ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: NormedCommGroup.tendsto_nhds_nhds
File: Mathlib/Analysis/Normed/Group/Basic.lean

Proof steps:
state_before: 𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
f : E → F
x : E
y : F
⊢ Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε
state_after: no goals
tactic: simp_rw [Metric.tendsto_nhds_nhds, dist_eq_norm_div]
Theorem: aemeasurable_const_vadd_iff
File: Mathlib/MeasureTheory/Group/Arithmetic.lean

Proof steps:
state_before: α✝ : Type u_1
M : Type u_2
β : Type u_3
α : Type u_4
inst✝⁹ : MeasurableSpace M
inst✝⁸ : MeasurableSpace β
inst✝⁷ : Monoid M
inst✝⁶ : MulAction M β
inst✝⁵ : MeasurableSMul M β
inst✝⁴ : MeasurableSpace α
f : α → β
μ : Measure α
G : Type u_5
inst✝³ : Group G
inst✝² : MeasurableSpace G
inst✝¹ : MulAction G β
inst✝ : MeasurableSMul G β
c : G
h : AEMeasurable (fun x => c • f x) μ
⊢ AEMeasurable f μ
state_after: no goals
tactic: simpa only [inv_smul_smul] using h.const_smul' c⁻¹
Theorem: Mathlib.Tactic.Coherence.insert_id_lhs
File: Mathlib/Tactic/CategoryTheory/Coherence.lean

Proof steps:
state_before: C✝ : Type u
inst✝² : Category.{v, u} C✝
inst✝¹ : MonoidalCategory C✝
C : Type u_1
inst✝ : Category.{u_2, u_1} C
X Y : C
f g : X ⟶ Y
w : f ≫ 𝟙 Y = g
⊢ f = g
state_after: no goals
tactic: simpa using w
Theorem: CategoryTheory.ShortComplex.π₁Toπ₂_comp_π₂Toπ₃
File: Mathlib/Algebra/Homology/ShortComplex/Basic.lean

Proof steps:
state_before: C : Type u_1
D : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{?u.30774, u_2} D
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
⊢ π₁Toπ₂ ≫ π₂Toπ₃ = 0
state_after: no goals
tactic: aesop_cat
Theorem: Finset.diffs_compls_eq_infs
File: Mathlib/Data/Finset/Sups.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
inst✝² : DecidableEq α
inst✝¹ : DecidableEq β
inst✝ : BooleanAlgebra α
s✝ s₁ s₂ t✝ t₁ t₂ u v : Finset α
a b c : α
s t : Finset α
⊢ s \\ tᶜˢ = s ⊼ t
state_after: no goals
tactic: rw [← infs_compls_eq_diffs, compls_compls]
Theorem: mem_nhdsSet_iff_forall
File: Mathlib/Topology/NhdsSet.lean

Proof steps:
state_before: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : Filter X
s t s₁ s₂ t₁ t₂ : Set X
x : X
⊢ s ∈ 𝓝ˢ t ↔ ∀ x ∈ t, s ∈ 𝓝 x
state_after: no goals
tactic: simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image]
Theorem: Representation.asAlgebraHom_of
File: Mathlib/RepresentationTheory/Basic.lean

Proof steps:
state_before: k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommSemiring k
inst✝² : Monoid G
inst✝¹ : AddCommMonoid V
inst✝ : Module k V
ρ : Representation k G V
g : G
⊢ ρ.asAlgebraHom ((of k G) g) = ρ g
state_after: no goals
tactic: simp only [MonoidAlgebra.of_apply, asAlgebraHom_single, one_smul]
Theorem: Real.fourierIntegral_iteratedFDeriv
File: Mathlib/Analysis/Fourier/FourierTransformDeriv.lean

