ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
Theorem: isOpen_iSup_iff
File: Mathlib/Topology/Order.lean

Proof steps:
state_before: α : Type u
ι : Sort v
t : ι → TopologicalSpace α
s : Set α
⊢ s ∈ {s | IsOpen s} ↔ s ∈ {x | ∀ (i : ι), IsOpen x}
state_after: no goals
tactic: simp [setOf_isOpen_iSup]
Theorem: Submodule.fst_map_fst
File: Mathlib/LinearAlgebra/Prod.lean

Proof steps:
state_before: R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p : Submodule R M
q : Submodule R M₂
⊢ map (LinearMap.fst R M M₂) (fst R M M₂) = ⊤
state_after: R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p : Submodule R M
q : Submodule R M₂
⊢ ⊤ ≤ map (LinearMap.fst R M M₂) (fst R M M₂)
tactic: rw [eq_top_iff]
state_before: R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p : Submodule R M
q : Submodule R M₂
⊢ ⊤ ≤ map (LinearMap.fst R M M₂) (fst R M M₂)
state_after: R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p : Submodule R M
q : Submodule R M₂
x : M
⊢ x ∈ map (LinearMap.fst R M M₂) (fst R M M₂)
tactic: rintro x -
state_before: R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p : Submodule R M
q : Submodule R M₂
x : M
⊢ x ∈ map (LinearMap.fst R M M₂) (fst R M M₂)
state_after: no goals
tactic: simp only [fst, comap_bot, mem_map, mem_ker, snd_apply, fst_apply,
    Prod.exists, exists_eq_left, exists_eq]
Theorem: algebraMap_apply
File: Mathlib/Topology/ContinuousFunction/Algebra.lean

Proof steps:
state_before: α : Type u_1
inst✝⁹ : TopologicalSpace α
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : TopologicalSemiring A
A₂ : Type u_4
inst✝³ : TopologicalSpace A₂
inst✝² : Semiring A₂
inst✝¹ : Algebra R A₂
inst✝ : TopologicalSemiring A₂
k : R
a : α
⊢ ((algebraMap R C(α, A)) k) a = k • 1
state_after: α : Type u_1
inst✝⁹ : TopologicalSpace α
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : TopologicalSemiring A
A₂ : Type u_4
inst✝³ : TopologicalSpace A₂
inst✝² : Semiring A₂
inst✝¹ : Algebra R A₂
inst✝ : TopologicalSemiring A₂
k : R
a : α
⊢ (k • 1) a = k • 1
tactic: rw [Algebra.algebraMap_eq_smul_one]
state_before: α : Type u_1
inst✝⁹ : TopologicalSpace α
R : Type u_2
inst✝⁸ : CommSemiring R
A : Type u_3
inst✝⁷ : TopologicalSpace A
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : TopologicalSemiring A
A₂ : Type u_4
inst✝³ : TopologicalSpace A₂
inst✝² : Semiring A₂
inst✝¹ : Algebra R A₂
inst✝ : TopologicalSemiring A₂
k : R
a : α
⊢ (k • 1) a = k • 1
state_after: no goals
tactic: rfl
Theorem: one_lt_finprod'
File: Mathlib/Algebra/BigOperators/Finprod.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
h' : ∃ i, 1 < f i
hf : (mulSupport f).Finite
⊢ 1 < ∏ᶠ (i : ι), f i
state_after: case intro
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
hf : (mulSupport f).Finite
i : ι
hi : 1 < f i
⊢ 1 < ∏ᶠ (i : ι), f i
tactic: rcases h' with ⟨i, hi⟩
state_before: case intro
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
hf : (mulSupport f).Finite
i : ι
hi : 1 < f i
⊢ 1 < ∏ᶠ (i : ι), f i
state_after: case intro
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
hf : (mulSupport f).Finite
i : ι
hi : 1 < f i
⊢ 1 < ∏ i ∈ hf.toFinset, f i
tactic: rw [finprod_eq_prod _ hf]
state_before: case intro
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
hf : (mulSupport f).Finite
i : ι
hi : 1 < f i
⊢ 1 < ∏ i ∈ hf.toFinset, f i
state_after: case intro
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
hf : (mulSupport f).Finite
i : ι
hi : 1 < f i
⊢ i ∈ hf.toFinset
tactic: refine Finset.one_lt_prod' (fun i _ ↦ h i) ⟨i, ?_, hi⟩
state_before: case intro
α : Type u_1
β : Type u_2
ι : Type u_3
G : Type u_4
M✝ : Type u_5
N : Type u_6
inst✝² : CommMonoid M✝
inst✝¹ : CommMonoid N
f✝ g : α → M✝
a b : α
s t : Set α
M : Type u_7
inst✝ : OrderedCancelCommMonoid M
f : ι → M
h : ∀ (i : ι), 1 ≤ f i
hf : (mulSupport f).Finite
i : ι
hi : 1 < f i
⊢ i ∈ hf.toFinset
state_after: no goals
tactic: simpa only [Finite.mem_toFinset, mem_mulSupport] using ne_of_gt hi
Theorem: IsGroupHom.inv_ker_one
File: Mathlib/Deprecated/Subgroup.lean

