1337xyz1337xyz/leandojo-benchmark4-v10 huggingface dataset:
Theorem: IsPGroup.to_le
File: Mathlib/GroupTheory/PGroup.lean

tactic: change ((Subgroup.inclusion hHK) a : G) = (Subgroup.inclusion hHK) b
state_before: p : ℕ
G : Type u_1
inst✝ : Group G
H K : Subgroup G
hK : IsPGroup p ↥K
hHK : H ≤ K
a b : ↥H
h : (Subgroup.inclusion hHK) a = (Subgroup.inclusion hHK) b
⊢ ↑a = ↑b
state_after: p : ℕ
G : Type u_1
inst✝ : Group G
H K : Subgroup G
hK : IsPGroup p ↥K
hHK : H ≤ K
a b : ↥H
h : (Subgroup.inclusion hHK) a = (Subgroup.inclusion hHK) b
⊢ ↑((Subgroup.inclusion hHK) a) = ↑((Subgroup.inclusion hHK) b)

tactic: apply Subtype.ext_iff.mp h
state_before: p : ℕ
G : Type u_1
inst✝ : Group G
H K : Subgroup G
hK : IsPGroup p ↥K
hHK : H ≤ K
a b : ↥H
h : (Subgroup.inclusion hHK) a = (Subgroup.inclusion hHK) b
⊢ ↑((Subgroup.inclusion hHK) a) = ↑((Subgroup.inclusion hHK) b)
state_after: no goals
Theorem: Filter.tendsto_bot
File: Mathlib/Order/Filter/Basic.lean

tactic: simp [Tendsto]
state_before: α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l✝ : Filter α
f : α → β
l : Filter β
⊢ Tendsto f ⊥ l
state_after: no goals
Theorem: SetTheory.PGame.neg_nim
File: Mathlib/SetTheory/Game/Nim.lean

tactic: induction' o using Ordinal.induction with o IH
state_before: o : Ordinal.{u_1}
⊢ -nim o = nim o
state_after: case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ -nim o = nim o

tactic: rw [nim_def]
state_before: case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ -nim o = nim o
state_after: case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (-mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
        nim (typein (fun x x_1 => x < x_1) o₂)) =
    mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
      nim (typein (fun x x_1 => x < x_1) o₂)

tactic: dsimp
state_before: case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (-mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
        nim (typein (fun x x_1 => x < x_1) o₂)) =
    mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
      nim (typein (fun x x_1 => x < x_1) o₂)
state_after: case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (mk (Quotient.out o).α (Quotient.out o).α (fun j => -nim (typein (fun x x_1 => x < x_1) j)) fun i =>
      -nim (typein (fun x x_1 => x < x_1) i)) =
    mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
      nim (typein (fun x x_1 => x < x_1) o₂)

tactic: congr <;> funext i <;> exact IH _ (Ordinal.typein_lt_self i)
state_before: case h
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (mk (Quotient.out o).α (Quotient.out o).α (fun j => -nim (typein (fun x x_1 => x < x_1) j)) fun i =>
      -nim (typein (fun x x_1 => x < x_1) i)) =
    mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ =>
      nim (typein (fun x x_1 => x < x_1) o₂)
state_after: no goals
Theorem: Ordinal.typein_top
File: Mathlib/SetTheory/Ordinal/Basic.lean

tactic: rcases f.down.1 h with ⟨b, rfl⟩
state_before: α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
α β : Type u_3
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
f : r ≺i s
x✝ : ↑{b | s b f.top}
a : β
h : a ∈ {b | s b f.top}
⊢ ∃ a_1,
    (RelEmbedding.codRestrict {b | s b f.top} { toRelEmbedding := f.toRelEmbedding, init' := ⋯ }.toRelEmbedding ⋯) a_1 =
      ⟨a, h⟩
state_after: case intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
α β : Type u_3
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
f : r ≺i s
x✝ : ↑{b | s b f.top}
b : α
h : f.toRelEmbedding b ∈ {b | s b f.top}
⊢ ∃ a,
    (RelEmbedding.codRestrict {b | s b f.top} { toRelEmbedding := f.toRelEmbedding, init' := ⋯ }.toRelEmbedding ⋯) a =
      ⟨f.toRelEmbedding b, h⟩

