom.coe_mk] | case refine_2
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹² : CommSemiring R
inst✝¹¹ : Nontrivial R
inst✝¹⁰ : StarRing R
inst✝⁹ : MetricSpace R
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : ContinuousStar R
inst✝⁶ : NonUnitalRing A
inst✝⁵ : StarRing A
inst✝⁴ : TopologicalSpace A
inst✝³ : Module R A
inst✝² : IsScalarTower ... | 5eff7aa8f4f5d67a |
TopCat.GlueData.rel_equiv | Mathlib/Topology/Gluing.lean | theorem rel_equiv : Equivalence D.Rel :=
⟨fun x => ⟨inv (D.f _ _) x.2, IsIso.inv_hom_id_apply (D.f x.fst x.fst) _,
-- Use `elementwise_of%` elaborator instead of `IsIso.inv_hom_id_apply` to work around
-- `ConcreteCategory`/`HasForget` mismatch:
by simp [elementwise_of% IsIso.inv_hom_id (D.f x.fst x.fst)]... | case mk.mk.mk.intro.intro.intro.intro
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (ConcreteCategory.hom (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst)) x = ⟨i, a⟩.snd
e₂ :
(ConcreteCategory.hom (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst)) ((ConcreteCategory.hom (D.t ⟨i, a⟩.fst ⟨j... | let z := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv ⟨⟨_, _⟩, e₂.trans e₃.symm⟩ | case mk.mk.mk.intro.intro.intro.intro
D : GlueData
i : D.J
a : ↑(D.U i)
j : D.J
b : ↑(D.U j)
k : D.J
c : ↑(D.U k)
x : ↑(D.V (⟨i, a⟩.fst, ⟨j, b⟩.fst))
e₁ : (ConcreteCategory.hom (D.f ⟨i, a⟩.fst ⟨j, b⟩.fst)) x = ⟨i, a⟩.snd
e₂ :
(ConcreteCategory.hom (D.f ⟨j, b⟩.fst ⟨i, a⟩.fst)) ((ConcreteCategory.hom (D.t ⟨i, a⟩.fst ⟨j... | ff5fea99544ad2bd |
ContinuousMap.sublattice_closure_eq_top | Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean | theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
(inf_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊓ g ∈ L)
(sup_mem : ∀ᵉ (f ∈ L) (g ∈ L), f ⊔ g ∈ L) (sep : L.SeparatesPointsStrongly) :
closure L = ⊤ | case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y... | let k : (L : Type _) :=
⟨xs.inf' xs_nonempty fun x => (h x : C(X, ℝ)),
Finset.inf'_mem _ inf_mem _ _ _ fun x _ => (h x).2⟩ | case pos
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
L : Set C(X, ℝ)
nA : L.Nonempty
inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L
sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L
sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y
f : C(X, ℝ)
ε : ℝ
pos : 0 < ε
nX : Nonempty X
g : X → X → C(X, ℝ)
hg : ∀ (x y... | fe3c7b7c90fed6eb |
FiberBundle.exists_trivialization_Icc_subset | Mathlib/Topology/FiberBundle/Basic.lean | theorem FiberBundle.exists_trivialization_Icc_subset [ConditionallyCompleteLinearOrder B]
[OrderTopology B] [FiberBundle F E] (a b : B) :
∃ e : Trivialization F (π F E), Icc a b ⊆ e.baseSet | B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivialization F TotalSpace.pr... | obtain ⟨d, hdcb, hd⟩ : ∃ d ∈ Ioc c b, Ico c d ⊆ ec.baseSet :=
(mem_nhdsGE_iff_exists_mem_Ioc_Ico_subset hlt).1
(mem_nhdsWithin_of_mem_nhds <| IsOpen.mem_nhds ec.open_baseSet (hec ⟨hc.1, le_rfl⟩)) | case intro.intro
B : Type u_2
F : Type u_3
inst✝⁶ : TopologicalSpace B
inst✝⁵ : TopologicalSpace F
E : B → Type u_5
inst✝⁴ : TopologicalSpace (TotalSpace F E)
inst✝³ : (b : B) → TopologicalSpace (E b)
inst✝² : ConditionallyCompleteLinearOrder B
inst✝¹ : OrderTopology B
inst✝ : FiberBundle F E
a b : B
ea : Trivializatio... | 0bdc8a2531150ce5 |
Set.preimage_eq | Mathlib/Data/Rel.lean | theorem preimage_eq (f : α → β) (s : Set β) : f ⁻¹' s = (Function.graph f).preimage s | α : Type u_1
β : Type u_2
f : α → β
s : Set β
⊢ f ⁻¹' s = (Function.graph f).preimage s | simp [Set.preimage, Rel.preimage, Rel.inv, flip, Rel.image] | no goals | a2db40cb9d7f6b63 |
IsTopologicalGroup.exists_mulInvClosureNhd | Mathlib/Topology/Algebra/OpenSubgroup.lean | @[to_additive]
lemma exists_mulInvClosureNhd {W : Set G} (WClopen : IsClopen W) :
∃ T, mulInvClosureNhd T W | case h
G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : S ∈ 𝓝 1
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen U
onememU : 1 ∈ U
⊢ mulInvClosureNhd (U ∩ U⁻¹) W | constructor | case h.nhd
G : Type u_1
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : IsTopologicalGroup G
inst✝ : CompactSpace G
W : Set G
WClopen : IsClopen W
S : Set G
Smemnhds : S ∈ 𝓝 1
mulclose : W * S ⊆ W
U : Set G
UsubS : U ⊆ S
Uopen : IsOpen U
onememU : 1 ∈ U
⊢ U ∩ U⁻¹ ∈ 𝓝 1
case h.inv
G : Type u_1
inst✝³ : Topologi... | 839b8de3a333177c |
Multiset.prod_eq_zero_iff | Mathlib/Algebra/BigOperators/Ring/Multiset.lean | @[simp] lemma prod_eq_zero_iff : s.prod = 0 ↔ (0 : α) ∈ s :=
