ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive
File: Mathlib/MeasureTheory/Decomposition/Jordan.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
state_after: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
hwuv : ↑s ((w ∩ u) ∆ (w ∩ v)) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
tactic: have hwuv : s ((w ∩ u) ∆ (w ∩ v)) = 0 := by
    refine
      subset_positive_null_set (hu.union hv) ((hw.inter hu).symmDiff (hw.inter hv))
        (hu.symmDiff hv) (restrict_le_restrict_union _ _ hu hsu hv hsv) hs
        Set.symmDiff_subset_union ?_
    rw [← Set.inter_symmDiff_distrib_left]
    exact Set.inter_subset_right
state_before: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
hwuv : ↑s ((w ∩ u) ∆ (w ∩ v)) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
state_after: case intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
hwuv : ↑s ((w ∩ u) ∆ (w ∩ v)) = 0
huv : ↑s ((w ∩ u) \ (w ∩ v)) = 0
hvu : ↑s ((w ∩ v) \ (w ∩ u)) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
tactic: obtain ⟨huv, hvu⟩ :=
    of_diff_eq_zero_of_symmDiff_eq_zero_positive (hw.inter hu) (hw.inter hv)
      (restrict_le_restrict_subset _ _ hu hsu (w.inter_subset_right))
      (restrict_le_restrict_subset _ _ hv hsv (w.inter_subset_right)) hwuv
state_before: case intro
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
s : SignedMeasure α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
u v w : Set α
hu : MeasurableSet u
hv : MeasurableSet v
hw : MeasurableSet w
hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u
hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v
hs : ↑s (u ∆ v) = 0
hwuv : ↑s ((w ∩ u) ∆ (w ∩ v)) = 0
huv : ↑s ((w ∩ u) \ (w ∩ v)) = 0
hvu : ↑s ((w ∩ v) \ (w ∩ u)) = 0
⊢ ↑s (w ∩ u) = ↑s (w ∩ v)
state_after: no goals
tactic: rw [← of_diff_of_diff_eq_zero (hw.inter hu) (hw.inter hv) hvu, huv, zero_add]
Theorem: cfc_comp_smul
File: Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean

Proof steps:
state_before: R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : StarRing R
inst✝¹³ : MetricSpace R
inst✝¹² : TopologicalSemiring R
inst✝¹¹ : ContinuousStar R
inst✝¹⁰ : TopologicalSpace A
inst✝⁹ : Ring A
inst✝⁸ : StarRing A
inst✝⁷ : Algebra R A
inst✝⁶ : ContinuousFunctionalCalculus R p
f✝ g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
inst✝⁵ : UniqueContinuousFunctionalCalculus R A
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : autoParam (ContinuousOn f ((fun x => s • x) '' spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝
⊢ cfc (fun x => f (s • x)) a = cfc f (s • a)
state_after: no goals
tactic: rw [cfc_comp' .., cfc_smul_id ..]
Theorem: List.sum_take_succ
File: Mathlib/Algebra/BigOperators/Group/List.lean

Proof steps:
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a : M
i : ℕ
p : i < [].length
⊢ (take (i + 1) []).prod = (take i []).prod * [][i]
state_after: no goals
tactic: cases p
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a h : M
t : List M
n : ℕ
p : n + 1 < (h :: t).length
⊢ (take (n + 1 + 1) (h :: t)).prod = (take (n + 1) (h :: t)).prod * (h :: t)[n + 1]
state_after: ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a h : M
t : List M
n : ℕ
p : n + 1 < (h :: t).length
⊢ (h :: take (n + 1) t).prod = (h :: take n t).prod * t[n]
tactic: dsimp
state_before: ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a h : M
t : List M
n : ℕ
p : n + 1 < (h :: t).length
⊢ (h :: take (n + 1) t).prod = (h :: take n t).prod * t[n]
state_after: no goals
tactic: rw [prod_cons, prod_cons, prod_take_succ t n (Nat.lt_of_succ_lt_succ p), mul_assoc]
Theorem: AddConstMapClass.map_nat
File: Mathlib/Algebra/AddConstMap/Basic.lean

