ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: CochainComplex.shiftShortComplexFunctor'_hom_app_τ₃
File: Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean

Proof steps:
state_before: C : Type u_1
inst✝¹ : Category.{?u.87, u_1} C
inst✝ : Preadditive C
n i j k i' j' k' : ℤ
hi : n + i = i'
hj : n + j = j'
hk : n + k = k'
K : CochainComplex C ℤ
⊢ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n ⋙ shortComplexFunctor' C (up ℤ) i j k).obj K ≅
    (shortComplexFunctor' C (up ℤ) i' j' k').obj K
state_after: C : Type u_1
inst✝¹ : Category.{?u.87, u_1} C
inst✝ : Preadditive C
n i j k i' j' k' : ℤ
hi : n + i = i'
hj : n + j = j'
hk : n + k = k'
K : CochainComplex C ℤ
⊢ ShortComplex.mk (n.negOnePow • K.d (i + n) (j + n)) (n.negOnePow • K.d (j + n) (k + n)) ⋯ ≅
    ShortComplex.mk (K.d i' j') (K.d j' k') ⋯
tactic: dsimp [shortComplexFunctor']
state_before: C : Type u_1
inst✝¹ : Category.{?u.87, u_1} C
inst✝ : Preadditive C
n i j k i' j' k' : ℤ
hi : n + i = i'
hj : n + j = j'
hk : n + k = k'
K : CochainComplex C ℤ
⊢ ShortComplex.mk (n.negOnePow • K.d (i + n) (j + n)) (n.negOnePow • K.d (j + n) (k + n)) ⋯ ≅
    ShortComplex.mk (K.d i' j') (K.d j' k') ⋯
state_after: no goals
tactic: exact ShortComplex.isoMk
      (n.negOnePow • ((shiftEval C n i i' hi).app K))
      ((shiftEval C n j j' hj).app K) (n.negOnePow • ((shiftEval C n k k' hk).app K))
Theorem: CategoryTheory.ComposableArrows.Mk₁.map_id
File: Mathlib/CategoryTheory/ComposableArrows.lean

Proof steps:
state_before: C : Type u_1
inst✝ : Category.{?u.9518, u_1} C
n m : ℕ
F G : ComposableArrows C n
X₀ X₁ : C
f : X₀ ⟶ X₁
i : Fin 2
⊢ i ≤ i
state_after: no goals
tactic: simp
Theorem: ENNReal.exists_inv_nat_lt
File: Mathlib/Data/ENNReal/Inv.lean

Proof steps:
state_before: a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ≥0∞
h : a ≠ 0
⊢ ∃ n, (↑n)⁻¹ < a⁻¹⁻¹
state_after: no goals
tactic: simp only [ENNReal.inv_lt_inv, ENNReal.exists_nat_gt (inv_ne_top.2 h)]
Theorem: leftAddCoset_eq_iff
File: Mathlib/GroupTheory/Coset.lean

Proof steps:
state_before: α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ x • ↑s = y • ↑s ↔ x⁻¹ * y ∈ s
state_after: α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ (∀ (x_1 : α), x_1 ∈ x • ↑s ↔ x_1 ∈ y • ↑s) ↔ x⁻¹ * y ∈ s
tactic: rw [Set.ext_iff]
state_before: α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ (∀ (x_1 : α), x_1 ∈ x • ↑s ↔ x_1 ∈ y • ↑s) ↔ x⁻¹ * y ∈ s
state_after: α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) ↔ x⁻¹ * y ∈ s
tactic: simp_rw [mem_leftCoset_iff, SetLike.mem_coe]
state_before: α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) ↔ x⁻¹ * y ∈ s
state_after: case mp
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ (∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s) → x⁻¹ * y ∈ s

case mpr
α : Type u_1
inst✝ : Group α
s : Subgroup α
x y : α
⊢ x⁻¹ * y ∈ s → ∀ (x_1 : α), x⁻¹ * x_1 ∈ s ↔ y⁻¹ * x_1 ∈ s
tactic: constructor
Theorem: prevD_eq
File: Mathlib/Algebra/Homology/Homotopy.lean