Quotient.inductionOn s fun l ↦ by rw [quot_mk_to_coe, prod_coe]; exact List.prod_eq_zero_iff
| α : Type u_2
inst✝² : CommMonoidWithZero α
inst✝¹ : NoZeroDivisors α
inst✝ : Nontrivial α
s : Multiset α
l : List α
⊢ l.prod = 0 ↔ 0 ∈ ↑l | exact List.prod_eq_zero_iff | no goals | a8ac0f2168c4fc46 |
IsFreeGroupoid.SpanningTree.endIsFree | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | /-- Given a free groupoid and an arborescence of its generating quiver, the vertex
group at the root is freely generated by loops coming from generating arrows
in the complement of the tree. -/
lemma endIsFree : IsFreeGroup (End (root' T)) :=
IsFreeGroup.ofUniqueLift ((wideSubquiverEquivSetTotal <| wideSubqui... | G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
fun a b e =... | rintro ⟨⟨a, b, e⟩, h⟩ | case mk.mk
G : Type u
inst✝³ : Groupoid G
inst✝² : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst✝ : Group X
f : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X
f' : Labelling (Generators G) X :=
... | 4742e81f46ca4ae4 |
Multiset.dvd_gcd | Mathlib/Algebra/GCDMonoid/Multiset.lean | theorem dvd_gcd {s : Multiset α} {a : α} : a ∣ s.gcd ↔ ∀ b ∈ s, a ∣ b :=
Multiset.induction_on s (by simp)
(by simp +contextual [or_imp, forall_and, dvd_gcd_iff])
| α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Multiset α
a : α
⊢ ∀ (a_1 : α) (s : Multiset α), (a ∣ s.gcd ↔ ∀ b ∈ s, a ∣ b) → (a ∣ (a_1 ::ₘ s).gcd ↔ ∀ b ∈ a_1 ::ₘ s, a ∣ b) | simp +contextual [or_imp, forall_and, dvd_gcd_iff] | no goals | d31ce47f486f81e6 |
Turing.ToPartrec.Code.exists_code | Mathlib/Computability/TMConfig.lean | theorem exists_code {n} {f : List.Vector ℕ n →. ℕ} (hf : Nat.Partrec' f) :
∃ c : Code, ∀ v : List.Vector ℕ n, c.eval v.1 = pure <$> f v | case neg
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ →. ℕ
n✝ : ℕ
f : List.Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n✝
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' v₀ : List ℕ
n : ℕ
h2 :
v' ∈
PFun.fix
(fun v =>
(cf.eval v).bind fun y =>
Part.some (if y.hea... | refine IH (n.succ::v.val) (by simp_all) _ rfl fun m h' => ?_ | case neg
n✝¹ : ℕ
f✝ : List.Vector ℕ n✝¹ →. ℕ
n✝ : ℕ
f : List.Vector ℕ (n✝ + 1) → ℕ
a✝ : Nat.Partrec' ↑f
cf : Code
v : List.Vector ℕ n✝
hf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]
v' v₀ : List ℕ
n : ℕ
h2 :
v' ∈
PFun.fix
(fun v =>
(cf.eval v).bind fun y =>
Part.some (if y.hea... | 01823c87e1740c3b |
ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
(h : ContDiffWithinAt 𝕜 n f s x) (hmn : m < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x | case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
m : ℕ
h : ContDiffWithinAt 𝕜 n f s x
hmn : ↑m < n
hs : UniqueDiffOn 𝕜 (... | have C : DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f t) t x :=
hu.differentiableOn_iteratedFDerivWithin (Nat.cast_lt.2 m.lt_succ_self) (hs.inter uo) x
⟨mem_insert _ _, xu⟩ | case intro.intro.intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type uF
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x : E
n : WithTop ℕ∞
m : ℕ
h : ContDiffWithinAt 𝕜 n f s x
hmn : ↑m < n
hs : UniqueDiffOn 𝕜 (... | 028f7d28b24d29ba |
ProbabilityTheory.IsMeasurableRatCDF.monotone_stieltjesFunctionAux | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | lemma IsMeasurableRatCDF.monotone_stieltjesFunctionAux (a : α) :
Monotone (IsMeasurableRatCDF.stieltjesFunctionAux f a) | α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
⊢ Nonempty { r' // y < ↑r' } | obtain ⟨r, hrx⟩ := exists_rat_gt y | case intro
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
r : ℚ
hrx : y < ↑r
⊢ Nonempty { r' // y < ↑r' } | b8b32d0abfad29b9 |
Real.continuousAt_rpow_of_ne | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | theorem continuousAt_rpow_of_ne (p : ℝ × ℝ) (hp : p.1 ≠ 0) :
ContinuousAt (fun p : ℝ × ℝ => p.1 ^ p.2) p | case inl
p : ℝ × ℝ
hp : p.1 < 0
⊢ ContinuousAt (fun p => p.1 ^ p.2) p | rw [continuousAt_congr (rpow_eq_nhds_of_neg hp)] | case inl
p : ℝ × ℝ
hp : p.1 < 0
⊢ ContinuousAt (fun x => rexp (log x.1 * x.2) * cos (x.2 * π)) p | 2e69ae2bc8a153ac |
NumberField.mixedEmbedding.volume_preserving_negAt | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | theorem volume_preserving_negAt [NumberField K] :
MeasurePreserving (negAt s) | case pos
K : Type u_1
inst✝¹ : Field K
s : Set { w // w.IsReal }
inst✝ : NumberField K
w : { w // w.IsReal }
hw : w ∈ s
⊢ MeasurePreserving (↑(neg ℝ).toLinearEquiv).toAddHom.1 volume volume | exact Measure.measurePreserving_neg _ | no goals | 299b1d1145ec3899 |