Proof steps:
state_before: F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
a : G
b : H
inst✝² : AddMonoidWithOne G
inst✝¹ : AddMonoidWithOne H
inst✝ : AddConstMapClass F G H 1 1
f : F
n : ℕ
⊢ f ↑n = f 0 + ↑n
state_after: no goals
tactic: simp
Theorem: CategoryTheory.ShortComplex.exact_iff_of_concreteCategory
File: Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean

Proof steps:
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : ConcreteCategory C
inst✝⁵ : HasForget₂ C Ab
inst✝⁴ : Preadditive C
inst✝³ : (forget₂ C Ab).Additive
inst✝² : (forget₂ C Ab).PreservesHomology
inst✝¹ : HasZeroObject C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ S.Exact ↔
    ∀ (x₂ : ↑((forget₂ C Ab).obj S.X₂)), ((forget₂ C Ab).map S.g) x₂ = 0 → ∃ x₁, ((forget₂ C Ab).map S.f) x₁ = x₂
state_after: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : ConcreteCategory C
inst✝⁵ : HasForget₂ C Ab
inst✝⁴ : Preadditive C
inst✝³ : (forget₂ C Ab).Additive
inst✝² : (forget₂ C Ab).PreservesHomology
inst✝¹ : HasZeroObject C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ (∀ (x₂ : ↑(S.map (forget₂ C Ab)).X₂), (S.map (forget₂ C Ab)).g x₂ = 0 → ∃ x₁, (S.map (forget₂ C Ab)).f x₁ = x₂) ↔
    ∀ (x₂ : ↑((forget₂ C Ab).obj S.X₂)), ((forget₂ C Ab).map S.g) x₂ = 0 → ∃ x₁, ((forget₂ C Ab).map S.f) x₁ = x₂
tactic: rw [S.exact_iff_exact_map_forget₂, ab_exact_iff]
state_before: C : Type u
inst✝⁷ : Category.{v, u} C
inst✝⁶ : ConcreteCategory C
inst✝⁵ : HasForget₂ C Ab
inst✝⁴ : Preadditive C
inst✝³ : (forget₂ C Ab).Additive
inst✝² : (forget₂ C Ab).PreservesHomology
inst✝¹ : HasZeroObject C
S : ShortComplex C
inst✝ : S.HasHomology
⊢ (∀ (x₂ : ↑(S.map (forget₂ C Ab)).X₂), (S.map (forget₂ C Ab)).g x₂ = 0 → ∃ x₁, (S.map (forget₂ C Ab)).f x₁ = x₂) ↔
    ∀ (x₂ : ↑((forget₂ C Ab).obj S.X₂)), ((forget₂ C Ab).map S.g) x₂ = 0 → ∃ x₁, ((forget₂ C Ab).map S.f) x₁ = x₂
state_after: no goals
tactic: rfl
Theorem: MeasureTheory.measurePreserving_finTwoArrow
File: Mathlib/MeasureTheory/Constructions/Pi.lean

