1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: ProbabilityTheory.condCDF_ae_eq
File: Mathlib/Probability/Kernel/Disintegration/CondCdf.lean

tactic: simp_rw [condCDF_eq_stieltjesOfMeasurableRat_unit_prod ρ]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
⊢ (fun a => ↑(condCDF ρ a) ↑r) =ᶠ[ae ρ.fst] fun a => (preCDF ρ r a).toReal
state_after: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
⊢ (fun a => ↑(stieltjesOfMeasurableRat (fun p r => (preCDF ρ r p.2).toReal) ⋯ ((), a)) ↑r) =ᶠ[ae ρ.fst] fun a =>
    (preCDF ρ r a).toReal

tactic: exact stieltjesOfMeasurableRat_ae_eq (isRatCondKernelCDF_preCDF ρ) () r
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
r : ℚ
⊢ (fun a => ↑(stieltjesOfMeasurableRat (fun p r => (preCDF ρ r p.2).toReal) ⋯ ((), a)) ↑r) =ᶠ[ae ρ.fst] fun a =>
    (preCDF ρ r a).toReal
state_after: no goals
Theorem: setOf_isPreconnected_eq_of_ordered
File: Mathlib/Topology/Order/IntermediateValue.lean

tactic: refine Subset.antisymm setOf_isPreconnected_subset_of_ordered ?_
state_before: α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
⊢ {s | IsPreconnected s} =
    range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})
state_after: α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
⊢ range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) ⊆
    {s | IsPreconnected s}

tactic: simp only [subset_def, forall_mem_range, uncurry, or_imp, forall_and, mem_union,
  mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true_iff,
  isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Ioo,
  isPreconnected_Ioi, isPreconnected_Iio, isPreconnected_Ici, isPreconnected_Iic,
  isPreconnected_univ, isPreconnected_empty]
state_before: α : Type u
β : Type v
γ : Type w
inst✝⁷ : ConditionallyCompleteLinearOrder α
inst✝⁶ : TopologicalSpace α
inst✝⁵ : OrderTopology α
inst✝⁴ : ConditionallyCompleteLinearOrder β
inst✝³ : TopologicalSpace β
inst✝² : OrderTopology β
inst✝¹ : Nonempty γ
inst✝ : DenselyOrdered α
a b : α
⊢ range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) ⊆
    {s | IsPreconnected s}
state_after: no goals
Theorem: PrincipalSeg.top_eq
File: Mathlib/Order/InitialSeg.lean

tactic: rw [Subsingleton.elim f (PrincipalSeg.equivLT e g)]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder γ t
e : r ≃r s
f : r ≺i t
g : s ≺i t
⊢ f.top = g.top
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder γ t
e : r ≃r s
f : r ≺i t
g : s ≺i t
⊢ (equivLT e g).top = g.top

tactic: rfl
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder γ t
e : r ≃r s
f : r ≺i t
g : s ≺i t
⊢ (equivLT e g).top = g.top
state_after: no goals
Theorem: inv_smul_le_iff_of_neg
File: Mathlib/Algebra/Order/Module/Defs.lean

tactic: rw [← smul_le_smul_iff_of_neg_left h, smul_inv_smul₀ h.ne]
state_before: α : Type u_1
β : Type u_2
a✝ a₁ a₂ : α
b b₁✝ b₂✝ : β
inst✝³ : LinearOrderedField α
inst✝² : OrderedAddCommGroup β
inst✝¹ : Module α β
a : α
b₁ b₂ : β
inst✝ : PosSMulMono α β
h : a < 0
⊢ a⁻¹ • b₁ ≤ b₂ ↔ a • b₂ ≤ b₁
state_after: no goals
Theorem: MeasureTheory.Measure.univ_pi_Ioc_ae_eq_Icc
File: Mathlib/MeasureTheory/Constructions/Pi.lean

tactic: rw [← pi_univ_Icc]
state_before: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NoAtoms (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i => Ioc (f i) (g i)) =ᶠ[ae (Measure.pi μ)] Icc f g
state_after: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NoAtoms (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i => Ioc (f i) (g i)) =ᶠ[ae (Measure.pi μ)] univ.pi fun i => Icc (f i) (g i)

tactic: exact pi_Ioc_ae_eq_pi_Icc
state_before: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NoAtoms (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i => Ioc (f i) (g i)) =ᶠ[ae (Measure.pi μ)] univ.pi fun i => Icc (f i) (g i)
state_after: no goals
Theorem: ContMDiffAt.div₀
File: Mathlib/Geometry/Manifold/Algebra/LieGroup.lean