tactic: exact ⟨b, rfl⟩
state_before: case intro
α✝ : Type u
β✝ : Type u_1
γ : Type u_2
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t : γ → γ → Prop
α β : Type u_3
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
f : r ≺i s
x✝ : ↑{b | s b f.top}
b : α
h : f.toRelEmbedding b ∈ {b | s b f.top}
⊢ ∃ a,
    (RelEmbedding.codRestrict {b | s b f.top} { toRelEmbedding := f.toRelEmbedding, init' := ⋯ }.toRelEmbedding ⋯) a =
      ⟨f.toRelEmbedding b, h⟩
state_after: no goals
Theorem: HomologicalComplex.dFrom_eq
File: Mathlib/Algebra/Homology/HomologicalComplex.lean

tactic: obtain rfl := c.next_eq' r
state_before: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i j : ι
r : c.Rel i j
⊢ C.dFrom i = C.d i j ≫ (C.xNextIso r).inv
state_after: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i : ι
r : c.Rel i (c.next i)
⊢ C.dFrom i = C.d i (c.next i) ≫ (C.xNextIso r).inv

tactic: exact (Category.comp_id _).symm
state_before: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C : HomologicalComplex V c
i : ι
r : c.Rel i (c.next i)
⊢ C.dFrom i = C.d i (c.next i) ≫ (C.xNextIso r).inv
state_after: no goals
Theorem: Complex.sin_pi_div_two_sub
File: Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean

tactic: simp [sub_eq_add_neg, sin_add]
state_before: x : ℂ
⊢ sin (↑π / 2 - x) = cos x
state_after: no goals
Theorem: MeasureTheory.Memℒp.toLp_congr
File: Mathlib/MeasureTheory/Function/LpSpace.lean

tactic: simp [toLp, hfg]
state_before: α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f g : α → E
hf : Memℒp f p μ
hg : Memℒp g p μ
hfg : f =ᶠ[ae μ] g
⊢ toLp f hf = toLp g hg
state_after: no goals
Theorem: IsPathConnected.union
File: Mathlib/Topology/Connected/PathConnected.lean

tactic: rcases hUV with ⟨x, xU, xV⟩
state_before: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
hUV : (U ∩ V).Nonempty
⊢ IsPathConnected (U ∪ V)
state_after: case intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
⊢ IsPathConnected (U ∪ V)

tactic: use x, Or.inl xU
state_before: case intro.intro
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
⊢ IsPathConnected (U ∪ V)
state_after: case right
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y

tactic: rintro y (yU | yV)
state_before: case right
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
⊢ ∀ {y : X}, y ∈ U ∪ V → JoinedIn (U ∪ V) x y
state_after: case right.inl
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y✝ z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yU : y ∈ U
⊢ JoinedIn (U ∪ V) x y

case right.inr
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y✝ z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yV : y ∈ V
⊢ JoinedIn (U ∪ V) x y

tactic: exact (hU.joinedIn x xU y yU).mono subset_union_left
state_before: case right.inl
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y✝ z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yU : y ∈ U
⊢ JoinedIn (U ∪ V) x y
state_after: no goals

tactic: exact (hV.joinedIn x xV y yV).mono subset_union_right
state_before: case right.inr
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x✝ y✝ z : X
ι : Type u_3
F U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yV : y ∈ V
⊢ JoinedIn (U ∪ V) x y
state_after: no goals
Theorem: Set.one_le_chainHeight_iff
File: Mathlib/Order/Height.lean

tactic: rw [← Nat.cast_one, Set.le_chainHeight_iff]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
s t : Set α
l : List α
a : α
⊢ 1 ≤ s.chainHeight ↔ s.Nonempty
state_after: α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
s t : Set α
l : List α
a : α
⊢ (∃ l ∈ s.subchain, l.length = 1) ↔ s.Nonempty