Ideal.map_mk_comap_factor | Mathlib/RingTheory/Ideal/Quotient/PowTransition.lean | lemma Ideal.map_mk_comap_factor [J.IsTwoSided] [K.IsTwoSided] (hIJ : J ≤ I) (hJK : K ≤ J) :
(I.map (mk J)).comap (factor hJK) = I.map (mk K) | case h.refine_2.intro.intro
R : Type u_1
inst✝² : Ring R
I J K : Ideal R
inst✝¹ : J.IsTwoSided
inst✝ : K.IsTwoSided
hIJ : J ≤ I
hJK : K ≤ J
x : R ⧸ K
h : x ∈ map (mk K) I
r : R
hr : r ∈ ↑I
eq : (mk K) r = x
⊢ x ∈ comap (factor hJK) (map (mk J) I) | simpa only [← eq] using mem_map_of_mem (mk J) hr | no goals | c5e0a4d45fe78c35 |
Polynomial.support_update | Mathlib/Algebra/Polynomial/Basic.lean | theorem support_update (p : R[X]) (n : ℕ) (a : R) [Decidable (a = 0)] :
support (p.update n a) = if a = 0 then p.support.erase n else insert n p.support | R : Type u
inst✝¹ : Semiring R
p : R[X]
n : ℕ
a : R
inst✝ : Decidable (a = 0)
⊢ (p.update n a).support = if a = 0 then p.support.erase n else insert n p.support | cases p | case ofFinsupp
R : Type u
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
toFinsupp✝ : R[ℕ]
⊢ ({ toFinsupp := toFinsupp✝ }.update n a).support =
if a = 0 then { toFinsupp := toFinsupp✝ }.support.erase n else insert n { toFinsupp := toFinsupp✝ }.support | 4d38e0e4d9b9c2de |
exteriorPower.pairingDual_apply_apply_eq_one_zero | Mathlib/LinearAlgebra/ExteriorPower/Pairing.lean | lemma pairingDual_apply_apply_eq_one_zero (a b : Fin n ↪o ι) (h : a ≠ b) :
pairingDual R M n (ιMulti _ _ (f ∘ a)) (ιMulti _ _ (x ∘ b)) = 0 | R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
σ : Equiv.Perm (Fin n)
x✝ : σ ∈ Finset.univ
h' : ¬∏ x_1 : Fin n, (f (a x_1)) (x (b (σ ... | have h'' := congr_fun this | R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : LinearOrder ι
x : ι → M
f : ι → Module.Dual R M
h₀ : ∀ ⦃i j : ι⦄, i ≠ j → (f i) (x j) = 0
n : ℕ
a b : Fin n ↪o ι
h : a ≠ b
σ : Equiv.Perm (Fin n)
x✝ : σ ∈ Finset.univ
h' : ¬∏ x_1 : Fin n, (f (a x_1)) (x (b (σ ... | fd3ea1bd40177425 |
Bimod.whisker_assoc_bimod | Mathlib/CategoryTheory/Monoidal/Bimod.lean | theorem whisker_assoc_bimod {W X Y Z : Mon_ C} (M : Bimod W X) {N N' : Bimod X Y} (f : N ⟶ N')
(P : Bimod Y Z) :
whiskerRight (whiskerLeft M f) P =
(associatorBimod M N P).hom ≫
whiskerLeft M (whiskerRight f P) ≫ (associatorBimod M N' P).inv | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M : Bimod W X
N N' : Bimod X Y
f :... | slice_lhs 1 2 => rw [← comp_whiskerRight, ι_colimMap, parallelPairHom_app_one] | case h.h
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon_ C
M : Bimod W X
N N' : Bimod X Y
f :... | 64d3d49872bb8a8d |
Vector.find?_toList | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem find?_toList (p : α → Bool) (v : Vector α n) :
v.toList.find? p = v.find? p | α : Type
n : Nat
p : α → Bool
v : Vector α n
⊢ List.find? p v.toList = find? p v | cases v | case mk
α : Type
n : Nat
p : α → Bool
toArray✝ : Array α
size_toArray✝ : toArray✝.size = n
⊢ List.find? p { toArray := toArray✝, size_toArray := size_toArray✝ }.toList =
find? p { toArray := toArray✝, size_toArray := size_toArray✝ } | 36dd87d502ce7afa |
LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable | Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean | theorem LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable
[FiniteDimensional F E] [Algebra.IsSeparable F E]
(h : LinearIndependent F v) : LinearIndependent F (v · ^ q ^ n) | case h.e'_4.h
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝¹ : FiniteDimensional F E
inst✝ : Algebra.IsSeparable F E
h : LinearIndependent F v
h' : LinearIndepOn F id (Set.range v)
ι' : Set E := h'.extend ⋯
b : Basis (↑ι') F E := Basis... | rw [Basis.extend_apply_self] | no goals | ad56485109349825 |
nhdsSet_univ | Mathlib/Topology/NhdsSet.lean | theorem nhdsSet_univ : 𝓝ˢ (univ : Set X) = ⊤ | X : Type u_1
inst✝ : TopologicalSpace X
⊢ 𝓝ˢ univ = ⊤ | rw [isOpen_univ.nhdsSet_eq, principal_univ] | no goals | b67f0727997ec829 |
Std.Tactic.BVDecide.BVExpr.bitblast.blastUdiv.denote_blastDivSubtractShift_r | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_blastDivSubtractShift_r (aig : AIG α) (assign : α → Bool) (lhs rhs : BitVec w)
(falseRef trueRef : AIG.Ref aig) (n d : AIG.RefVec aig w) (wn wr : Nat)
(q r : AIG.RefVec aig w) (qbv rbv : BitVec w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, n.get idx hidx, assign⟧ = lhs.getLsbD idx)
(h... | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀... | rw [BVPred.mkUlt_denote_eq (lhs := rbv.shiftConcat (lhs.getLsbD (wn - 1))) (rhs := rhs)] | α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
