1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: CategoryTheory.Subobject.factorThru_mk_self
File: Mathlib/CategoryTheory/Subobject/FactorThru.lean

tactic: ext
state_before: C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
f : X ⟶ Y
inst✝ : Mono f
⊢ (mk f).factorThru f ⋯ = (underlyingIso f).inv
state_after: case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
f : X ⟶ Y
inst✝ : Mono f
⊢ (mk f).factorThru f ⋯ ≫ (mk f).arrow = (underlyingIso f).inv ≫ (mk f).arrow

tactic: simp
state_before: case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
f : X ⟶ Y
inst✝ : Mono f
⊢ (mk f).factorThru f ⋯ ≫ (mk f).arrow = (underlyingIso f).inv ≫ (mk f).arrow
state_after: no goals
Theorem: SetTheory.PGame.moveLeft_nim
File: Mathlib/SetTheory/Game/Nim.lean

tactic: simp
state_before: o : Ordinal.{u_1}
i : ↑(Set.Iio o)
⊢ (nim o).moveLeft (toLeftMovesNim i) = nim ↑i
state_after: no goals
Theorem: MeasureTheory.MeasurePreserving.symm
File: Mathlib/Dynamics/Ergodic/MeasurePreserving.lean

tactic: rw [← h.map_eq, map_map e.symm.measurable e.measurable, e.symm_comp_self, map_id]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μa✝ : Measure α
μb✝ : Measure β
μc : Measure γ
μd : Measure δ
e : α ≃ᵐ β
μa : Measure α
μb : Measure β
h : MeasurePreserving (⇑e) μa μb
⊢ map (⇑e.symm) μb = μa
state_after: no goals
Theorem: intervalIntegral.norm_integral_le_of_norm_le_const_ae
File: Mathlib/MeasureTheory/Integral/IntervalIntegral.lean

tactic: rw [norm_integral_eq_norm_integral_Ioc]
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ ‖∫ (x : ℝ) in a..b, f x‖ ≤ C * |b - a|
state_after: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ ‖∫ (x : ℝ) in Ι a b, f x ∂volume‖ ≤ C * |b - a|

tactic: convert norm_setIntegral_le_of_norm_le_const_ae'' _ measurableSet_Ioc h using 1
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ ‖∫ (x : ℝ) in Ι a b, f x ∂volume‖ ≤ C * |b - a|
state_after: case h.e'_4
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ C * |b - a| = C * (volume (Ioc (min a b) (max a b))).toReal

ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ volume (Ioc (min a b) (max a b)) < ⊤

tactic: rw [Real.volume_Ioc, max_sub_min_eq_abs, ENNReal.toReal_ofReal (abs_nonneg _)]
state_before: case h.e'_4
ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ C * |b - a| = C * (volume (Ioc (min a b) (max a b))).toReal
state_after: no goals

tactic: simp only [Real.volume_Ioc, ENNReal.ofReal_lt_top]
state_before: ι : Type u_1
𝕜 : Type u_2
E : Type u_3
F : Type u_4
A : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f✝ g : ℝ → E
μ : Measure ℝ
a b C : ℝ
f : ℝ → E
h : ∀ᵐ (x : ℝ), x ∈ Ι a b → ‖f x‖ ≤ C
⊢ volume (Ioc (min a b) (max a b)) < ⊤
state_after: no goals
Theorem: CategoryTheory.Pretriangulated.Opposite.isomorphic_distinguished
File: Mathlib/CategoryTheory/Triangulated/Opposite.lean

tactic: simp only [mem_distinguishedTriangles_iff] at hT₁ ⊢
state_before: C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ : Triangle Cᵒᵖ
hT₁ : T₁ ∈ distinguishedTriangles C
T₂ : Triangle Cᵒᵖ
e : T₂ ≅ T₁
⊢ T₂ ∈ distinguishedTriangles C
state_after: C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ T₂ : Triangle Cᵒᵖ
e : T₂ ≅ T₁
hT₁ : ((triangleOpEquivalence C).inverse.obj T₁).unop ∈ Pretriangulated.distinguishedTriangles
⊢ ((triangleOpEquivalence C).inverse.obj T₂).unop ∈ Pretriangulated.distinguishedTriangles