tactic: simp only [length_eq_one, @and_comm (_ ∈ _), @eq_comm _ _ [_], exists_exists_eq_and,
  singleton_mem_subchain_iff, Set.Nonempty]
state_before: α : Type u_1
β : Type u_2
inst✝¹ : LT α
inst✝ : LT β
s t : Set α
l : List α
a : α
⊢ (∃ l ∈ s.subchain, l.length = 1) ↔ s.Nonempty
state_after: no goals
Theorem: Finpartition.isUniform_one
File: Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean

tactic: rw [IsUniform, mul_one, Nat.cast_le]
state_before: α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε δ : 𝕜
u v : Finset α
⊢ P.IsUniform G 1
state_after: α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε δ : 𝕜
u v : Finset α
⊢ (P.nonUniforms G 1).card ≤ P.parts.card * (P.parts.card - 1)

tactic: refine (card_filter_le _
  (fun uv => ¬SimpleGraph.IsUniform G 1 (Prod.fst uv) (Prod.snd uv))).trans ?_
state_before: α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε δ : 𝕜
u v : Finset α
⊢ (P.nonUniforms G 1).card ≤ P.parts.card * (P.parts.card - 1)
state_after: α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε δ : 𝕜
u v : Finset α
⊢ P.parts.offDiag.card ≤ P.parts.card * (P.parts.card - 1)

tactic: rw [offDiag_card, Nat.mul_sub_left_distrib, mul_one]
state_before: α : Type u_1
𝕜 : Type u_2
inst✝² : LinearOrderedField 𝕜
inst✝¹ : DecidableEq α
A : Finset α
P : Finpartition A
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε δ : 𝕜
u v : Finset α
⊢ P.parts.offDiag.card ≤ P.parts.card * (P.parts.card - 1)
state_after: no goals
Theorem: Set.toFinset_image
File: Mathlib/Data/Fintype/Basic.lean

tactic: simp
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
s✝ t : Set α
inst✝² : DecidableEq β
f : α → β
s : Set α
inst✝¹ : Fintype ↑s
inst✝ : Fintype ↑(f '' s)
⊢ ↑(f '' s).toFinset = ↑(Finset.image f s.toFinset)
state_after: no goals
Theorem: Multiset.toDFinsupp_inter
File: Mathlib/Data/DFinsupp/Multiset.lean

tactic: ext i
state_before: α : Type u_1
β : α → Type u_2
inst✝ : DecidableEq α
s✝ t✝ s t : Multiset α
⊢ toDFinsupp (s ∩ t) = toDFinsupp s ⊓ toDFinsupp t
state_after: case h
α : Type u_1
β : α → Type u_2
inst✝ : DecidableEq α
s✝ t✝ s t : Multiset α
i : α
⊢ (toDFinsupp (s ∩ t)) i = (toDFinsupp s ⊓ toDFinsupp t) i

tactic: simp [inf_eq_min]
state_before: case h
α : Type u_1
β : α → Type u_2
inst✝ : DecidableEq α
s✝ t✝ s t : Multiset α
i : α
⊢ (toDFinsupp (s ∩ t)) i = (toDFinsupp s ⊓ toDFinsupp t) i
state_after: no goals
Theorem: CategoryTheory.Functor.pi'_eval
File: Mathlib/CategoryTheory/Pi/Basic.lean

tactic: apply Functor.ext
state_before: I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ pi' f ⋙ Pi.eval C i = f i
state_after: case h_map
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ autoParam (∀ (X Y : A) (f_1 : X ⟶ Y), (pi' f ⋙ Pi.eval C i).map f_1 = eqToHom ⋯ ≫ (f i).map f_1 ≫ eqToHom ⋯) _auto✝

case h_obj
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ ∀ (X : A), (pi' f ⋙ Pi.eval C i).obj X = (f i).obj X