assign : α → Bool
lhs rhs : BitVec w
falseRef trueRef : aig.Ref
n d : aig.RefVec w
wn wr : Nat
q r : aig.RefVec w
qbv rbv : BitVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := n.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀... | 249a96ab3a68ac7e |
lemma₁ | Mathlib/NumberTheory/LSeries/SumCoeff.lean | theorem lemma₁ (hlim : Tendsto (fun n : ℕ ↦ (∑ k ∈ Icc 1 n, f k) / n) atTop (𝓝 l))
{s : ℝ} (hs : 1 < s) :
IntegrableOn (fun t : ℝ ↦ (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * (t : ℂ) ^ (-(s : ℂ) - 1)) (Set.Ici 1) | f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
s : ℝ
hs : 1 < s
h₁ : LocallyIntegrableOn (fun t => (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume
⊢ (fun t => ∑ k ∈ Icc 1 ⌊t⌋₊, f k) =O[atTop] fun t => t | refine IsBigO.trans_isEquivalent ?_ isEquivalent_nat_floor | f : ℕ → ℂ
l : ℂ
hlim : Tendsto (fun n => (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)
s : ℝ
hs : 1 < s
h₁ : LocallyIntegrableOn (fun t => (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * ↑t ^ (-↑s - 1)) (Set.Ici 1) volume
⊢ (fun t => ∑ k ∈ Icc 1 ⌊t⌋₊, f k) =O[atTop] fun x => ↑⌊x⌋₊ | 44984ad91259849f |
Filter.tendsto_const_mul_atTop_iff_pos | Mathlib/Order/Filter/AtTopBot/Field.lean | theorem tendsto_const_mul_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
Tendsto (fun x => r * f x) l atTop ↔ 0 < r | case intro.intro
α : Type u_1
β : Type u_2
inst✝¹ : LinearOrderedSemifield α
l : Filter β
f : β → α
r : α
inst✝ : l.NeBot
h : Tendsto f l atTop
hrf : Tendsto (fun x => r * f x) l atTop
hr : r ≤ 0
x : β
hx : 0 ≤ f x
hrx : 0 < r * f x
⊢ False | exact (mul_nonpos_of_nonpos_of_nonneg hr hx).not_lt hrx | no goals | f67b1f5dc7971625 |
AlgebraicGeometry.isCompact_basicOpen | Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean | theorem isCompact_basicOpen (X : Scheme) {U : X.Opens} (hU : IsCompact (U : Set X))
(f : Γ(X, U)) : IsCompact (X.basicOpen f : Set X) | case intro.intro
X : Scheme
U : X.Opens
hU : IsCompact ↑U
f : ↑Γ(X, U)
s : Set ↑X.affineOpens
hs : s.Finite
e : ↑U = ⋃ i ∈ s, ↑↑i
g : ↑s → ↑X.affineOpens := fun V => ⟨↑↑V ⊓ X.basicOpen f, ⋯⟩
this : Finite ↑s
⊢ ⋃ i ∈ s, ↑↑i ∩ ↑(X.toRingedSpace.basicOpen f) = ⋃ i ∈ Set.range g, ↑↑i | apply le_antisymm <;> apply Set.iUnion₂_subset | case intro.intro.a.h
X : Scheme
U : X.Opens
hU : IsCompact ↑U
f : ↑Γ(X, U)
s : Set ↑X.affineOpens
hs : s.Finite
e : ↑U = ⋃ i ∈ s, ↑↑i
g : ↑s → ↑X.affineOpens := fun V => ⟨↑↑V ⊓ X.basicOpen f, ⋯⟩
this : Finite ↑s
⊢ ∀ i ∈ s, ↑↑i ∩ ↑(X.toRingedSpace.basicOpen f) ⊆ ⋃ i ∈ Set.range g, ↑↑i
case intro.intro.a.h
X : Scheme
U ... | 85d4cb2bab8ba3ba |
ENNReal.lintegral_Lp_add_le | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | theorem lintegral_Lp_add_le {p : ℝ} {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hg : AEMeasurable g μ)
(hp1 : 1 ≤ p) :
(∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
(∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) | α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤
hg_top : ¬∫⁻ (a : α), g a ^ p ∂μ = ⊤
h1 : ¬p = 1
⊢ p ≠ 1 | exact h1 | no goals | 917d466c6455bcc8 |
exists_prime_orderOf_dvd_card | Mathlib/GroupTheory/Perm/Cycle/Type.lean | theorem _root_.exists_prime_orderOf_dvd_card {G : Type*} [Group G] [Fintype G] (p : ℕ)
[hp : Fact p.Prime] (hdvd : p ∣ Fintype.card G) : ∃ x : G, orderOf x = p | G : Type u_3
inst✝¹ : Group G
inst✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ∣ Fintype.card G
hp' : p - 1 ≠ 0
Scard : p ∣ Fintype.card ↑(vectorsProdEqOne G p)
f : ℕ → ↑(vectorsProdEqOne G p) → ↑(vectorsProdEqOne G p) := fun k v => VectorsProdEqOne.rotate v k
hf1 : ∀ (v : ↑(vectorsProdEqOne G p)), f 0 v = v
hf... | rw [hf2, add_tsub_cancel_of_le hp.out.one_lt.le, hf3] | no goals | 8d70eb3bee101171 |
Monoid.exponent_prod | Mathlib/GroupTheory/Exponent.lean | theorem Monoid.exponent_prod {M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂] :
exponent (M₁ × M₂) = lcm (exponent M₁) (exponent M₂) | case refine_1
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
⊢ exponent (M₁ × M₂) ∣ GCDMonoid.lcm (exponent M₁) (exponent M₂) | refine exponent_dvd_of_forall_pow_eq_one fun g ↦ ?_ | case refine_1
M₁ : Type u_1
M₂ : Type u_2
inst✝¹ : Monoid M₁
inst✝ : Monoid M₂
g : M₁ × M₂
⊢ g ^ GCDMonoid.lcm (exponent M₁) (exponent M₂) = 1 | 6a9d76694de00ba8 |
refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set | Mathlib/Topology/Compactness/Paracompact.lean | theorem refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set [WeaklyLocallyCompactSpace X]
[SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : ∀ x, ι x → Prop} {B : ∀ x, ι x → Set X}
{s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)) :
∃ (α : Type v) (c : α → X) (r : ∀ a, ι... | case refine_2
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := CompactExhaustion.cho... | rcases mem_iUnion₂.1 (hT _ ⟨hKcov x, hx⟩) with ⟨⟨c, hc⟩, hcT, hcx⟩ | case refine_2.intro.mk.intro
X : Type v
inst✝³ : TopologicalSpace X
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : SigmaCompactSpace X
inst✝ : T2Space X
ι : X → Type u
p : (x : X) → ι x → Prop
B : (x : X) → ι x → Set X
s : Set X
hs : IsClosed s
hB : ∀ x ∈ s, (𝓝 x).HasBasis (p x) (B x)
K' : CompactExhaustion X := Compac... | d7d8360667f4b953 |
HNNExtension.NormalWord.of_smul_eq_smul | Mathlib/GroupTheory/HNNExtension.lean | theorem of_smul_eq_smul (g : G) (w : NormalWord d) :
(of g : HNNExtension G A B φ) • w = g • w | G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g : G
w : NormalWord d
⊢ of g • w = g • w | simp [instHSMul, SMul.smul, MulAction.toEndHom] | no goals | c79a48fcf81ba796 |
Std.Tactic.BVDecide.BVExpr.bitblast.denote_blastUdiv | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Lemmas/Operations/Udiv.lean | theorem denote_blastUdiv (aig : AIG α) (lhs rhs : BitVec w) (assign : α → Bool)
(input : BinaryRefVec aig w)
(hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.lhs.get idx hidx, assign⟧ = lhs.getLsbD idx)
(hright : ∀ (idx : Nat) (hidx : idx < w), ⟦aig, input.rhs.get idx hidx, assign⟧ = rhs.getLsbD i... | case htrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := a... | rw [AIG.LawfulOperator.denote_mem_prefix (f := BVPred.mkEq)] | case htrue
α : Type
inst✝¹ : Hashable α
inst✝ : DecidableEq α
w : Nat
aig : AIG α
lhs rhs : BitVec w
assign : α → Bool
input : aig.BinaryRefVec w
hleft : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := aig, ref := input.lhs.get idx hidx }⟧ = lhs.getLsbD idx
hright : ∀ (idx : Nat) (hidx : idx < w), ⟦assign, { aig := a... | 1e06923b53c4a62f |
CategoryTheory.Abelian.epiWithInjectiveKernel_iff | Mathlib/CategoryTheory/Abelian/EpiWithInjectiveKernel.lean | /-- A morphism `g : X ⟶ Y` is epi with an injective kernel iff there exists a morphism
`f : I ⟶ X` with `I` injective such that `f ≫ g = 0` and
the short complex `I ⟶ X ⟶ Y` has a splitting. -/
lemma epiWithInjectiveKernel_iff {X Y : C} (g : X ⟶ Y) :
epiWithInjectiveKernel g ↔ ∃ (I : C) (_ : Injective I) (f : I ⟶ X... | case mpr.intro.intro.intro.intro.intro
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
X Y : C
g : X ⟶ Y
I : C
w✝ : Injective I
f : I ⟶ X
w : f ≫ g = 0
σ : (ShortComplex.mk f g w).Splitting
this : IsSplitEpi g
e : I ≅ kernel g := ⋯.fIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (ShortComple... | exact ⟨inferInstance, Injective.of_iso e inferInstance⟩ | no goals | 0c2777975f72cb4f |
RootPairing.Base.eq_one_or_neg_one_of_mem_support_of_smul_mem | Mathlib/LinearAlgebra/RootSystem/Base.lean | lemma eq_one_or_neg_one_of_mem_support_of_smul_mem [Finite ι] [CharZero R]
[NoZeroSMulDivisors ℤ M] [NoZeroSMulDivisors ℤ N]
(i : ι) (h : i ∈ b.support) (t : R) (ht : t • P.root i ∈ range P.root) :
t = 1 ∨ t = - 1 | case intro.intro.intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
b : P.Base
inst✝³ : Finite ι
inst✝² : CharZero R
inst✝¹ : NoZeroSMulDivisors ℤ M
inst✝ : NoZeroSMulDivisors ℤ N
i... | norm_cast at this | case intro.intro.intro
ι : Type u_1
R : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : RootPairing ι R M N
b : P.Base
inst✝³ : Finite ι
inst✝² : CharZero R
inst✝¹ : NoZeroSMulDivisors ℤ M
inst✝ : NoZeroSMulDivisors ℤ N
i... | 439980ec01794b6b |
AlgebraNorm.extends_norm | Mathlib/Analysis/Normed/Algebra/Norm.lean | theorem extends_norm (hf1 : f 1 = 1) (a : R) : f (algebraMap R S a) = ‖a‖ | R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
hf1 : f 1 = 1
a : R
⊢ f ((algebraMap R S) a) = ‖a‖ | rw [Algebra.algebraMap_eq_smul_one] | R : Type u_1
inst✝² : SeminormedCommRing R
S : Type u_2
inst✝¹ : Ring S
inst✝ : Algebra R S
f : AlgebraNorm R S
hf1 : f 1 = 1
a : R
⊢ f (a • 1) = ‖a‖ | 431c6dca7cb059de |
Complex.convexHull_reProdIm | Mathlib/Analysis/Complex/Convex.lean | /-- A version of `convexHull_prod` for `Set.reProdIm`. -/
lemma convexHull_reProdIm (s t : Set ℝ) :
convexHull ℝ (s ×ℂ t) = convexHull ℝ s ×ℂ convexHull ℝ t :=
calc
convexHull ℝ (equivRealProdLm ⁻¹' (s ×ˢ t)) = equivRealProdLm ⁻¹' convexHull ℝ (s ×ˢ t) | s t : Set ℝ
⊢ ⇑equivRealProdLm ⁻¹' (convexHull ℝ) s ×ˢ (convexHull ℝ) t = (convexHull ℝ) s ×ℂ (convexHull ℝ) t | rfl | no goals | 592c11d91a158a4b |