tactic: exact Pretriangulated.isomorphic_distinguished _ hT₁ _
  ((triangleOpEquivalence C).inverse.mapIso e).unop.symm
state_before: C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : HasShift C ℤ
inst✝³ : HasZeroObject C
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
T₁ T₂ : Triangle Cᵒᵖ
e : T₂ ≅ T₁
hT₁ : ((triangleOpEquivalence C).inverse.obj T₁).unop ∈ Pretriangulated.distinguishedTriangles
⊢ ((triangleOpEquivalence C).inverse.obj T₂).unop ∈ Pretriangulated.distinguishedTriangles
state_after: no goals
Theorem: decide_True'
File: Mathlib/Init/Logic.lean

tactic: simp
state_before: a b c d : Prop
p : ?m.8557
h : Decidable True
⊢ decide True = true
state_after: no goals
Theorem: Set.offDiag_nonempty
File: Mathlib/Data/Set/Prod.lean

tactic: simp [offDiag, Set.Nonempty, Set.Nontrivial]
state_before: α : Type u_1
s t : Set α
x : α × α
a : α
⊢ s.offDiag.Nonempty ↔ s.Nontrivial
state_after: no goals
Theorem: bernsteinPolynomial.sum
File: Mathlib/RingTheory/Polynomial/Bernstein.lean

tactic: rw [add_pow]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν = (X + (1 - X)) ^ n
state_after: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ m ∈ Finset.range (n + 1), X ^ m * (1 - X) ^ (n - m) * ↑(n.choose m)

tactic: simp only [bernsteinPolynomial, mul_comm, mul_assoc, mul_left_comm]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν =
    ∑ m ∈ Finset.range (n + 1), X ^ m * (1 - X) ^ (n - m) * ↑(n.choose m)
state_after: no goals

tactic: simp
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
⊢ (X + (1 - X)) ^ n = 1
state_after: no goals
Theorem: CategoryTheory.Abelian.FunctorCategory.coimageImageComparison_app'
File: Mathlib/CategoryTheory/Abelian/FunctorCategory.lean

tactic: simp only [coimageImageComparison_app, Iso.hom_inv_id_assoc, Iso.hom_inv_id, Category.assoc,
  Category.comp_id]
state_before: C : Type u
inst✝² : Category.{v, u} C
D : Type w
inst✝¹ : Category.{z, w} D
inst✝ : Abelian D
F G : C ⥤ D
α : F ⟶ G
X : C
⊢ (coimageImageComparison α).app X = (coimageObjIso α X).hom ≫ coimageImageComparison (α.app X) ≫ (imageObjIso α X).inv
state_after: no goals
Theorem: diff_subset_closure_iff
File: Mathlib/Topology/Basic.lean

tactic: rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure]
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✝ : TopologicalSpace X
⊢ s \ t ⊆ closure t ↔ s ⊆ closure t
state_after: no goals
Theorem: Associates.FactorSet.sup_add_inf_eq_add
File: Mathlib/RingTheory/UniqueFactorizationDomain.lean

tactic: simp
state_before: α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableEq (Associates α)
b : FactorSet α
⊢ ⊤ ⊔ b + ⊤ ⊓ b = ⊤ + b
state_after: no goals

tactic: simp
state_before: α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableEq (Associates α)
a : FactorSet α
⊢ a ⊔ ⊤ + a ⊓ ⊤ = a + ⊤
state_after: no goals

tactic: rw [← WithTop.coe_sup, ← WithTop.coe_inf, ← WithTop.coe_add, ← WithTop.coe_add,
  WithTop.coe_eq_coe]
state_before: α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableEq (Associates α)
a b : Multiset { a // Irreducible a }
⊢ ↑a ⊔ ↑b + ↑a ⊓ ↑b = ↑a + ↑b
state_after: α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableEq (Associates α)
a b : Multiset { a // Irreducible a }
⊢ a ⊔ b + a ⊓ b = a + b

tactic: exact Multiset.union_add_inter _ _
state_before: α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : DecidableEq (Associates α)
a b : Multiset { a // Irreducible a }
⊢ a ⊔ b + a ⊓ b = a + b
state_after: no goals
Theorem: Metric.sphere_union_ball
File: Mathlib/Topology/MetricSpace/Pseudo/Defs.lean

