ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae
File: Mathlib/MeasureTheory/Integral/SetIntegral.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
C : ℝ
hs : μ s < ⊤
hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
C : ℝ
hs : (μ.restrict s) univ < ⊤
hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * ((μ.restrict s) univ).toReal
tactic: rw [← Measure.restrict_apply_univ] at *
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
C : ℝ
hs : (μ.restrict s) univ < ⊤
hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * ((μ.restrict s) univ).toReal
state_after: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
C : ℝ
hs : (μ.restrict s) univ < ⊤
hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C
this : IsFiniteMeasure (μ.restrict s)
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * ((μ.restrict s) univ).toReal
tactic: haveI : IsFiniteMeasure (μ.restrict s) := ⟨hs⟩
state_before: X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
C : ℝ
hs : (μ.restrict s) univ < ⊤
hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C
this : IsFiniteMeasure (μ.restrict s)
⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * ((μ.restrict s) univ).toReal
state_after: no goals
tactic: exact norm_integral_le_of_norm_le_const hC
Theorem: CompHausLike.pullback.condition
File: Mathlib/Topology/Category/CompHausLike/Limits.lean

Proof steps:
state_before: P : TopCat → Prop
X Y B : CompHausLike P
f : X ⟶ B
g : Y ⟶ B
inst✝ : HasExplicitPullback f g
⊢ fst f g ≫ f = snd f g ≫ g
state_after: case w.mk
P : TopCat → Prop
X Y B : CompHausLike P
f : X ⟶ B
g : Y ⟶ B
inst✝ : HasExplicitPullback f g
val✝ : ↑X.toTop × ↑Y.toTop
h : val✝ ∈ {xy | f xy.1 = g xy.2}
⊢ (fst f g ≫ f) ⟨val✝, h⟩ = (snd f g ≫ g) ⟨val✝, h⟩
tactic: ext ⟨_,h⟩
state_before: case w.mk
P : TopCat → Prop
X Y B : CompHausLike P
f : X ⟶ B
g : Y ⟶ B
inst✝ : HasExplicitPullback f g
val✝ : ↑X.toTop × ↑Y.toTop
h : val✝ ∈ {xy | f xy.1 = g xy.2}
⊢ (fst f g ≫ f) ⟨val✝, h⟩ = (snd f g ≫ g) ⟨val✝, h⟩
state_after: no goals
tactic: exact h
Theorem: ENNReal.rpow_le_rpow_of_exponent_le
File: Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean

Proof steps:
state_before: x : ℝ≥0∞
y z : ℝ
hx : 1 ≤ x
hyz : y ≤ z
⊢ x ^ y ≤ x ^ z
state_after: case top
y z : ℝ
hyz : y ≤ z
hx : 1 ≤ ⊤
⊢ ⊤ ^ y ≤ ⊤ ^ z

case coe
y z : ℝ
hyz : y ≤ z
x✝ : ℝ≥0
hx : 1 ≤ ↑x✝
⊢ ↑x✝ ^ y ≤ ↑x✝ ^ z
tactic: cases x
Theorem: List.toFinset_eq
File: Mathlib/Data/Finset/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
l l' : List α
a : α
n : l.Nodup
⊢ (↑l).Nodup
state_after: no goals
tactic: rwa [Multiset.coe_nodup]
Theorem: Pell.Solution₁.sign_y_zpow_eq_sign_of_x_pos_of_y_pos
File: Mathlib/NumberTheory/Pell.lean

Proof steps:
state_before: d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℤ
⊢ (a ^ n).y.sign = n.sign
state_after: case ofNat.zero
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
⊢ (a ^ Int.ofNat 0).y.sign = (Int.ofNat 0).sign

case ofNat.succ
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
⊢ (a ^ Int.ofNat (n + 1)).y.sign = (Int.ofNat (n + 1)).sign

case negSucc
d : ℤ
a : Solution₁ d
hax : 0 < a.x
hay : 0 < a.y
n : ℕ
⊢ (a ^ Int.negSucc n).y.sign = (Int.negSucc n).sign
tactic: rcases n with ((_ | n) | n)
Theorem: MvPolynomial.support_X_pow
File: Mathlib/Algebra/MvPolynomial/Basic.lean