Proof steps:
state_before: ι : Type u_1
ι' : Type u_2
α✝ : ι → Type u_3
inst✝³ : Fintype ι
m✝¹ : (i : ι) → OuterMeasure (α✝ i)
m✝ : (i : ι) → MeasurableSpace (α✝ i)
μ✝ : (i : ι) → Measure (α✝ i)
inst✝² : ∀ (i : ι), SigmaFinite (μ✝ i)
inst✝¹ : Fintype ι'
α : Type u
m : MeasurableSpace α
μ : Measure α
inst✝ : SigmaFinite μ
⊢ MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (Measure.pi fun x => μ) (μ.prod μ)
state_after: no goals
tactic: simpa only [Matrix.vec_single_eq_const, Matrix.vecCons_const] using
    measurePreserving_finTwoArrow_vec μ μ
Theorem: neg_div'
File: Mathlib/Algebra/Field/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
K : Type u_3
inst✝¹ : DivisionMonoid K
inst✝ : HasDistribNeg K
a✝ b✝ a b : K
⊢ -(b / a) = -b / a
state_after: no goals
tactic: simp [neg_div]
Theorem: ClosedEmbedding.sigmaCompactSpace
File: Mathlib/Topology/Compactness/SigmaCompact.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
ι : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
s t : Set X
inst✝ : SigmaCompactSpace X
e : Y → X
he : ClosedEmbedding e
⊢ ⋃ n, (fun n => e ⁻¹' compactCovering X n) n = univ
state_after: no goals
tactic: rw [← preimage_iUnion, iUnion_compactCovering, preimage_univ]
Theorem: Nat.pow_pred_clog_lt_self
File: Mathlib/Data/Nat/Log.lean

Proof steps:
state_before: b : ℕ
hb : 1 < b
x : ℕ
hx : 1 < x
⊢ b ^ (clog b x).pred < x
state_after: b : ℕ
hb : 1 < b
x : ℕ
hx : 1 < x
⊢ (clog b x).pred < clog b x
tactic: rw [← not_le, le_pow_iff_clog_le hb, not_le]
state_before: b : ℕ
hb : 1 < b
x : ℕ
hx : 1 < x
⊢ (clog b x).pred < clog b x
state_after: no goals
tactic: exact pred_lt (clog_pos hb hx).ne'
Theorem: fourier_eval_zero
File: Mathlib/Analysis/Fourier/AddCircle.lean

Proof steps:
state_before: T : ℝ
n : ℤ
⊢ (fourier n) 0 = 1
state_after: no goals
tactic: rw [← QuotientAddGroup.mk_zero, fourier_coe_apply, Complex.ofReal_zero, mul_zero,
    zero_div, Complex.exp_zero]
Theorem: Group.covariant_iff_contravariant
File: Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean

Proof steps:
state_before: M : Type u_1
N : Type u_2
μ : M → N → N
r : N → N → Prop
inst✝¹ : CovariantClass M N μ r
inst✝ : Group N
⊢ Covariant N N (fun x x_1 => x * x_1) r ↔ Contravariant N N (fun x x_1 => x * x_1) r
state_after: case refine_1
M : Type u_1
N : Type u_2
μ : M → N → N
r : N → N → Prop
inst✝¹ : CovariantClass M N μ r
inst✝ : Group N
h : Covariant N N (fun x x_1 => x * x_1) r
a b c : N
bc : r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)
⊢ r b c

case refine_2
M : Type u_1
N : Type u_2
μ : M → N → N
r : N → N → Prop
inst✝¹ : CovariantClass M N μ r
inst✝ : Group N
h : Contravariant N N (fun x x_1 => x * x_1) r
a b c : N
bc : r b c
⊢ r ((fun x x_1 => x * x_1) a b) ((fun x x_1 => x * x_1) a c)
tactic: refine ⟨fun h a b c bc ↦ ?_, fun h a b c bc ↦ ?_⟩
Theorem: Ordinal.div_opow_log_pos
File: Mathlib/SetTheory/Ordinal/Exponential.lean

Proof steps:
state_before: b o : Ordinal.{u_1}
ho : o ≠ 0
⊢ 0 < o / b ^ log b o
state_after: case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ 0 < o / 0 ^ log 0 o

case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log b o
tactic: rcases eq_zero_or_pos b with (rfl | hb)
Theorem: ProbabilityTheory.gaussianPDFReal_nonneg
File: Mathlib/Probability/Distributions/Gaussian.lean

Proof steps:
state_before: μ : ℝ
v : ℝ≥0
x : ℝ
⊢ 0 ≤ gaussianPDFReal μ v x
state_after: μ : ℝ
v : ℝ≥0
x : ℝ
⊢ 0 ≤ (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))
tactic: rw [gaussianPDFReal]
state_before: μ : ℝ
v : ℝ≥0
x : ℝ
⊢ 0 ≤ (√(2 * π * ↑v))⁻¹ * rexp (-(x - μ) ^ 2 / (2 * ↑v))
state_after: no goals
tactic: positivity
Theorem: Int.bodd_add_div2
File: Mathlib/Data/Int/Bitwise.lean