Proof steps:
state_before: G : Type u_1
H : Type u_2
A : Type u_3
a✝ a₁ a₂ b✝ c : G
inst✝¹ : Group G
inst✝ : Group H
f : G → H
hf : IsGroupHom f
a b : G
h : f a = f b
⊢ f (a * b⁻¹) = 1
state_after: G : Type u_1
H : Type u_2
A : Type u_3
a✝ a₁ a₂ b✝ c : G
inst✝¹ : Group G
inst✝ : Group H
f : G → H
hf : IsGroupHom f
a b : G
h : f a = f b
this : f a * (f b)⁻¹ = 1
⊢ f (a * b⁻¹) = 1
tactic: have : f a * (f b)⁻¹ = 1 := by rw [h, mul_right_inv]
state_before: G : Type u_1
H : Type u_2
A : Type u_3
a✝ a₁ a₂ b✝ c : G
inst✝¹ : Group G
inst✝ : Group H
f : G → H
hf : IsGroupHom f
a b : G
h : f a = f b
this : f a * (f b)⁻¹ = 1
⊢ f (a * b⁻¹) = 1
state_after: no goals
tactic: rwa [← hf.map_inv, ← hf.map_mul] at this
Theorem: Prefunctor.IsCovering.of_comp_left
File: Mathlib/Combinatorics/Quiver/Covering.lean

Proof steps:
state_before: U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type u_3
inst✝ : Quiver W
ψ : V ⥤q W
hφ : φ.IsCovering
hφψ : (φ ⋙q ψ).IsCovering
φsur : Surjective φ.obj
⊢ ψ.IsCovering
state_after: case refine_1.intro
U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type u_3
inst✝ : Quiver W
ψ : V ⥤q W
hφ : φ.IsCovering
hφψ : (φ ⋙q ψ).IsCovering
φsur : Surjective φ.obj
u : U
⊢ Bijective (ψ.star (φ.obj u))

case refine_2.intro
U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type u_3
inst✝ : Quiver W
ψ : V ⥤q W
hφ : φ.IsCovering
hφψ : (φ ⋙q ψ).IsCovering
φsur : Surjective φ.obj
u : U
⊢ Bijective (ψ.costar (φ.obj u))
tactic: refine ⟨fun v => ?_, fun v => ?_⟩ <;> obtain ⟨u, rfl⟩ := φsur v
state_before: case refine_1.intro
U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type u_3
inst✝ : Quiver W
ψ : V ⥤q W
hφ : φ.IsCovering
hφψ : (φ ⋙q ψ).IsCovering
φsur : Surjective φ.obj
u : U
⊢ Bijective (ψ.star (φ.obj u))

case refine_2.intro
U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type u_3
inst✝ : Quiver W
ψ : V ⥤q W
hφ : φ.IsCovering
hφψ : (φ ⋙q ψ).IsCovering
φsur : Surjective φ.obj
u : U
⊢ Bijective (ψ.costar (φ.obj u))
state_after: no goals
tactic: exacts [(Bijective.of_comp_iff _ (hφ.star_bijective u)).mp (hφψ.star_bijective u),
    (Bijective.of_comp_iff _ (hφ.costar_bijective u)).mp (hφψ.costar_bijective u)]
Theorem: MeasureTheory.measure_union_add_inter'
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
hs : MeasurableSet s
t : Set α
⊢ μ (s ∪ t) + μ (s ∩ t) = μ s + μ t
state_after: no goals
tactic: rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]
Theorem: IsCyclotomicExtension.splits_X_pow_sub_one
File: Mathlib/NumberTheory/Cyclotomic/Basic.lean

