ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: MeasureTheory.measure_lt_top_of_isCompact_of_isMulLeftInvariant
File: Mathlib/MeasureTheory/Group/Measure.lean

Proof steps:
state_before: 𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
U : Set G
hU : IsOpen U
h'U : U.Nonempty
h : μ U ≠ ⊤
K : Set G
hK : IsCompact K
⊢ μ K < ⊤
state_after: 𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
U : Set G
hU : IsOpen U
h'U : (interior U).Nonempty
h : μ U ≠ ⊤
K : Set G
hK : IsCompact K
⊢ μ K < ⊤
tactic: rw [← hU.interior_eq] at h'U
state_before: 𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
U : Set G
hU : IsOpen U
h'U : (interior U).Nonempty
h : μ U ≠ ⊤
K : Set G
hK : IsCompact K
⊢ μ K < ⊤
state_after: case intro
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
U : Set G
hU : IsOpen U
h'U : (interior U).Nonempty
h : μ U ≠ ⊤
K : Set G
hK : IsCompact K
t : Finset G
hKt : K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' U
⊢ μ K < ⊤
tactic: obtain ⟨t, hKt⟩ : ∃ t : Finset G, K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U :=
    compact_covered_by_mul_left_translates hK h'U
state_before: case intro
𝕜 : Type u_1
G : Type u_2
H : Type u_3
inst✝⁶ : MeasurableSpace G
inst✝⁵ : MeasurableSpace H
inst✝⁴ : TopologicalSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : μ.IsMulLeftInvariant
U : Set G
hU : IsOpen U
h'U : (interior U).Nonempty
h : μ U ≠ ⊤
K : Set G
hK : IsCompact K
t : Finset G
hKt : K ⊆ ⋃ g ∈ t, (fun h => g * h) ⁻¹' U
⊢ μ K < ⊤
state_after: no goals
tactic: calc
    μ K ≤ μ (⋃ (g : G) (_ : g ∈ t), (fun h : G => g * h) ⁻¹' U) := measure_mono hKt
    _ ≤ ∑ g ∈ t, μ ((fun h : G => g * h) ⁻¹' U) := measure_biUnion_finset_le _ _
    _ = Finset.card t * μ U := by simp only [measure_preimage_mul, Finset.sum_const, nsmul_eq_mul]
    _ < ∞ := ENNReal.mul_lt_top (ENNReal.natCast_ne_top _) h
Theorem: CategoryTheory.Limits.colimitHomIsoLimitYoneda_hom_comp_π
File: Mathlib/CategoryTheory/Limits/IndYoneda.lean

Proof steps:
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ (colimitHomIsoLimitYoneda F A).hom ≫ limit.π (F.op ⋙ yoneda.obj A) (op i) = (coyoneda.map (colimit.ι F i).op).app A
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ (coyonedaOpColimitIsoLimitCoyoneda F).hom.app A ≫
      (limitObjIsoLimitCompEvaluation (F.op ⋙ coyoneda) A).hom ≫ limit.π (F.op ⋙ yoneda.obj A) (op i) =
    (coyoneda.map (colimit.ι F i).op).app A
tactic: simp only [colimitHomIsoLimitYoneda, Iso.trans_hom, Iso.app_hom, Category.assoc]
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ (coyonedaOpColimitIsoLimitCoyoneda F).hom.app A ≫
      (limitObjIsoLimitCompEvaluation (F.op ⋙ coyoneda) A).hom ≫ limit.π (F.op ⋙ yoneda.obj A) (op i) =
    (coyoneda.map (colimit.ι F i).op).app A
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ (coyonedaOpColimitIsoLimitCoyoneda F).hom.app A ≫ (limit.π (F.op ⋙ coyoneda) (op i)).app A =
    (coyoneda.map (colimit.ι F i).op).app A
tactic: erw [limitObjIsoLimitCompEvaluation_hom_π]
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ (coyonedaOpColimitIsoLimitCoyoneda F).hom.app A ≫ (limit.π (F.op ⋙ coyoneda) (op i)).app A =
    (coyoneda.map (colimit.ι F i).op).app A
state_after: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ ((coyonedaOpColimitIsoLimitCoyoneda F).hom ≫ limit.π (F.op ⋙ coyoneda) (op i)).app A =
    (coyoneda.map (colimit.ι F i).op).app A
tactic: change ((coyonedaOpColimitIsoLimitCoyoneda F).hom ≫ _).app A = _
state_before: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
I : Type v₁
inst✝² : Category.{v₂, v₁} I
F : I ⥤ C
inst✝¹ : HasColimit F
inst✝ : HasLimitsOfShape Iᵒᵖ (Type u₂)
A : C
i : I
⊢ ((coyonedaOpColimitIsoLimitCoyoneda F).hom ≫ limit.π (F.op ⋙ coyoneda) (op i)).app A =
    (coyoneda.map (colimit.ι F i).op).app A
state_after: no goals
tactic: rw [coyonedaOpColimitIsoLimitCoyoneda_hom_comp_π]
Theorem: Multiset.card_filter_le_iff
File: Mathlib/Data/Multiset/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type v
γ : Type u_2
p : α → Prop
inst✝¹ : DecidablePred p
s : Multiset α
P : α → Prop
inst✝ : DecidablePred P
n : ℕ
⊢ card (filter P s) ≤ n ↔ ∀ s' ≤ s, n < card s' → ∃ a ∈ s', ¬P a
state_after: case mp
α : Type u_1
β : Type v
γ : Type u_2
p : α → Prop
inst✝¹ : DecidablePred p
s : Multiset α
P : α → Prop
inst✝ : DecidablePred P
n : ℕ
⊢ card (filter P s) ≤ n → ∀ s' ≤ s, n < card s' → ∃ a ∈ s', ¬P a

