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 |
|---|---|---|---|---|---|---|
Submodule.linearMap_eq_iff_of_eq_span | Mathlib/LinearAlgebra/Span/Basic.lean | lemma linearMap_eq_iff_of_eq_span {V : Submodule R M} (f g : V →ₗ[R] N)
{S : Set M} (hV : V = span R S) :
f = g ↔ ∀ (s : S), f ⟨s, by simpa only [hV] using subset_span (by simp)⟩ =
g ⟨s, by simpa only [hV] using subset_span (by simp)⟩ | case mpr
R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_9
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
V : Submodule R M
f g : ↥V →ₗ[R] N
S : Set M
hV : V = span R S
h : ∀ (s : ↑S), f ⟨↑s, ⋯⟩ = g ⟨↑s, ⋯⟩
⊢ f = g | subst hV | case mpr
R : Type u_1
M : Type u_4
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
N : Type u_9
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
S : Set M
f g : ↥(span R S) →ₗ[R] N
h : ∀ (s : ↑S), f ⟨↑s, ⋯⟩ = g ⟨↑s, ⋯⟩
⊢ f = g | 24eeafbeb19e0c60 |
Acc.rec_eq_recC | Mathlib/.lake/packages/batteries/Batteries/WF.lean | theorem rec_eq_recC : @Acc.rec = @Acc.recC | case h.h.h.h.h.h.intro
α : Sort u_1
r : α → α → Prop
motive : (a : α) → Acc r a → Sort u_2
intro : (x : α) → (h : ∀ (y : α), r y x → Acc r y) → ((y : α) → (a : r y x) → motive y ⋯) → motive x ⋯
a x : α
h : ∀ (y : α), r y x → Acc r y
ih : ∀ (y : α) (a : r y x), rec intro ⋯ = Acc.recC intro ⋯
⊢ (intro x h fun y a => rec ... | congr | case h.h.h.h.h.h.intro.e_h_ih
α : Sort u_1
r : α → α → Prop
motive : (a : α) → Acc r a → Sort u_2
intro : (x : α) → (h : ∀ (y : α), r y x → Acc r y) → ((y : α) → (a : r y x) → motive y ⋯) → motive x ⋯
a x : α
h : ∀ (y : α), r y x → Acc r y
ih : ∀ (y : α) (a : r y x), rec intro ⋯ = Acc.recC intro ⋯
⊢ (fun y a => rec int... | c46fa129d81ee764 |
Algebra.FormallyEtale.of_isSeparable | Mathlib/RingTheory/Etale/Field.lean | lemma of_isSeparable [Algebra.IsSeparable K L] : FormallyEtale K L | case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScala... | have H : ∀ x y : L, ∃ α : L, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯ := by
intro x y
have : FiniteDimensional K K⟮x, y⟯ := by
apply IntermediateField.finiteDimensional_adjoin
intro x _; exact (Algebra.IsSeparable.isSeparable K x).isIntegral
have := IsSeparable.of_algHom _ _ (IsScalarTower.toAlgHom K (K⟮x, y⟯) L)
obtain ⟨⟨α... | case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScala... | 1efc01fc91428f3c |
AlgebraicGeometry.compactSpace_of_universallyClosed | Mathlib/AlgebraicGeometry/Morphisms/UniversallyClosed.lean | /-- If `X` is universally closed over a field, then `X` is quasi-compact. -/
lemma compactSpace_of_universallyClosed
{K} [Field K] (f : X ⟶ Spec (.of K)) [UniversallyClosed f] : CompactSpace X | X : Scheme
K : Type u
inst✝¹ : Field K
f : X ⟶ Spec (CommRingCat.of K)
inst✝ : UniversallyClosed f
𝒰 : X.OpenCover := X.affineCover
U : 𝒰.J → X.Opens := fun i => Scheme.Hom.opensRange (𝒰.map i)
T : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.J K))
q : T ⟶ Spec (CommRingCat.of K) := Spec.map (CommRingCat.ofHom C)... | exact isOpen_iUnion fun i ↦ (fT.continuous.1 _ (Ti i).2).inter (p.continuous.1 _ (U i).2) | no goals | 4d25dbfc4433be01 |
CategoryTheory.Adjunction.rightAdjointUniq_trans_app | Mathlib/CategoryTheory/Adjunction/Unique.lean | theorem rightAdjointUniq_trans_app {F : C ⥤ D} {G G' G'' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
(adj3 : F ⊣ G'') (x : D) :
(rightAdjointUniq adj1 adj2).hom.app x ≫ (rightAdjointUniq adj2 adj3).hom.app x =
(rightAdjointUniq adj1 adj3).hom.app x | C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F : C ⥤ D
G G' G'' : D ⥤ C
adj1 : F ⊣ G
adj2 : F ⊣ G'
adj3 : F ⊣ G''
x : D
⊢ (adj1.rightAdjointUniq adj2).hom.app x ≫ (adj2.rightAdjointUniq adj3).hom.app x =
(adj1.rightAdjointUniq adj3).hom.app x | rw [← rightAdjointUniq_trans adj1 adj2 adj3] | C : Type u_1
D : Type u_2
inst✝¹ : Category.{u_3, u_1} C
inst✝ : Category.{u_4, u_2} D
F : C ⥤ D
G G' G'' : D ⥤ C
adj1 : F ⊣ G
adj2 : F ⊣ G'
adj3 : F ⊣ G''
x : D
⊢ (adj1.rightAdjointUniq adj2).hom.app x ≫ (adj2.rightAdjointUniq adj3).hom.app x =
((adj1.rightAdjointUniq adj2).hom ≫ (adj2.rightAdjointUniq adj3).hom).... | a0f670618f6a4526 |
sum_Ico_pow | Mathlib/NumberTheory/Bernoulli.lean | theorem sum_Ico_pow (n p : ℕ) :
(∑ k ∈ Ico 1 (n + 1), (k : ℚ) ^ p) =
∑ i ∈ range (p + 1), bernoulli' i * (p + 1).choose i * (n : ℚ) ^ (p + 1 - i) / (p + 1) | n p : ℕ
f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ
hle : 1 ≤ n + 1
hne : ↑p + 1 + 1 ≠ 0
h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ ... | ring | no goals | 0112bf36c164f141 |
Finset.prod_dvd_of_isRelPrime | Mathlib/RingTheory/Coprime/Lemmas.lean | theorem Finset.prod_dvd_of_isRelPrime :
(t : Set I).Pairwise (IsRelPrime on s) → (∀ i ∈ t, s i ∣ z) → (∏ x ∈ t, s x) ∣ z | α : Type u_2
I : Type u_1
inst✝¹ : CommMonoid α
inst✝ : DecompositionMonoid α
z : α
s : I → α
t : Finset I
a : I
r : Finset I
har : a ∉ r
ih : (↑r).Pairwise (IsRelPrime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z
Hs : (↑(insert a r)).Pairwise (IsRelPrime on s)
Hs1 : ∀ i ∈ insert a r, s i ∣ z
aux1 : a ∈ ↑(insert a r)
... | simp only [Finset.coe_insert, Set.subset_insert] | no goals | 83dc5ba78bc0ac42 |
MeasureTheory.Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | theorem Integrable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E} (hf : Integrable f μ)
(hf_zero : ∀ s, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0) : f =ᵐ[μ] 0 | α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
hf_Lp : MemLp f 1 μ
f_Lp : ↥(Lp E 1 μ) := MemLp.toLp f hf_Lp
⊢ f =ᶠ[... | have hf_f_Lp : f =ᵐ[μ] f_Lp := (MemLp.coeFn_toLp hf_Lp).symm | α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : Integrable f μ
hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0
hf_Lp : MemLp f 1 μ
f_Lp : ↥(Lp E 1 μ) := MemLp.toLp f hf_Lp
hf_f_Lp... | 167db4153e687a4a |
Matrix.mem_subfield_of_mul_eq_one_of_mem_subfield_right | Mathlib/LinearAlgebra/Matrix/BaseChange.lean | lemma mem_subfield_of_mul_eq_one_of_mem_subfield_right
(h_mem : ∀ i j, A i j ∈ K) (i : n) (j : m) :
B i j ∈ K | m : Type u_1
n : Type u_2
L : Type u_3
inst✝³ : Finite m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : Field L
e : m ≃ n
K : Subfield L
A : Matrix m n L
B : Matrix n m L
hAB : A * B = 1
h_mem : ∀ (i : m) (j : n), A i j ∈ K
i : n
j : m
val✝ : Fintype m
A' : Matrix m m ↥K := of fun i j => ⟨A.submatrix id (⇑e) i j, ⋯⟩... | have h_det : (A.submatrix id e).det = K.subtype A'.det := by
simp [A', K.subtype.map_det, map, submatrix] | m : Type u_1
n : Type u_2
L : Type u_3
inst✝³ : Finite m
inst✝² : Fintype n
inst✝¹ : DecidableEq m
inst✝ : Field L
e : m ≃ n
K : Subfield L
A : Matrix m n L
B : Matrix n m L
hAB : A * B = 1
h_mem : ∀ (i : m) (j : n), A i j ∈ K
i : n
j : m
val✝ : Fintype m
A' : Matrix m m ↥K := of fun i j => ⟨A.submatrix id (⇑e) i j, ⋯⟩... | 8d2ecd23f1ae3281 |
CategoryTheory.NatTrans.whiskerLeft_app_tensor_app | Mathlib/CategoryTheory/Monoidal/Category.lean | @[reassoc]
lemma whiskerLeft_app_tensor_app {X' Y' : J} (f : X' ⟶ Y') (X : J) :
F.obj X ◁ G.map f ≫ (α.app X ⊗ β.app Y') =
(α.app X ⊗ β.app X') ≫ F'.obj X ◁ G'.map f | J : Type u_1
inst✝² : Category.{u_3, u_1} J
C : Type u_2
inst✝¹ : Category.{u_4, u_2} C
inst✝ : MonoidalCategory C
F G F' G' : J ⥤ C
α : F ⟶ F'
β : G ⟶ G'
X' Y' : J
f : X' ⟶ Y'
X : J
⊢ F.obj X ◁ G.map f ≫ (α.app X ⊗ β.app Y') = (α.app X ⊗ β.app X') ≫ F'.obj X ◁ G'.map f | simpa using tensor_naturality α β (𝟙 X) f | no goals | ef50c4ae0a50466e |
MeasureTheory.uniformIntegrable_of' | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | theorem uniformIntegrable_of' [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ i, StronglyMeasurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UniformIntegrable f p μ | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofRe... | simp_rw [NNReal.val_eq_coe, ENNReal.ofReal_coe_nnreal, mul_comm] | α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup β