tactic: rw [union_comm, ball_union_sphere]
state_before: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
x y z : α
δ ε ε₁ ε₂ : ℝ
s : Set α
⊢ sphere x ε ∪ ball x ε = closedBall x ε
state_after: no goals
Theorem: LSeries.term_neg_apply
File: Mathlib/NumberTheory/LSeries/Linearity.lean

tactic: rw [term_neg, Pi.neg_apply]
state_before: f : ℕ → ℂ
s : ℂ
n : ℕ
⊢ term (-f) s n = -term f s n
state_after: no goals
Theorem: IsLowerSet.thickening'
File: Mathlib/Analysis/Normed/Order/UpperLower.lean

tactic: rw [← ball_mul_one]
state_before: α : Type u_1
ι : Type u_2
inst✝ : NormedOrderedGroup α
s : Set α
hs : IsLowerSet s
ε : ℝ
⊢ IsLowerSet (thickening ε s)
state_after: α : Type u_1
ι : Type u_2
inst✝ : NormedOrderedGroup α
s : Set α
hs : IsLowerSet s
ε : ℝ
⊢ IsLowerSet (ball 1 ε * s)

tactic: exact hs.mul_left
state_before: α : Type u_1
ι : Type u_2
inst✝ : NormedOrderedGroup α
s : Set α
hs : IsLowerSet s
ε : ℝ
⊢ IsLowerSet (ball 1 ε * s)
state_after: no goals
Theorem: Nat.coprime_pow_right_iff
File: Mathlib/Data/Nat/GCD/Basic.lean

tactic: rw [Nat.coprime_comm, coprime_pow_left_iff hn, Nat.coprime_comm]
state_before: n : ℕ
hn : 0 < n
a b : ℕ
⊢ a.Coprime (b ^ n) ↔ a.Coprime b
state_after: no goals
Theorem: CancelDenoms.pow_subst
File: Mathlib/Tactic/CancelDenoms/Core.lean

tactic: rw [← h2, ← h1, mul_pow, mul_assoc]
state_before: α : Type u_1
inst✝ : CommRing α
n e1 t1 k l : α
e2 : ℕ
h1 : n * e1 = t1
h2 : l * n ^ e2 = k
⊢ k * e1 ^ e2 = l * t1 ^ e2
state_after: no goals
Theorem: MeasureTheory.OuterMeasure.coe_iSup
File: Mathlib/MeasureTheory/OuterMeasure/Operations.lean

tactic: simp
state_before: α : Type u_1
β : Type u_2
m : OuterMeasure α
ι : Sort u_3
f : ι → OuterMeasure α
s : Set α
⊢ (⨆ i, f i) s = (⨆ i, ⇑(f i)) s
state_after: no goals
Theorem: CategoryTheory.IsHomLift.of_commSq
File: Mathlib/CategoryTheory/FiberedCategory/HomLift.lean

tactic: subst ha hb
state_before: 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
R S : 𝒮
a b : 𝒳
f : R ⟶ S
φ : a ⟶ b
ha : p.obj a = R
hb : p.obj b = S
h : CommSq (p.map φ) (eqToHom ha) (eqToHom hb) f
⊢ p.IsHomLift f φ
state_after: 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
f : p.obj a ⟶ p.obj b
h : CommSq (p.map φ) (eqToHom ⋯) (eqToHom ⋯) f
⊢ p.IsHomLift f φ

tactic: obtain rfl : f = p.map φ := by simpa using h.1.symm
state_before: 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
f : p.obj a ⟶ p.obj b
h : CommSq (p.map φ) (eqToHom ⋯) (eqToHom ⋯) f
⊢ p.IsHomLift f φ
state_after: 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
h : CommSq (p.map φ) (eqToHom ⋯) (eqToHom ⋯) (p.map φ)
⊢ p.IsHomLift (p.map φ) φ

tactic: infer_instance
state_before: 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
h : CommSq (p.map φ) (eqToHom ⋯) (eqToHom ⋯) (p.map φ)
⊢ p.IsHomLift (p.map φ) φ
state_after: no goals