case mpr
α : Type u_1
β : Type v
γ : Type u_2
p : α → Prop
inst✝¹ : DecidablePred p
s : Multiset α
P : α → Prop
inst✝ : DecidablePred P
n : ℕ
⊢ (∀ s' ≤ s, n < card s' → ∃ a ∈ s', ¬P a) → card (filter P s) ≤ n
tactic: fconstructor
Theorem: Int.le_add_one_iff
File: Mathlib/Data/Int/Defs.lean

Proof steps:
state_before: a b c d m n : ℤ
⊢ m ≤ n + 1 ↔ m ≤ n ∨ m = n + 1
state_after: no goals
tactic: omega
Theorem: CategoryTheory.Limits.Bicone.toBinaryBiconeFunctor_obj_fst
File: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean

Proof steps:
state_before: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.left ≫ b.π WalkingPair.left = 𝟙 X
state_after: no goals
tactic: simp [Bicone.ι_π]
state_before: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.right ≫ b.π WalkingPair.left = 0
state_after: no goals
tactic: simp [Bicone.ι_π]
state_before: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.left ≫ b.π WalkingPair.right = 0
state_after: no goals
tactic: simp [Bicone.ι_π]
state_before: J : Type w
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
X Y : C
b : Bicone (pairFunction X Y)
⊢ b.ι WalkingPair.right ≫ b.π WalkingPair.right = 𝟙 Y
state_after: no goals
tactic: simp [Bicone.ι_π]
Theorem: mem_segment_iff_div
File: Mathlib/Analysis/Convex/Segment.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝² : LinearOrderedSemifield 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y z : E
⊢ x ∈ [y-[𝕜]z] ↔ ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x
state_after: case mp
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝² : LinearOrderedSemifield 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y z : E
⊢ x ∈ [y-[𝕜]z] → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x

case mpr
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
ι : Type u_5
π : ι → Type u_6
inst✝² : LinearOrderedSemifield 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
x y z : E
⊢ (∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ 0 < a + b ∧ (a / (a + b)) • y + (b / (a + b)) • z = x) → x ∈ [y-[𝕜]z]
tactic: constructor
Theorem: MeasureTheory.IsSetSemiring.sUnion_union_sUnion_diffFinset₀_of_subset
File: Mathlib/MeasureTheory/SetSemiring.lean

Proof steps:
state_before: α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
hI_ss : ∀ t ∈ I, t ⊆ s
⊢ ⋃₀ ↑I ∪ ⋃₀ ↑(hC.diffFinset₀ hs hI) = s
state_after: no goals
tactic: conv_rhs => rw [← union_diff_cancel (Set.sUnion_subset hI_ss : ⋃₀ ↑I ⊆ s),
    hC.diff_sUnion_eq_sUnion_diffFinset₀ hs hI]
Theorem: CommGrp.uliftFunctor_map
File: Mathlib/Algebra/Category/Grp/Basic.lean