p : ℝ≥0∞
f : ι → α → β
inst✝ : IsFiniteMeasure μ
hp : 1 ≤ p
hp' : p ≠ ⊤
hf : ∀ (i : ι), StronglyMeasurable (f i)
h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofRe... | 539fc25109976788 |
FirstOrder.Language.Term.card_sigma | Mathlib/ModelTheory/Encoding.lean | theorem card_sigma : #(Σn, L.Term (α ⊕ (Fin n))) = max ℵ₀ #(α ⊕ (Σi, L.Functions i)) | case refine_2.refine_2
L : Language
α : Type u'
⊢ Function.Injective (Sum.elim (fun i => ⟨0, var (Sum.inl i)⟩) fun F => ⟨1, func F.snd fun x => var (Sum.inr 0)⟩) | rintro (a | a) (b | b) h | case refine_2.refine_2.inl.inl
L : Language
α : Type u'
a b : α
h :
Sum.elim (fun i => ⟨0, var (Sum.inl i)⟩) (fun F => ⟨1, func F.snd fun x => var (Sum.inr 0)⟩) (Sum.inl a) =
Sum.elim (fun i => ⟨0, var (Sum.inl i)⟩) (fun F => ⟨1, func F.snd fun x => var (Sum.inr 0)⟩) (Sum.inl b)
⊢ Sum.inl a = Sum.inl b
case refi... | 76dcd067702ce776 |
Types.Pushout.equivalence_rel' | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | lemma equivalence_rel' [Mono f] : _root_.Equivalence (Rel' f g) where
refl := Rel'.refl
symm h := h.symm
trans | case inr_inl.inr
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
inst✝ : Mono f
s✝ : S
z₂ : X₂
hyz : ∃ s, f s✝ = f s ∧ z₂ = g s
⊢ Rel' f g (Sum.inr (g s✝)) (Sum.inr z₂) | obtain ⟨s, hs, rfl⟩ := hyz | case inr_inl.inr.intro.intro
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
inst✝ : Mono f
s✝ s : S
hs : f s✝ = f s
⊢ Rel' f g (Sum.inr (g s✝)) (Sum.inr (g s)) | 4e0c3f5d1e7c96ff |
BitVec.getLsbD_neg | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Bitblast.lean | theorem getLsbD_neg {i : Nat} {x : BitVec w} :
getLsbD (-x) i =
(getLsbD x i ^^ decide (i < w) && decide (∃ j < i, getLsbD x j = true)) | w : Nat
x : BitVec w
this hi : 0 < w
⊢ (2 ^ w - 1 - x.toNat) % 1 = 0 | omega | no goals | c6eec1fd2118f217 |
Std.Tactic.BVDecide.LRAT.Internal.DefaultFormula.reduce_postcondition | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/LRAT/Internal/Formula/RupAddSound.lean | theorem reduce_postcondition {n : Nat} (c : DefaultClause n) (assignment : Array Assignment) :
(reduce c assignment = reducedToEmpty → Incompatible (PosFin n) c assignment) ∧
(∀ l : Literal (PosFin n), reduce c assignment = reducedToUnit l → ∀ (p : (PosFin n) → Bool), p ⊨ assignment → p ⊨ c → p ⊨ l) | case intro.right.intro.inr.intro.intro
n : Nat
c : DefaultClause n
assignment : Array Assignment
c_arr : Array (Literal (PosFin n)) := List.toArray c.clause
c_clause_rw : c.clause = c_arr.toList
motive : Nat → ReduceResult (PosFin n) → Prop := ReducePostconditionInductionMotive c_arr assignment
h_base : motive 0 reduce... | exact of_decide_eq_true pc2 | no goals | bc01533c0ab5cdb8 |
MeasureTheory.Measure.IsEverywherePos.IsGdelta_of_isMulLeftInvariant | Mathlib/MeasureTheory/Measure/EverywherePos.lean | /-- If a compact closed set is everywhere positive with respect to a left-invariant measure on a
topological group, then it is a Gδ set. This is nontrivial, as there is no second-countability or
metrizability assumption in the statement, so a general compact closed set has no reason to be
a countable intersection of op... | G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk... | apply IsOpen.mem_nhds (by simpa using h'k.smul _) | G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk... | 5dba91f75a485a07 |
mem_pairSelfAdjointMatricesSubmodule | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | theorem mem_pairSelfAdjointMatricesSubmodule :
A₁ ∈ pairSelfAdjointMatricesSubmodule J J₂ ↔ Matrix.IsAdjointPair J J₂ A₁ A₁ | case mp.intro.intro
R : Type u_1
n : Type u_11
inst✝² : CommRing R
inst✝¹ : Fintype n
J J₂ A₁ : Matrix n n R
inst✝ : DecidableEq n
f : Module.End R (n → R)
hf : ((toLinearMap₂' R) J).IsPairSelfAdjoint ((toLinearMap₂' R) J₂) ⇑f
hA : toMatrix' f = A₁
hf' : f = toLin' A₁
⊢ J.IsAdjointPair J₂ A₁ A₁ | rw [hf'] at hf | case mp.intro.intro
R : Type u_1
n : Type u_11
inst✝² : CommRing R
inst✝¹ : Fintype n
J J₂ A₁ : Matrix n n R
inst✝ : DecidableEq n
f : Module.End R (n → R)
hf : ((toLinearMap₂' R) J).IsPairSelfAdjoint ((toLinearMap₂' R) J₂) ⇑(toLin' A₁)
hA : toMatrix' f = A₁
hf' : f = toLin' A₁
⊢ J.IsAdjointPair J₂ A₁ A₁ | 313378ad504ebecc |
Lean.Data.AC.Context.evalList_insert | Mathlib/.lake/packages/lean4/src/lean/Init/Data/AC.lean | theorem Context.evalList_insert
(ctx : Context α)
(h : Commutative ctx.op)
(x : Nat)
(xs : List Nat)
: evalList α ctx (insert x xs) = evalList α ctx (x::xs) | case single.isFalse
α : Sort u_1
ctx : Context α
h : Commutative ctx.op
x a✝ : Nat
h✝ : ¬x < a✝
⊢ evalList α ctx [a✝, x] = evalList α ctx [x, a✝] | simp [evalList, h.1, EvalInformation.evalOp] | no goals | 37aa11cde795cd0c |
CategoryTheory.Pretriangulated.Triangle.mor₁_eq_zero_iff_mono₂ | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | lemma mor₁_eq_zero_iff_mono₂ : T.mor₁ = 0 ↔ Mono T.mor₂ | case mp
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ T.mor₁ = 0 → Mono T.mor₂ | intro h | case mp
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h : T.mor₁ = 0
⊢ Mono T.mor₂ | ab45c565c7b058da |
Icc_subset_segment | Mathlib/Analysis/Convex/Segment.lean | theorem Icc_subset_segment : Icc x y ⊆ [x -[𝕜] y] | 𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
x y : 𝕜
⊢ Icc x y ⊆ [x-[𝕜]y] | rintro z ⟨hxz, hyz⟩ | case intro
𝕜 : Type u_1
inst✝ : LinearOrderedField 𝕜
x y z : 𝕜
hxz : x ≤ z
hyz : z ≤ y
⊢ z ∈ [x-[𝕜]y] | c179504b2c4995a3 |
differentiableAt_finCons | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | theorem differentiableAt_finCons :
DifferentiableAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) x ↔
DifferentiableAt 𝕜 φ x ∧ DifferentiableAt 𝕜 φs x | 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
x : E
n : ℕ
F' : Fin n.succ → Type u_6
inst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)
inst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)
φ : E → F' 0
φs : E → (i : Fin n) → F' i.succ
⊢ (Differenti... | simp only [Fin.cons_zero, Fin.cons_succ] | no goals | 2ad45d5dc2b04b6f |
AlgebraicGeometry.morphismRestrict_app | Mathlib/AlgebraicGeometry/Restrict.lean | theorem morphismRestrict_app {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V : U.toScheme.Opens) :
(f ∣_ U).app V = f.app (U.ι ''ᵁ V) ≫
X.presheaf.map (eqToHom (image_morphismRestrict_preimage f U V)).op | X Y : Scheme
f : X ⟶ Y
U : Y.Opens
V : (↑U).Opens
this :
Y.presheaf.map (homOfLE ⋯).op ≫ Hom.appLE (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) ((f ∣_ U) ⁻¹ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ⋯ =
(Hom.appLE f (U.ι ''ᵁ V) (f ⁻¹ᵁ U.ι ''ᵁ V) ⋯ ≫ X.presheaf.map (homOfLE ⋯).op) ≫
X.presheaf.map ((Hom.opensFunctor (f ⁻¹ᵁ U).ι).map (eqToHom ⋯))... | rw [e] | no goals | 9d85b88464a65d5b |
FirstOrder.Language.BoundedFormula.isPrenex_toPrenexImp | Mathlib/ModelTheory/Complexity.lean | theorem isPrenex_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ) (hψ : IsPrenex ψ) :
IsPrenex (φ.toPrenexImp ψ) | case all
L : Language
α : Type u'
n : ℕ
φ : L.BoundedFormula α n
n✝ : ℕ
φ✝ : L.BoundedFormula α (n✝ + 1)
h✝ : φ✝.IsPrenex
ih1 : ∀ {ψ : L.BoundedFormula α (n✝ + 1)}, ψ.IsPrenex → (φ✝.toPrenexImp ψ).IsPrenex
ψ : L.BoundedFormula α n✝
hψ : ψ.IsPrenex
⊢ ((∀'φ✝).toPrenexImp ψ).IsPrenex | exact (ih1 hψ.liftAt).ex | no goals | 6a9982774a927dfc |
Set.mem_dite_empty_right | Mathlib/Data/Set/Basic.lean | theorem mem_dite_empty_right (p : Prop) [Decidable p] (t : p → Set α) (x : α) :
(x ∈ if h : p then t h else ∅) ↔ ∃ h : p, x ∈ t h | α : Type u
p : Prop
inst✝ : Decidable p
t : p → Set α
x : α
⊢ (x ∈ if h : p then t h else ∅) ↔ ∃ (h : p), x ∈ t h | simp only [mem_dite, mem_empty_iff_false, imp_false, not_not] | α : Type u
p : Prop
inst✝ : Decidable p
t : p → Set α
x : α
⊢ (∀ (h : p), x ∈ t h) ∧ p ↔ ∃ (h : p), x ∈ t h | 24957136cdb53276 |