Proof steps:
state_before: n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ Splits (algebraMap K L) (X ^ ↑n - 1)
state_after: n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ Splits (RingHom.id L) (X ^ ↑n - 1)
tactic: rw [← splits_id_iff_splits, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
    Polynomial.map_X]
state_before: n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
⊢ Splits (RingHom.id L) (X ^ ↑n - 1)
state_after: case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
z : L
hz : IsPrimitiveRoot z ↑n
⊢ Splits (RingHom.id L) (X ^ ↑n - 1)
tactic: obtain ⟨z, hz⟩ := ((isCyclotomicExtension_iff _ _ _).1 H).1 hS
state_before: case intro
n : ℕ+
S T : Set ℕ+
A : Type u
B : Type v
K : Type w
L : Type z
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra A B
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
H : IsCyclotomicExtension S K L
hS : n ∈ S
z : L
hz : IsPrimitiveRoot z ↑n
⊢ Splits (RingHom.id L) (X ^ ↑n - 1)
state_after: no goals
tactic: exact X_pow_sub_one_splits hz
Theorem: natDegree_minpolyDiv_succ
File: Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean

Proof steps:
state_before: R : Type u_2
K : Type ?u.58132
L : Type ?u.58135
S : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra K L
x : S
hx : IsIntegral R x
inst✝ : Nontrivial S
⊢ (minpolyDiv R x).natDegree + 1 = (minpoly R x).natDegree
state_after: R : Type u_2
K : Type ?u.58132
L : Type ?u.58135
S : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra K L
x : S
hx : IsIntegral R x
inst✝ : Nontrivial S
⊢ (minpolyDiv R x).natDegree + 1 = (minpolyDiv R x).natDegree + (X - C x).natDegree

R : Type u_2
K : Type ?u.58132
L : Type ?u.58135
S : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Algebra K L
x : S
hx : IsIntegral R x
inst✝ : Nontrivial S
⊢ (minpolyDiv R x).leadingCoeff * (X - C x).leadingCoeff ≠ 0
tactic: rw [← (minpoly.monic hx).natDegree_map (algebraMap R S), ← minpolyDiv_spec, natDegree_mul']
Theorem: FirstOrder.Language.Term.relabelEquiv_apply
File: Mathlib/ModelTheory/Syntax.lean

Proof steps:
state_before: L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
g : α ≃ β
t : L.Term α
⊢ relabel (⇑g.symm) (relabel (⇑g) t) = t
state_after: no goals
tactic: simp
state_before: L : Language
L' : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : L.Structure M
inst✝¹ : L.Structure N
inst✝ : L.Structure P
α : Type u'
β : Type v'
γ : Type u_3
g : α ≃ β
t : L.Term β
⊢ relabel (⇑g) (relabel (⇑g.symm) t) = t
state_after: no goals
tactic: simp
Theorem: Finset.sum_piecewise
File: Mathlib/Algebra/BigOperators/Group/Finset.lean

Proof steps:
state_before: ι : Type u_1
κ : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
s✝ s₁ s₂ : Finset α
a : α
f✝ g✝ : α → β
inst✝¹ : CommMonoid β
inst✝ : DecidableEq α
s t : Finset α
f g : α → β
⊢ ∏ x ∈ s, t.piecewise f g x = (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \ t, g x
state_after: no goals
tactic: erw [prod_ite, filter_mem_eq_inter, ← sdiff_eq_filter]
Theorem: Polynomial.separable_X_add_C
File: Mathlib/FieldTheory/Separable.lean