Proof steps:
state_before: R : Type u
S₁ : Type v
S₂ : Type w
S₃ : Type x
σ : Type u_1
a a' a₁ a₂ : R
e : ℕ
n✝ m : σ
s✝ : σ →₀ ℕ
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S₁
p q : MvPolynomial σ R
inst✝ : Nontrivial R
s : σ
n : ℕ
⊢ (X s ^ n).support = {Finsupp.single s n}
state_after: no goals
tactic: classical
    rw [X_pow_eq_monomial, support_monomial, if_neg (one_ne_zero' R)]
Theorem: Real.doublingGamma_eq_Gamma
File: Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean

Proof steps:
state_before: s : ℝ
hs : 0 < s
⊢ s.doublingGamma = Gamma s
state_after: s : ℝ
hs : 0 < s
y : ℝ
hy : 0 < y
⊢ 0 < y.doublingGamma
tactic: refine
    eq_Gamma_of_log_convex doublingGamma_log_convex_Ioi
      (fun {y} hy => doublingGamma_add_one y hy.ne') (fun {y} hy => ?_) doublingGamma_one hs
state_before: s : ℝ
hs : 0 < s
y : ℝ
hy : 0 < y
⊢ 0 < y.doublingGamma
state_after: no goals
tactic: apply_rules [mul_pos, Gamma_pos_of_pos, add_pos, inv_pos_of_pos, rpow_pos_of_pos, two_pos,
    one_pos, sqrt_pos_of_pos pi_pos]
Theorem: Complex.hasDerivAt_exp
File: Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
⊢ HasDerivAt cexp (cexp x) x
state_after: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
⊢ (fun h => cexp (x + h) - cexp x - h • cexp x) =o[𝓝 0] fun h => h
tactic: rw [hasDerivAt_iff_isLittleO_nhds_zero]
state_before: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
⊢ (fun h => cexp (x + h) - cexp x - h • cexp x) =o[𝓝 0] fun h => h
state_after: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ (fun h => cexp (x + h) - cexp x - h • cexp x) =o[𝓝 0] fun h => h
tactic: have : (1 : ℕ) < 2 := by norm_num
state_before: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ (fun h => cexp (x + h) - cexp x - h • cexp x) =o[𝓝 0] fun h => h
state_after: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ ∀ᶠ (x_1 : ℂ) in 𝓝 0, ‖cexp (x + x_1) - cexp x - x_1 • cexp x‖ ≤ ‖cexp x‖ * ‖x_1 ^ 2‖
tactic: refine (IsBigO.of_bound ‖exp x‖ ?_).trans_isLittleO (isLittleO_pow_id this)
state_before: 𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ ∀ᶠ (x_1 : ℂ) in 𝓝 0, ‖cexp (x + x_1) - cexp x - x_1 • cexp x‖ ≤ ‖cexp x‖ * ‖x_1 ^ 2‖
state_after: case h
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ ∀ a ∈ Metric.ball 0 1, ‖cexp (x + a) - cexp x - a • cexp x‖ ≤ ‖cexp x‖ * ‖a ^ 2‖
tactic: filter_upwards [Metric.ball_mem_nhds (0 : ℂ) zero_lt_one]
state_before: case h
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ ∀ a ∈ Metric.ball 0 1, ‖cexp (x + a) - cexp x - a • cexp x‖ ≤ ‖cexp x‖ * ‖a ^ 2‖
state_after: case h
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ ∀ (a : ℂ), ‖a‖ < 1 → ‖cexp (x + a) - cexp x - a • cexp x‖ ≤ ‖cexp x‖ * ‖a‖ ^ 2
tactic: simp only [Metric.mem_ball, dist_zero_right, norm_pow]
state_before: case h
𝕜 : Type u_1
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedAlgebra 𝕜 ℂ
x : ℂ
this : 1 < 2
⊢ ∀ (a : ℂ), ‖a‖ < 1 → ‖cexp (x + a) - cexp x - a • cexp x‖ ≤ ‖cexp x‖ * ‖a‖ ^ 2
state_after: no goals
tactic: exact fun z hz => exp_bound_sq x z hz.le
Theorem: Set.univ_pi_update
File: Mathlib/Data/Set/Prod.lean