Proof steps:
state_before: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
N : ℕ∞
hf : ContDiff ℝ N f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) volume
n : ℕ
hn : ↑n ≤ N
⊢ 𝓕 (iteratedFDeriv ℝ n f) = fun w => fourierPowSMulRight (-innerSL ℝ) (𝓕 f) w n
state_after: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
N : ℕ∞
hf : ContDiff ℝ N f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) volume
n : ℕ
hn : ↑n ≤ N
⊢ 𝓕 (iteratedFDeriv ℝ n f) = fun w => fourierPowSMulRight (-(innerSL ℝ).flip) (𝓕 f) w n
tactic: rw [← innerSL_real_flip V]
state_before: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℂ E
V : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : FiniteDimensional ℝ V
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : V → E
N : ℕ∞
hf : ContDiff ℝ N f
h'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedFDeriv ℝ n f) volume
n : ℕ
hn : ↑n ≤ N
⊢ 𝓕 (iteratedFDeriv ℝ n f) = fun w => fourierPowSMulRight (-(innerSL ℝ).flip) (𝓕 f) w n
state_after: no goals
tactic: exact VectorFourier.fourierIntegral_iteratedFDeriv (innerSL ℝ) hf h'f hn
Theorem: UniqueFactorizationMonoid.normalizedFactors_zero
File: Mathlib/RingTheory/UniqueFactorizationDomain.lean

Proof steps:
state_before: α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
⊢ normalizedFactors 0 = 0
state_after: no goals
tactic: simp [normalizedFactors, factors]
Theorem: SimpleGraph.induce_union_connected
File: Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean

Proof steps:
state_before: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
s t : Set V
sconn : (induce s G).Connected
tconn : (induce t G).Connected
sintert : (s ∩ t).Nonempty
⊢ (induce (s ∪ t) G).Connected
state_after: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
s t : Set V
sconn : (⊤.induce s).Connected
tconn : (⊤.induce t).Connected
sintert : (s ∩ t).Nonempty
⊢ (⊤.induce (s ∪ t)).Connected
tactic: rw [connected_induce_iff] at sconn tconn ⊢
state_before: V : Type u
V' : Type v
G : SimpleGraph V
G' : SimpleGraph V'
s t : Set V
sconn : (⊤.induce s).Connected
tconn : (⊤.induce t).Connected
sintert : (s ∩ t).Nonempty
⊢ (⊤.induce (s ∪ t)).Connected
state_after: no goals
tactic: exact Subgraph.induce_union_connected sconn tconn sintert
Theorem: Monoid.Coprod.induction_on
File: Mathlib/GroupTheory/Coprod/Basic.lean

Proof steps:
state_before: M : Type u_1
N : Type u_2
M' : Type u_3
N' : Type u_4
P : Type u_5
inst✝⁴ : MulOneClass M
inst✝³ : MulOneClass N
inst✝² : MulOneClass M'
inst✝¹ : MulOneClass N'
inst✝ : MulOneClass P
C : M ∗ N → Prop
m : M ∗ N
inl : ∀ (m : M), C (Monoid.Coprod.inl m)
inr : ∀ (n : N), C (Monoid.Coprod.inr n)
mul : ∀ (x y : M ∗ N), C x → C y → C (x * y)
⊢ C 1
state_after: no goals
tactic: simpa using inl 1
Theorem: Asymptotics.isBigO_iff_eventually
File: Mathlib/Analysis/Asymptotics/Asymptotics.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
⊢ (∀ᶠ (c : ℝ) in atTop, IsBigOWith c l f g') ↔ ∀ᶠ (c : ℝ) in atTop, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g' x‖
state_after: no goals
tactic: simp only [IsBigOWith_def]
Theorem: ThreeGPFree.eq_right
File: Mathlib/Combinatorics/Additive/AP/Three/Defs.lean

Proof steps:
state_before: F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝ : CancelCommMonoid α
s : Set α
a : α
hs : ThreeGPFree s
⊢ ∀ ⦃a : α⦄, a ∈ s → ∀ ⦃b : α⦄, b ∈ s → ∀ ⦃c : α⦄, c ∈ s → a * c = b * b → b = c
state_after: F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝ : CancelCommMonoid α
s : Set α
a✝ : α
hs : ThreeGPFree s
a : α
ha : a ∈ s
b : α
hb : b ∈ s
c : α
hc : c ∈ s
habc : a * c = b * b
⊢ b = c
tactic: rintro a ha b hb c hc habc
state_before: F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝ : CancelCommMonoid α
s : Set α
a✝ : α
hs : ThreeGPFree s
a : α
ha : a ∈ s
b : α
hb : b ∈ s
c : α
hc : c ∈ s
habc : a * c = b * b
⊢ b = c
state_after: F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝ : CancelCommMonoid α
s : Set α
a✝ : α
hs : ThreeGPFree s
a : α
ha : a ∈ s
c : α
hc : c ∈ s
hb : a ∈ s
habc : a * c = a * a
⊢ a = c
tactic: obtain rfl := hs ha hb hc habc
state_before: F : Type u_1
α : Type u_2
β : Type u_3
𝕜 : Type u_4
E : Type u_5
inst✝ : CancelCommMonoid α
s : Set α
a✝ : α
hs : ThreeGPFree s
a : α
ha : a ∈ s
c : α
hc : c ∈ s
hb : a ∈ s
habc : a * c = a * a
⊢ a = c
state_after: no goals
tactic: simpa using habc.symm
Theorem: linearIndependent_le_span_finset
File: Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean

Proof steps:
state_before: R : Type u
M : Type v
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι✝ : Type w
ι' : Type w'
inst✝ : StrongRankCondition R
ι : Type u_1
v : ι → M
i : LinearIndependent R v
w : Finset M
s : span R ↑w = ⊤
⊢ #ι ≤ ↑w.card
state_after: no goals
tactic: simpa only [Finset.coe_sort_coe, Fintype.card_coe] using linearIndependent_le_span v i w s
Theorem: MeasureTheory.nndist_integral_add_measure_le_lintegral
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✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑(nndist (∫ (x : α), f x ∂μ) (∫ (x : α), f x ∂(μ + ν))) ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂ν
state_after: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑‖∫ (x : α), f x ∂ν‖₊ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂ν
tactic: rw [integral_add_measure h₁ h₂, nndist_comm, nndist_eq_nnnorm, add_sub_cancel_left]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_4
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NormedSpace 𝕜 E
inst✝⁷ : SMulCommClass ℝ 𝕜 E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : CompleteSpace F
G : Type u_5
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
f✝ g : α → E
m : MeasurableSpace α
μ : Measure α
X : Type u_6
inst✝¹ : TopologicalSpace X
inst✝ : FirstCountableTopology X
ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑‖∫ (x : α), f x ∂ν‖₊ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂ν
state_after: no goals
tactic: exact ennnorm_integral_le_lintegral_ennnorm _
Theorem: ValuationSubring.monotone_mapOfLE
File: Mathlib/RingTheory/Valuation/ValuationSubring.lean

Proof steps:
state_before: K : Type u
inst✝ : Field K
A R S : ValuationSubring K
h : R ≤ S
⊢ Monotone ⇑(R.mapOfLE S h)
state_after: case mk.mk.intro
K : Type u
inst✝ : Field K
A R S : ValuationSubring K
h : R ≤ S
a✝² : R.ValueGroup
a✝¹ : K
b✝ : R.ValueGroup
a✝ : K
a : ↥R
ha : a • a✝ = a✝¹
⊢ (R.mapOfLE S h) (Quot.mk Setoid.r a✝¹) ≤ (R.mapOfLE S h) (Quot.mk Setoid.r a✝)
tactic: rintro ⟨⟩ ⟨⟩ ⟨a, ha⟩
state_before: case mk.mk.intro
K : Type u
inst✝ : Field K
A R S : ValuationSubring K
h : R ≤ S
a✝² : R.ValueGroup
a✝¹ : K
b✝ : R.ValueGroup
a✝ : K
a : ↥R
ha : a • a✝ = a✝¹
⊢ (R.mapOfLE S h) (Quot.mk Setoid.r a✝¹) ≤ (R.mapOfLE S h) (Quot.mk Setoid.r a✝)
state_after: no goals
tactic: exact ⟨R.inclusion S h a, ha⟩
Theorem: CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_inv_right
File: Mathlib/AlgebraicTopology/SimplicialObject.lean