CantorScheme.ClosureAntitone.map_of_vanishingDiam | Mathlib/Topology/MetricSpace/CantorScheme.lean | theorem ClosureAntitone.map_of_vanishingDiam [CompleteSpace α] (hdiam : VanishingDiam A)
(hanti : ClosureAntitone A) (hnonempty : ∀ l, (A l).Nonempty) : (inducedMap A).1 = univ | β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
⊢ x ∈ (inducedMap A).fst | have umem : ∀ n m : ℕ, n ≤ m → u m ∈ A (res x n) := by
have : Antitone fun n : ℕ => A (res x n) := by
refine antitone_nat_of_succ_le ?_
intro n
apply hanti.antitone
intro n m hnm
exact this hnm (hu _) | β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
⊢ x ∈ (inducedMap A).fst | 84475af12dd43d16 |
Ideal.finsuppTotal_apply_eq_of_fintype | Mathlib/RingTheory/Ideal/Operations.lean | theorem finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
finsuppTotal ι M I v f = ∑ i, (f i : R) • v i | case h
ι : Type u_1
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I : Ideal R
v : ι → M
inst✝ : Fintype ι
f : ι →₀ ↥I
⊢ ∀ (i : ι), ↑0 • v i = 0 | exact fun _ => zero_smul _ _ | no goals | b52934c6ae1d2ce2 |
MeasureTheory.integrableOn_image_iff_integrableOn_abs_deriv_smul | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem integrableOn_image_iff_integrableOn_abs_deriv_smul {s : Set ℝ} {f : ℝ → ℝ} {f' : ℝ → ℝ}
(hs : MeasurableSet s) (hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (hf : InjOn f s)
(g : ℝ → F) : IntegrableOn g (f '' s) ↔ IntegrableOn (fun x => |f' x| • g (f x)) s | F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set ℝ
f f' : ℝ → ℝ
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
hf : InjOn f s
g : ℝ → F
⊢ IntegrableOn g (f '' s) volume ↔ IntegrableOn (fun x => |f' x| • g (f x)) s volume | simpa only [det_one_smulRight] using
integrableOn_image_iff_integrableOn_abs_det_fderiv_smul volume hs
(fun x hx => (hf' x hx).hasFDerivWithinAt) hf g | no goals | cfe539c64d05a471 |
AffineMap.pi_ext_zero | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | theorem pi_ext_zero (h : ∀ i x, f (Pi.single i x) = g (Pi.single i x)) (h₂ : f 0 = g 0) :
f = g | case h₁.h
k : Type u_2
V2 : Type u_5
P2 : Type u_6
inst✝⁷ : Ring k
inst✝⁶ : AddCommGroup V2
inst✝⁵ : AffineSpace V2 P2
inst✝⁴ : Module k V2
ι : Type u_9
φv : ι → Type u_10
inst✝³ : (i : ι) → AddCommGroup (φv i)
inst✝² : (i : ι) → Module k (φv i)
inst✝¹ : Finite ι
inst✝ : DecidableEq ι
f g : ((i : ι) → φv i) →ᵃ[k] P2
h ... | rw [vadd_eq_add, add_zero] at s₂ s₃ | case h₁.h
k : Type u_2
V2 : Type u_5
P2 : Type u_6
inst✝⁷ : Ring k
inst✝⁶ : AddCommGroup V2
inst✝⁵ : AffineSpace V2 P2
inst✝⁴ : Module k V2
ι : Type u_9
φv : ι → Type u_10
inst✝³ : (i : ι) → AddCommGroup (φv i)
inst✝² : (i : ι) → Module k (φv i)
inst✝¹ : Finite ι
inst✝ : DecidableEq ι
f g : ((i : ι) → φv i) →ᵃ[k] P2
h ... | f4a501a2d9ab4d4a |
Turing.TM2to1.tr_respects_aux | Mathlib/Computability/TuringMachine.lean | theorem tr_respects_aux {q v T k} {S : ∀ k, List (Γ k)}
(hT : ∀ k, ListBlank.map (proj k) T = ListBlank.mk ((S k).map some).reverse)
(o : StAct K Γ σ k)
(IH : ∀ {v : σ} {S : ∀ k : K, List (Γ k)} {T : ListBlank (∀ k, Option (Γ k))},
(∀ k, ListBlank.map (proj k) T = ListBlank.mk ((S k).map some).reverse... | K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
q : TM2.Stmt Γ Λ σ
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct K Γ σ k
IH :
∀ {v : σ} {S ... | obtain ⟨T', hT', hrun⟩ := tr_respects_aux₂ (Λ := Λ) hT o | case intro.intro
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
q : TM2.Stmt Γ Λ σ
v : σ
T : ListBlank ((i : K) → Option (Γ i))
k : K
S : (k : K) → List (Γ k)
hT : ∀ (k : K), ListBlank.map (proj k) T = ListBlank.mk (List.map some (S k)).reverse
o : StAct K Γ σ k
IH ... | 87971184416ca292 |
SimpleGraph.deleteEdges_edgeSet | Mathlib/Combinatorics/SimpleGraph/Basic.lean | @[simp] lemma deleteEdges_edgeSet (G G' : SimpleGraph V) : G.deleteEdges G'.edgeSet = G \ G' | case Adj.h.h.a
V : Type u
G G' : SimpleGraph V
x✝¹ x✝ : V
⊢ (G.deleteEdges G'.edgeSet).Adj x✝¹ x✝ ↔ (G \ G').Adj x✝¹ x✝ | simp | no goals | 01becce2541a1a09 |
image_subset_closure_compl_image_compl_of_isOpen | Mathlib/Topology/ExtremallyDisconnected.lean | /-- Lemma 2.1 in [Gleason, *Projective topological spaces*][gleason1958]:
if $\rho$ is a continuous surjection from a topological space $E$ to a topological space $A$
satisfying the "Zorn subset condition", then $\rho(G)$ is contained in
the closure of $A \setminus \rho(E \setminus G)$ for any open set $G$ of $E$. -/
l... | case neg.intro.intro
A E : Type u