Proof steps:
state_before: R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
a : R
⊢ (X + C a).Separable
state_after: R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
a : R
⊢ IsCoprime (X + C a) 1
tactic: rw [separable_def, derivative_add, derivative_X, derivative_C, add_zero]
state_before: R : Type u
inst✝¹ : CommSemiring R
S : Type v
inst✝ : CommSemiring S
a : R
⊢ IsCoprime (X + C a) 1
state_after: no goals
tactic: exact isCoprime_one_right
Theorem: Set.Icc_diff_Ico_same
File: Mathlib/Order/Interval/Set/Basic.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝ : PartialOrder α
a b c : α
h : a ≤ b
⊢ Icc a b \ Ico a b = {b}
state_after: no goals
tactic: rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <| right_mem_Icc.2 h)]
Theorem: MeasureTheory.Measure.sum_apply_eq_zero'
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
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
μ : ι → Measure α
s : Set α
hs : MeasurableSet s
⊢ (sum μ) s = 0 ↔ ∀ (i : ι), (μ i) s = 0
state_after: no goals
tactic: simp [hs]
Theorem: ULift.isOpen_iff
File: Mathlib/Topology/Constructions.lean

Proof steps:
state_before: X : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
inst✝ : TopologicalSpace X
s : Set (ULift.{v, u} X)
⊢ IsOpen s ↔ IsOpen (up ⁻¹' s)
state_after: no goals
tactic: rw [ULift.topologicalSpace, ← Equiv.ulift_apply, ← Equiv.ulift.coinduced_symm, ← isOpen_coinduced]
Theorem: Finset.weightedVSubOfPoint_apply
File: Mathlib/LinearAlgebra/AffineSpace/Combination.lean

Proof steps:
state_before: k : Type u_1
V : Type u_2
P : Type u_3
inst✝² : Ring k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
S : AffineSpace V P
ι : Type u_4
s : Finset ι
ι₂ : Type u_5
s₂ : Finset ι₂
w : ι → k
p : ι → P
b : P
⊢ (s.weightedVSubOfPoint p b) w = ∑ i ∈ s, w i • (p i -ᵥ b)
state_after: no goals
tactic: simp [weightedVSubOfPoint, LinearMap.sum_apply]
Theorem: Ideal.ramificationIdx_lt
File: Mathlib/NumberTheory/RamificationInertia.lean

Proof steps:
state_before: R : Type u
inst✝¹ : CommRing R
S : Type v
inst✝ : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
n : ℕ
hgt : ¬map f p ≤ P ^ n
⊢ ramificationIdx f p P < n
state_after: case zero
R : Type u
inst✝¹ : CommRing R
S : Type v
inst✝ : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
hgt : ¬map f p ≤ P ^ 0
⊢ ramificationIdx f p P < 0

case succ
R : Type u
inst✝¹ : CommRing R
S : Type v
inst✝ : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
n : ℕ
hgt : ¬map f p ≤ P ^ (n + 1)
⊢ ramificationIdx f p P < n + 1
tactic: cases' n with n n
Theorem: EuclideanGeometry.Sphere.secondInter_zero
File: Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean

Proof steps:
state_before: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
⊢ s.secondInter p 0 = p
state_after: no goals
tactic: simp [Sphere.secondInter]
Theorem: CategoryTheory.extendFan_π_app
File: Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean

Proof steps:
state_before: J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan fun i => f i.succ
c₂ : BinaryFan (f 0) c₁.pt
⊢ (b : Fin (n + 1)) → c₂.pt ⟶ f b
state_after: case refine_1
J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan fun i => f i.succ
c₂ : BinaryFan (f 0) c₁.pt
⊢ c₂.pt ⟶ f 0

case refine_2
J : Type v
inst✝² : SmallCategory J
C : Type u
inst✝¹ : Category.{v, u} C
D : Type u'
inst✝ : Category.{v', u'} D
n : ℕ
f : Fin (n + 1) → C
c₁ : Fan fun i => f i.succ
c₂ : BinaryFan (f 0) c₁.pt
⊢ (i : Fin n) → c₂.pt ⟶ f i.succ
tactic: refine Fin.cases ?_ ?_
Theorem: WeierstrassCurve.Jacobian.nonsingular_of_equiv
File: Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean

Proof steps:
state_before: R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → R
h : P ≈ Q
⊢ W'.Nonsingular P ↔ W'.Nonsingular Q
state_after: case intro
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
Q : Fin 3 → R
u : Rˣ
⊢ W'.Nonsingular ((fun m => m • Q) u) ↔ W'.Nonsingular Q
tactic: rcases h with ⟨u, rfl⟩
state_before: case intro
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
Q : Fin 3 → R
u : Rˣ
⊢ W'.Nonsingular ((fun m => m • Q) u) ↔ W'.Nonsingular Q
state_after: no goals
tactic: exact nonsingular_smul Q u.isUnit
Theorem: FormalMultilinearSeries.radius_eq_liminf
File: Mathlib/Analysis/Analytic/RadiusLiminf.lean