Proof steps:
state_before: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f✝ g : C ⟶ D
h k : D ⟶ E
i : ι
f : (i j : ι) → C.X i ⟶ D.X j
j j' : ι
w : c.Rel j' j
⊢ (prevD j) f = f j j' ≫ D.d j' j
state_after: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f✝ g : C ⟶ D
h k : D ⟶ E
i : ι
f : (i j : ι) → C.X i ⟶ D.X j
j : ι
w : c.Rel (c.prev j) j
⊢ (prevD j) f = f j (c.prev j) ≫ D.d (c.prev j) j
tactic: obtain rfl := c.prev_eq' w
state_before: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f✝ g : C ⟶ D
h k : D ⟶ E
i : ι
f : (i j : ι) → C.X i ⟶ D.X j
j : ι
w : c.Rel (c.prev j) j
⊢ (prevD j) f = f j (c.prev j) ≫ D.d (c.prev j) j
state_after: no goals
tactic: rfl
Theorem: CategoryTheory.isIso_of_fully_faithful
File: Mathlib/CategoryTheory/Functor/FullyFaithful.lean

Proof steps:
state_before: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.14273, u_1} E
F : C ⥤ D
inst✝² : F.Full
inst✝¹ : F.Faithful
X Y : C
f : X ⟶ Y
inst✝ : IsIso (F.map f)
⊢ F.map (f ≫ F.preimage (inv (F.map f))) = F.map (𝟙 X)
state_after: no goals
tactic: simp
state_before: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
E : Type u_1
inst✝³ : Category.{?u.14273, u_1} E
F : C ⥤ D
inst✝² : F.Full
inst✝¹ : F.Faithful
X Y : C
f : X ⟶ Y
inst✝ : IsIso (F.map f)
⊢ F.map (F.preimage (inv (F.map f)) ≫ f) = F.map (𝟙 Y)
state_after: no goals
tactic: simp
Theorem: isClosed_iUnion_of_finite
File: Mathlib/Topology/Basic.lean

Proof steps:
state_before: X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s✝ s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
inst✝ : Finite ι
s : ι → Set X
h : ∀ (i : ι), IsClosed (s i)
⊢ IsClosed (⋃ i, s i)
state_after: X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s✝ s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
inst✝ : Finite ι
s : ι → Set X
h : ∀ (i : ι), IsOpen (s i)ᶜ
⊢ IsOpen (⋂ i, (s i)ᶜ)
tactic: simp only [← isOpen_compl_iff, compl_iUnion] at *
state_before: X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s✝ s₁ s₂ t : Set X
p p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
inst✝ : Finite ι
s : ι → Set X
h : ∀ (i : ι), IsOpen (s i)ᶜ
⊢ IsOpen (⋂ i, (s i)ᶜ)
state_after: no goals
tactic: exact isOpen_iInter_of_finite h
Theorem: AddCommute.list_sum_right
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
l : List M
y : M
h : ∀ x ∈ l, Commute y x
⊢ Commute y l.prod
state_after: case nil
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 y : M
h : ∀ x ∈ [], Commute y x
⊢ Commute y [].prod

case cons
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 y z : M
l : List M
IH : (∀ x ∈ l, Commute y x) → Commute y l.prod
h : ∀ x ∈ z :: l, Commute y x
⊢ Commute y (z :: l).prod
tactic: induction' l with z l IH
Theorem: IsRelPrime.ne_zero_or_ne_zero
File: Mathlib/Algebra/GroupWithZero/Divisibility.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : CommMonoidWithZero α
x y : α
inst✝ : Nontrivial α
h : IsRelPrime x y
⊢ x = 0 → y ≠ 0
state_after: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : Nontrivial α
h : IsRelPrime 0 0
⊢ False
tactic: rintro rfl rfl
state_before: α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : Nontrivial α
h : IsRelPrime 0 0
⊢ False
state_after: no goals
tactic: exact not_isRelPrime_zero_zero h
Theorem: Real.continuousAt_rpow_of_ne
File: Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean

Proof steps:
state_before: p : ℝ × ℝ
hp : p.1 ≠ 0
⊢ ContinuousAt (fun p => p.1 ^ p.2) p
state_after: p : ℝ × ℝ
hp : p.1 < 0 ∨ p.1 > 0
⊢ ContinuousAt (fun p => p.1 ^ p.2) p
tactic: rw [ne_iff_lt_or_gt] at hp
Theorem: Filter.tendsto_div_const_atTop_of_pos
File: Mathlib/Order/Filter/AtTopBot.lean

Proof steps:
state_before: ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝ : LinearOrderedSemifield α
l : Filter β
f : β → α
r c : α
n : ℕ
hr : 0 < r
⊢ Tendsto (fun x => f x / r) l atTop ↔ Tendsto f l atTop
state_after: no goals
tactic: simpa only [div_eq_mul_inv] using tendsto_mul_const_atTop_of_pos (inv_pos.2 hr)
Theorem: HomologicalComplex₂.D₁_totalShift₁XIso_hom
File: Mathlib/Algebra/Homology/TotalComplexShift.lean

Proof steps:
state_before: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
⊢ ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'
state_after: case pos
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : (up ℤ).Rel n₀ n₁
⊢ ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'

case neg
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
x y : ℤ
inst✝ : K.HasTotal (up ℤ)
n₀ n₁ n₀' n₁' : ℤ
h₀ : n₀ + x = n₀'
h₁ : n₁ + x = n₁'
h : ¬(up ℤ).Rel n₀ n₁
⊢ ((shiftFunctor₁ C x).obj K).D₁ (up ℤ) n₀ n₁ ≫ (K.totalShift₁XIso x n₁ n₁' h₁).hom =
    x.negOnePow • (K.totalShift₁XIso x n₀ n₀' h₀).hom ≫ K.D₁ (up ℤ) n₀' n₁'
tactic: by_cases h : (up ℤ).Rel n₀ n₁
Theorem: Multiset.Nat.antidiagonal_succ
File: Mathlib/Data/Multiset/NatAntidiagonal.lean

Proof steps:
state_before: n : ℕ
⊢ antidiagonal (n + 1) = (0, n + 1) ::ₘ map (Prod.map Nat.succ id) (antidiagonal n)
state_after: no goals
tactic: simp only [antidiagonal, List.Nat.antidiagonal_succ, map_coe, cons_coe]
Theorem: hasStrictFDerivAt_list_prod_attach'
File: Mathlib/Analysis/Calculus/FDeriv/Mul.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F : Type u_3
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁶ : NormedAddCommGroup G'
inst✝⁵ : NormedSpace 𝕜 G'
f✝ f₀ f₁ g✝ : E → F
f'✝ f₀' f₁' g'✝ e : E →L[𝕜] F
x✝ : E
s t : Set E
L L₁ L₂ : Filter E
ι : Type u_6
𝔸 : Type u_7
𝔸' : Type u_8
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedCommRing 𝔸'
inst✝² : NormedAlgebra 𝕜 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → E → 𝔸
f' : ι → E →L[𝕜] 𝔸
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ ∀ (i : Fin l.length), i ∈ Finset.univ ↔ (finCongr ⋯) i ∈ Finset.univ
state_after: no goals
tactic: simp
state_before: 𝕜 : Type u_1
inst✝¹³ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace 𝕜 E
F : Type u_3
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace 𝕜 F
G : Type u_4
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace 𝕜 G
G' : Type u_5
inst✝⁶ : NormedAddCommGroup G'
inst✝⁵ : NormedSpace 𝕜 G'
f✝ f₀ f₁ g✝ : E → F
f'✝ f₀' f₁' g'✝ e : E →L[𝕜] F
x✝ : E
s t : Set E
L L₁ L₂ : Filter E
ι : Type u_6
𝔸 : Type u_7
𝔸' : Type u_8
inst✝⁴ : NormedRing 𝔸
inst✝³ : NormedCommRing 𝔸'
inst✝² : NormedAlgebra 𝕜 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝔸'
u : Finset ι
f : ι → E → 𝔸
f' : ι → E →L[𝕜] 𝔸
g : ι → E → 𝔸'
g' : ι → E →L[𝕜] 𝔸'
inst✝ : DecidableEq ι
l : List ι
x : { i // i ∈ l } → 𝔸
⊢ ∀ i ∈ Finset.univ,
    (List.map x (List.take (↑i) l.attach)).prod •
        (proj (l.attach.get (Fin.cast ⋯ i))).smulRight (List.map x (List.drop (↑i).succ l.attach)).prod =
      (List.map x (List.take (↑((finCongr ⋯) i)) l.attach)).prod •
        (proj (l.attach.get ((finCongr ⋯) i))).smulRight (List.map x (List.drop (↑((finCongr ⋯) i)).succ l.attach)).prod
state_after: no goals
tactic: simp
Theorem: AddCommGroup.modEq_comm
File: Mathlib/Algebra/ModEq.lean