Proof steps:
state_before: ι : Type u_1
α : ι → Type u_2
β✝ : ι → Type u_3
s s₁ s₂ : Set ι
t✝ t₁ t₂ : (i : ι) → Set (α i)
i✝ : ι
inst✝ : DecidableEq ι
β : ι → Type u_4
i : ι
f : (j : ι) → α j
a : α i
t : (j : ι) → α j → Set (β j)
⊢ (univ.pi fun j => t j (update f i a j)) = {x | x i ∈ t i a} ∩ {i}ᶜ.pi fun j => t j (f j)
state_after: no goals
tactic: rw [compl_eq_univ_diff, ← pi_update_of_mem (mem_univ _)]
Theorem: MeasureTheory.measure_inter_conull'
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ht : μ (s \ t) = 0
⊢ μ (s ∩ t) = μ s
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ht : μ (s \ t) = 0
⊢ μ (s ∩ tᶜ) = 0
tactic: rw [← diff_compl, measure_diff_null']
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m : MeasurableSpace α
μ μ₁ μ₂ : Measure α
s s₁ s₂ t : Set α
ht : μ (s \ t) = 0
⊢ μ (s ∩ tᶜ) = 0
state_after: no goals
tactic: rwa [← diff_eq]
Theorem: Finset.inr_mem_sumLexLift
File: Mathlib/Data/Sum/Interval.lean

Proof steps:
state_before: α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁ ⊕ α₂
b : β₁ ⊕ β₂
c : γ₁ ⊕ γ₂
c₂ : γ₂
⊢ inr c₂ ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔
    (∃ a₁ b₂, a = inl a₁ ∧ b = inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨ ∃ a₂ b₂, a = inr a₂ ∧ b = inr b₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: no goals
tactic: simp [mem_sumLexLift]
Theorem: CharP.natCast_eq_natCast
File: Mathlib/Algebra/CharP/Defs.lean

Proof steps:
state_before: R : Type u_1
inst✝² : AddMonoidWithOne R
p : ℕ
inst✝¹ : CharP R p
a b : ℕ
inst✝ : IsRightCancelAdd R
⊢ ↑a = ↑b ↔ a ≡ b [MOD p]
state_after: case inr
R : Type u_1
inst✝² : AddMonoidWithOne R
p : ℕ
inst✝¹ : CharP R p
a b : ℕ
inst✝ : IsRightCancelAdd R
this :
  ∀ (R : Type u_1) [inst : AddMonoidWithOne R] (p : ℕ) [inst_1 : CharP R p] {a b : ℕ} [inst_2 : IsRightCancelAdd R],
    a ≤ b → (↑a = ↑b ↔ a ≡ b [MOD p])
hle : ¬a ≤ b
⊢ ↑a = ↑b ↔ a ≡ b [MOD p]