tactic: simpa [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀)
state_before: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹² : TopologicalSpace H
E : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_4
inst✝⁹ : TopologicalSpace G
inst✝⁸ : ChartedSpace H G
inst✝⁷ : GroupWithZero G
inst✝⁶ : SmoothInv₀ I G
inst✝⁵ : SmoothMul I G
E' : Type u_5
inst✝⁴ : NormedAddCommGroup E'
inst✝³ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝² : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝¹ : TopologicalSpace M
inst✝ : ChartedSpace H' M
f g : M → G
s : Set M
a : M
n : ℕ∞
hf : ContMDiffAt I' I n f a
hg : ContMDiffAt I' I n g a
h₀ : g a ≠ 0
⊢ ContMDiffAt I' I n (f / g) a
state_after: no goals
Theorem: CategoryTheory.Limits.PushoutCocone.epi_of_isColimitMkIdId
File: Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean

tactic: rcases PushoutCocone.IsColimit.desc' t _ _ eq with ⟨_, rfl, rfl⟩
state_before: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
W X Y Z✝ : C
f✝ : X ⟶ Y
g✝ : X ⟶ Z✝
f : X ⟶ Y
t : IsColimit (mk (𝟙 Y) (𝟙 Y) ⋯)
Z : C
g h : Y ⟶ Z
eq : f ≫ g = f ≫ h
⊢ g = h
state_after: case mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
W X Y Z✝ : C
f✝ : X ⟶ Y
g : X ⟶ Z✝
f : X ⟶ Y
t : IsColimit (mk (𝟙 Y) (𝟙 Y) ⋯)
Z : C
val✝ : (mk (𝟙 Y) (𝟙 Y) ⋯).pt ⟶ Z
eq : f ≫ (mk (𝟙 Y) (𝟙 Y) ⋯).inl ≫ val✝ = f ≫ (mk (𝟙 Y) (𝟙 Y) ⋯).inr ≫ val✝
⊢ (mk (𝟙 Y) (𝟙 Y) ⋯).inl ≫ val✝ = (mk (𝟙 Y) (𝟙 Y) ⋯).inr ≫ val✝

tactic: rfl
state_before: case mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
W X Y Z✝ : C
f✝ : X ⟶ Y
g : X ⟶ Z✝
f : X ⟶ Y
t : IsColimit (mk (𝟙 Y) (𝟙 Y) ⋯)
Z : C
val✝ : (mk (𝟙 Y) (𝟙 Y) ⋯).pt ⟶ Z
eq : f ≫ (mk (𝟙 Y) (𝟙 Y) ⋯).inl ≫ val✝ = f ≫ (mk (𝟙 Y) (𝟙 Y) ⋯).inr ≫ val✝
⊢ (mk (𝟙 Y) (𝟙 Y) ⋯).inl ≫ val✝ = (mk (𝟙 Y) (𝟙 Y) ⋯).inr ≫ val✝
state_after: no goals
Theorem: tprod_eq_one_of_not_multipliable
File: Mathlib/Topology/Algebra/InfiniteSum/Defs.lean

tactic: simp [tprod_def, h]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f g : β → α
a b : α
s : Finset β
h : ¬Multipliable f
⊢ ∏' (b : β), f b = 1
state_after: no goals
Theorem: IsSelfAdjoint.eq_smul_self_of_isLocalExtrOn
File: Mathlib/Analysis/InnerProductSpace/Rayleigh.lean

tactic: letI := InnerProductSpace.rclikeToReal 𝕜 E
state_before: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
⊢ T x₀ = ↑(T.rayleighQuotient x₀) • x₀
state_after: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
this : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
⊢ T x₀ = ↑(T.rayleighQuotient x₀) • x₀

tactic: let hSA := hT.isSymmetric.restrictScalars.toSelfAdjoint.prop
state_before: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
this : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
⊢ T x₀ = ↑(T.rayleighQuotient x₀) • x₀
state_after: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
this : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
hSA : ↑⋯.toSelfAdjoint ∈ selfAdjoint (E →L[ℝ] E) :=
  Subtype.prop (LinearMap.IsSymmetric.restrictScalars (isSymmetric hT)).toSelfAdjoint
⊢ T x₀ = ↑(T.rayleighQuotient x₀) • x₀

tactic: exact hSA.eq_smul_self_of_isLocalExtrOn_real hextr
state_before: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsSelfAdjoint T
x₀ : E
hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
this : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
hSA : ↑⋯.toSelfAdjoint ∈ selfAdjoint (E →L[ℝ] E) :=
  Subtype.prop (LinearMap.IsSymmetric.restrictScalars (isSymmetric hT)).toSelfAdjoint
⊢ T x₀ = ↑(T.rayleighQuotient x₀) • x₀
state_after: no goals
Theorem: MeasureTheory.FiniteMeasure.restrict_mass
File: Mathlib/MeasureTheory/Measure/FiniteMeasure.lean