Proof steps:
state_before: X : CommGrp
⊢ { obj := fun X => of (ULift.{v, u} ↑X),
          map := fun {X Y} f =>
            ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
      (𝟙 X) =
    𝟙
      ({ obj := fun X => of (ULift.{v, u} ↑X),
            map := fun {X Y} f =>
              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.obj
        X)
state_after: no goals
tactic: rfl
state_before: X Y Z : CommGrp
f : X ⟶ Y
g : Y ⟶ Z
⊢ { obj := fun X => of (ULift.{v, u} ↑X),
          map := fun {X Y} f =>
            ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
      (f ≫ g) =
    { obj := fun X => of (ULift.{v, u} ↑X),
            map := fun {X Y} f =>
              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
        f ≫
      { obj := fun X => of (ULift.{v, u} ↑X),
            map := fun {X Y} f =>
              ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom)) }.map
        g
state_after: no goals
tactic: rfl
Theorem: RestrictGenTopology.isClosed_iff
File: Mathlib/Topology/RestrictGenTopology.lean

Proof steps:
state_before: X : Type u_1
inst✝ : TopologicalSpace X
S : Set (Set X)
t : Set X
x : X
hS : RestrictGenTopology S
⊢ IsClosed t ↔ ∀ s ∈ S, IsClosed (Subtype.val ⁻¹' t)
state_after: no goals
tactic: simp only [← isOpen_compl_iff, hS.isOpen_iff, preimage_compl]
Theorem: Ordinal.cof_eq_one_iff_is_succ
File: Mathlib/SetTheory/Cardinal/Cofinality.lean

Proof steps:
state_before: α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
⊢ ∃ a, type r = succ a
state_after: case intro.intro
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = (type r).cof
⊢ ∃ a, type r = succ a
tactic: rcases cof_eq r with ⟨S, hl, e⟩
state_before: case intro.intro
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = (type r).cof
⊢ ∃ a, type r = succ a
state_after: case intro.intro
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
⊢ ∃ a, type r = succ a
tactic: rw [z] at e
state_before: case intro.intro
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
⊢ ∃ a, type r = succ a
state_after: case intro.intro.intro
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
⊢ ∃ a, type r = succ a
tactic: cases' mk_ne_zero_iff.1 (by rw [e]; exact one_ne_zero) with a
state_before: case intro.intro.intro
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
⊢ ∃ a, type r = succ a
state_after: case intro.intro.intro.refine_1
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
⊢ ↑{b | r b ↑a} ⊕ PUnit.{u + 1} → α

case intro.intro.intro.refine_2
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
x y : ↑{b | r b ↑a} ⊕ PUnit.{u + 1}
⊢ Sum.Lex (Subrel r {b | r b ↑a}) EmptyRelation x y → r (?intro.intro.intro.refine_1 x) (?intro.intro.intro.refine_1 y)

case intro.intro.intro.refine_3
α✝ : Type u_1
r✝ : α✝ → α✝ → Prop
o : Ordinal.{u}
α : Type u
r : α → α → Prop
x✝ : IsWellOrder α r
z : (type r).cof = 1
S : Set α
hl : Unbounded r S
e : #↑S = 1
a : ↑S
x : α
⊢ ∃ a_1, (RelEmbedding.ofMonotone ?intro.intro.intro.refine_1 ⋯) a_1 = x
tactic: refine
        ⟨typein r a,
          Eq.symm <|
            Quotient.sound
              ⟨RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ fun x y => ?_) fun x => ?_⟩⟩
state_before: α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
x✝ : ∃ a, o = succ a
a : Ordinal.{u}
e : o = succ a
⊢ o.cof = 1
state_after: no goals
tactic: simp [e]
Theorem: Encodable.decode₂_encode
File: Mathlib/Logic/Encodable/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : Encodable α
a : α
⊢ decode₂ α (encode a) = some a
state_after: case a
α : Type u_1
β : Type u_2
inst✝ : Encodable α
a a✝ : α
⊢ a✝ ∈ decode₂ α (encode a) ↔ a✝ ∈ some a
tactic: ext
state_before: case a
α : Type u_1
β : Type u_2
inst✝ : Encodable α
a a✝ : α
⊢ a✝ ∈ decode₂ α (encode a) ↔ a✝ ∈ some a
state_after: no goals
tactic: simp [mem_decode₂, eq_comm, decode₂_eq_some]
Theorem: MonotoneOn.intervalIntegrable
File: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean

Proof steps:
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
μ : Measure ℝ
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : ConditionallyCompleteLinearOrder E
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
u : ℝ → E
a b : ℝ
hu : MonotoneOn u [[a, b]]
⊢ IntervalIntegrable u μ a b
state_after: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
μ : Measure ℝ
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : ConditionallyCompleteLinearOrder E
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
u : ℝ → E
a b : ℝ
hu : MonotoneOn u [[a, b]]
⊢ IntegrableOn u (Ι a b) μ
tactic: rw [intervalIntegrable_iff]
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝⁴ : NormedAddCommGroup E
μ : Measure ℝ
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : ConditionallyCompleteLinearOrder E
inst✝¹ : OrderTopology E
inst✝ : SecondCountableTopology E
u : ℝ → E
a b : ℝ
hu : MonotoneOn u [[a, b]]
⊢ IntegrableOn u (Ι a b) μ
state_after: no goals
tactic: exact (hu.integrableOn_isCompact isCompact_uIcc).mono_set Ioc_subset_Icc_self
Theorem: SeminormedAddGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_zero
File: Mathlib/Analysis/Normed/Group/Basic.lean

Proof steps:
state_before: 𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝³ : SeminormedGroup E
inst✝² : SeminormedGroup F
inst✝¹ : SeminormedGroup G
s✝ : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
inst✝ : FunLike 𝓕 E F
f : ι → κ → G
s : Set κ
l : Filter ι
⊢ UniformCauchySeqOn f l s ↔ TendstoUniformlyOn (fun n z => f n.1 z / f n.2 z) 1 (l ×ˢ l) s
state_after: no goals
tactic: rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter,
    uniformCauchySeqOn_iff_uniformCauchySeqOnFilter,
    SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one]
Theorem: MeasureTheory.L1.integral_eq_norm_posPart_sub
File: Mathlib/MeasureTheory/Integral/Bochner.lean

Proof steps:
state_before: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : SMulCommClass ℝ 𝕜 E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
f : ↥(Lp ℝ 1 μ)
⊢ integral f = ‖Lp.posPart f‖ - ‖Lp.negPart f‖
state_after: case refine_1
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : SMulCommClass ℝ 𝕜 E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
f : ↥(Lp ℝ 1 μ)
⊢ Continuous integral

case refine_2
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : SMulCommClass ℝ 𝕜 E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
f : ↥(Lp ℝ 1 μ)
⊢ Continuous fun x => ‖Lp.posPart x‖ - ‖Lp.negPart x‖

case refine_3
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : SMulCommClass ℝ 𝕜 E
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace E
f : ↥(Lp ℝ 1 μ)
⊢ ∀ (a : { x // x ∈ ↑(simpleFunc ℝ 1 μ) }), (fun f => integral f = ‖Lp.posPart f‖ - ‖Lp.negPart f‖) ↑a
tactic: refine @isClosed_property _ _ _ ((↑) : (α →₁ₛ[μ] ℝ) → α →₁[μ] ℝ)
      (fun f : α →₁[μ] ℝ => integral f = ‖Lp.posPart f‖ - ‖Lp.negPart f‖)
      (simpleFunc.denseRange one_ne_top) (isClosed_eq ?_ ?_) ?_ f
Theorem: IntervalIntegrable.mul_continuousOn
File: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean

Proof steps:
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedRing A
f✝ g✝ : ℝ → E
a b : ℝ
μ : Measure ℝ
f g : ℝ → A
hf : IntervalIntegrable f μ a b
hg : ContinuousOn g [[a, b]]
⊢ IntervalIntegrable (fun x => f x * g x) μ a b
state_after: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedRing A
f✝ g✝ : ℝ → E
a b : ℝ
μ : Measure ℝ
f g : ℝ → A
hf : IntegrableOn f (Ι a b) μ
hg : ContinuousOn g [[a, b]]
⊢ IntegrableOn (fun x => f x * g x) (Ι a b) μ
tactic: rw [intervalIntegrable_iff] at hf ⊢
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedRing A
f✝ g✝ : ℝ → E
a b : ℝ
μ : Measure ℝ
f g : ℝ → A
hf : IntegrableOn f (Ι a b) μ
hg : ContinuousOn g [[a, b]]
⊢ IntegrableOn (fun x => f x * g x) (Ι a b) μ
state_after: no goals
tactic: exact hf.mul_continuousOn_of_subset hg measurableSet_Ioc isCompact_uIcc Ioc_subset_Icc_self
Theorem: DistribMulActionHom.toLinearMap_injective
File: Mathlib/Algebra/Module/LinearMap/Defs.lean