R✝ : Type u_1
inst✝³ : AddMonoidWithOne R✝
p✝ a✝ b✝ : ℕ
R : Type u_1
inst✝² : AddMonoidWithOne R
p : ℕ
inst✝¹ : CharP R p
a b : ℕ
inst✝ : IsRightCancelAdd R
hle : a ≤ b
⊢ ↑a = ↑b ↔ a ≡ b [MOD p]
tactic: wlog hle : a ≤ b
state_before: R✝ : Type u_1
inst✝³ : AddMonoidWithOne R✝
p✝ a✝ b✝ : ℕ
R : Type u_1
inst✝² : AddMonoidWithOne R
p : ℕ
inst✝¹ : CharP R p
a b : ℕ
inst✝ : IsRightCancelAdd R
hle : a ≤ b
⊢ ↑a = ↑b ↔ a ≡ b [MOD p]
state_after: no goals
tactic: rw [Nat.modEq_iff_dvd' hle, ← cast_eq_zero_iff R p (b - a),
    ← add_right_cancel_iff (G := R) (a := a) (b := b - a), zero_add, ← Nat.cast_add,
    Nat.sub_add_cancel hle, eq_comm]
Theorem: CstarRing.norm_coe_unitary
File: Mathlib/Analysis/CstarAlgebra/Basic.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
α : Type u_3
inst✝³ : NormedRing E
inst✝² : StarRing E
inst✝¹ : CstarRing E
inst✝ : Nontrivial E
U : ↥(unitary E)
⊢ ‖↑U‖ = 1
state_after: no goals
tactic: rw [← sq_eq_sq (norm_nonneg _) zero_le_one, one_pow 2, sq, ← CstarRing.norm_star_mul_self,
    unitary.coe_star_mul_self, CstarRing.norm_one]
Theorem: MeasureTheory.compl_mem_ae_iff
File: Mathlib/MeasureTheory/OuterMeasure/AE.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : FunLike F (Set α) ℝ≥0∞
inst✝ : OuterMeasureClass F α
μ : F
s✝ t s : Set α
⊢ sᶜ ∈ ae μ ↔ μ s = 0
state_after: no goals
tactic: simp only [mem_ae_iff, compl_compl]
Theorem: PMF.filter_apply_ne_zero_iff
File: Mathlib/Probability/ProbabilityMassFunction/Constructions.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
p : PMF α
s : Set α
h : ∃ a ∈ s, a ∈ p.support
a : α
⊢ (p.filter s h) a ≠ 0 ↔ a ∈ s ∧ a ∈ p.support
state_after: no goals
tactic: rw [Ne, filter_apply_eq_zero_iff, not_or, Classical.not_not, Classical.not_not]
Theorem: Sum.Lex.Icc_inl_inl
File: Mathlib/Data/Sum/Interval.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : OrderTop α
inst✝² : OrderBot β
inst✝¹ : LocallyFiniteOrder α
inst✝ : LocallyFiniteOrder β
a a₁ a₂ : α
b b₁ b₂ : β
⊢ Icc (inlₗ a₁) (inlₗ a₂) = map (Embedding.inl.trans toLex.toEmbedding) (Icc a₁ a₂)
state_after: α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : OrderTop α
inst✝² : OrderBot β
inst✝¹ : LocallyFiniteOrder α
inst✝ : LocallyFiniteOrder β
a a₁ a₂ : α
b b₁ b₂ : β
⊢ Icc (inlₗ a₁) (inlₗ a₂) = map toLex.toEmbedding (map Embedding.inl (Icc a₁ a₂))
tactic: rw [← Finset.map_map]
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : OrderTop α
inst✝² : OrderBot β
inst✝¹ : LocallyFiniteOrder α
inst✝ : LocallyFiniteOrder β
a a₁ a₂ : α
b b₁ b₂ : β
⊢ Icc (inlₗ a₁) (inlₗ a₂) = map toLex.toEmbedding (map Embedding.inl (Icc a₁ a₂))
state_after: no goals
tactic: rfl
Theorem: normEDS_neg
File: Mathlib/NumberTheory/EllipticDivisibilitySequence.lean