tactic: simpa using h.1.symm
state_before: 𝒮 : Type u₁
𝒳 : Type u₂
inst✝¹ : Category.{v₁, u₂} 𝒳
inst✝ : Category.{v₂, u₁} 𝒮
p : 𝒳 ⥤ 𝒮
a b : 𝒳
φ : a ⟶ b
f : p.obj a ⟶ p.obj b
h : CommSq (p.map φ) (eqToHom ⋯) (eqToHom ⋯) f
⊢ f = p.map φ
state_after: no goals
Theorem: Fin.insertNth_le_iff
File: Mathlib/Data/Fin/Tuple/Basic.lean

tactic: simp [Pi.le_def, forall_iff_succAbove i]
state_before: m n : ℕ
α : Fin (n + 1) → Type u
β : Type v
inst✝ : (i : Fin (n + 1)) → Preorder (α i)
i : Fin (n + 1)
x : α i
p : (j : Fin n) → α (i.succAbove j)
q : (j : Fin (n + 1)) → α j
⊢ i.insertNth x p ≤ q ↔ x ≤ q i ∧ p ≤ fun j => q (i.succAbove j)
state_after: no goals
Theorem: gaugeSeminorm_ball_one
File: Mathlib/Analysis/Convex/Gauge.lean

tactic: rw [Seminorm.ball_zero_eq]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s t : Set E
x : E
a : ℝ
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
hs₀ : Balanced 𝕜 s
hs₁ : Convex ℝ s
hs₂ : Absorbent ℝ s
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : IsOpen s
⊢ (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s t : Set E
x : E
a : ℝ
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
hs₀ : Balanced 𝕜 s
hs₁ : Convex ℝ s
hs₂ : Absorbent ℝ s
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : IsOpen s
⊢ {y | (gaugeSeminorm hs₀ hs₁ hs₂) y < 1} = s

tactic: exact gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
s t : Set E
x : E
a : ℝ
inst✝⁴ : RCLike 𝕜
inst✝³ : Module 𝕜 E
inst✝² : IsScalarTower ℝ 𝕜 E
hs₀ : Balanced 𝕜 s
hs₁ : Convex ℝ s
hs₂ : Absorbent ℝ s
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hs : IsOpen s
⊢ {y | (gaugeSeminorm hs₀ hs₁ hs₂) y < 1} = s
state_after: no goals
Theorem: MeasureTheory.StronglyMeasurable.tendsto_approxBounded_ae
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

tactic: filter_upwards [hf_bound] with x hfx using tendsto_approxBounded_of_norm_le hf hfx
state_before: α : Type u_1
β✝ : Type u_2
γ : Type u_3
ι : Type u_4
inst✝³ : Countable ι
f✝ g : α → β✝
inst✝² : TopologicalSpace β✝
β : Type u_5
f : α → β
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedSpace ℝ β
m m0 : MeasurableSpace α
μ : Measure α
hf : StronglyMeasurable f
c : ℝ
hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => ↑(hf.approxBounded c n) x) atTop (𝓝 (f x))
state_after: no goals
Theorem: Finset.infs_compls_eq_diffs
File: Mathlib/Data/Finset/Sups.lean

tactic: ext
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: case a
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
a✝ : α
⊢ a✝ ∈ s ⊼ tᶜˢ ↔ a✝ ∈ s \\ t

tactic: simp [sdiff_eq]
state_before: case a
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
a✝ : α
⊢ a✝ ∈ s ⊼ tᶜˢ ↔ a✝ ∈ s \\ t
state_after: case a
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
a✝ : α
⊢ (∃ a ∈ s, ∃ b, bᶜ ∈ t ∧ a ⊓ b = a✝) ↔ ∃ a ∈ s, ∃ b ∈ t, a ⊓ bᶜ = a✝

tactic: aesop
state_before: case a
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
a✝ : α
⊢ (∃ a ∈ s, ∃ b, bᶜ ∈ t ∧ a ⊓ b = a✝) ↔ ∃ a ∈ s, ∃ b ∈ t, a ⊓ bᶜ = a✝
state_after: no goals
Theorem: Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le
File: Mathlib/Analysis/Calculus/MeanValue.lean