tactic: intro _ _ _
state_before: case h_map
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ autoParam (∀ (X Y : A) (f_1 : X ⟶ Y), (pi' f ⋙ Pi.eval C i).map f_1 = eqToHom ⋯ ≫ (f i).map f_1 ≫ eqToHom ⋯) _auto✝
state_after: case h_map
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
X✝ Y✝ : A
f✝ : X✝ ⟶ Y✝
⊢ (pi' f ⋙ Pi.eval C i).map f✝ = eqToHom ⋯ ≫ (f i).map f✝ ≫ eqToHom ⋯

tactic: simp
state_before: case h_map
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
X✝ Y✝ : A
f✝ : X✝ ⟶ Y✝
⊢ (pi' f ⋙ Pi.eval C i).map f✝ = eqToHom ⋯ ≫ (f i).map f✝ ≫ eqToHom ⋯
state_after: no goals

tactic: intro _
state_before: case h_obj
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
⊢ ∀ (X : A), (pi' f ⋙ Pi.eval C i).obj X = (f i).obj X
state_after: case h_obj
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
X✝ : A
⊢ (pi' f ⋙ Pi.eval C i).obj X✝ = (f i).obj X✝

tactic: rfl
state_before: case h_obj
I : Type w₀
J : Type w₁
C : I → Type u₁
inst✝² : (i : I) → Category.{v₁, u₁} (C i)
D : I → Type u₂
inst✝¹ : (i : I) → Category.{v₂, u₂} (D i)
A : Type u₃
inst✝ : Category.{v₃, u₃} A
f : (i : I) → A ⥤ C i
i : I
X✝ : A
⊢ (pi' f ⋙ Pi.eval C i).obj X✝ = (f i).obj X✝
state_after: no goals
Theorem: ProbabilityTheory.cdf_measure_stieltjesFunction
File: Mathlib/Probability/Cdf.lean

tactic: refine (cdf f.measure).eq_of_measure_of_tendsto_atBot f ?_ (tendsto_cdf_atBot _) hf0
state_before: f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
⊢ cdf f.measure = f
state_after: f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
⊢ (cdf f.measure).measure = f.measure

tactic: have h_prob : IsProbabilityMeasure f.measure :=
  ⟨by rw [f.measure_univ hf0 hf1, sub_zero, ENNReal.ofReal_one]⟩
state_before: f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
⊢ (cdf f.measure).measure = f.measure
state_after: f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
h_prob : IsProbabilityMeasure f.measure
⊢ (cdf f.measure).measure = f.measure

tactic: exact measure_cdf f.measure
state_before: f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
h_prob : IsProbabilityMeasure f.measure
⊢ (cdf f.measure).measure = f.measure
state_after: no goals

tactic: rw [f.measure_univ hf0 hf1, sub_zero, ENNReal.ofReal_one]
state_before: f : StieltjesFunction
hf0 : Tendsto (↑f) atBot (𝓝 0)
hf1 : Tendsto (↑f) atTop (𝓝 1)
⊢ f.measure univ = 1
state_after: no goals
Theorem: div_div
File: Mathlib/Algebra/Group/Basic.lean

tactic: simp
state_before: α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivisionCommMonoid α
a b c d : α
⊢ a / b / c = a / (b * c)
state_after: no goals
Theorem: Seminorm.smul_ball_zero
File: Mathlib/Analysis/Seminorm.lean

tactic: ext
state_before: R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk : k ≠ 0
⊢ k • p.ball 0 r = p.ball 0 (‖k‖ * r)
state_after: case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk : k ≠ 0
x✝ : E
⊢ x✝ ∈ k • p.ball 0 r ↔ x✝ ∈ p.ball 0 (‖k‖ * r)