Proof steps:
state_before: n : ℕ
⊢ (bif (↑n).bodd then 1 else 0) + 2 * (↑n).div2 = ↑n
state_after: n : ℕ
⊢ ↑(bif (↑n).bodd then 1 else 0) + 2 * (↑n).div2 = ↑n
tactic: rw [show (cond (bodd n) 1 0 : ℤ) = (cond (bodd n) 1 0 : ℕ) by cases bodd n <;> rfl]
state_before: n : ℕ
⊢ ↑(bif (↑n).bodd then 1 else 0) + 2 * (↑n).div2 = ↑n
state_after: no goals
tactic: exact congr_arg ofNat n.bodd_add_div2
state_before: n : ℕ
⊢ (bif -[n+1].bodd then 1 else 0) + 2 * -[n+1].div2 = -[n+1]
state_after: n : ℕ
⊢ (bif -[n+1].bodd then 1 else 0) + 2 * -[n+1].div2 = -[(bif n.bodd then 1 else 0) + 2 * n.div2+1]
tactic: refine Eq.trans ?_ (congr_arg negSucc n.bodd_add_div2)
state_before: n : ℕ
⊢ (bif -[n+1].bodd then 1 else 0) + 2 * -[n+1].div2 = -[(bif n.bodd then 1 else 0) + 2 * n.div2+1]
state_after: n : ℕ
⊢ (bif !n.bodd then 1 else 0) + 2 * -[n+1].div2 = -[(bif n.bodd then 1 else 0) + 2 * n.div2+1]
tactic: dsimp [bodd]
state_before: n : ℕ
⊢ (bif !n.bodd then 1 else 0) + 2 * -[n+1].div2 = -[(bif n.bodd then 1 else 0) + 2 * n.div2+1]
state_after: case false
n : ℕ
⊢ 1 + 2 * -[n.div2+1] = -[0 + 2 * n.div2+1]

case true
n : ℕ
⊢ 0 + 2 * -[n.div2+1] = -[1 + 2 * n.div2+1]
tactic: cases Nat.bodd n <;> dsimp [cond, not, div2, Int.mul]
Theorem: Nat.strongRecOn'_beta
File: Mathlib/Data/Nat/Defs.lean

Proof steps:
state_before: a b c d m n k : ℕ
p q : ℕ → Prop
P : ℕ → Sort u_1
h : (n : ℕ) → ((m : ℕ) → m < n → P m) → P n
⊢ n.strongRecOn' h = h n fun m x => m.strongRecOn' h
state_after: a b c d m n k : ℕ
p q : ℕ → Prop
P : ℕ → Sort u_1
h : (n : ℕ) → ((m : ℕ) → m < n → P m) → P n
⊢ Nat.strongRec' h n = h n fun m x => Nat.strongRec' h m
tactic: simp only [strongRecOn']
state_before: a b c d m n k : ℕ
p q : ℕ → Prop
P : ℕ → Sort u_1
h : (n : ℕ) → ((m : ℕ) → m < n → P m) → P n
⊢ Nat.strongRec' h n = h n fun m x => Nat.strongRec' h m
state_after: no goals
tactic: rw [Nat.strongRec']
Theorem: TopologicalSpace.Opens.functor_obj_map_obj
File: Mathlib/Topology/Category/TopCat/Opens.lean