Proof steps:
state_before: R : Type u
S : Type v
inst✝¹ : CommRing R
inst✝ : CommRing S
W : ℤ → R
f : R →+* S
b c d : R
n : ℤ
⊢ normEDS b c d (-n) = -normEDS b c d n
state_after: no goals
tactic: simp only [normEDS, preNormEDS_neg, neg_mul, even_neg]
Theorem: measurableSet_of_differentiableAt
File: Mathlib/Analysis/Calculus/FDeriv/Measurable.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
⊢ MeasurableSet {x | DifferentiableAt 𝕜 f x}
state_after: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
this : IsComplete univ
⊢ MeasurableSet {x | DifferentiableAt 𝕜 f x}
tactic: have : IsComplete (univ : Set (E →L[𝕜] F)) := complete_univ
state_before: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
this : IsComplete univ
⊢ MeasurableSet {x | DifferentiableAt 𝕜 f x}
state_after: case h.e'_3.h.e'_2.h.a
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
this : IsComplete univ
x✝ : E
⊢ DifferentiableAt 𝕜 f x✝ ↔ DifferentiableAt 𝕜 f x✝ ∧ fderiv 𝕜 f x✝ ∈ univ
tactic: convert measurableSet_of_differentiableAt_of_isComplete 𝕜 f this
state_before: case h.e'_3.h.e'_2.h.a
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : CompleteSpace F
this : IsComplete univ
x✝ : E
⊢ DifferentiableAt 𝕜 f x✝ ↔ DifferentiableAt 𝕜 f x✝ ∧ fderiv 𝕜 f x✝ ∈ univ
state_after: no goals
tactic: simp
Theorem: MeasureTheory.lintegral_union
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
f : α → ℝ≥0∞
A B : Set α
hB : MeasurableSet B
hAB : Disjoint A B
⊢ ∫⁻ (a : α) in A ∪ B, f a ∂μ = ∫⁻ (a : α) in A, f a ∂μ + ∫⁻ (a : α) in B, f a ∂μ
state_after: no goals
tactic: rw [restrict_union hAB hB, lintegral_add_measure]
Theorem: SeminormFamily.withSeminorms_iff_nhds_eq_iInf
File: Mathlib/Analysis/LocallyConvex/WithSeminorms.lean

Proof steps:
state_before: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
t : TopologicalSpace E
inst✝ : TopologicalAddGroup E
p : SeminormFamily 𝕜 E ι
⊢ WithSeminorms p ↔ 𝓝 0 = ⨅ i, comap (⇑(p i)) (𝓝 0)
state_after: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
t : TopologicalSpace E
inst✝ : TopologicalAddGroup E
p : SeminormFamily 𝕜 E ι
⊢ WithSeminorms p ↔ 𝓝 0 = AddGroupFilterBasis.toFilterBasis.filter
tactic: rw [← p.filter_eq_iInf]
state_before: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
t : TopologicalSpace E
inst✝ : TopologicalAddGroup E
p : SeminormFamily 𝕜 E ι
⊢ WithSeminorms p ↔ 𝓝 0 = AddGroupFilterBasis.toFilterBasis.filter
state_after: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
t : TopologicalSpace E
inst✝ : TopologicalAddGroup E
p : SeminormFamily 𝕜 E ι
h : WithSeminorms p
⊢ 𝓝 0 = AddGroupFilterBasis.toFilterBasis.filter
tactic: refine ⟨fun h => ?_, p.withSeminorms_of_nhds⟩
state_before: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝⁴ : NormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Nonempty ι
t : TopologicalSpace E
inst✝ : TopologicalAddGroup E
p : SeminormFamily 𝕜 E ι
h : WithSeminorms p
⊢ 𝓝 0 = AddGroupFilterBasis.toFilterBasis.filter
state_after: no goals
tactic: exact AddGroupFilterBasis.nhds_zero_eq _
Theorem: PartENat.eq_top_iff_forall_lt
File: Mathlib/Data/Nat/PartENat.lean

Proof steps:
state_before: x : PartENat
⊢ x = ⊤ ↔ ∀ (n : ℕ), ↑n < x
state_after: case mp
x : PartENat
⊢ x = ⊤ → ∀ (n : ℕ), ↑n < x

case mpr
x : PartENat
⊢ (∀ (n : ℕ), ↑n < x) → x = ⊤
tactic: constructor
Theorem: MeasureTheory.integral_undef
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✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
f : α → G
h : ¬Integrable f μ
⊢ ∫ (a : α), f a ∂μ = 0
state_after: case pos
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
f : α → G
h : ¬Integrable f μ
hG : CompleteSpace G
⊢ ∫ (a : α), f a ∂μ = 0

case neg
α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : SMulCommClass ℝ 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : CompleteSpace F
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
f : α → G
h : ¬Integrable f μ
hG : ¬CompleteSpace G
⊢ ∫ (a : α), f a ∂μ = 0
tactic: by_cases hG : CompleteSpace G
Theorem: Turing.TM2to1.tr_respects_aux₂
File: Mathlib/Computability/TuringMachine.lean