tactic: simp
state_before: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
𝕜 : Type u_3
G : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f f' : 𝕜 → G
s : Set 𝕜
x✝ y : 𝕜
C : ℝ≥0
hs : Convex ℝ s
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C
x : 𝕜
hx : x ∈ s
⊢ ‖smulRight 1 (f' x)‖₊ ≤ ‖f' x‖₊
state_after: no goals
Theorem: cauchySeq_tendsto_of_isComplete
File: Mathlib/Topology/UniformSpace/Cauchy.lean

tactic: rwa [image_univ, range_subset_iff]
state_before: α : Type u
β : Type v
uniformSpace : UniformSpace α
inst✝ : Preorder β
K : Set α
h₁ : IsComplete K
u : β → α
h₂ : ∀ (n : β), u n ∈ K
h₃ : CauchySeq u
⊢ u '' univ ⊆ K
state_after: no goals
Theorem: FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_changeOrigin
File: Mathlib/Analysis/Analytic/CPolynomial.lean

tactic: rw [add_comm n k]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝¹ pf pg : FormalMultilinearSeries 𝕜 E F
x✝¹ : E
r✝ r' : ℝ≥0∞
n✝ m : ℕ
p✝ : FormalMultilinearSeries 𝕜 E F
x y : E
r R : ℝ≥0
p : FormalMultilinearSeries 𝕜 E F
n k : ℕ
hn : ∀ (m : ℕ), n + k ≤ m → p m = 0
x✝ : ℕ
hm : n ≤ x✝
⊢ n + k ≤ k + x✝
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝¹ pf pg : FormalMultilinearSeries 𝕜 E F
x✝¹ : E
r✝ r' : ℝ≥0∞
n✝ m : ℕ
p✝ : FormalMultilinearSeries 𝕜 E F
x y : E
r R : ℝ≥0
p : FormalMultilinearSeries 𝕜 E F
n k : ℕ
hn : ∀ (m : ℕ), n + k ≤ m → p m = 0
x✝ : ℕ
hm : n ≤ x✝
⊢ k + n ≤ k + x✝

tactic: apply add_le_add_left hm
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f g : E → F
p✝¹ pf pg : FormalMultilinearSeries 𝕜 E F
x✝¹ : E
r✝ r' : ℝ≥0∞
n✝ m : ℕ
p✝ : FormalMultilinearSeries 𝕜 E F
x y : E
r R : ℝ≥0
p : FormalMultilinearSeries 𝕜 E F
n k : ℕ
hn : ∀ (m : ℕ), n + k ≤ m → p m = 0
x✝ : ℕ
hm : n ≤ x✝
⊢ k + n ≤ k + x✝
state_after: no goals
Theorem: Nat.le_three_of_sqrt_eq_one
File: Mathlib/Data/Nat/Defs.lean

tactic: rw [h]
state_before: a b c d m n k : ℕ
p q : ℕ → Prop
h : n.sqrt = 1
⊢ n.sqrt < 2
state_after: a b c d m n k : ℕ
p q : ℕ → Prop
h : n.sqrt = 1
⊢ 1 < 2

tactic: decide
state_before: a b c d m n k : ℕ
p q : ℕ → Prop
h : n.sqrt = 1
⊢ 1 < 2
state_after: no goals
Theorem: MvPowerSeries.smul_inv
File: Mathlib/RingTheory/MvPowerSeries/Inverse.lean

tactic: simp [smul_eq_C_mul, mul_comm]
state_before: σ : Type u_1
R : Type u_2
k : Type u_3
inst✝ : Field k
r : k
φ : MvPowerSeries σ k
⊢ (r • φ)⁻¹ = r⁻¹ • φ⁻¹
state_after: no goals
Theorem: Multiset.lcm_add
File: Mathlib/Algebra/GCDMonoid/Multiset.lean

tactic: simp [lcm]
state_before: α : Type u_1
inst✝¹ : CancelCommMonoidWithZero α
inst✝ : NormalizedGCDMonoid α
s₁ s₂ : Multiset α
⊢ (s₁ + s₂).lcm = fold GCDMonoid.lcm (GCDMonoid.lcm 1 1) (s₁ + s₂)
state_after: no goals
Theorem: tendsto_comp_of_locally_uniform_limit
File: Mathlib/Topology/UniformSpace/UniformConvergence.lean