Proof steps:
state_before: C : Type u
inst✝ : Category.{v, u} C
X : Augmented C
⊢ X.rightOp.leftOp.right = X.right
state_after: no goals
tactic: aesop_cat
Theorem: norm_sq_eq_add_norm_sq_projection
File: Mathlib/Analysis/InnerProductSpace/Projection.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : _root_.RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
x : E
S : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection S
⊢ ‖x‖ ^ 2 = ‖↑((orthogonalProjection S) x) + ↑((orthogonalProjection Sᗮ) x)‖ ^ 2
state_after: no goals
tactic: rw [orthogonalProjection_add_orthogonalProjection_orthogonal]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : _root_.RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
x : E
S : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection S
⊢ ‖↑((orthogonalProjection S) x) + ↑((orthogonalProjection Sᗮ) x)‖ ^ 2 =
    ‖(orthogonalProjection S) x‖ ^ 2 + ‖(orthogonalProjection Sᗮ) x‖ ^ 2
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : _root_.RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
x : E
S : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection S
⊢ ‖↑((orthogonalProjection S) x) + ↑((orthogonalProjection Sᗮ) x)‖ *
      ‖↑((orthogonalProjection S) x) + ↑((orthogonalProjection Sᗮ) x)‖ =
    ‖(orthogonalProjection S) x‖ * ‖(orthogonalProjection S) x‖ +
      ‖(orthogonalProjection Sᗮ) x‖ * ‖(orthogonalProjection Sᗮ) x‖
tactic: simp only [sq]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : _root_.RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
x : E
S : Submodule 𝕜 E
inst✝ : HasOrthogonalProjection S
⊢ ‖↑((orthogonalProjection S) x) + ↑((orthogonalProjection Sᗮ) x)‖ *
      ‖↑((orthogonalProjection S) x) + ↑((orthogonalProjection Sᗮ) x)‖ =
    ‖(orthogonalProjection S) x‖ * ‖(orthogonalProjection S) x‖ +
      ‖(orthogonalProjection Sᗮ) x‖ * ‖(orthogonalProjection Sᗮ) x‖
state_after: no goals
tactic: exact norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero _ _ <|
        (S.mem_orthogonal _).1 (orthogonalProjection Sᗮ x).2 _ (orthogonalProjection S x).2
Theorem: Set.strictMonoOn_projIcc
File: Mathlib/Order/Interval/Set/ProjIcc.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a b : α
h : a ≤ b
x✝ x : α
hx : x ∈ Icc a b
y : α
hy : y ∈ Icc a b
hxy : x < y
⊢ projIcc a b h x < projIcc a b h y
state_after: no goals
tactic: simpa only [projIcc_of_mem, hx, hy]
Theorem: Finset.insert_union_comm
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
s✝ t✝ u v : Finset α
a✝ b : α
s t : Finset α
a : α
⊢ insert a s ∪ t = s ∪ insert a t
state_after: no goals
tactic: rw [insert_union, union_insert]
Theorem: NonUnitalStarAlgebra.mem_bot
File: Mathlib/Algebra/Star/NonUnitalSubalgebra.lean

Proof steps:
state_before: F : Type v'
R' : Type u'
R : Type u
A : Type v
B : Type w
C : Type w'
inst✝¹⁶ : CommSemiring R
inst✝¹⁵ : StarRing R
inst✝¹⁴ : NonUnitalSemiring A
inst✝¹³ : StarRing A
inst✝¹² : Module R A
inst✝¹¹ : IsScalarTower R A A
inst✝¹⁰ : SMulCommClass R A A
inst✝⁹ : StarModule R A
inst✝⁸ : NonUnitalSemiring B
inst✝⁷ : StarRing B
inst✝⁶ : Module R B
inst✝⁵ : IsScalarTower R B B
inst✝⁴ : SMulCommClass R B B
inst✝³ : StarModule R B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F R A B
inst✝ : NonUnitalStarAlgHomClass F R A B
x : A
⊢ x ∈ NonUnitalAlgebra.adjoin R (∅ ∪ star ∅) ↔ x = 0
state_after: no goals
tactic: rw [Set.star_empty, Set.union_empty, NonUnitalAlgebra.adjoin_empty, NonUnitalAlgebra.mem_bot]
Theorem: RatFunc.coe_C
File: Mathlib/RingTheory/LaurentSeries.lean

Proof steps:
state_before: R : Type u_1
F : Type u
inst✝ : Field F
p q : F[X]
f g : RatFunc F
r : F
⊢ ↑(C r) = HahnSeries.C r
state_after: R : Type u_1
F : Type u
inst✝ : Field F
p q : F[X]
f g : RatFunc F
r : F
⊢ (ofPowerSeries ℤ F) ((PowerSeries.C F) r) = HahnSeries.C r
tactic: rw [coe_num_denom, num_C, denom_C, Polynomial.coe_C, -- Porting note: removed `coe_C`
    Polynomial.coe_one,
    PowerSeries.coe_one, div_one]
state_before: R : Type u_1
F : Type u
inst✝ : Field F
p q : F[X]
f g : RatFunc F
r : F
⊢ (ofPowerSeries ℤ F) ((PowerSeries.C F) r) = HahnSeries.C r
state_after: no goals
tactic: simp only [algebraMap_eq_C, ofPowerSeries_C, C_apply]
Theorem: StarConvex.smul_mem
File: Mathlib/Analysis/Convex/Star.lean