Proof steps:
state_before: K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
k : K
q : TM1.Stmt Γ' Λ' σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
⊢ let v' := stVar v (S k) o;
  let Sk' := stWrite v (S k) o;
  let S' := update S k Sk';
  ∃ L',
    (∀ (k : K), ListBlank.map (proj k) L' = ListBlank.mk (List.map some (S' k)).reverse) ∧
      TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom L))) =
        TM1.stepAux q v' ((Tape.move Dir.right)^[(S' k).length] (Tape.mk' ∅ (addBottom L')))
state_after: K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
k : K
q : TM1.Stmt Γ' Λ' σ
v : σ
S : (k : K) → List (Γ k)
L : ListBlank ((k : K) → Option (Γ k))
hL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse
o : StAct k
⊢ ∃ L',
    (∀ (k_1 : K),
        ListBlank.map (proj k_1) L' = ListBlank.mk (List.map some (update S k (stWrite v (S k) o) k_1)).reverse) ∧
      TM1.stepAux (trStAct q o) v ((Tape.move Dir.right)^[(S k).length] (Tape.mk' ∅ (addBottom L))) =
        TM1.stepAux q (stVar v (S k) o) ((Tape.move Dir.right)^[(stWrite v (S k) o).length] (Tape.mk' ∅ (addBottom L')))
tactic: simp only [Function.update_same]
Theorem: IsClosedMap.comp
File: Mathlib/Topology/Maps/Basic.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : IsClosedMap g
hf : IsClosedMap f
⊢ IsClosedMap (g ∘ f)
state_after: X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : IsClosedMap g
hf : IsClosedMap f
s : Set X
hs : IsClosed s
⊢ IsClosed (g ∘ f '' s)
tactic: intro s hs
state_before: X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : IsClosedMap g
hf : IsClosedMap f
s : Set X
hs : IsClosed s
⊢ IsClosed (g ∘ f '' s)
state_after: X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : IsClosedMap g
hf : IsClosedMap f
s : Set X
hs : IsClosed s
⊢ IsClosed (g '' (f '' s))
tactic: rw [image_comp]
state_before: X : Type u_1
Y : Type u_2
Z : Type u_3
ι : Type u_4
f : X → Y
g : Y → Z
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
hg : IsClosedMap g
hf : IsClosedMap f
s : Set X
hs : IsClosed s
⊢ IsClosed (g '' (f '' s))
state_after: no goals
tactic: exact hg _ (hf _ hs)
Theorem: FreeAddGroup.Red.sizeof_of_step
File: Mathlib/GroupTheory/FreeGroup/Basic.lean

Proof steps:
state_before: α : Type u
L L₁ L₂ L₃ L₄ L1 L2 : List (α × Bool)
x : α
b : Bool
⊢ sizeOf (L1 ++ L2) < sizeOf (L1 ++ (x, b) :: (x, !b) :: L2)
state_after: no goals
tactic: induction L1 with
    | nil =>
      -- dsimp [sizeOf]
      dsimp
      simp only [Bool.sizeOf_eq_one]

      have H :
        1 + (1 + 1) + (1 + (1 + 1) + sizeOf L2) =
          sizeOf L2 + (1 + ((1 + 1) + (1 + 1) + 1)) := by
        ac_rfl
      rw [H]
      apply Nat.lt_add_of_pos_right
      apply Nat.lt_add_right
      apply Nat.zero_lt_one
    | cons hd tl ih =>
      dsimp
      exact Nat.add_lt_add_left ih _
Theorem: Int.ceil_sub_one
File: Mathlib/Algebra/Order/Floor.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a✝ a : α
⊢ ⌈a - 1⌉ = ⌈a⌉ - 1
state_after: no goals
tactic: rw [eq_sub_iff_add_eq, ← ceil_add_one, sub_add_cancel]