Proof steps:
state_before: α : Type u_1
inst✝ : AddCommGroup α
p a a₁ a₂ b b₁ b₂ c : α
n : ℕ
z : ℤ
⊢ (∃ b_1, b - a = (Equiv.symm (Equiv.neg ℤ)) b_1 • p) ↔ b ≡ a [PMOD p]
state_after: no goals
tactic: simp [ModEq, ← neg_eq_iff_eq_neg]
Theorem: FirstOrder.Language.Substructure.fg_def
File: Mathlib/ModelTheory/FinitelyGenerated.lean

Proof steps:
state_before: L : Language
M : Type u_1
inst✝ : L.Structure M
N : L.Substructure M
⊢ (∃ S, S.Finite ∧ (closure L).toFun S = N) → N.FG
state_after: case intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
t' : Set M
h : t'.Finite
⊢ ((closure L).toFun t').FG
tactic: rintro ⟨t', h, rfl⟩
state_before: case intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
t' : Set M
h : t'.Finite
⊢ ((closure L).toFun t').FG
state_after: case intro.intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
t : Finset M
h : (↑t).Finite
⊢ ((closure L).toFun ↑t).FG
tactic: rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
state_before: case intro.intro.intro
L : Language
M : Type u_1
inst✝ : L.Structure M
t : Finset M
h : (↑t).Finite
⊢ ((closure L).toFun ↑t).FG
state_after: no goals
tactic: exact ⟨t, rfl⟩
Theorem: CategoryTheory.Limits.bicone_ι_π_self
File: Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean

Proof steps:
state_before: J : Type w
C : Type uC
inst✝³ : Category.{uC', uC} C
inst✝² : HasZeroMorphisms C
D : Type uD
inst✝¹ : Category.{uD', uD} D
inst✝ : HasZeroMorphisms D
F : J → C
B : Bicone F
j : J
⊢ B.ι j ≫ B.π j = 𝟙 (F j)
state_after: no goals
tactic: simpa using B.ι_π j j
Theorem: ProbabilityTheory.iIndepFun.cgf_sum
File: Mathlib/Probability/Moments.lean

Proof steps:
state_before: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ cgf (∑ i ∈ s, X i) μ t = ∑ i ∈ s, cgf (X i) μ t
state_after: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ log (mgf (∑ i ∈ s, X i) μ t) = ∑ x ∈ s, log (mgf (X x) μ t)
tactic: simp_rw [cgf]
state_before: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ log (mgf (∑ i ∈ s, X i) μ t) = ∑ x ∈ s, log (mgf (X x) μ t)
state_after: Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
⊢ log (mgf (∑ i ∈ s, X i) μ t) = log (∏ i ∈ s, mgf (X i) μ t)

Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X✝ : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ
inst✝ : IsProbabilityMeasure μ
X : ι → Ω → ℝ
h_indep : iIndepFun (fun i => inferInstance) X μ
h_meas : ∀ (i : ι), Measurable (X i)
s : Finset ι
h_int : ∀ i ∈ s, Integrable (fun ω => rexp (t * X i ω)) μ
j : ι
hj : j ∈ s
⊢ mgf (X j) μ t ≠ 0
tactic: rw [← log_prod _ _ fun j hj => ?_]
Theorem: Algebra.adjoin_eq_span
File: Mathlib/RingTheory/Adjoin/Basic.lean

Proof steps:
state_before: R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ Subalgebra.toSubmodule (adjoin R s) = span R ↑(Submonoid.closure s)
state_after: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ Subalgebra.toSubmodule (adjoin R s) ≤ span R ↑(Submonoid.closure s)

case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ span R ↑(Submonoid.closure s) ≤ Subalgebra.toSubmodule (adjoin R s)
tactic: apply le_antisymm
state_before: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ span R ↑(Submonoid.closure s) ≤ Subalgebra.toSubmodule (adjoin R s)
state_after: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ ↑(Submonoid.closure s) ⊆ ↑(Subalgebra.toSubmodule (adjoin R s))
tactic: refine span_le.2 ?_
state_before: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ ↑(Submonoid.closure s) ⊆ ↑(Subalgebra.toSubmodule (adjoin R s))
state_after: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ Submonoid.closure s ≤ (adjoin R s).toSubmonoid
tactic: change Submonoid.closure s ≤ (adjoin R s).toSubsemiring.toSubmonoid
state_before: case a
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s t : Set A
⊢ Submonoid.closure s ≤ (adjoin R s).toSubmonoid
state_after: no goals
tactic: exact Submonoid.closure_le.2 subset_adjoin
Theorem: PosNum.pred'_to_nat
File: Mathlib/Data/Num/Lemmas.lean

Proof steps:
state_before: α : Type u_1
n : PosNum
⊢ (↑n.pred').succ = ↑n
state_after: no goals
tactic: rw [pred'_to_nat n, Nat.succ_pred_eq_of_pos (to_nat_pos n)]
state_before: α : Type u_1
n : PosNum
this : (↑n.pred').succ = ↑n
h : ↑1 = ↑n
⊢ ↑(Num.casesOn 0 1 bit1) = (↑n + ↑n).pred
state_after: α : Type u_1
n : PosNum
this : (↑n.pred').succ = ↑n
h : ↑1 = ↑n
⊢ ↑(Num.casesOn 0 1 bit1) = (↑1 + ↑1).pred
tactic: rw [← to_nat_inj.1 h]
state_before: α : Type u_1
n : PosNum
this : (↑n.pred').succ = ↑n
h : ↑1 = ↑n
⊢ ↑(Num.casesOn 0 1 bit1) = (↑1 + ↑1).pred
state_after: no goals
tactic: rfl
state_before: α : Type u_1
n : PosNum
this : (↑n.pred').succ = ↑n
p : PosNum
h : (↑p).succ = ↑n
⊢ ↑(Num.casesOn (pos p) 1 bit1) = (↑n + ↑n).pred
state_after: α : Type u_1
n : PosNum
this : (↑n.pred').succ = ↑n
p : PosNum
h : (↑p).succ = ↑n
⊢ ↑(Num.casesOn (pos p) 1 bit1) = ((↑p).succ + (↑p).succ).pred
tactic: rw [← h]
state_before: α : Type u_1
n : PosNum
this : (↑n.pred').succ = ↑n
p : PosNum
h : (↑p).succ = ↑n
⊢ ↑(Num.casesOn (pos p) 1 bit1) = ((↑p).succ + (↑p).succ).pred
state_after: no goals
tactic: exact (Nat.succ_add p p).symm
Theorem: Rat.round_cast
File: Mathlib/Data/Rat/Floor.lean