tactic: simp only [mass, restrict_apply μ A MeasurableSet.univ, univ_inter]
state_before: Ω : Type u_1
inst✝⁴ : MeasurableSpace Ω
R : Type u_2
inst✝³ : SMul R ℝ≥0
inst✝² : SMul R ℝ≥0∞
inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞
inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞
μ : FiniteMeasure Ω
A : Set Ω
⊢ (μ.restrict A).mass = μ A
state_after: no goals
Theorem: MeasureTheory.lintegral_map
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

tactic: erw [lintegral_eq_iSup_eapprox_lintegral hf, lintegral_eq_iSup_eapprox_lintegral (hf.comp hg)]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
⊢ ∫⁻ (a : β), f a ∂Measure.map g μ = ∫⁻ (a : α), f (g a) ∂μ
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
⊢ ⨆ n, (eapprox f n).lintegral (Measure.map g μ) = ⨆ n, (eapprox (f ∘ g) n).lintegral μ

tactic: congr with n : 1
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
⊢ ⨆ n, (eapprox f n).lintegral (Measure.map g μ) = ⨆ n, (eapprox (f ∘ g) n).lintegral μ
state_after: case e_s.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
⊢ (eapprox f n).lintegral (Measure.map g μ) = (eapprox (f ∘ g) n).lintegral μ

tactic: convert SimpleFunc.lintegral_map _ hg
state_before: case e_s.h
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
⊢ (eapprox f n).lintegral (Measure.map g μ) = (eapprox (f ∘ g) n).lintegral μ
state_after: case h.e'_3.h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
⊢ eapprox (f ∘ g) n = (eapprox f n).comp g hg

tactic: ext1 x
state_before: case h.e'_3.h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
⊢ eapprox (f ∘ g) n = (eapprox f n).comp g hg
state_after: case h.e'_3.h.e'_3.H
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
x : α
⊢ ↑(eapprox (f ∘ g) n) x = ↑((eapprox f n).comp g hg) x

tactic: simp only [eapprox_comp hf hg, coe_comp]
state_before: case h.e'_3.h.e'_3.H
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
mβ : MeasurableSpace β
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
x : α
⊢ ↑(eapprox (f ∘ g) n) x = ↑((eapprox f n).comp g hg) x
state_after: no goals
Theorem: CochainComplex.HomComplex.Cochain.ofHoms_zero
File: Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean

tactic: aesop_cat
state_before: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
⊢ (ofHoms fun p => 0) = 0
state_after: no goals
Theorem: MeasureTheory.HasFiniteIntegral.restrict
File: Mathlib/MeasureTheory/Function/L1Space.lean

tactic: refine lt_of_le_of_lt ?_ h
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : MeasurableSpace δ
inst✝² : NormedAddCommGroup β
inst✝¹ : NormedAddCommGroup γ
E : Type u_5
inst✝ : NormedAddCommGroup E
f : α → E
h : HasFiniteIntegral f μ
s : Set α
⊢ HasFiniteIntegral f (μ.restrict s)
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : MeasurableSpace δ
inst✝² : NormedAddCommGroup β
inst✝¹ : NormedAddCommGroup γ
E : Type u_5
inst✝ : NormedAddCommGroup E
f : α → E
h : HasFiniteIntegral f μ
s : Set α
⊢ ∫⁻ (a : α) in s, ↑‖f a‖₊ ∂μ ≤ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ

tactic: convert lintegral_mono_set (μ := μ) (s := s) (t := univ) (f := fun x ↦ ↑‖f x‖₊) (subset_univ s)
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : MeasurableSpace δ
inst✝² : NormedAddCommGroup β
inst✝¹ : NormedAddCommGroup γ
E : Type u_5
inst✝ : NormedAddCommGroup E
f : α → E
h : HasFiniteIntegral f μ
s : Set α
⊢ ∫⁻ (a : α) in s, ↑‖f a‖₊ ∂μ ≤ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ
state_after: case h.e'_4.h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : MeasurableSpace δ
inst✝² : NormedAddCommGroup β
inst✝¹ : NormedAddCommGroup γ
E : Type u_5
inst✝ : NormedAddCommGroup E
f : α → E
h : HasFiniteIntegral f μ
s : Set α
⊢ μ = μ.restrict univ