Proof steps:
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : OrderedRing 𝕜
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
x y : E
s t✝ : Set E
hs : StarConvex 𝕜 0 s
hx : x ∈ s
t : 𝕜
ht₀ : 0 ≤ t
ht₁ : t ≤ 1
⊢ t • x ∈ s
state_after: no goals
tactic: simpa using hs.add_smul_mem (by simpa using hx) ht₀ ht₁
Theorem: AddSubgroup.isComplement_singleton_left
File: Mathlib/GroupTheory/Complement.lean

Proof steps:
state_before: G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
g : G
⊢ IsComplement {g} S ↔ S = univ
state_after: G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
g : G
h : IsComplement {g} S
x : G
x✝ : x ∈ ⊤
⊢ x ∈ S
tactic: refine
    ⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => (congr_arg _ h).mpr isComplement_singleton_univ⟩
state_before: G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
g : G
h : IsComplement {g} S
x : G
x✝ : x ∈ ⊤
⊢ x ∈ S
state_after: case intro.mk.mk.mk
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
x : G
x✝ : x ∈ ⊤
z : G
h : IsComplement {z} S
y : G
property✝ : y ∈ S
hy : (fun x => ↑x.1 * ↑x.2) (⟨z, ⋯⟩, ⟨y, property✝⟩) = z * x
⊢ x ∈ S
tactic: obtain ⟨⟨⟨z, rfl : z = g⟩, y, _⟩, hy⟩ := h.2 (g * x)
state_before: case intro.mk.mk.mk
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
x : G
x✝ : x ∈ ⊤
z : G
h : IsComplement {z} S
y : G
property✝ : y ∈ S
hy : (fun x => ↑x.1 * ↑x.2) (⟨z, ⋯⟩, ⟨y, property✝⟩) = z * x
⊢ x ∈ S
state_after: no goals
tactic: rwa [← mul_left_cancel hy]
Theorem: MeasureTheory.Measure.rnDeriv_eq_zero
File: Mathlib/MeasureTheory/Decomposition/Lebesgue.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ✝ ν✝ μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
⊢ μ.rnDeriv ν =ᶠ[ae ν] 0 ↔ μ ⟂ₘ ν
state_after: no goals
tactic: rw [← withDensity_rnDeriv_eq_zero, withDensity_eq_zero_iff (measurable_rnDeriv _ _).aemeasurable]
Theorem: Rat.ofScientific_false_def
File: .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean

Proof steps:
state_before: m e : Nat
⊢ Rat.ofScientific m false e = ↑(m * 10 ^ e)
state_after: m e : Nat
⊢ (if false = true then normalize (↑m) (10 ^ e) ⋯ else ↑(m * 10 ^ e)) = ↑(m * 10 ^ e)
tactic: unfold Rat.ofScientific
state_before: m e : Nat
⊢ (if false = true then normalize (↑m) (10 ^ e) ⋯ else ↑(m * 10 ^ e)) = ↑(m * 10 ^ e)
state_after: no goals
tactic: rfl
Theorem: EuclideanDomain.dvd_gcd
File: Mathlib/Algebra/EuclideanDomain/Basic.lean

Proof steps:
state_before: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
a b c x✝¹ : R
x✝ : c ∣ 0
H : c ∣ x✝¹
⊢ c ∣ gcd 0 x✝¹
state_after: no goals
tactic: simpa only [gcd_zero_left] using H
state_before: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
a✝ b✝ c a b : R
x✝ : a ≠ 0
IH : c ∣ b % a → c ∣ a → c ∣ gcd (b % a) a
ca : c ∣ a
cb : c ∣ b
⊢ c ∣ gcd a b
state_after: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
a✝ b✝ c a b : R
x✝ : a ≠ 0
IH : c ∣ b % a → c ∣ a → c ∣ gcd (b % a) a
ca : c ∣ a
cb : c ∣ b
⊢ c ∣ gcd (b % a) a
tactic: rw [gcd_val]
state_before: R : Type u
inst✝¹ : EuclideanDomain R
inst✝ : DecidableEq R
a✝ b✝ c a b : R
x✝ : a ≠ 0
IH : c ∣ b % a → c ∣ a → c ∣ gcd (b % a) a
ca : c ∣ a
cb : c ∣ b
⊢ c ∣ gcd (b % a) a
state_after: no goals
tactic: exact IH ((dvd_mod_iff ca).2 cb) ca
Theorem: CategoryTheory.Functor.initial_of_exists_of_isCofiltered
File: Mathlib/CategoryTheory/Filtered/Final.lean