HasFPowerSeriesWithinOnBall.hasFDerivWithinAt | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | theorem HasFPowerSeriesWithinOnBall.hasFDerivWithinAt [CompleteSpace F]
(h : HasFPowerSeriesWithinOnBall f p s x r)
{y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) (h'y : x + y ∈ insert x s) :
HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1))
(insert x s) (x + y) | case inl
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWithinOnBall f... | convert (h.changeOrigin hy h'y).hasFPowerSeriesWithinAt.hasFDerivWithinAt | case h.e'_13.h.e'_4
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0∞
f : E → F
x : E
s : Set E
inst✝ : CompleteSpace F
h : HasFPowerSeriesWit... | 3c2f2e0982cd7521 |
BitVec.cast_append_right | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem cast_append_right (h : w + v = w + v') (x : BitVec w) (y : BitVec v) :
(x ++ y).cast h = x ++ y.cast (by omega) | case pred.isTrue.isFalse
w v v' : Nat
h : w + v = w + v'
x : BitVec w
y : BitVec v
i✝ : Nat
a✝ : i✝ < w + v'
h✝¹ : i✝ < v
h✝ : ¬i✝ < v'
⊢ y.getLsbD i✝ = x.getLsbD (i✝ - v') | omega | no goals | 8022bd10e5d4bd8f |
SimplexCategoryGenRel.σ_comp_σ_nat | Mathlib/AlgebraicTopology/SimplexCategory/GeneratorsRelations/Basic.lean | /-- A version of σ_comp_σ with indices in ℕ satisfying relevant inequalities. -/
lemma σ_comp_σ_nat {n} (i j : ℕ) (hi : i < n + 1) (hj : j < n + 1) (H : i ≤ j) :
σ ⟨i, by omega⟩ ≫ σ ⟨j, hj⟩ = σ ⟨j + 1, by omega⟩ ≫ σ ⟨i, hi⟩ :=
σ_comp_σ (n := n) (i := ⟨i, by omega⟩) (j := ⟨j, by omega⟩) (by simpa)
| n i j : ℕ
hi : i < n + 1
hj : j < n + 1
H : i ≤ j
⊢ ⟨i, ⋯⟩ ≤ ⟨j, ⋯⟩ | simpa | no goals | 3d6c941c80fcfe51 |
Finpartition.IsEquipartition.exists_partPreservingEquiv | Mathlib/Order/Partition/Equipartition.lean | theorem IsEquipartition.exists_partPreservingEquiv (hP : P.IsEquipartition) : ∃ f : s ≃ Fin #s,
∀ a b : s, P.part a = P.part b ↔ f a % #P.parts = f b % #P.parts | case h
α : Type u_1
inst✝ : DecidableEq α
s : Finset α
P : Finpartition s
hP : P.IsEquipartition
f : { x // x ∈ s } ≃ (t : { x // x ∈ P.parts }) × Fin #↑t
hf : ∀ (a b : { x // x ∈ s }), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst
g : { x // x ∈ P.parts } ≃ Fin #P.parts
hg : ∀ (t : { x // x ∈ P.parts }), #↑t = #s / #P... | simp_rw [z', z, Equiv.ofBijective_apply, hf a b, Nat.mul_add_mod,
Nat.mod_eq_of_lt (gl a), Nat.mod_eq_of_lt (gl b), Fin.val_eq_val, g.apply_eq_iff_eq] | no goals | 9cb0450224c5f721 |
inf_sq_eq_mul_div_mabs_div | Mathlib/Algebra/Order/Group/Unbundled/Abs.lean | @[to_additive two_nsmul_inf_eq_add_sub_abs_sub]
lemma inf_sq_eq_mul_div_mabs_div (a b : α) : (a ⊓ b) ^ 2 = a * b / |b / a|ₘ | α : Type u_1
inst✝² : Lattice α
inst✝¹ : CommGroup α
inst✝ : MulLeftMono α
a b : α
⊢ (a ⊓ b) ^ 2 = a * b / mabs (b / a) | rw [← inf_mul_sup a b, ← sup_div_inf_eq_mabs_div, div_eq_mul_inv, div_eq_mul_inv, mul_inv_rev,
inv_inv, mul_assoc, mul_inv_cancel_comm_assoc, ← pow_two] | no goals | 613f81dc4e51e047 |
Subgroup.exists_finiteIndex_of_leftCoset_cover | Mathlib/GroupTheory/CosetCover.lean | theorem exists_finiteIndex_of_leftCoset_cover : ∃ k ∈ s, (H k).FiniteIndex | case neg
G : Type u_1
inst✝ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ¬⋃ i ∈ Finset.filter (fun x => H x = H j) s, g i • ↑(H i) = Set.univ
i : ι
hi : i ∈ s
left✝ : H i ≠ H j
hfi : (H i).FiniteIndex
⊢ ∃ k ∈ s, (H k).FiniteIndex | exact ⟨i, hi, hfi⟩ | no goals | a157e4700b8dbc0e |
OreLocalization.add_smul | Mathlib/RingTheory/OreLocalization/Ring.lean | theorem add_smul (y z : R[S⁻¹]) (x : X[S⁻¹]) :
(y + z) • x = y • x + z • x | case c.c.c.mk.mk.intro.mk.mk.intro
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra * ↑s₃ ∈ S
h... | rcases oreDivSMulChar' ((sb * ra) • r₃) r₁ (sb * sa * s₂) s₁ with ⟨rc, sc, hc, hc'⟩ | case c.c.c.mk.mk.intro.mk.mk.intro.mk.mk.intro
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra... | 79dc4eb2119c8abc |
IsExposed.sInter | Mathlib/Analysis/Convex/Exposed.lean | theorem sInter [ContinuousAdd 𝕜] {F : Finset (Set E)} (hF : F.Nonempty)
(hAF : ∀ B ∈ F, IsExposed 𝕜 A B) : IsExposed 𝕜 A (⋂₀ F) | case insert.inr
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommMonoid E
inst✝² : TopologicalSpace E
inst✝¹ : Module 𝕜 E
A : Set E
inst✝ : ContinuousAdd 𝕜
C : Set E
F : Finset (Set E)
a✝ : C ∉ F
hF' : F.Nonempty → (∀ B ∈ F, IsExposed 𝕜 A B) → IsExposed 𝕜 A (⋂₀ ↑F)
hF ... | exact (hAF C (Finset.mem_insert_self C F)).inter
(hF' hFnemp fun B hB => hAF B (Finset.mem_insert_of_mem hB)) | no goals | ab06a23c6f395397 |
Subgroup.normal_of_index_eq_minFac_card | Mathlib/GroupTheory/IndexNormal.lean | theorem normal_of_index_eq_minFac_card (hHp : H.index = (Nat.card G).minFac) :
H.Normal | G : Type u_1
inst✝ : Group G
H : Subgroup G
hHp : H.index = (Nat.card G).minFac
hG0 : ¬Nat.card G = 0
hG1 : ¬Nat.card G = 1
this : H.normalCore.relindex H = 1
⊢ H.Normal | convert H.normalCore_normal | case h.e'_3
G : Type u_1
inst✝ : Group G
H : Subgroup G
hHp : H.index = (Nat.card G).minFac
hG0 : ¬Nat.card G = 0
hG1 : ¬Nat.card G = 1
this : H.normalCore.relindex H = 1
⊢ H = H.normalCore | cf69968f09f09288 |
Filter.map_val_atTop_of_Ici_subset | Mathlib/Order/Filter/AtTopBot/Basic.lean | theorem map_val_atTop_of_Ici_subset [Preorder α] [IsDirected α (· ≤ ·)] {a : α} {s : Set α}
(h : Ici a ⊆ s) : map ((↑) : s → α) atTop = atTop | α : Type u_3
inst✝¹ : Preorder α
inst✝ : IsDirected α fun x1 x2 => x1 ≤ x2
a : α
s : Set α
h : Ici a ⊆ s
⊢ map Subtype.val atTop = atTop | choose f hl hr using exists_ge_ge (α := α) | α : Type u_3
inst✝¹ : Preorder α
inst✝ : IsDirected α fun x1 x2 => x1 ≤ x2
a : α
s : Set α
h : Ici a ⊆ s
f : α → α → α
hl : ∀ (a b : α), a ≤ f a b
hr : ∀ (a b : α), b ≤ f a b
⊢ map Subtype.val atTop = atTop | b72dbf58416ab246 |
AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | theorem ofRestrict_invApp {C : Type*} [Category C] (X : PresheafedSpace C) {Y : TopCat}
{f : Y ⟶ TopCat.of X.carrier} (h : IsOpenEmbedding f) (U : Opens (X.restrict h).carrier) :
(PresheafedSpace.IsOpenImmersion.ofRestrict X h).invApp _ U = 𝟙 _ | C : Type u_1
inst✝ : Category.{u_2, u_1} C
X : PresheafedSpace C
Y : TopCat
f : Y ⟶ TopCat.of ↑↑X
h : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
U : Opens ↑↑(X.restrict h)
⊢ invApp (X.ofRestrict h) U = 𝟙 ((X.restrict h).presheaf.obj (op U)) | delta invApp | C : Type u_1
inst✝ : Category.{u_2, u_1} C
X : PresheafedSpace C
Y : TopCat
f : Y ⟶ TopCat.of ↑↑X
h : IsOpenEmbedding ⇑(ConcreteCategory.hom f)
U : Opens ↑↑(X.restrict h)
⊢ (X.restrict h).presheaf.map (eqToHom ⋯) ≫ inv ((X.ofRestrict h).c.app (op ((opensFunctor (X.ofRestrict h)).obj U))) =
𝟙 ((X.restrict h).preshe... | 22681e993020e5c3 |
continuousAt_const_cpow' | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | theorem continuousAt_const_cpow' {a b : ℂ} (h : b ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b | case pos
a b : ℂ
h : b ≠ 0
ha : a = 0
⊢ ContinuousAt 0 b | exact continuousAt_const | no goals | 1c89bcd007cebda3 |
Algebra.IsPushout.symm | Mathlib/RingTheory/IsTensorProduct.lean | theorem Algebra.IsPushout.symm (h : Algebra.IsPushout R S R' S') : Algebra.IsPushout R R' S S' | case refine_1
R : Type u_1
S : Type v₃
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
R' : Type u_6
S' : Type u_7
inst✝⁷ : CommSemiring R'
inst✝⁶ : CommSemiring S'
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R' S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : IsSca... | simp only [smul_zero, map_zero] | no goals | ee0cd1db0e501e91 |
LSeries.tendsto_cpow_mul_atTop | Mathlib/NumberTheory/LSeries/Injectivity.lean | /-- If the coefficients `f m` of an L-series are zero for `m ≤ n` and the L-series converges
at some point, then `f (n+1)` is the limit of `(n+1)^x * LSeries f x` as `x → ∞`. -/
lemma LSeries.tendsto_cpow_mul_atTop {f : ℕ → ℂ} {n : ℕ} (h : ∀ m ≤ n, f m = 0)
(ha : abscissaOfAbsConv f < ⊤):