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN... | have nonempty : (G ∩ ρ⁻¹' N).Nonempty := ⟨e, mem_inter he <| mem_preimage.mpr hN⟩ | case neg.intro.intro
A E : Type u
inst✝¹ : TopologicalSpace A
inst✝ : TopologicalSpace E
ρ : E → A
ρ_cont : Continuous ρ
ρ_surj : Surjective ρ
zorn_subset : ∀ (E₀ : Set E), E₀ ≠ univ → IsClosed E₀ → ρ '' E₀ ≠ univ
G : Set E
hG : IsOpen G
G_empty : ¬G = ∅
N : Set A
N_open : IsOpen N
e : E
he : e ∈ G
ha : ρ e ∈ ρ '' G
hN... | 7e7582b54cb8739a |
ruzsaSzemerediNumberNat_asymptotic_lower_bound | Mathlib/Combinatorics/Extremal/RuzsaSzemeredi.lean | theorem ruzsaSzemerediNumberNat_asymptotic_lower_bound :
(fun n ↦ n ^ 2 * exp (-4 * sqrt (log n)) : ℕ → ℝ) =O[atTop]
fun n ↦ (ruzsaSzemerediNumberNat n : ℝ) | x : ℕ
hx : x ≥ 15
⊢ ↑x ≤ 12 * ↑((x - 3) / 6) | norm_cast | x : ℕ
hx : x ≥ 15
⊢ x ≤ 12 * ((x - 3) / 6) | 72990b3a12cc99c5 |
LieSubmodule.lieIdeal_oper_eq_linear_span' | Mathlib/Algebra/Lie/IdealOperations.lean | theorem lieIdeal_oper_eq_linear_span' [LieModule R L M] :
(↑⁅I, N⁆ : Submodule R M) = Submodule.span R { ⁅x, n⁆ | (x ∈ I) (n ∈ N) } | case e_s.h.mpr
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
N : LieSubmodule R L M
inst✝¹ : LieAlgebra R L
I : LieIdeal R L
inst✝ : LieModule R L M
m : M
⊢ m ∈ {x | ∃ x_1 ∈ I, ∃ n ∈ N, ⁅x_1, n⁆ = x} → m ∈ {x | ∃ x_1 n, ⁅↑x... | rintro ⟨x, hx, n, hn, rfl⟩ | case e_s.h.mpr.intro.intro.intro.intro
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
N : LieSubmodule R L M
inst✝¹ : LieAlgebra R L
I : LieIdeal R L
inst✝ : LieModule R L M
x : L
hx : x ∈ I
n : M
hn : n ∈ N
⊢ ⁅x, n⁆ ∈ {x | ... | 36507cbeea573138 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.insertUnitInvariant_insertUnit | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddResult.lean | theorem insertUnitInvariant_insertUnit {n : Nat} (assignments0 : Array Assignment)
(assignments0_size : assignments0.size = n) (units : Array (Literal (PosFin n)))
(assignments : Array Assignment) (assignments_size : assignments.size = n)
(foundContradiction : Bool) (l : Literal (PosFin n)) :
InsertUnit... | case neg
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < as... | exfalso | case neg
n : Nat
assignments0 : Array Assignment
assignments0_size : assignments0.size = n
units : Array (Literal (PosFin n))
assignments : Array Assignment
assignments_size : assignments.size = n
foundContradiction : Bool
l : Literal (PosFin n)
i : Fin n
i_in_bounds : ↑i < assignments.size
l_in_bounds : l.fst.val < as... | 4e9908103f3f6d5f |
Polynomial.reflect_mul_induction | Mathlib/Algebra/Polynomial/Reverse.lean | theorem reflect_mul_induction (cf cg : ℕ) :
∀ N O : ℕ,
∀ f g : R[X],
#f.support ≤ cf.succ →
#g.support ≤ cg.succ →
f.natDegree ≤ N →
g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g | R : Type u_1
inst✝ : Semiring R
cf cg : ℕ
⊢ ∀ (N O : ℕ) (f g : R[X]),
#f.support ≤ cf.succ →
#g.support ≤ cg.succ → f.natDegree ≤ N → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g | induction' cf with cf hcf | case zero
R : Type u_1
inst✝ : Semiring R
cg : ℕ
⊢ ∀ (N O : ℕ) (f g : R[X]),
#f.support ≤ Nat.succ 0 →
#g.support ≤ cg.succ → f.natDegree ≤ N → g.natDegree ≤ O → reflect (N + O) (f * g) = reflect N f * reflect O g
case succ
R : Type u_1
inst✝ : Semiring R
cg cf : ℕ
hcf :
∀ (N O : ℕ) (f g : R[X]),
#f.su... | 0a2af005637b9810 |
Vitali.exists_disjoint_covering_ae | Mathlib/MeasureTheory/Covering/Vitali.lean | theorem exists_disjoint_covering_ae
[PseudoMetricSpace α] [MeasurableSpace α] [OpensMeasurableSpace α]
[SecondCountableTopology α] (μ : Measure α) [IsLocallyFiniteMeasure μ] (s : Set α) (t : Set ι)
(C : ℝ≥0) (r : ι → ℝ) (c : ι → α) (B : ι → Set α) (hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a))
(μB : ∀ a ∈... | α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (cl... | refine ⟨1, fun r' hr' => ?_⟩ | α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (cl... | 4466d2fc3e80cb3f |
CharTwo.of_one_ne_zero_of_two_eq_zero | Mathlib/Algebra/CharP/Two.lean | theorem of_one_ne_zero_of_two_eq_zero (h₁ : (1 : R) ≠ 0) (h₂ : (2 : R) = 0) : CharP R 2 where
cast_eq_zero_iff' n | case inr
R : Type u_1
inst✝ : AddMonoidWithOne R
h₁ : 1 ≠ 0
h₂ : 2 = 0
n : ℕ
hn : Odd n
⊢ ¬↑n = 0 | rwa [natCast_eq_one_of_odd_of_two_eq_zero hn h₂] | no goals | 64f3abafce47a6cd |
MvPolynomial.degrees_rename | Mathlib/Algebra/MvPolynomial/Degrees.lean | theorem degrees_rename (f : σ → τ) (φ : MvPolynomial σ R) :
(rename f φ).degrees ⊆ φ.degrees.map f | R : Type u
σ : Type u_1
τ : Type u_2
inst✝ : CommSemiring R
f : σ → τ