Proof steps:
state_before: α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
x : ℚ
⊢ round ↑x = round x
state_after: α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
x : ℚ
this : ↑(x + 1 / 2) = ↑x + 1 / 2
⊢ round ↑x = round x
tactic: have : ((x + 1 / 2 : ℚ) : α) = x + 1 / 2 := by simp
state_before: α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : FloorRing α
x : ℚ
this : ↑(x + 1 / 2) = ↑x + 1 / 2
⊢ round ↑x = round x
state_after: no goals
tactic: rw [round_eq, round_eq, ← this, floor_cast]
Theorem: MeasureTheory.Measure.restrict_union_add_inter'
File: Mathlib/MeasureTheory/Measure/Restrict.lean

Proof steps:
state_before: R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t✝ : Set α
hs : MeasurableSet s
t : Set α
⊢ μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t
state_after: no goals
tactic: simpa only [union_comm, inter_comm, add_comm] using restrict_union_add_inter t hs
Theorem: MeasureTheory.lintegral_count
File: Mathlib/MeasureTheory/Integral/Lebesgue.lean

Proof steps:
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✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
⊢ ∫⁻ (a : α), f a ∂count = ∑' (a : α), f a
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✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a
tactic: rw [count, lintegral_sum_measure]
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✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a
state_after: case e_f
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a
tactic: congr
state_before: case e_f
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
f : α → ℝ≥0∞
⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a
state_after: no goals
tactic: exact funext fun a => lintegral_dirac a f
Theorem: Finset.image₂_union_left
File: Mathlib/Data/Finset/NAry.lean

Proof steps:
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁸ : DecidableEq α'
inst✝⁷ : DecidableEq β'
inst✝⁶ : DecidableEq γ
inst✝⁵ : DecidableEq γ'
inst✝⁴ : DecidableEq δ
inst✝³ : DecidableEq δ'
inst✝² : DecidableEq ε
inst✝¹ : DecidableEq ε'
f f' : α → β → γ
g g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
inst✝ : DecidableEq α
⊢ ↑(image₂ f (s ∪ s') t) = ↑(image₂ f s t ∪ image₂ f s' t)
state_after: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁸ : DecidableEq α'
inst✝⁷ : DecidableEq β'
inst✝⁶ : DecidableEq γ
inst✝⁵ : DecidableEq γ'
inst✝⁴ : DecidableEq δ
inst✝³ : DecidableEq δ'
inst✝² : DecidableEq ε
inst✝¹ : DecidableEq ε'
f f' : α → β → γ
g g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
inst✝ : DecidableEq α
⊢ image2 f (↑s ∪ ↑s') ↑t = image2 f ↑s ↑t ∪ image2 f ↑s' ↑t
tactic: push_cast
state_before: α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁸ : DecidableEq α'
inst✝⁷ : DecidableEq β'
inst✝⁶ : DecidableEq γ
inst✝⁵ : DecidableEq γ'
inst✝⁴ : DecidableEq δ
inst✝³ : DecidableEq δ'
inst✝² : DecidableEq ε
inst✝¹ : DecidableEq ε'
f f' : α → β → γ
g g' : α → β → γ → δ
s s' : Finset α
t t' : Finset β
u u' : Finset γ
a a' : α
b b' : β
c : γ
inst✝ : DecidableEq α
⊢ image2 f (↑s ∪ ↑s') ↑t = image2 f ↑s ↑t ∪ image2 f ↑s' ↑t
state_after: no goals
tactic: exact image2_union_left
Theorem: Real.Angle.two_nsmul_toReal_eq_two_mul_sub_two_pi
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

Proof steps:
state_before: θ : Angle
⊢ (2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal
state_after: θ : Angle
⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal
tactic: nth_rw 1 [← coe_toReal θ]
state_before: θ : Angle
⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal
state_after: θ : Angle
⊢ π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal
tactic: rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_sub_two_pi_iff, Set.mem_Ioc]
state_before: θ : Angle
⊢ π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ 3 * π ↔ π / 2 < θ.toReal
state_after: no goals
tactic: exact
    ⟨fun h => by linarith, fun h =>
      ⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