Proof steps:
state_before: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Semiring R
inst✝³ : Module R M
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : Module R M₃
σ : R →+* S
f g : M →ₑ+[↑σ] M₂
h : ↑f = ↑g
⊢ f = g
state_after: case a
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Semiring R
inst✝³ : Module R M
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : Module R M₃
σ : R →+* S
f g : M →ₑ+[↑σ] M₂
h : ↑f = ↑g
m : M
⊢ f m = g m
tactic: ext m
state_before: case a
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
R₃ : Type u_4
k : Type u_5
S : Type u_6
S₃ : Type u_7
T : Type u_8
M : Type u_9
M₁ : Type u_10
M₂ : Type u_11
M₃ : Type u_12
N₁ : Type u_13
N₂ : Type u_14
N₃ : Type u_15
ι : Type u_16
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid M₂
inst✝⁵ : AddCommMonoid M₃
inst✝⁴ : Semiring R
inst✝³ : Module R M
inst✝² : Semiring S
inst✝¹ : Module S M₂
inst✝ : Module R M₃
σ : R →+* S
f g : M →ₑ+[↑σ] M₂
h : ↑f = ↑g
m : M
⊢ f m = g m
state_after: no goals
tactic: exact LinearMap.congr_fun h m
Theorem: AffineMap.hasStrictDerivAt
File: Mathlib/Analysis/Calculus/Deriv/AffineMap.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 →ᵃ[𝕜] E
a b : E
L : Filter 𝕜
s : Set 𝕜
x : 𝕜
⊢ HasStrictDerivAt (⇑f) (f.linear 1) x
state_after: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 →ᵃ[𝕜] E
a b : E
L : Filter 𝕜
s : Set 𝕜
x : 𝕜
⊢ HasStrictDerivAt (⇑f.linear + fun x => f 0) (f.linear 1) x
tactic: rw [f.decomp]
state_before: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 →ᵃ[𝕜] E
a b : E
L : Filter 𝕜
s : Set 𝕜
x : 𝕜
⊢ HasStrictDerivAt (⇑f.linear + fun x => f 0) (f.linear 1) x
state_after: no goals
tactic: exact f.linear.hasStrictDerivAt.add_const (f 0)
Theorem: IsCompact.smul_set
File: Mathlib/Topology/Algebra/MulAction.lean

Proof steps:
state_before: M : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : SMul M X
inst✝ : ContinuousSMul M X
f : Y → M
g : Y → X
b : Y
s : Set Y
k : Set M
u : Set X
hk : IsCompact k
hu : IsCompact u
⊢ IsCompact (k • u)
state_after: M : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : SMul M X
inst✝ : ContinuousSMul M X
f : Y → M
g : Y → X
b : Y
s : Set Y
k : Set M
u : Set X
hk : IsCompact k
hu : IsCompact u
⊢ IsCompact ((fun x => x.1 • x.2) '' k ×ˢ u)
tactic: rw [← Set.image_smul_prod]
state_before: M : Type u_1
X : Type u_2
Y : Type u_3
α : Type u_4
inst✝⁴ : TopologicalSpace M
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : SMul M X
inst✝ : ContinuousSMul M X
f : Y → M
g : Y → X
b : Y
s : Set Y
k : Set M
u : Set X
hk : IsCompact k
hu : IsCompact u
⊢ IsCompact ((fun x => x.1 • x.2) '' k ×ˢ u)
state_after: no goals
tactic: exact IsCompact.image (hk.prod hu) continuous_smul
Theorem: geom_sum_two
File: Mathlib/Algebra/GeomSum.lean

Proof steps:
state_before: α : Type u
inst✝ : Semiring α
x : α
⊢ ∑ i ∈ range 2, x ^ i = x + 1
state_after: no goals
tactic: simp [geom_sum_succ']