tactic: rw [mem_smul_set_iff_inv_smul_mem₀ hk, p.mem_ball_zero, p.mem_ball_zero, map_smul_eq_mul,
  norm_inv, ← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hk), mul_comm]
state_before: case h
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p✝ : Seminorm 𝕜 E
A B : Set E
a : 𝕜
r✝ : ℝ
x : E
p : Seminorm 𝕜 E
k : 𝕜
r : ℝ
hk : k ≠ 0
x✝ : E
⊢ x✝ ∈ k • p.ball 0 r ↔ x✝ ∈ p.ball 0 (‖k‖ * r)
state_after: no goals
Theorem: Real.toNNReal_lt_natCast'
File: Mathlib/Data/NNReal/Basic.lean

tactic: simpa [pos_iff_ne_zero] using toNNReal_lt_toNNReal_iff' (r := r) (p := n)
state_before: n : ℕ
r : ℝ
⊢ r.toNNReal < ↑n ↔ r < ↑n ∧ n ≠ 0
state_after: no goals
Theorem: RatFunc.num_zero
File: Mathlib/FieldTheory/RatFunc/Basic.lean

tactic: convert num_div' (0 : K[X]) one_ne_zero <;> simp
state_before: K : Type u
inst✝ : Field K
⊢ num 0 = 0
state_after: no goals
Theorem: Mathlib.Meta.NormNum.isNat_natSucc
File: Mathlib/Tactic/NormNum/Basic.lean

tactic: simp
state_before: n✝ : ℕ
⊢ (↑n✝).succ = ↑n✝.succ
state_after: no goals
Theorem: IsSeparatedMap.eq_of_comp_eq
File: Mathlib/Topology/SeparatedMap.lean

tactic: apply (IsClopen.eq_univ ⟨sep.isClosed_eqLocus h₁ h₂ he, inj.isOpen_eqLocus h₁ h₂ he⟩ ⟨a, ha⟩).symm
  ▸ Set.mem_univ a'
state_before: X : Type u_3
Y : ?m.21392
A✝ : Type ?u.21401
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace A✝
E : Type u_1
A : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace A
p : E → X
sep : IsSeparatedMap p
inj : IsLocallyInjective p
s : Set A
hs : IsPreconnected s
g g₁ g₂ : A → E
inst✝ : PreconnectedSpace A
h₁ : Continuous g₁
h₂ : Continuous g₂
he : p ∘ g₁ = p ∘ g₂
a : A
ha : g₁ a = g₂ a
a' : A
⊢ g₁ a' = g₂ a'
state_after: no goals
Theorem: sub_pow_expChar_of_commute
File: Mathlib/Algebra/CharP/ExpChar.lean

tactic: cases' hR with _ _ hprime _
state_before: R : Type u
inst✝ : Ring R
q : ℕ
hR : ExpChar R q
x y : R
h : Commute x y
⊢ (x - y) ^ q = x ^ q - y ^ q
state_after: case zero
R : Type u
inst✝¹ : Ring R
x y : R
h : Commute x y
inst✝ : CharZero R
⊢ (x - y) ^ 1 = x ^ 1 - y ^ 1

case prime
R : Type u
inst✝ : Ring R
q : ℕ
x y : R
h : Commute x y
hprime : Nat.Prime q
hchar✝ : CharP R q
⊢ (x - y) ^ q = x ^ q - y ^ q

tactic: haveI := Fact.mk hprime
state_before: case prime
R : Type u
inst✝ : Ring R
q : ℕ
x y : R
h : Commute x y
hprime : Nat.Prime q
hchar✝ : CharP R q
⊢ (x - y) ^ q = x ^ q - y ^ q
state_after: case prime
R : Type u
inst✝ : Ring R
q : ℕ
x y : R
h : Commute x y
hprime : Nat.Prime q
hchar✝ : CharP R q
this : Fact (Nat.Prime q)
⊢ (x - y) ^ q = x ^ q - y ^ q

tactic: exact sub_pow_char_of_commute R x y h
state_before: case prime
R : Type u
inst✝ : Ring R
q : ℕ
x y : R
h : Commute x y
hprime : Nat.Prime q
hchar✝ : CharP R q
this : Fact (Nat.Prime q)
⊢ (x - y) ^ q = x ^ q - y ^ q
state_after: no goals