Proof steps:
state_before: X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑f
U : Opens ↑Y
⊢ hf.functor.obj ((map f).obj U) = hf.functor.obj ⊤ ⊓ U
state_after: case h.h
X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑f
U : Opens ↑Y
x✝ : ↑Y
⊢ x✝ ∈ ↑(hf.functor.obj ((map f).obj U)) ↔ x✝ ∈ ↑(hf.functor.obj ⊤ ⊓ U)
tactic: ext
state_before: case h.h
X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑f
U : Opens ↑Y
x✝ : ↑Y
⊢ x✝ ∈ ↑(hf.functor.obj ((map f).obj U)) ↔ x✝ ∈ ↑(hf.functor.obj ⊤ ⊓ U)
state_after: case h.h.mp
X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑f
U : Opens ↑Y
x✝ : ↑Y
⊢ x✝ ∈ ↑(hf.functor.obj ((map f).obj U)) → x✝ ∈ ↑(hf.functor.obj ⊤ ⊓ U)

case h.h.mpr
X Y : TopCat
f : X ⟶ Y
hf : IsOpenMap ⇑f
U : Opens ↑Y
x✝ : ↑Y
⊢ x✝ ∈ ↑(hf.functor.obj ⊤ ⊓ U) → x✝ ∈ ↑(hf.functor.obj ((map f).obj U))
tactic: constructor
Theorem: BoxIntegral.HasIntegral.congr_ae
File: Mathlib/Analysis/BoxIntegral/Integrability.lean

Proof steps:
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
y : E
f g : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : HasIntegral I l f μ.toBoxAdditive.toSMul y
hfg : f =ᶠ[ae (μ.restrict ↑I)] g
hl : l.bRiemann = false
⊢ HasIntegral I l g μ.toBoxAdditive.toSMul y
state_after: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
y : E
f g : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : HasIntegral I l f μ.toBoxAdditive.toSMul y
hfg : f =ᶠ[ae (μ.restrict ↑I)] g
hl : l.bRiemann = false
this : g - f =ᶠ[ae (μ.restrict ↑I)] 0
⊢ HasIntegral I l g μ.toBoxAdditive.toSMul y
tactic: have : g - f =ᵐ[μ.restrict I] 0 := hfg.mono fun x hx => sub_eq_zero.2 hx.symm
state_before: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
y : E
f g : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : HasIntegral I l f μ.toBoxAdditive.toSMul y
hfg : f =ᶠ[ae (μ.restrict ↑I)] g
hl : l.bRiemann = false
this : g - f =ᶠ[ae (μ.restrict ↑I)] 0
⊢ HasIntegral I l g μ.toBoxAdditive.toSMul y
state_after: no goals
tactic: simpa using hf.add (HasIntegral.of_aeEq_zero this hl)
Theorem: openSegment_eq_image'
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✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
⊢ openSegment 𝕜 x y = (fun θ => x + θ • (y - x)) '' Ioo 0 1
state_after: case h.e'_3.h
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✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ x + a✝¹ • (y - x) = (1 - a✝¹) • x + a✝¹ • y
tactic: convert openSegment_eq_image 𝕜 x y using 2
state_before: case h.e'_3.h
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✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ x + a✝¹ • (y - x) = (1 - a✝¹) • x + a✝¹ • y
state_after: case h.e'_3.h
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✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ x + (a✝¹ • y - a✝¹ • x) = x - a✝¹ • x + a✝¹ • y
tactic: simp only [smul_sub, sub_smul, one_smul]
state_before: case h.e'_3.h
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✝⁵ : OrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : AddCommGroup F
inst✝² : AddCommGroup G
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
a✝¹ : 𝕜
a✝ : a✝¹ ∈ Ioo 0 1
⊢ x + (a✝¹ • y - a✝¹ • x) = x - a✝¹ • x + a✝¹ • y
state_after: no goals
tactic: abel
Theorem: MeasureTheory.FiniteMeasure.tendsto_normalize_iff_tendsto
File: Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean

Proof steps:
state_before: Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass) ↔
    Tendsto μs F (𝓝 μ)
state_after: case mp
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass) →
    Tendsto μs F (𝓝 μ)

case mpr
Ω : Type u_1
inst✝² : Nonempty Ω
m0 : MeasurableSpace Ω
μ : FiniteMeasure Ω
inst✝¹ : TopologicalSpace Ω
inst✝ : OpensMeasurableSpace Ω
γ : Type u_2
F : Filter γ
μs : γ → FiniteMeasure Ω
nonzero : μ ≠ 0
⊢ Tendsto μs F (𝓝 μ) →
    Tendsto (fun i => (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass)
tactic: constructor
Theorem: OrdinalApprox.lfpApprox_le_of_mem_fixedPoints
File: Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean

Proof steps:
state_before: α : Type u
inst✝ : CompleteLattice α
f : α →o α
x a : α
h_a : a ∈ fixedPoints ⇑f
h_le_init : x ≤ a
i : Ordinal.{u}
⊢ lfpApprox f x i ≤ a
state_after: no goals
tactic: induction i using Ordinal.induction with
  | h i IH =>
    unfold lfpApprox
    apply sSup_le
    simp only [exists_prop]
    intro y h_y
    simp only [Set.mem_union, Set.mem_setOf_eq, Set.mem_singleton_iff] at h_y
    cases h_y with
    | inl h_y =>
      let ⟨j, h_j_lt, h_j⟩ := h_y
      rw [← h_j, ← h_a]
      apply f.monotone'
      exact IH j h_j_lt
    | inr h_y =>
      rw [h_y]
      exact h_le_init
Theorem: Function.support_extend_zero
File: Mathlib/Algebra/Group/Support.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
A : Type u_3
B : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
inst✝² : One M
inst✝¹ : One N
inst✝ : One P
f : α → M'
g : α → N
hf : Injective f
⊢ f '' mulSupport g ⊆ mulSupport (extend f g 1)
state_after: case intro.intro
α : Type u_1
β : Type u_2
A : Type u_3
B : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
inst✝² : One M
inst✝¹ : One N
inst✝ : One P
f : α → M'
g : α → N
hf : Injective f
x : α
hx : x ∈ mulSupport g
⊢ f x ∈ mulSupport (extend f g 1)
tactic: rintro _ ⟨x, hx, rfl⟩
state_before: case intro.intro
α : Type u_1
β : Type u_2
A : Type u_3
B : Type u_4
M : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
inst✝² : One M
inst✝¹ : One N
inst✝ : One P
f : α → M'
g : α → N
hf : Injective f
x : α
hx : x ∈ mulSupport g
⊢ f x ∈ mulSupport (extend f g 1)
state_after: no goals
tactic: rwa [mem_mulSupport, hf.extend_apply]
Theorem: Ideal.toCharacterSpace_apply_eq_zero_of_mem
File: Mathlib/Analysis/CstarAlgebra/GelfandDuality.lean

Proof steps:
state_before: A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ I.toCharacterSpace a = 0
state_after: A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ (CharacterSpace.equivAlgHom.symm ((↑(NormedRing.algEquivComplexOfComplete ⋯).symm).comp (Quotient.mkₐ ℂ I))) a = 0
tactic: unfold Ideal.toCharacterSpace
state_before: A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ (CharacterSpace.equivAlgHom.symm ((↑(NormedRing.algEquivComplexOfComplete ⋯).symm).comp (Quotient.mkₐ ℂ I))) a = 0
state_after: A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ ⋯.some = 0
tactic: simp only [CharacterSpace.equivAlgHom_symm_coe, AlgHom.coe_comp, AlgHom.coe_coe,
    Quotient.mkₐ_eq_mk, Function.comp_apply, NormedRing.algEquivComplexOfComplete_symm_apply]
state_before: A : Type u_1
inst✝³ : NormedCommRing A
inst✝² : NormedAlgebra ℂ A
inst✝¹ : CompleteSpace A
I : Ideal A
inst✝ : I.IsMaximal
a : A
ha : a ∈ I
⊢ ⋯.some = 0
state_after: no goals
tactic: exact Set.eq_of_mem_singleton (Set.singleton_nonempty (0 : ℂ)).some_mem