Tendsto (fun x : ℝ ↦ (... | f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑... | refine (summable_mul_left_iff <| natCast_add_one_cpow_ne_zero n _).mpr <|
LSeriesSummable_of_abscissaOfAbsConv_lt_re ?_ | f : ℕ → ℂ
n : ℕ
h : ∀ m ≤ n, f m = 0
ha : abscissaOfAbsConv f < ⊤
y : ℝ
hay : abscissaOfAbsConv f < ↑y
hyt : ↑y < ⊤
F : ℝ → ℕ → ℂ := fun x => {m | n + 1 < m}.indicator fun m => f m / (↑m / (↑n + 1)) ^ ↑x
hF₀ : ∀ (x : ℝ) {m : ℕ}, m ≤ n + 1 → F x m = 0
hF : ∀ (x : ℝ) {m : ℕ}, m ≠ n + 1 → F x m = (↑n + 1) ^ ↑x * term f (↑... | 0f9b595cc43bb900 |
IsBaseChange.ofEquiv | Mathlib/RingTheory/IsTensorProduct.lean | theorem IsBaseChange.ofEquiv (e : M ≃ₗ[R] N) : IsBaseChange R e.toLinearMap | case h.refl.h
R : Type u_1
M : Type v₁
N : Type v₂
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid N
inst✝² : CommSemiring R
inst✝¹ : Module R M
inst✝ : Module R N
e : M ≃ₗ[R] N
Q : Type (max v₁ v₂ u_1)
I₁ : AddCommMonoid Q
I₂ : Module R Q
I₄ : IsScalarTower R R Q
y : N →ₗ[R] Q
x✝ : N
⊢ y x✝ = ((↑R y ∘ₗ ↑e) ∘ₗ ↑e.symm)... | simp | no goals | 4bf0cf3eb1d464b0 |
CategoryTheory.regularTopology.exists_effectiveEpi_iff_mem_induced | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | theorem exists_effectiveEpi_iff_mem_induced (X : C) (S : Sieve X) :
(∃ (Y : C) (π : Y ⟶ X),
EffectiveEpi π ∧ S.arrows π) ↔
(S ∈ F.inducedTopology (regularTopology _) X) | case h.e'_6
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{u_3, u_1} C
inst✝⁶ : Category.{u_4, u_2} D
F : C ⥤ D
inst✝⁵ : F.PreservesEffectiveEpis
inst✝⁴ : F.ReflectsEffectiveEpis
inst✝³ : F.Full
inst✝² : F.Faithful
inst✝¹ : F.EffectivelyEnough
inst✝ : Preregular D
X : C
S : Sieve X
hS : S ∈ (F.inducedTopology (regularTop... | exact F.map_injective (by simp) | no goals | b6a574b353856e83 |
Submodule.orthogonal_orthogonal | Mathlib/Analysis/InnerProductSpace/Projection.lean | theorem Submodule.orthogonal_orthogonal [HasOrthogonalProjection K] : Kᗮᗮ = K | case h.mp.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection K
y : E
hy : y ∈ K
z : E
hz : z ∈ Kᗮ
⊢ y + z ∈ Kᗮᗮ → y + z ∈ K | intro hv | case h.mp.intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection K
y : E
hy : y ∈ K
z : E
hz : z ∈ Kᗮ
hv : y + z ∈ Kᗮᗮ
⊢ y + z ∈ K | 7fddae50e8a8e4a3 |
Nat.nth_comp_of_strictMono | Mathlib/Data/Nat/Nth.lean | lemma nth_comp_of_strictMono {n : ℕ} {f : ℕ → ℕ} (hf : StrictMono f)
(h0 : ∀ k, p k → k ∈ Set.range f) (h : ∀ hfi : (setOf p).Finite, n < hfi.toFinset.card) :
f (nth (fun i ↦ p (f i)) n) = nth p n | case intro
p : ℕ → Prop
f : ℕ → ℕ
hf : StrictMono f
h0 : ∀ (k : ℕ), p k → k ∈ Set.range f
hs : ∀ {p' : ℕ → Prop}, (∀ (k : ℕ), p' k → k ∈ Set.range f) → f '' {i | p' (f i)} = setOf p'
h : p (nth p 0)
t : ℕ
ht : f t = nth p 0
⊢ {i | p (f i)}.Nonempty | exact ⟨t, Set.mem_setOf_eq ▸ ht ▸ h⟩ | no goals | 03d0809f8aba723f |
MeasureTheory.Lp.ae_eq_of_forall_setIntegral_eq' | Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean | theorem Lp.ae_eq_of_forall_setIntegral_eq' (hm : m ≤ m0) (f g : Lp E' p μ) (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn f s μ)
(hg_int_finite : ∀ s, MeasurableSet[m] s → μ s < ∞ → IntegrableOn g s μ)
(hfg : ∀ s : Set α, MeasurableSet[m] s → μ s <... | α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f g : ↥(Lp E' p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ⊤
hf_int_finite : ∀ (s : Set α)... | exact Lp.ae_eq_zero_of_forall_setIntegral_eq_zero' 𝕜 hm (f - g) hp_ne_zero hp_ne_top hfg_int hfg'
<| (hf_meas.sub hg_meas).congr (Lp.coeFn_sub f g).symm | no goals | 6a930985da20f571 |
Std.DHashMap.Internal.List.getEntry?_eraseKey_of_false | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem getEntry?_eraseKey_of_false [BEq α] [PartialEquivBEq α] {l : List ((a : α) × β a)}
{k a : α} (hka : (k == a) = false) : getEntry? a (eraseKey k l) = getEntry? a l | case false.true
α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
k a : α
hka : (k == a) = false
k' : α
v' : β k'
t : List ((a : α) × β a)
ih : getEntry? a (eraseKey k t) = getEntry? a t
h' : (k' == k) = false
h'' : (k' == a) = true
⊢ getEntry? a (⟨k', v'⟩ :: eraseKey k t) = getEntry? a (⟨k', v'⟩ :: t) | rw [getEntry?_cons_of_true h'', getEntry?_cons_of_true h''] | no goals | 91f3eedba4da607e |
Compactum.continuous_of_hom | Mathlib/Topology/Category/Compactum.lean | theorem continuous_of_hom {X Y : Compactum} (f : X ⟶ Y) : Continuous f | X Y : Compactum
f : X ⟶ Y
x : X.A
g : Ultrafilter X.A
h : ↑g ≤ 𝓝 x
⊢ ↑g ≤ 𝓝 x | apply h | no goals | cfc32a0269fac9c9 |
Continuous.integrableAt_nhds | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | theorem Continuous.integrableAt_nhds [TopologicalSpace α] [SecondCountableTopologyEither α E]
[OpensMeasurableSpace α] {μ : Measure α} [IsLocallyFiniteMeasure μ] {f : α → E}
(hf : Continuous f) (a : α) : IntegrableAtFilter f (𝓝 a) μ | α : Type u_1
E : Type u_4
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous f
a : α
⊢ IntegrableAtFilter f (𝓝[univ] a) μ | exact hf.continuousOn.integrableAt_nhdsWithin MeasurableSet.univ (mem_univ a) | no goals | 61ea7baddfe6310e |
PowerSeries.divided_by_X_pow_orderMul | Mathlib/RingTheory/PowerSeries/Order.lean | theorem divided_by_X_pow_orderMul {f g : R⟦X⟧} (hf : f ≠ 0) (hg : g ≠ 0) :
divided_by_X_pow_order hf * divided_by_X_pow_order hg =
divided_by_X_pow_order (mul_ne_zero hf hg) | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R⟦X⟧
hf : f ≠ 0
hg : g ≠ 0
df : ℕ := f.order.lift ⋯
⊢ divided_by_X_pow_order hf * divided_by_X_pow_order hg = divided_by_X_pow_order ⋯ | set dg := g.order.lift (order_finite_iff_ne_zero.mpr hg) | R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
f g : R⟦X⟧
hf : f ≠ 0
hg : g ≠ 0
df : ℕ := f.order.lift ⋯
dg : ℕ := g.order.lift ⋯
⊢ divided_by_X_pow_order hf * divided_by_X_pow_order hg = divided_by_X_pow_order ⋯ | ee1cc582fae5b885 |
Set.mul_subset_range | Mathlib/Algebra/Group/Pointwise/Set/Basic.lean | @[to_additive]
lemma mul_subset_range {s t : Set β} (hs : s ⊆ range m) (ht : t ⊆ range m) : s * t ⊆ range m | F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Mul α
inst✝² : Mul β
inst✝¹ : FunLike F α β
inst✝ : MulHomClass F α β
m : F
s t : Set β
hs : s ⊆ range ⇑m
ht : t ⊆ range ⇑m
⊢ s * t ⊆ range ⇑m | rintro _ ⟨a, ha, b, hb, rfl⟩ | case intro.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
inst✝³ : Mul α
inst✝² : Mul β
inst✝¹ : FunLike F α β
inst✝ : MulHomClass F α β
m : F
s t : Set β
hs : s ⊆ range ⇑m
ht : t ⊆ range ⇑m
a : β
ha : a ∈ s
b : β
hb : b ∈ t
⊢ (fun x1 x2 => x1 * x2) a b ∈ range ⇑m | 294bb22a50ef29ba |
Batteries.RBNode.Ordered.upperBound?_least_ub | Mathlib/.lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | theorem Ordered.upperBound?_least_ub [@TransCmp α cmp] [IsCut cmp cut] (h : Ordered cmp t)
(hub : ∀ {x}, ub = some x → t.All (cmpLT cmp · x)) :
t.upperBound? cut ub = some x → y ∈ t → cut x = .lt → cmp y x = .lt → cut y = .gt | case nil
α : Type u_1
cmp : α → α → Ordering
cut : α → Ordering
x y : α
inst✝¹ : TransCmp cmp
inst✝ : IsCut cmp cut
ub : Option α
h : Ordered cmp nil
hub : ∀ {x : α}, ub = some x → All (fun x_1 => cmpLT cmp x_1 x) nil
⊢ upperBound? cut nil ub = some x → y ∈ nil → cut x = Ordering.lt → cmp y x = Ordering.lt → cut y = Or... | nofun | no goals | 730b8c71238d69a7 |
LinearIsometry.im_apply_eq_im | Mathlib/Analysis/Complex/Isometry.lean | theorem LinearIsometry.im_apply_eq_im {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) :
z + conj z = f z + conj (f z) | f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this : ‖f z - 1‖ = ‖z - 1‖
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z) | apply_fun fun x => x ^ 2 at this | f : ℂ →ₗᵢ[ℝ] ℂ
h : f 1 = 1
z : ℂ
this : ‖f z - 1‖ ^ 2 = ‖z - 1‖ ^ 2
⊢ z + (starRingEnd ℂ) z = f z + (starRingEnd ℂ) (f z) | 859e8ff608f3767d |
BitVec.toInt_neg_of_ne_intMin | Mathlib/.lake/packages/lean4/src/lean/Init/Data/BitVec/Lemmas.lean | theorem toInt_neg_of_ne_intMin {x : BitVec w} (rs : x ≠ intMin w) :