φ : MvPolynomial σ R
⊢ ((rename f) φ).degrees ⊆ Multiset.map f φ.degrees | classical
intro i
rw [mem_degrees, Multiset.mem_map]
rintro ⟨d, hd, hi⟩
obtain ⟨x, rfl, hx⟩ := coeff_rename_ne_zero _ _ _ hd
simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi
rw [sum_apply, Finsupp.sum] at hi
contrapose! hi
rw [Finset.sum_eq_zero]
intro j hj
simp only [exists_prop, mem_degrees] at hi
special... | no goals | b4bc66e6c91de3f2 |
LinearMap.finrank_range_add_finrank_ker | Mathlib/LinearAlgebra/FiniteDimensional.lean | theorem finrank_range_add_finrank_ker [FiniteDimensional K V] (f : V →ₗ[K] V₂) :
finrank K (LinearMap.range f) + finrank K (LinearMap.ker f) = finrank K V | K : Type u
V : Type v
inst✝⁵ : DivisionRing K
inst✝⁴ : AddCommGroup V
inst✝³ : Module K V
V₂ : Type v'
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
inst✝ : FiniteDimensional K V
f : V →ₗ[K] V₂
⊢ finrank K (V ⧸ ker f) + finrank K ↥(ker f) = finrank K V | exact Submodule.finrank_quotient_add_finrank _ | no goals | 4e1d011075378c2d |
CompleteOrthogonalIdempotents.iff_ortho_complete | Mathlib/RingTheory/Idempotents.lean | /-- If a family is complete orthogonal, it consists of idempotents. -/
lemma CompleteOrthogonalIdempotents.iff_ortho_complete :
CompleteOrthogonalIdempotents e ↔ Pairwise (e · * e · = 0) ∧ ∑ i, e i = 1 | R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : Fintype I
⊢ CompleteOrthogonalIdempotents e ↔ (Pairwise fun x1 x2 => e x1 * e x2 = 0) ∧ ∑ i : I, e i = 1 | rw [completeOrthogonalIdempotents_iff, orthogonalIdempotents_iff, and_assoc, and_iff_right_of_imp] | R : Type u_1
inst✝¹ : Semiring R
I : Type u_3
e : I → R
inst✝ : Fintype I
⊢ (Pairwise fun x1 x2 => e x1 * e x2 = 0) ∧ ∑ i : I, e i = 1 → ∀ (i : I), IsIdempotentElem (e i) | 5c7c10bb3cc35a64 |
AffineSubspace.wSameSide_and_wOppSide_iff | Mathlib/Analysis/Convex/Side.lean | theorem wSameSide_and_wOppSide_iff {s : AffineSubspace R P} {x y : P} :
s.WSameSide x y ∧ s.WOppSide x y ↔ x ∈ s ∨ y ∈ s | case mp
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
⊢ s.WSameSide x y ∧ s.WOppSide x y → x ∈ s ∨ y ∈ s | rintro ⟨hs, ho⟩ | case mp.intro
R : Type u_1
V : Type u_2
P : Type u_4
inst✝³ : LinearOrderedField R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hs : s.WSameSide x y
ho : s.WOppSide x y
⊢ x ∈ s ∨ y ∈ s | f4466fd45c8d23c6 |
MeasureTheory.MemLp.eLpNormEssSup_indicator_norm_ge_eq_zero | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem MemLp.eLpNormEssSup_indicator_norm_ge_eq_zero (hf : MemLp f ∞ μ)
(hmeas : StronglyMeasurable f) :
∃ M : ℝ, eLpNormEssSup ({ x | M ≤ ‖f x‖₊ }.indicator f) μ = 0 | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ⊤ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ⊤
⊢ eLpNormEssSup ({x | (eLpNorm f ⊤ μ + 1).toReal ≤ ↑‖f x‖₊}.indicator f) μ = 0 | rw [eLpNormEssSup_indicator_eq_eLpNormEssSup_restrict] | α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : MemLp f ⊤ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ⊤
⊢ eLpNormEssSup f (μ.restrict {x | (eLpNorm f ⊤ μ + 1).toReal ≤ ↑‖f x‖₊}) = 0
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
ins... | ee15b826545be751 |
Valuation.map_one_add_of_lt | Mathlib/RingTheory/Valuation/Basic.lean | theorem map_one_add_of_lt (h : v x < 1) : v (1 + x) = 1 | R : Type u_3
Γ₀ : Type u_4
inst✝¹ : Ring R
inst✝ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
x : R
h : v x < v 1
⊢ v (1 + x) = 1 | simpa only [v.map_one] using v.map_add_eq_of_lt_left h | no goals | 7e61d6fa588dc2ee |
WellFounded.prod_lex_of_wellFoundedOn_fiber | Mathlib/Order/WellFoundedSet.lean | theorem WellFounded.prod_lex_of_wellFoundedOn_fiber (hα : WellFounded (rα on f))
(hβ : ∀ a, (f ⁻¹' {a}).WellFoundedOn (rβ on g)) :
WellFounded (Prod.Lex rα rβ on fun c => (f c, g c)) | α : Type u_2
β : Type u_3
γ : Type u_4
rα : α → α → Prop
rβ : β → β → Prop
f : γ → α
g : γ → β
hα : WellFounded (rα on f)
hβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)
⊢ WellFounded (Prod.Lex rα rβ on fun c => (f c, g c)) | refine ((psigma_lex (wellFoundedOn_range.2 hα) fun a => hβ a).onFun
(f := fun c => ⟨⟨_, c, rfl⟩, c, rfl⟩)).mono fun c c' h => ?_ | α : Type u_2
β : Type u_3
γ : Type u_4
rα : α → α → Prop
rβ : β → β → Prop
f : γ → α
g : γ → β
hα : WellFounded (rα on f)
hβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)
c c' : γ
h : (Prod.Lex rα rβ on fun c => (f c, g c)) c c'
⊢ ((PSigma.Lex (fun a b => rα ↑a ↑b) fun a a_1 b => (rβ on g) ↑a_1 ↑b) on fun c => ⟨⟨f c... | 56092b86f3486016 |
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