tactic: exact Measure.restrict_univ.symm
state_before: case h.e'_4.h.e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝³ : MeasurableSpace δ
inst✝² : NormedAddCommGroup β
inst✝¹ : NormedAddCommGroup γ
E : Type u_5
inst✝ : NormedAddCommGroup E
f : α → E
h : HasFiniteIntegral f μ
s : Set α
⊢ μ = μ.restrict univ
state_after: no goals
Theorem: Ideal.sup_pow_eq_top
File: Mathlib/RingTheory/Ideal/Operations.lean

tactic: rw [← Finset.card_range n, ← Finset.prod_const]
state_before: R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ I ⊔ J ^ n = ⊤
state_after: R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ I ⊔ ∏ _x ∈ Finset.range n, J = ⊤

tactic: exact sup_prod_eq_top fun _ _ => h
state_before: R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
n : ℕ
h : I ⊔ J = ⊤
⊢ I ⊔ ∏ _x ∈ Finset.range n, J = ⊤
state_after: no goals
Theorem: Set.mulIndicator_compl_mul_self_apply
File: Mathlib/Algebra/Group/Indicator.lean

tactic: simp [ha]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
M : Type u_4
N : Type u_5
inst✝ : MulOneClass M
s✝ t : Set α
f✝ g : α → M
a✝ : α
s : Set α
f : α → M
a : α
ha : a ∈ s
⊢ sᶜ.mulIndicator f a * s.mulIndicator f a = f a
state_after: no goals

tactic: simp [ha]
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
M : Type u_4
N : Type u_5
inst✝ : MulOneClass M
s✝ t : Set α
f✝ g : α → M
a✝ : α
s : Set α
f : α → M
a : α
ha : a ∉ s
⊢ sᶜ.mulIndicator f a * s.mulIndicator f a = f a
state_after: no goals
Theorem: Mathlib.Meta.NormNum.isNat_minFac_4
File: Mathlib/Tactic/NormNum/Prime.lean

tactic: refine ⟨(Nat.prime_def_minFac.mp ?_).2⟩
state_before: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
⊢ IsNat n.minFac n
state_after: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
⊢ Nat.Prime n

tactic: rw [Nat.prime_def_le_sqrt]
state_before: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
⊢ Nat.Prime n
state_after: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
⊢ 2 ≤ n ∧ ∀ (m : ℕ), 2 ≤ m → m ≤ n.sqrt → ¬m ∣ n

tactic: refine ⟨h1.one_lt, λ m hm hmn h2mn ↦ ?_⟩
state_before: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
⊢ 2 ≤ n ∧ ∀ (m : ℕ), 2 ≤ m → m ≤ n.sqrt → ¬m ∣ n
state_after: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
m : ℕ
hm : 2 ≤ m
hmn : m ≤ n.sqrt
h2mn : m ∣ n
⊢ False

tactic: exact lt_irrefl m <| calc
  m ≤ sqrt n   := hmn
  _ < k        := sqrt_lt.mpr (ble_eq_false.mp h2)
  _ ≤ n.minFac := h1.2.2
  _ ≤ m        := Nat.minFac_le_of_dvd hm h2mn
state_before: n k : ℕ
h1 : MinFacHelper n k
h2 : (k * k).ble n = false
m : ℕ
hm : 2 ≤ m
hmn : m ≤ n.sqrt
h2mn : m ∣ n
⊢ False
state_after: no goals
Theorem: MeasureTheory.isStoppingTime_of_measurableSet_eq
File: Mathlib/Probability/Process/Stopping.lean

tactic: intro i
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
⊢ IsStoppingTime f τ
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
⊢ MeasurableSet {ω | τ ω ≤ i}

tactic: rw [show {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} by ext; simp]
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
⊢ MeasurableSet {ω | τ ω ≤ i}
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
⊢ MeasurableSet (⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k})

tactic: refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
⊢ MeasurableSet (⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k})
state_after: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i k : ι
hk : k ∈ fun k => Preorder.toLE.1 k i
⊢ MeasurableSet {ω | τ ω = k}

tactic: exact f.mono hk _ (hτ k)
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i k : ι
hk : k ∈ fun k => Preorder.toLE.1 k i
⊢ MeasurableSet {ω | τ ω = k}
state_after: no goals

tactic: ext
state_before: Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
⊢ {ω | τ ω ≤ i} = ⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k}
state_after: case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
x✝ : Ω
⊢ x✝ ∈ {ω | τ ω ≤ i} ↔ x✝ ∈ ⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k}