(-x).toInt = -(x.toInt) | case neg
w : Nat
x : BitVec w
rs : ¬x.toNat = 2 ^ (w - 1) % 2 ^ w
x_zero : ¬x = 0
w_0 : ¬w = 0
⊢ (-x).toInt = -x.toInt | have : 0 < w := by omega | case neg
w : Nat
x : BitVec w
rs : ¬x.toNat = 2 ^ (w - 1) % 2 ^ w
x_zero : ¬x = 0
w_0 : ¬w = 0
this : 0 < w
⊢ (-x).toInt = -x.toInt | fd71256140c0ecff |
NumberField.FinitePlace.prod_eq_inv_abs_norm | Mathlib/NumberTheory/NumberField/ProductFormula.lean | theorem FinitePlace.prod_eq_inv_abs_norm {x : K} (h_x_nezero : x ≠ 0) :
∏ᶠ w : FinitePlace K, w x = |(Algebra.norm ℚ) x|⁻¹ | case intro.intro.intro.e_a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a b : 𝓞 K
h_x_nezero : (algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b ≠ 0
hb : b ≠ 0
ha : a ≠ 0
hb₀ : ↑((Algebra.norm ℤ) b) ≠ 0
⊢ ↑((Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b)) * ↑((Algebra.norm ℤ) b) = ↑((Algebr... | norm_cast | case intro.intro.intro.e_a
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
a b : 𝓞 K
h_x_nezero : (algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b ≠ 0
hb : b ≠ 0
ha : a ≠ 0
hb₀ : ↑((Algebra.norm ℤ) b) ≠ 0
⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b) * ↑((Algebra.norm ℤ) b) = ↑((Algebra.n... | bc2653900b49a69c |
MeasureTheory.measure_iInter_eq_iInf_measure_iInter_le | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | theorem measure_iInter_eq_iInf_measure_iInter_le {α ι : Type*} {_ : MeasurableSpace α}
{μ : Measure α} [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
{f : ι → Set α} (h : ∀ i, NullMeasurableSet (f i) μ) (hfin : ∃ i, μ (f i) ≠ ∞) :
μ (⋂ i, f i) = ⨅ i, μ (⋂ j ≤ i, f j) | α : Type u_8
ι : Type u_9
x✝ : MeasurableSpace α
μ : Measure α
inst✝² : Countable ι
inst✝¹ : Preorder ι
inst✝ : IsDirected ι fun x1 x2 => x1 ≤ x2
f : ι → Set α
h : ∀ (i : ι), NullMeasurableSet (f i) μ
hfin : ∃ i, μ (f i) ≠ ⊤
⊢ μ (⋂ i, f i) = μ (⋂ i', ⋂ i, ⋂ (_ : i' ≤ i), f i') | exact congrArg μ <| iInter_congr fun i ↦ (biInf_const nonempty_Ici).symm | no goals | 47f3a8ea5c0e90b2 |
Fin.mul_one' | Mathlib/Data/Fin/Basic.lean | theorem mul_one' [NeZero n] (k : Fin n) : k * 1 = k | n : ℕ
inst✝ : NeZero n
k : Fin n
⊢ k * 1 = k | rcases n with - | n | case zero
inst✝ : NeZero 0
k : Fin 0
⊢ k * 1 = k
case succ
n : ℕ
inst✝ : NeZero (n + 1)
k : Fin (n + 1)
⊢ k * 1 = k | 6151942578d89d80 |
solvableByRad.induction2 | Mathlib/FieldTheory/AbelRuffini.lean | theorem induction2 {α β γ : solvableByRad F E} (hγ : γ ∈ F⟮α, β⟯) (hα : P α) (hβ : P β) : P γ | case a.a
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
Splits (algebraMap F (p * q).SplittingField) (... | erw [Polynomial.aeval_subalgebra_coe (minpoly F γ)] | case a.a
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
α β γ : ↥(solvableByRad F E)
hγ : γ ∈ F⟮α, β⟯
hα : P α
hβ : P β
p : F[X] := minpoly F α
q : F[X] := minpoly F β
hpq :
Splits (algebraMap F (p * q).SplittingField) (minpoly F α) ∧
Splits (algebraMap F (p * q).SplittingField) (... | baad06864f8e9f42 |
Polynomial.content_X_mul | Mathlib/RingTheory/Polynomial/Content.lean | theorem content_X_mul {p : R[X]} : content (X * p) = content p | case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : IsDomain R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h : (X * p).support = Finset.map { toFun := Nat.succ, inj' := Nat.succ_injective } p.support
a : ℕ
⊢ (p * X).coeff a.succ = p.coeff a | simp [coeff_mul_X] | no goals | ac071a5a6c3a3683 |
Stream'.Seq.mem_rec_on | Mathlib/Data/Seq/Seq.lean | theorem mem_rec_on {C : Seq α → Prop} {a s} (M : a ∈ s)
(h1 : ∀ b s', a = b ∨ C s' → C (cons b s')) : C s | case intro.zero
α : Type u
C : Seq α → Prop
a : α
h1 : ∀ (b : α) (s' : Seq α), a = b ∨ C s' → C (cons b s')
s : Seq α
e : some a = ↑s 0
TH : s = cons a s.tail
⊢ C (cons a s.tail) | apply h1 _ _ (Or.inl rfl) | no goals | 6a5946c9fe3560f9 |
MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_range | Mathlib/Probability/Martingale/BorelCantelli.lean | theorem Martingale.bddAbove_range_iff_bddBelow_range [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
(hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) | case h.mp.intro
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.ra... | simpa only [Pi.neg_apply, Set.mem_range, neg_inj] | no goals | 2fa09fbce73a9a3f |
Fin.snoc_update | Mathlib/Data/Fin/Tuple/Basic.lean | theorem snoc_update : snoc (update p i y) x = update (snoc p x) i.castSucc y | n : ℕ
α : Fin (n + 1) → Sort u_1
x : α (last n)
p : (i : Fin n) → α i.castSucc
i : Fin n
y : α i.castSucc
j : Fin (n + 1)
h : ↑j < n
h' : j = i.castSucc
C1 : α i.castSucc = α j
E1 : update (snoc p x) i.castSucc y j = cast C1 y
C2 : α i.castSucc = α (j.castLT h).castSucc
⊢ update p i y (j.castLT h) = cast C2 y | have : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y := by simp | n : ℕ
α : Fin (n + 1) → Sort u_1
x : α (last n)
p : (i : Fin n) → α i.castSucc
i : Fin n
y : α i.castSucc
j : Fin (n + 1)
h : ↑j < n
h' : j = i.castSucc
C1 : α i.castSucc = α j
E1 : update (snoc p x) i.castSucc y j = cast C1 y
C2 : α i.castSucc = α (j.castLT h).castSucc
this : update p (j.castLT h) (cast C2 y) (j.castL... | 4eef9dec140e05c3 |
ProbabilityTheory.Kernel.lintegral_compProd' | Mathlib/Probability/Kernel/Composition/CompProd.lean | theorem lintegral_compProd' (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
[IsSFiniteKernel η] (a : α) {f : β → γ → ℝ≥0∞} (hf : Measurable (Function.uncurry f)) :
∫⁻ bc, f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ b, ∫⁻ c, f b c ∂η (a, b) ∂κ a | case e_s.h.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f✝ : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f✝)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := Simple... | change
∫⁻ x, (f : β × γ → ℝ≥0∞) x + f' x ∂(κ ⊗ₖ η) a =
∫⁻ b, ∫⁻ c : γ, f (b, c) + f' (b, c) ∂η (a, b) ∂κ a | case e_s.h.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f✝ : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f✝)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := Simple... | c37ea276d042f595 |
Nat.log2_eq_log_two | Mathlib/Data/Nat/Log.lean | lemma log2_eq_log_two {n : ℕ} : Nat.log2 n = Nat.log 2 n | case inr.H
n : ℕ
hn : n ≠ 0
⊢ ∀ (c : ℕ), c ≤ n.log2 ↔ c ≤ log 2 n | intro m | case inr.H
n : ℕ
hn : n ≠ 0
m : ℕ
⊢ m ≤ n.log2 ↔ m ≤ log 2 n | ac9a91dd8edca788 |
RegularExpression.mul_rmatch_iff | Mathlib/Computability/RegularExpressions.lean | theorem mul_rmatch_iff (P Q : RegularExpression α) (x : List α) :
(P * Q).rmatch x ↔ ∃ t u : List α, x = t ++ u ∧ P.rmatch t ∧ Q.rmatch u | α : Type u_1
inst✝ : DecidableEq α
P Q : RegularExpression α
x : List α
⊢ (P * Q).rmatch x = true ↔ ∃ t u, x = t ++ u ∧ P.rmatch t = true ∧ Q.rmatch u = true | induction' x with a x ih generalizing P Q | case nil
α : Type u_1
inst✝ : DecidableEq α
P Q : RegularExpression α
⊢ (P * Q).rmatch [] = true ↔ ∃ t u, [] = t ++ u ∧ P.rmatch t = true ∧ Q.rmatch u = true
case cons
α : Type u_1
inst✝ : DecidableEq α
a : α
x : List α
ih : ∀ (P Q : RegularExpression α), (P * Q).rmatch x = true ↔ ∃ t u, x = t ++ u ∧ P.rmatch t = true... | eb4ea08dacfd0649 |
AdicCompletion.map_injective | Mathlib/RingTheory/AdicCompletion/Exactness.lean | theorem map_injective {f : M →ₗ[R] N} (hf : Function.Injective f) :
Function.Injective (map I f) | R : Type u
inst✝⁶ : CommRing R
I : Ideal R
M : Type u
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u
inst✝³ : AddCommGroup N
inst✝² : Module R N
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
k : ℕ
hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ range f = I ^ (n - k) • (I ^ k • ⊤ ⊓ ran... | simpa using congrArg (fun x ↦ x.val (n + k)) hx | no goals | 527bc8a2a4cc44f8 |
PartialEquiv.trans_symm_eq_symm_trans_symm | Mathlib/Logic/Equiv/PartialEquiv.lean | theorem trans_symm_eq_symm_trans_symm : (e.trans e').symm = e'.symm.trans e.symm | α : Type u_1