Proof steps:
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
⊢ p.radius = liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop
state_after: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
⊢ p.radius = liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop
tactic: have :
    ∀ (r : ℝ≥0) {n : ℕ},
      0 < n → ((r : ℝ≥0∞) ≤ 1 / ↑(‖p n‖₊ ^ (1 / (n : ℝ))) ↔ ‖p n‖₊ * r ^ n ≤ 1) := by
    intro r n hn
    have : 0 < (n : ℝ) := Nat.cast_pos.2 hn
    conv_lhs =>
      rw [one_div, ENNReal.le_inv_iff_mul_le, ← ENNReal.coe_mul, ENNReal.coe_le_one_iff, one_div, ←
        NNReal.rpow_one r, ← mul_inv_cancel this.ne', NNReal.rpow_mul, ← NNReal.mul_rpow, ←
        NNReal.one_rpow n⁻¹, NNReal.rpow_le_rpow_iff (inv_pos.2 this), mul_comm,
        NNReal.rpow_natCast]
state_before: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
⊢ p.radius = liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop
state_after: case a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
r : ℝ≥0
hr : ↑r < p.radius
⊢ ↑r ≤ liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop

case a
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)
r : ℝ≥0
hr : ↑r < liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop
⊢ ↑r ≤ p.radius
tactic: apply le_antisymm <;> refine ENNReal.le_of_forall_nnreal_lt fun r hr => ?_
Theorem: Filter.Tendsto.subseq_mem_entourage
File: Mathlib/Topology/UniformSpace/Cauchy.lean

Proof steps:
state_before: α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
a : α
hu : Tendsto u atTop (𝓝 a)
⊢ ∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)
state_after: case intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
a : α
hu : Tendsto u atTop (𝓝 a)
n : ℕ
hn : ∀ b ≥ n, b ∈ u ⁻¹' ball a (Prod.swap ⁻¹' V 0)
⊢ ∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)
tactic: rcases mem_atTop_sets.1 (hu (ball_mem_nhds a (symm_le_uniformity <| hV 0))) with ⟨n, hn⟩
state_before: case intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
a : α
hu : Tendsto u atTop (𝓝 a)
n : ℕ
hn : ∀ b ≥ n, b ∈ u ⁻¹' ball a (Prod.swap ⁻¹' V 0)
⊢ ∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)
state_after: case intro.intro.intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
a : α
hu : Tendsto u atTop (𝓝 a)
n : ℕ
hn : ∀ b ≥ n, b ∈ u ⁻¹' ball a (Prod.swap ⁻¹' V 0)
φ : ℕ → ℕ
φ_mono : StrictMono φ
hφV : ∀ (n_1 : ℕ), ((u ∘ fun a => a + n) (φ (n_1 + 1)), (u ∘ fun a => a + n) (φ n_1)) ∈ V (n_1 + 1)
⊢ ∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)
tactic: rcases (hu.comp (tendsto_add_atTop_nat n)).cauchySeq.subseq_mem fun n => hV (n + 1) with
    ⟨φ, φ_mono, hφV⟩
state_before: case intro.intro.intro
α : Type u
β : Type v
uniformSpace : UniformSpace α
V : ℕ → Set (α × α)
hV : ∀ (n : ℕ), V n ∈ 𝓤 α
u : ℕ → α
a : α
hu : Tendsto u atTop (𝓝 a)
n : ℕ
hn : ∀ b ≥ n, b ∈ u ⁻¹' ball a (Prod.swap ⁻¹' V 0)
φ : ℕ → ℕ
φ_mono : StrictMono φ
hφV : ∀ (n_1 : ℕ), ((u ∘ fun a => a + n) (φ (n_1 + 1)), (u ∘ fun a => a + n) (φ n_1)) ∈ V (n_1 + 1)
⊢ ∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)
state_after: no goals
tactic: exact ⟨fun k => φ k + n, φ_mono.add_const _, hn _ le_add_self, hφV⟩