tactic: rw [← continuousWithinAt_univ] at h
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousAt f x
hg : Tendsto g p (𝓝 x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
state_after: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝 x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))

tactic: rw [← nhdsWithin_univ] at hunif hg
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝 x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
state_after: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝[univ] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))

tactic: exact tendsto_comp_of_locally_uniform_limit_within h hg hunif
state_before: α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f univ x
hg : Tendsto g p (𝓝[univ] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
state_after: no goals
Theorem: isEmpty_sigma
File: Mathlib/Logic/IsEmpty.lean

tactic: simp only [← not_nonempty_iff, nonempty_sigma, not_exists]
state_before: α✝ : Sort u_1
β : Sort u_2
γ : Sort u_3
α : Type u_5
E : α → Type u_4
⊢ IsEmpty (Sigma E) ↔ ∀ (a : α), IsEmpty (E a)
state_after: no goals
Theorem: ProbabilityTheory.variance_zero
File: Mathlib/Probability/Variance.lean

tactic: simp only [variance, evariance_zero, ENNReal.zero_toReal]
state_before: Ω : Type u_1
m : MeasurableSpace Ω
X : Ω → ℝ
μ✝ μ : Measure Ω
⊢ variance 0 μ = 0
state_after: no goals
Theorem: RatFunc.denom_zero
File: Mathlib/FieldTheory/RatFunc/Basic.lean

tactic: convert denom_div (0 : K[X]) one_ne_zero <;> simp
state_before: K : Type u
inst✝ : Field K
⊢ denom 0 = 1
state_after: no goals
Theorem: List.length_range
File: .lake/packages/batteries/Batteries/Data/List/Lemmas.lean

tactic: simp only [range_eq_range', length_range']
state_before: n : Nat
⊢ (range n).length = n
state_after: no goals
Theorem: MeasureTheory.setLaverage_eq
File: Mathlib/MeasureTheory/Integral/Average.lean

tactic: rw [laverage_eq, restrict_apply_univ]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
μ ν : Measure α
s✝ t : Set α
f✝ g f : α → ℝ≥0∞
s : Set α
⊢ ⨍⁻ (x : α) in s, f x ∂μ = (∫⁻ (x : α) in s, f x ∂μ) / μ s
state_after: no goals
Theorem: Nat.odd_mul_odd_div_two
File: Mathlib/Data/Nat/ModEq.lean

tactic: simp_all
state_before: m✝ n✝ a b c d m n : ℕ
hm1 : m % 2 = 1
hn1 : n % 2 = 1
h : m = 0
⊢ False
state_after: no goals

tactic: simp_all
state_before: m✝ n✝ a b c d m n : ℕ
hm1 : m % 2 = 1
hn1 : n % 2 = 1
hm0 : 0 < m
h : n = 0
⊢ False
state_after: no goals

tactic: dsimp
state_before: m✝ n✝ a b c d m n : ℕ
hm1 : m % 2 = 1
hn1 : n % 2 = 1
hm0 : 0 < m
hn0 : 0 < n
⊢ (fun x => 2 * x) (m * n / 2) = (fun x => 2 * x) (m * (n / 2) + m / 2)
state_after: m✝ n✝ a b c d m n : ℕ
hm1 : m % 2 = 1
hn1 : n % 2 = 1
hm0 : 0 < m
hn0 : 0 < n
⊢ 2 * (m * n / 2) = 2 * (m * (n / 2) + m / 2)

tactic: rw [mul_add, two_mul_odd_div_two hm1, mul_left_comm, two_mul_odd_div_two hn1,
  two_mul_odd_div_two (Nat.odd_mul_odd hm1 hn1), mul_tsub, mul_one, ←
  add_tsub_assoc_of_le (succ_le_of_lt hm0),
  tsub_add_cancel_of_le (le_mul_of_one_le_right (Nat.zero_le _) hn0)]
state_before: m✝ n✝ a b c d m n : ℕ
hm1 : m % 2 = 1
hn1 : n % 2 = 1
hm0 : 0 < m
hn0 : 0 < n
⊢ 2 * (m * n / 2) = 2 * (m * (n / 2) + m / 2)
state_after: no goals