β : Type u_2
γ : Type u_3
e : PartialEquiv α β
e' : PartialEquiv β γ
⊢ (e.trans e').symm = e'.symm.trans e.symm | cases e | case mk
α : Type u_1
β : Type u_2
γ : Type u_3
e' : PartialEquiv β γ
toFun✝ : α → β
invFun✝ : β → α
source✝ : Set α
target✝ : Set β
map_source'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → toFun✝ x ∈ target✝
map_target'✝ : ∀ ⦃x : β⦄, x ∈ target✝ → invFun✝ x ∈ source✝
left_inv'✝ : ∀ ⦃x : α⦄, x ∈ source✝ → invFun✝ (toFun✝ x) = x
right_in... | 3205c45671bbb84a |
Orientation.oangle_eq_of_angle_eq_of_sign_eq | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
(h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
(hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).s... | have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hsyz | V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).s... | ec88cb80dbeb7f16 |
Int.exists_strictMono | Mathlib/Order/Monotone/Basic.lean | theorem exists_strictMono : ∃ f : ℤ → α, StrictMono f | case intro.intro.intro.intro.negSucc.zero
α : Type u
inst✝³ : Preorder α
inst✝² : Nonempty α
inst✝¹ : NoMinOrder α
inst✝ : NoMaxOrder α
inhabited_h : Inhabited α
f : ℕ → α
hf : StrictMono f
hf₀ : f 0 = default
g : ℕ → α
hg : StrictAnti g
hg₀ : g 0 = default
⊢ (casesOn -[0+1] f fun n => g (n + 1)) < casesOn (-[0+1] + 1)... | show g 1 < f 0 | case intro.intro.intro.intro.negSucc.zero
α : Type u
inst✝³ : Preorder α
inst✝² : Nonempty α
inst✝¹ : NoMinOrder α
inst✝ : NoMaxOrder α
inhabited_h : Inhabited α
f : ℕ → α
hf : StrictMono f
hf₀ : f 0 = default
g : ℕ → α
hg : StrictAnti g
hg₀ : g 0 = default
⊢ g 1 < f 0 | c85bc10b850ad184 |
Multiset.rel_cons_left | Mathlib/Data/Multiset/ZeroCons.lean | theorem rel_cons_left {a as bs} :
Rel r (a ::ₘ as) bs ↔ ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs' | case mp.cons.inl.intro
α : Type u_1
β : Type v
r : α → β → Prop
a : α
bs✝ : Multiset β
m : Multiset α
b : β
bs : Multiset β
as : Multiset α
ha'b : r a b
h : Rel r as bs
ih : ∀ {as_1 : Multiset α}, a ::ₘ as_1 = as → ∃ b bs', r a b ∧ Rel r as_1 bs' ∧ bs = b ::ₘ bs'
hm : a ::ₘ as = a ::ₘ as
⊢ ∃ b_1 bs', r a b_1 ∧ Rel r as... | exact ⟨b, bs, ha'b, h, rfl⟩ | no goals | 3e8e36661971236c |
SchwartzMap.fourierTransformCLE_symm_apply | Mathlib/Analysis/Distribution/FourierSchwartz.lean | @[simp] lemma fourierTransformCLE_symm_apply (f : 𝓢(V, E)) :
(fourierTransformCLE 𝕜).symm f = 𝓕⁻ f | 𝕜 : Type u_1
inst✝¹⁰ : RCLike 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
inst✝ : ... | ext x | case h
𝕜 : Type u_1
inst✝¹⁰ : RCLike 𝕜
E : Type u_2
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℂ E
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : SMulCommClass ℂ 𝕜 E
V : Type u_3
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
i... | 0f400df34e80e58d |
thickening_cthickening | Mathlib/Analysis/NormedSpace/Pointwise.lean | theorem thickening_cthickening (hε : 0 < ε) (hδ : 0 ≤ δ) (s : Set E) :
thickening ε (cthickening δ s) = thickening (ε + δ) s | case inr
E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
δ ε : ℝ
hε : 0 < ε
hδ✝ : 0 ≤ δ
s : Set E
hδ : 0 < δ
⊢ thickening ε (cthickening δ s) = thickening (ε + δ) s | rw [← closure_thickening hδ, thickening_closure, thickening_thickening hε hδ] | no goals | 3c3e83a56fd150de |
Finset.card_preimage | Mathlib/Data/Finset/Preimage.lean | lemma card_preimage (s : Finset β) (f : α → β) (hf) [DecidablePred (· ∈ Set.range f)] :
(s.preimage f hf).card = {x ∈ s | x ∈ Set.range f}.card :=
card_nbij f (by simp) (by simpa) (fun b hb ↦ by aesop)
| α : Type u
β : Type v
s : Finset β
f : α → β
hf : InjOn f (f ⁻¹' ↑s)
inst✝ : DecidablePred fun x => x ∈ Set.range f
⊢ ∀ a ∈ s.preimage f hf, f a ∈ filter (fun x => x ∈ Set.range f) s | simp | no goals | 50414e7ebbc23122 |
Std.DHashMap.Internal.List.containsKey_of_perm | Mathlib/.lake/packages/lean4/src/lean/Std/Data/DHashMap/Internal/List/Associative.lean | theorem containsKey_of_perm [BEq α] [PartialEquivBEq α] {l l' : List ((a : α) × β a)} {k : α}
(h : Perm l l') : containsKey k l = containsKey k l' | α : Type u
β : α → Type v
inst✝¹ : BEq α
inst✝ : PartialEquivBEq α
l l' : List ((a : α) × β a)
k : α
p p' : (a : α) × β a
l✝ : List ((a : α) × β a)
⊢ (p'.fst == k || (p.fst == k || containsKey k l✝)) = (p.fst == k || (p'.fst == k || containsKey k l✝)) | simp only [← Bool.or_assoc, Bool.or_comm] | no goals | 70651f11be1009a2 |
PredOrder.nhdsLE | Mathlib/Topology/Order/OrderClosed.lean | theorem PredOrder.nhdsLE [PredOrder α] : 𝓝[≤] b = pure b | α : Type u
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : ClosedIicTopology α
b : α
inst✝ : PredOrder α
⊢ 𝓝[≤] b = pure b | rw [← Iio_insert, nhdsWithin_insert, PredOrder.nhdsLT, sup_bot_eq] | no goals | eab7a6955ad24145 |
Vector.empty_append | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Vector/Lemmas.lean | theorem empty_append (as : Vector α n) : (#v[] : Vector α 0) ++ as = as.cast (by omega) | case mk
α : Type u_1
as : Array α
⊢ { toArray := #[], size_toArray := ⋯ } ++ { toArray := as, size_toArray := ⋯ } =
Vector.cast ⋯ { toArray := as, size_toArray := ⋯ } | simp | no goals | 714540f82f37bf51 |
Ideal.Filtration.Stable.of_le | Mathlib/RingTheory/Filtration.lean | theorem Stable.of_le [IsNoetherianRing R] [Module.Finite R M] (hF : F.Stable)
{F' : I.Filtration M} (hf : F' ≤ F) : F'.Stable | R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
I : Ideal R
F : I.Filtration M
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R M
hF : F.submodule.FG
F' : I.Filtration M
hf : F' ≤ F
this : ∀ s ≤ F.submodule, s.FG
⊢ F'.submodule.FG | exact this _ (OrderHomClass.mono (submoduleInfHom M I) hf) | no goals | 3d09736b8efb7e8b |
Fin.foldlM_eq_foldlM_finRange | Mathlib/.lake/packages/batteries/Batteries/Data/Fin/Fold.lean | theorem foldlM_eq_foldlM_finRange [Monad m] (f : α → Fin n → m α) (x) :
foldlM n f x = (List.finRange n).foldlM f x | m : Type u_1 → Type u_2
α : Type u_1
n : Nat
inst✝ : Monad m
f : α → Fin n → m α
x : α
⊢ foldlM n f x = List.foldlM f x (List.finRange n) | induction n generalizing x with
| zero => simp
| succ n ih =>
rw [foldlM_succ, List.finRange_succ, List.foldlM_cons]
congr; funext
rw [List.foldlM_map, ih] | no goals | f31314f041b60b3d |
Filter.exists_seq_monotone_tendsto_atTop_atTop | Mathlib/Order/Filter/AtTopBot/CountablyGenerated.lean | theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [Preorder α] [Nonempty α]
[IsDirected α (· ≤ ·)] [(atTop : Filter α).IsCountablyGenerated] :
∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop | case intro.refine_1
α : Type u_3
inst✝³ : Preorder α
inst✝² : Nonempty α
inst✝¹ : IsDirected α fun x1 x2 => x1 ≤ x2
inst✝ : atTop.IsCountablyGenerated
ys : ℕ → α
h : Tendsto ys atTop atTop
c : α → α → α
hleft : ∀ (a b : α), a ≤ c a b
hright : ∀ (a b : α), b ≤ c a b
xs : ℕ → α := fun n => List.foldl (fun x n => c x (ys ... | refine monotone_nat_of_le_succ fun n ↦ ?_ | case intro.refine_1
α : Type u_3
inst✝³ : Preorder α
inst✝² : Nonempty α
inst✝¹ : IsDirected α fun x1 x2 => x1 ≤ x2
inst✝ : atTop.IsCountablyGenerated
ys : ℕ → α
h : Tendsto ys atTop atTop
c : α → α → α
hleft : ∀ (a b : α), a ≤ c a b
hright : ∀ (a b : α), b ≤ c a b
xs : ℕ → α := fun n => List.foldl (fun x n => c x (ys ... | cfd85fba236c098c |
MeasureTheory.Measure.compProd_const | Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean | /-- The composition product of a measure and a constant kernel is the product between the two
measures. -/
@[simp]
lemma compProd_const {ν : Measure β} [SFinite μ] [SFinite ν] :
μ ⊗ₘ (Kernel.const α ν) = μ.prod ν | case h
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
ν : Measure β
inst✝¹ : SFinite μ
inst✝ : SFinite ν
s : Set (α × β)
hs : MeasurableSet s
⊢ (μ ⊗ₘ Kernel.const α ν) s = (μ.prod ν) s | simp_rw [compProd_apply hs, prod_apply hs, Kernel.const_apply] | no goals | 078663d59dfc4640 |