tactic: simp
state_before: case h
Ω : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace Ω
inst✝¹ : Preorder ι
inst✝ : Countable ι
f : Filtration ι m
τ : Ω → ι
hτ : ∀ (i : ι), MeasurableSet {ω | τ ω = i}
i : ι
x✝ : Ω
⊢ x✝ ∈ {ω | τ ω ≤ i} ↔ x✝ ∈ ⋃ k, ⋃ (_ : k ≤ i), {ω | τ ω = k}
state_after: no goals
Theorem: HNNExtension.lift_of
File: Mathlib/GroupTheory/HNNExtension.lean

tactic: simp [lift, of]
state_before: G : Type u_1
inst✝² : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
H : Type u_2
inst✝¹ : Group H
M : Type u_3
inst✝ : Monoid M
f : G →* H
x : H
hx : ∀ (a : ↥A), x * f ↑a = f ↑(φ a) * x
g : G
⊢ (lift f x hx) (of g) = f g
state_after: no goals
Theorem: LinearMap.mulLeft_one
File: Mathlib/Algebra/Algebra/Bilinear.lean

tactic: ext
state_before: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
⊢ mulLeft R 1 = id
state_after: case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x✝ : A
⊢ (mulLeft R 1) x✝ = id x✝

tactic: simp
state_before: case h
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x✝ : A
⊢ (mulLeft R 1) x✝ = id x✝
state_after: no goals
Theorem: PowerSeries.coeff_coe
File: Mathlib/RingTheory/LaurentSeries.lean

tactic: cases i
state_before: R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
i : ℤ
⊢ ((ofPowerSeries ℤ R) f).coeff i = if i < 0 then 0 else (coeff R i.natAbs) f
state_after: case ofNat
R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
a✝ : ℕ
⊢ ((ofPowerSeries ℤ R) f).coeff (Int.ofNat a✝) = if Int.ofNat a✝ < 0 then 0 else (coeff R (Int.ofNat a✝).natAbs) f

case negSucc
R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
a✝ : ℕ
⊢ ((ofPowerSeries ℤ R) f).coeff (Int.negSucc a✝) = if Int.negSucc a✝ < 0 then 0 else (coeff R (Int.negSucc a✝).natAbs) f

tactic: rw [Int.ofNat_eq_coe, coeff_coe_powerSeries, if_neg (Int.natCast_nonneg _).not_lt,
  Int.natAbs_ofNat]
state_before: case ofNat
R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
a✝ : ℕ
⊢ ((ofPowerSeries ℤ R) f).coeff (Int.ofNat a✝) = if Int.ofNat a✝ < 0 then 0 else (coeff R (Int.ofNat a✝).natAbs) f
state_after: no goals

tactic: rw [ofPowerSeries_apply, embDomain_notin_image_support, if_pos (Int.negSucc_lt_zero _)]
state_before: case negSucc
R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
a✝ : ℕ
⊢ ((ofPowerSeries ℤ R) f).coeff (Int.negSucc a✝) = if Int.negSucc a✝ < 0 then 0 else (coeff R (Int.negSucc a✝).natAbs) f
state_after: case negSucc
R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
a✝ : ℕ
⊢ Int.negSucc a✝ ∉ ⇑{ toFun := Nat.cast, inj' := ⋯, map_rel_iff' := ⋯ } '' (toPowerSeries.symm f).support

tactic: simp only [not_exists, RelEmbedding.coe_mk, Set.mem_image, not_and, Function.Embedding.coeFn_mk,
  Ne, toPowerSeries_symm_apply_coeff, mem_support, imp_true_iff,
  not_false_iff]
state_before: case negSucc
R : Type u_1
R' : Type u_2
inst✝¹ : Semiring R
inst✝ : Ring R'
f g : R⟦X⟧
f' g' : R'⟦X⟧
a✝ : ℕ
⊢ Int.negSucc a✝ ∉ ⇑{ toFun := Nat.cast, inj' := ⋯, map_rel_iff' := ⋯ } '' (toPowerSeries.symm f).support
state_after: no goals
Theorem: CategoryTheory.e_assoc'
File: Mathlib/CategoryTheory/Enriched/Basic.lean

tactic: rw [← e_assoc V W X Y Z, Iso.hom_inv_id_assoc]
state_before: V : Type v
inst✝² : Category.{w, v} V
inst✝¹ : MonoidalCategory V
C : Type u₁
inst✝ : EnrichedCategory V C
W X Y Z : C
⊢ (α_ (EnrichedCategory.Hom W X) (EnrichedCategory.Hom X Y) (EnrichedCategory.Hom Y Z)).hom ≫
      EnrichedCategory.Hom W X ◁ eComp V X Y Z ≫ eComp V W X Z =
    eComp V W X Y ▷ EnrichedCategory.Hom Y Z ≫ eComp V W Y Z
state_after: no goals