tactic: simp only [pow_one]
state_before: case zero
R : Type u
inst✝¹ : Ring R
x y : R
h : Commute x y
inst✝ : CharZero R
⊢ (x - y) ^ 1 = x ^ 1 - y ^ 1
state_after: no goals
Theorem: Filter.map_le_map_iff
File: Mathlib/Order/Filter/Basic.lean

tactic: rw [map_le_iff_le_comap, comap_map hm]
state_before: α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l f✝ f₁ f₂ : Filter α
g✝ g₁ g₂ : Filter β
m✝ : α → β
m' : β → γ
s : Set α
t : Set β
f g : Filter α
m : α → β
hm : Injective m
⊢ map m f ≤ map m g ↔ f ≤ g
state_after: no goals
Theorem: QuotSMulTop.map_comp_mkQ
File: Mathlib/RingTheory/QuotSMulTop.lean

tactic: ext
state_before: R : Type u_2
inst✝⁶ : CommRing R
r : R
M : Type u_1
M' : Type u_3
M'' : Type ?u.46471
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : AddCommGroup M''
inst✝ : Module R M''
f : M →ₗ[R] M'
⊢ (map r) f ∘ₗ (r • ⊤).mkQ = (r • ⊤).mkQ ∘ₗ f
state_after: case h
R : Type u_2
inst✝⁶ : CommRing R
r : R
M : Type u_1
M' : Type u_3
M'' : Type ?u.46471
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : AddCommGroup M''
inst✝ : Module R M''
f : M →ₗ[R] M'
x✝ : M
⊢ ((map r) f ∘ₗ (r • ⊤).mkQ) x✝ = ((r • ⊤).mkQ ∘ₗ f) x✝

tactic: rfl
state_before: case h
R : Type u_2
inst✝⁶ : CommRing R
r : R
M : Type u_1
M' : Type u_3
M'' : Type ?u.46471
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : AddCommGroup M'
inst✝² : Module R M'
inst✝¹ : AddCommGroup M''
inst✝ : Module R M''
f : M →ₗ[R] M'
x✝ : M
⊢ ((map r) f ∘ₗ (r • ⊤).mkQ) x✝ = ((r • ⊤).mkQ ∘ₗ f) x✝
state_after: no goals
Theorem: PMF.toMeasure_bindOnSupport_apply
File: Mathlib/Probability/ProbabilityMassFunction/Monad.lean

tactic: simp only [toMeasure_apply_eq_toOuterMeasure_apply _ _ hs, toOuterMeasure_bindOnSupport_apply]
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
p : PMF α
f : (a : α) → a ∈ p.support → PMF β
s : Set β
inst✝ : MeasurableSpace β
hs : MeasurableSet s
⊢ (p.bindOnSupport f).toMeasure s = ∑' (a : α), p a * if h : p a = 0 then 0 else (f a h).toMeasure s
state_after: no goals
Theorem: PiTensorProduct.toDualContinuousMultilinearMap_le_projectiveSeminorm
File: Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean

tactic: simp only [toDualContinuousMultilinearMap, LinearMap.coe_mk, AddHom.coe_mk]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : ⨂[𝕜] (i : ι), E i
⊢ ‖(toDualContinuousMultilinearMap F) x‖ ≤ projectiveSeminorm x
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : ⨂[𝕜] (i : ι), E i
⊢ ‖((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯‖ ≤
    projectiveSeminorm x

tactic: apply LinearMap.mkContinuous_norm_le _ (apply_nonneg _ _)
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : ⨂[𝕜] (i : ι), E i
⊢ ‖((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯‖ ≤
    projectiveSeminorm x
state_after: no goals
Theorem: Cardinal.lsub_lt_ord_lift_of_isRegular
File: Mathlib/SetTheory/Cardinal/Cofinality.lean

tactic: rwa [hc.cof_eq]
state_before: α : Type u_1
r : α → α → Prop
ι : Type u
f : ι → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
⊢ lift.{v, u} #ι < c.ord.cof
state_after: no goals