mul_nonneg_of_three | Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean | theorem mul_nonneg_of_three [ExistsAddOfLE α] [MulPosStrictMono α] [PosMulStrictMono α]
[AddLeftMono α] [AddLeftReflectLE α]
(a b c : α) : 0 ≤ a * b ∨ 0 ≤ b * c ∨ 0 ≤ c * a | α : Type u
inst✝⁶ : Semiring α
inst✝⁵ : LinearOrder α
inst✝⁴ : ExistsAddOfLE α
inst✝³ : MulPosStrictMono α
inst✝² : PosMulStrictMono α
inst✝¹ : AddLeftMono α
inst✝ : AddLeftReflectLE α
a b c : α
⊢ (0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) ∨ (0 ≤ b ∧ 0 ≤ c ∨ b ≤ 0 ∧ c ≤ 0) ∨ 0 ≤ c ∧ 0 ≤ a ∨ c ≤ 0 ∧ a ≤ 0 | have or_a := le_total 0 a | α : Type u
inst✝⁶ : Semiring α
inst✝⁵ : LinearOrder α
inst✝⁴ : ExistsAddOfLE α
inst✝³ : MulPosStrictMono α
inst✝² : PosMulStrictMono α
inst✝¹ : AddLeftMono α
inst✝ : AddLeftReflectLE α
a b c : α
or_a : 0 ≤ a ∨ a ≤ 0
⊢ (0 ≤ a ∧ 0 ≤ b ∨ a ≤ 0 ∧ b ≤ 0) ∨ (0 ≤ b ∧ 0 ≤ c ∨ b ≤ 0 ∧ c ≤ 0) ∨ 0 ≤ c ∧ 0 ≤ a ∨ c ≤ 0 ∧ a ≤ 0 | 4c34bdb62bdba8b8 |
FirstOrder.ACF_models_genericPolyMapSurjOnOfInjOn_of_prime | Mathlib/FieldTheory/AxGrothendieck.lean | theorem ACF_models_genericPolyMapSurjOnOfInjOn_of_prime [Finite ι]
{p : ℕ} (hp : p.Prime) (φ : ring.Formula (α ⊕ ι)) (mons : ι → Finset (ι →₀ ℕ)) :
Theory.ACF p ⊨ᵇ genericPolyMapSurjOnOfInjOn φ mons | ι : Type u_1
α : Type u_2
inst✝¹ : Finite α
inst✝ : Finite ι
p : ℕ
hp : Nat.Prime p
φ : ring.Formula (α ⊕ ι)
mons : ι → Finset (ι →₀ ℕ)
this✝ : Fact (Nat.Prime p)
this : CompatibleRing (AlgebraicClosure (ZMod p)) := compatibleRingOfRing (AlgebraicClosure (ZMod p))
⊢ Theory.ACF p ⊨ᵇ genericPolyMapSurjOnOfInjOn φ mons | have : CharP (AlgebraicClosure (ZMod p)) p :=
charP_of_injective_algebraMap
(RingHom.injective (algebraMap (ZMod p) (AlgebraicClosure (ZMod p)))) p | ι : Type u_1
α : Type u_2
inst✝¹ : Finite α
inst✝ : Finite ι
p : ℕ
hp : Nat.Prime p
φ : ring.Formula (α ⊕ ι)
mons : ι → Finset (ι →₀ ℕ)
this✝¹ : Fact (Nat.Prime p)
this✝ : CompatibleRing (AlgebraicClosure (ZMod p)) := compatibleRingOfRing (AlgebraicClosure (ZMod p))
this : CharP (AlgebraicClosure (ZMod p)) p
⊢ Theory.A... | 9d0baa5e16d2fbab |
Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne | Mathlib/RingTheory/DedekindDomain/PID.lean | theorem Ideal.eq_span_singleton_of_mem_of_not_mem_sq_of_not_mem_prime_ne {P : Ideal R}
(hP : P.IsPrime) [IsDedekindDomain R] {x : R} (x_mem : x ∈ P) (hxP2 : x ∉ P ^ 2)
(hxQ : ∀ Q : Ideal R, IsPrime Q → Q ≠ P → x ∉ Q) : P = Ideal.span {x} | R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
hP : P.IsPrime
inst✝ : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ P ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
this : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
⊢ x ≠ 0 | rintro rfl | R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
hP : P.IsPrime
inst✝ : IsDedekindDomain R
this : DecidableEq (Ideal R) := Classical.decEq (Ideal R)
x_mem : 0 ∈ P
hxP2 : 0 ∉ P ^ 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → 0 ∉ Q
⊢ False | 2ba74b55b2935b6c |
Types.pushoutCocone_inl_eq_inr_iff_of_isColimit | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | lemma pushoutCocone_inl_eq_inr_iff_of_isColimit {c : PushoutCocone f g} (hc : IsColimit c)
(h₁ : Function.Injective f) (x₁ : X₁) (x₂ : X₂) :
c.inl x₁ = c.inr x₂ ↔ ∃ (s : S), f s = x₁ ∧ g s = x₂ | S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
c : PushoutCocone f g
hc : IsColimit c
h₁ : Function.Injective f
x₁ : X₁
x₂ : X₂
⊢ ∀ (j : WalkingSpan),
c.ι.app j ≫ (hc.coconePointUniqueUpToIso (Pushout.isColimitCocone f g)).hom = (Pushout.cocone f g).ι.app j | simp | no goals | 19aa63a1e336c260 |
Finset.gcd_eq_zero_iff | Mathlib/Algebra/GCDMonoid/Finset.lean | theorem gcd_eq_zero_iff : s.gcd f = 0 ↔ ∀ x : β, x ∈ s → f x = 0 | case mpr
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Finset β
f : β → α
h : ∀ x ∈ s, f x = 0
a : α
as : a ∈ Multiset.map f s.val
⊢ a = 0 | rw [Multiset.mem_map] at as | case mpr
α : Type u_2
β : Type u_3
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s : Finset β
f : β → α
h : ∀ x ∈ s, f x = 0
a : α
as : ∃ a_1 ∈ s.val, f a_1 = a
⊢ a = 0 | c2b830459b9b3360 |
LieAlgebra.IsKilling.eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul | Mathlib/Algebra/Lie/Weights/RootSystem.lean | lemma eq_neg_one_or_eq_zero_or_eq_one_of_eq_smul
(hα : α.IsNonZero) (k : K) (h : (β : H → K) = k • α) :
k = -1 ∨ k = 0 ∨ k = 1 | case inr.inl.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H✝ : LieSubalgebra K L
inst✝¹ : H✝.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H✝) L
α β : Weight K (↥H✝) L
hα : α.IsNonZero
k : K
h : ... | rw [chainTopCoeff_zero_right α hα, chainBotCoeff_zero_right α hα, Nat.cast_one] at this | case inr.inl.intro
K : Type u_1
L : Type u_2
inst✝⁷ : Field K
inst✝⁶ : CharZero K
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra K L
inst✝³ : IsKilling K L
inst✝² : FiniteDimensional K L
H✝ : LieSubalgebra K L
inst✝¹ : H✝.IsCartanSubalgebra
inst✝ : IsTriangularizable K (↥H✝) L
α β : Weight K (↥H✝) L
hα : α.IsNonZero
k : K
h : ... | 1279a8c2c7f69b11 |
map_preNormEDS' | Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | lemma map_preNormEDS' (n : ℕ) : f (preNormEDS' b c d n) = preNormEDS' (f b) (f c) (f d) n | R : Type u
inst✝¹ : CommRing R
b c d : R
S : Type v
inst✝ : CommRing S
f : R →+* S
n : ℕ
⊢ f (preNormEDS' b c d n) = preNormEDS' (f b) (f c) (f d) n | induction n using normEDSRec' with
| zero => rw [preNormEDS'_zero, map_zero, preNormEDS'_zero]
| one => rw [preNormEDS'_one, map_one, preNormEDS'_one]
| two => rw [preNormEDS'_two, map_one, preNormEDS'_two]
| three => rw [preNormEDS'_three, preNormEDS'_three]
| four => rw [preNormEDS'_four, preNormEDS'_four]
| _ _ ih =... | no goals | 1dd850ac8d082b25 |
isNonarchimedean_smoothingFun | Mathlib/Analysis/Normed/Ring/SmoothingSeminorm.lean | theorem isNonarchimedean_smoothingFun (hμ1 : μ 1 ≤ 1) (hna : IsNonarchimedean μ) :
IsNonarchimedean (smoothingFun μ) | case intro.intro.intro.intro.h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n =>... | intro ε hε | case intro.intro.intro.intro.h
R : Type u_1
inst✝ : CommRing R
μ : RingSeminorm R
hμ1 : μ 1 ≤ 1
hna : IsNonarchimedean ⇑μ
x y : R
hn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)
mu : ℕ → ℕ := fun n => _root_.mu μ hn n
nu : ℕ → ℕ := fun n => n - mu n
hnu : nu = fun n =>... | 003e5c0c3a2046fe |
Algebra.norm_eq_of_ringEquiv | Mathlib/RingTheory/Norm/Basic.lean | lemma norm_eq_of_ringEquiv {A B C : Type*} [CommRing A] [CommRing B] [Ring C]
[Algebra A C] [Algebra B C] (e : A ≃+* B) (he : (algebraMap B C).comp e = algebraMap A C)
(x : C) :
e (Algebra.norm A x) = Algebra.norm B x | case pos.intro.intro
A : Type u_8
B : Type u_9
C : Type u_10
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Ring C
inst✝¹ : Algebra A C
inst✝ : Algebra B C
e : A ≃+* B
he : (algebraMap B C).comp ↑e = algebraMap A C
x : C
s : Finset C
b : Basis { x // x ∈ s } B C
this✝ : Algebra A B := (↑e).toAlgebra
this : IsScalarTo... | congr | case pos.intro.intro.e_M
A : Type u_8
B : Type u_9
C : Type u_10
inst✝⁴ : CommRing A
inst✝³ : CommRing B
inst✝² : Ring C
inst✝¹ : Algebra A C
inst✝ : Algebra B C
e : A ≃+* B
he : (algebraMap B C).comp ↑e = algebraMap A C
x : C
s : Finset C
b : Basis { x // x ∈ s } B C
this✝ : Algebra A B := (↑e).toAlgebra
this : IsScal... | ba2b8fbc7e66511d |
isIntegral_discr_mul_of_mem_traceDual | Mathlib/RingTheory/DedekindDomain/Different.lean | /-- If `b` is an `A`-integral basis of `L` with discriminant `b`, then `d • a * x` is integral over
`A` for all `a ∈ I` and `x ∈ Iᵛ`. -/
lemma isIntegral_discr_mul_of_mem_traceDual
(I : Submodule B L) {ι} [DecidableEq ι] [Fintype ι]
{b : Basis ι K L} (hb : ∀ i, IsIntegral A (b i))
{a x : L} (ha : a ∈ I) (... | case h.h.h.isTrue
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ ... | rw [mul_assoc] | case h.h.h.isTrue
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ ... | 14fab2851c4d1928 |