Proof steps:
state_before: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : IsCofilteredOrEmpty C
h₁ : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)
h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d), ∃ c' t, F.map t ≫ s = F.map t ≫ s'
⊢ F.Initial
state_after: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : IsCofilteredOrEmpty C
h₁ : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)
h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d), ∃ c' t, F.map t ≫ s = F.map t ≫ s'
⊢ ∀ (d : D), IsCofiltered (CostructuredArrow F d)
tactic: suffices ∀ d, IsCofiltered (CostructuredArrow F d) from
    initial_of_isCofiltered_costructuredArrow F
state_before: C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : IsCofilteredOrEmpty C
h₁ : ∀ (d : D), ∃ c, Nonempty (F.obj c ⟶ d)
h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d), ∃ c' t, F.map t ≫ s = F.map t ≫ s'
⊢ ∀ (d : D), IsCofiltered (CostructuredArrow F d)
state_after: no goals
tactic: exact isCofiltered_costructuredArrow_of_isCofiltered_of_exists F h₁ h₂
Theorem: ENNReal.toReal_ofReal_eq_iff
File: Mathlib/Data/ENNReal/Basic.lean

Proof steps:
state_before: α : Type u_1
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ
h : (ENNReal.ofReal a).toReal = a
⊢ 0 ≤ a
state_after: α : Type u_1
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ
h : (ENNReal.ofReal a).toReal = a
⊢ 0 ≤ (ENNReal.ofReal a).toReal
tactic: rw [← h]
state_before: α : Type u_1
a✝ b c d : ℝ≥0∞
r p q : ℝ≥0
a : ℝ
h : (ENNReal.ofReal a).toReal = a
⊢ 0 ≤ (ENNReal.ofReal a).toReal
state_after: no goals
tactic: exact toReal_nonneg
Theorem: SimpleGraph.induceHom_comp
File: Mathlib/Combinatorics/SimpleGraph/Maps.lean

Proof steps:
state_before: V : Type u_1
W : Type u_2
X : Type u_3
G : SimpleGraph V
G' : SimpleGraph W
u v : V
G'' : SimpleGraph X
s : Set V
t : Set W
r : Set X
φ : G →g G'
φst : Set.MapsTo (⇑φ) s t
ψ : G' →g G''
ψtr : Set.MapsTo (⇑ψ) t r
⊢ (induceHom ψ ψtr).comp (induceHom φ φst) = induceHom (ψ.comp φ) ⋯
state_after: case h.a
V : Type u_1
W : Type u_2
X : Type u_3
G : SimpleGraph V
G' : SimpleGraph W
u v : V
G'' : SimpleGraph X
s : Set V
t : Set W
r : Set X
φ : G →g G'
φst : Set.MapsTo (⇑φ) s t
ψ : G' →g G''
ψtr : Set.MapsTo (⇑ψ) t r
x : ↑s
⊢ ↑(((induceHom ψ ψtr).comp (induceHom φ φst)) x) = ↑((induceHom (ψ.comp φ) ⋯) x)
tactic: ext x
state_before: case h.a
V : Type u_1
W : Type u_2
X : Type u_3
G : SimpleGraph V
G' : SimpleGraph W
u v : V
G'' : SimpleGraph X
s : Set V
t : Set W
r : Set X
φ : G →g G'
φst : Set.MapsTo (⇑φ) s t
ψ : G' →g G''
ψtr : Set.MapsTo (⇑ψ) t r
x : ↑s
⊢ ↑(((induceHom ψ ψtr).comp (induceHom φ φst)) x) = ↑((induceHom (ψ.comp φ) ⋯) x)
state_after: no goals
tactic: rfl