Ordnode.emem_iff_mem_toList | Mathlib/Data/Ordmap/Ordset.lean | theorem emem_iff_mem_toList {x : α} {t} : Emem x t ↔ x ∈ toList t | α : Type u_1
x : α
t : Ordnode α
⊢ Emem x t ↔ x ∈ t.toList | unfold Emem | α : Type u_1
x : α
t : Ordnode α
⊢ Any (Eq x) t ↔ x ∈ t.toList | 4820649edff5963c |
MeasureTheory.SimpleFunc.integral_piecewise_zero | Mathlib/MeasureTheory/Integral/BochnerL1.lean | theorem integral_piecewise_zero {m : MeasurableSpace α} (f : α →ₛ F) (μ : Measure α) {s : Set α}
(hs : MeasurableSet s) : (piecewise s hs f 0).integral μ = f.integral (μ.restrict s) | case refine_2
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
y : F
hy : y ∈ filter (fun x => x ≠ 0) f.range
⊢ (μ (⇑(piecewise s hs f 0) ⁻¹' {y})).toReal • y = ((μ.restrict s) (⇑f ⁻¹' {y})).toReal • y | dsimp | case refine_2
α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
f : α →ₛ F
μ : Measure α
s : Set α
hs : MeasurableSet s
y : F
hy : y ∈ filter (fun x => x ≠ 0) f.range
⊢ (μ (s.piecewise (⇑f) 0 ⁻¹' {y})).toReal • y = ((μ.restrict s) (⇑f ⁻¹' {y})).toReal • y | 81410ec8ec0b4f37 |
SimpleGraph.Walk.IsPath.getVert_injOn | Mathlib/Combinatorics/SimpleGraph/Path.lean | lemma IsPath.getVert_injOn {p : G.Walk u v} (hp : p.IsPath) :
Set.InjOn p.getVert {i | i ≤ p.length} | case pos
V : Type u
G : SimpleGraph V
u✝ v✝ v w u : V
h : G.Adj v w
p : G.Walk w u
ihp :
p.IsPath → ∀ ⦃n : ℕ⦄, n ∈ {i | i ≤ p.length} → ∀ ⦃m : ℕ⦄, m ∈ {i | i ≤ p.length} → p.getVert n = p.getVert m → n = m
hp : (cons h p).IsPath
n : ℕ
hn : n ≤ p.length + 1
m : ℕ
hm : m ≤ p.length + 1
hnm : (cons h p).getVert n = (con... | aesop | no goals | a5208a8e99283bf1 |
Finset.sum_Ico_by_parts | Mathlib/Algebra/BigOperators/Module.lean | theorem sum_Ico_by_parts (hmn : m < n) :
∑ i ∈ Ico m n, f i • g i =
f (n - 1) • G n - f m • G m - ∑ i ∈ Ico m (n - 1), (f (i + 1) - f i) • G (i + 1) | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range i, g i = ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i
h₂ :
∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range (i + 1), g i =
∑ i ∈ Ico m (n - 1)... | have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by
intro i
rw [sub_smul]
abel | R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : ℕ → R
g : ℕ → M
m n : ℕ
hmn : m < n
h₁ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range i, g i = ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i
h₂ :
∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range (i + 1), g i =
∑ i ∈ Ico m (n - 1)... | 35456e90ec4d4807 |
Types.Pushout.equivalence_rel' | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | lemma equivalence_rel' [Mono f] : _root_.Equivalence (Rel' f g) where
refl := Rel'.refl
symm h := h.symm
trans | case inl_inr
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
inst✝ : Mono f
z : X₁ ⊕ X₂
s : S
hyz : Rel' f g (Sum.inr (g s)) z
⊢ Rel' f g (Sum.inl (f s)) z | obtain z₁|z₂ := z | case inl_inr.inl
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
inst✝ : Mono f
s : S
z₁ : X₁
hyz : Rel' f g (Sum.inr (g s)) (Sum.inl z₁)
⊢ Rel' f g (Sum.inl (f s)) (Sum.inl z₁)
case inl_inr.inr
S X₁ X₂ : Type u
f : S ⟶ X₁
g : S ⟶ X₂
inst✝ : Mono f
s : S
z₂ : X₂
hyz : Rel' f g (Sum.inr (g s)) (Sum.inr z₂)
⊢ Rel' f g (Sum.inl (... | 4e0c3f5d1e7c96ff |
Std.Tactic.BVDecide.BVExpr.bitblast.go_decl_eq | Mathlib/.lake/packages/lean4/src/lean/Std/Tactic/BVDecide/Bitblast/BVExpr/Circuit/Impl/Expr.lean | theorem bitblast.go_decl_eq (aig : AIG BVBit) (expr : BVExpr w) :
∀ (idx : Nat) (h1) (h2), (go aig expr).val.aig.decls[idx]'h2 = aig.decls[idx]'h1 | case shiftLeft.h2
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.si... | apply Nat.lt_of_lt_of_le h1 | case shiftLeft.h2
w idx m✝ n✝ : Nat
lhs : BVExpr m✝
rhs : BVExpr n✝
lih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig lhs).val.aig.decls.size),
(go aig lhs).val.aig.decls[idx] = aig.decls[idx]
rih :
∀ (aig : AIG BVBit) (h1 : idx < aig.decls.size) (h2 : idx < (go aig rhs).val.aig.decls.si... | f92988d5a1595b39 |
Nat.add_left_eq_self | Mathlib/.lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean | theorem add_left_eq_self {a b : Nat} : a + b = b ↔ a = 0 | a b : Nat
⊢ a + b = b ↔ a = 0 | omega | no goals | 12d0bc9ae86a01c6 |
Function.Periodic.image_uIcc | Mathlib/Algebra/Field/Periodic.lean | theorem Periodic.image_uIcc [LinearOrderedAddCommGroup α] [Archimedean α] (h : Periodic f c)
(hc : c ≠ 0) (a : α) : f '' uIcc a (a + c) = range f | α : Type u_1
β : Type u_2
f : α → β
c : α
inst✝¹ : LinearOrderedAddCommGroup α
inst✝ : Archimedean α
h : Periodic f c
hc : c ≠ 0
a : α
⊢ f '' uIcc a (a + c) = range f | cases hc.lt_or_lt with
| inl hc =>
rw [uIcc_of_ge (add_le_of_nonpos_right hc.le), ← h.neg.image_Icc (neg_pos.2 hc) (a + c),
add_neg_cancel_right]
| inr hc => rw [uIcc_of_le (le_add_of_nonneg_right hc.le), h.image_Icc hc] | no goals | c5696b4a193e3464 |
MeasureTheory.lintegral_abs_det_fderiv_le_addHaar_image_aux1 | Mathlib/MeasureTheory/Function/Jacobian.lean | theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) {ε : ℝ≥0} (εpos : 0 < ε) :
(∫⁻ x in s, ENNReal.ofReal |(f' x).det| ∂μ) ≤ μ (f '' s) + 2 * ε * μ s | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εp... | rw [← Real.dist_eq] | E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
s : Set E
f : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hf : InjOn f s
ε : ℝ≥0
εp... | 10b49d97c45d9730 |
Int.cast_div | Mathlib/Data/Int/Cast/Field.lean | theorem cast_div [DivisionRing α] {m n : ℤ} (n_dvd : n ∣ m) (hn : (n : α) ≠ 0) :
((m / n : ℤ) : α) = m / n | case intro
α : Type u_1
inst✝ : DivisionRing α
n : ℤ
hn : ↑n ≠ 0
k : ℤ
⊢ ↑(n * k / n) = ↑(n * k) / ↑n | have : n ≠ 0 := by rintro rfl; simp at hn | case intro
α : Type u_1
inst✝ : DivisionRing α
n : ℤ
hn : ↑n ≠ 0
k : ℤ
this : n ≠ 0
⊢ ↑(n * k / n) = ↑(n * k) / ↑n | f1d951eaaf9d3b09 |
HahnSeries.coeff_mul_single_add | Mathlib/RingTheory/HahnSeries/Multiplication.lean | theorem coeff_mul_single_add [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeries Γ R} {a : Γ}
{b : Γ} : (x * single b r).coeff (a + b) = x.coeff a * r | Γ : Type u_1
R : Type u_3
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
r : R
x : HahnSeries Γ R
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ ∑ x_1 ∈ addAntidiagonal ⋯ ⋯ (a + b), x.coeff x_1.1 * ((single b) r).coeff x_1.2 =
∑ ij ∈ {(a, b)}, x.coeff ij.1 * ((single b) r).coeff ij.2 | apply sum_congr _ fun _ _ => rfl | Γ : Type u_1
R : Type u_3
inst✝¹ : OrderedCancelAddCommMonoid Γ
inst✝ : NonUnitalNonAssocSemiring R
r : R
x : HahnSeries Γ R
a b : Γ
hr : ¬r = 0
hx : ¬x.coeff a = 0
⊢ addAntidiagonal ⋯ ⋯ (a + b) = {(a, b)} | 32af46b617a94f1a |
ProbabilityTheory.strong_law_ae_simpleFunc_comp | Mathlib/Probability/StrongLaw.lean | /-- Preliminary lemma for the strong law of large numbers for vector-valued random variables:
the composition of the random variables with a simple function satisfies the strong law of large
numbers. -/
lemma strong_law_ae_simpleFunc_comp (X : ℕ → Ω → E) (h' : Measurable (X 0))
(hindep : Pairwise ((IndepFun · · μ) ... | case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ),... | have I : indicator s (Function.const E c) = (fun x ↦ (indicator s (1 : E → ℝ) x) • c) := by
ext
rw [← indicator_smul_const_apply]
congr! 1
ext
simp | case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 => IndepFun x1 x2 μ) on X)
hident : ∀ (i : ℕ),... | 547645d227ed719a |
Nat.ceil_le_two_mul | Mathlib/Algebra/Order/Floor.lean | lemma ceil_le_two_mul (ha : 2⁻¹ ≤ a) : ⌈a⌉₊ ≤ 2 * a :=
ceil_le_mul one_lt_two (by norm_num at ha ⊢; exact ha)
| α : Type u_2
inst✝¹ : LinearOrderedField α
inst✝ : FloorSemiring α
a : α
ha : 2⁻¹ ≤ a
⊢ ↑⌈(2 - 1)⁻¹⌉₊ / 2 ≤ a | norm_num at ha ⊢ | α : Type u_2
inst✝¹ : LinearOrderedField α
inst✝ : FloorSemiring α
a : α
ha : 1 / 2 ≤ a
⊢ 1 / 2 ≤ a | 1a8bd32882b6bba8 |
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