tasksource/leandojo huggingface dataset:
Theorem: Complex.int_cast_im
File: Mathlib/Data/Complex/Basic.lean

tactic: rw [← ofReal_int_cast, ofReal_im]
state_before: n : ℤ
⊢ (↑n).im = 0
Theorem: Std.RBNode.Path.zoom_fill'
File: Std/Data/RBMap/Alter.lean

tactic: induction t generalizing path with
| nil => rfl
| node _ _ _ _ iha ihb => unfold zoom; split <;> [apply iha; apply ihb; rfl]
state_before: α : Type u_1
cut : α → Ordering
t : RBNode α
path : Path α
⊢ fill' (zoom cut t path) = fill path t

tactic: rfl
state_before: case nil
α : Type u_1
cut : α → Ordering
path : Path α
⊢ fill' (zoom cut nil path) = fill path nil

tactic: unfold zoom
state_before: case node
α : Type u_1
cut : α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
iha : ∀ (path : Path α), fill' (zoom cut l✝ path) = fill path l✝
ihb : ∀ (path : Path α), fill' (zoom cut r✝ path) = fill path r✝
path : Path α
⊢ fill' (zoom cut (node c✝ l✝ v✝ r✝) path) = fill path (node c✝ l✝ v✝ r✝)

tactic: split <;> [apply iha; apply ihb; rfl]
state_before: case node
α : Type u_1
cut : α → Ordering
c✝ : RBColor
l✝ : RBNode α
v✝ : α
r✝ : RBNode α
iha : ∀ (path : Path α), fill' (zoom cut l✝ path) = fill path l✝
ihb : ∀ (path : Path α), fill' (zoom cut r✝ path) = fill path r✝
path : Path α
⊢ fill'
      (match cut v✝ with
      | Ordering.lt => zoom cut l✝ (left c✝ path v✝ r✝)
      | Ordering.gt => zoom cut r✝ (right c✝ l✝ v✝ path)
      | Ordering.eq => (node c✝ l✝ v✝ r✝, path)) =
    fill path (node c✝ l✝ v✝ r✝)
Theorem: nhds_nil
File: Mathlib/Topology/List.lean

tactic: rw [nhds_list, List.traverse_nil _]
state_before: α : Type u_1
β : Type ?u.3791
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
⊢ 𝓝 [] = pure []
Theorem: connectedSpace_iff_connectedComponent
File: Mathlib/Topology/Connected.lean

tactic: constructor
state_before: α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
⊢ ConnectedSpace α ↔ ∃ x, connectedComponent x = univ

tactic: rintro ⟨⟨x⟩⟩
state_before: case mp
α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
⊢ ConnectedSpace α → ∃ x, connectedComponent x = univ

tactic: exact
  ⟨x, eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)⟩
state_before: case mp.mk.intro
α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
toPreconnectedSpace✝ : PreconnectedSpace α
x : α
⊢ ∃ x, connectedComponent x = univ

tactic: rintro ⟨x, h⟩
state_before: case mpr
α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
⊢ (∃ x, connectedComponent x = univ) → ConnectedSpace α

tactic: haveI : PreconnectedSpace α :=
  ⟨by rw [← h]; exact isPreconnected_connectedComponent⟩
state_before: case mpr.intro
α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
h : connectedComponent x = univ
⊢ ConnectedSpace α

tactic: exact ⟨⟨x⟩⟩
state_before: case mpr.intro
α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
h : connectedComponent x = univ
this : PreconnectedSpace α
⊢ ConnectedSpace α

tactic: rw [← h]
state_before: α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
h : connectedComponent x = univ
⊢ IsPreconnected univ

tactic: exact isPreconnected_connectedComponent
state_before: α : Type u
β : Type v
ι : Type ?u.63292
π : ι → Type ?u.63297
inst✝ : TopologicalSpace α
s t u v : Set α
x : α
h : connectedComponent x = univ
⊢ IsPreconnected (connectedComponent x)
Theorem: PGame.zero_fuzzy_neg_iff
File: Mathlib/SetTheory/Game/PGame.lean

tactic: rw [← neg_fuzzy_iff, neg_zero]
state_before: x : PGame
⊢ 0 ‖ -x ↔ 0 ‖ x
Theorem: Set.Finite.toFinset_union
File: Mathlib/Data/Set/Finite.lean

tactic: ext
state_before: α : Type u
β : Type v
ι : Sort w
γ : Type x
s t : Set α
a : α
hs✝ : Set.Finite s
ht✝ : Set.Finite t
inst✝ : DecidableEq α
hs : Set.Finite s
ht : Set.Finite t
h : Set.Finite (s ∪ t)
⊢ Finite.toFinset h = Finite.toFinset hs ∪ Finite.toFinset ht

tactic: simp
state_before: case a
α : Type u
β : Type v
ι : Sort w
γ : Type x
s t : Set α
a : α
hs✝ : Set.Finite s
ht✝ : Set.Finite t
inst✝ : DecidableEq α
hs : Set.Finite s
ht : Set.Finite t
h : Set.Finite (s ∪ t)
a✝ : α
⊢ a✝ ∈ Finite.toFinset h ↔ a✝ ∈ Finite.toFinset hs ∪ Finite.toFinset ht
Theorem: div_lt_one_iff
File: Mathlib/Algebra/Order/Field/Basic.lean

tactic: rcases lt_trichotomy b 0 with (hb | rfl | hb)
state_before: ι : Type ?u.170288
α : Type u_1
β : Type ?u.170294
inst✝ : LinearOrderedField α
a b c d : α
n : ℤ
⊢ a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a

tactic: simp [hb, hb.not_lt, hb.ne, div_lt_one_of_neg]
state_before: case inl
ι : Type ?u.170288
α : Type u_1
β : Type ?u.170294
inst✝ : LinearOrderedField α
a b c d : α
n : ℤ
hb : b < 0
⊢ a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a

tactic: simp [zero_lt_one]
state_before: case inr.inl
ι : Type ?u.170288
α : Type u_1
β : Type ?u.170294
inst✝ : LinearOrderedField α
a c d : α
n : ℤ
⊢ a / 0 < 1 ↔ 0 < 0 ∧ a < 0 ∨ 0 = 0 ∨ 0 < 0 ∧ 0 < a

tactic: simp [hb, hb.not_lt, div_lt_one, hb.ne.symm]
state_before: case inr.inr
ι : Type ?u.170288
α : Type u_1
β : Type ?u.170294
inst✝ : LinearOrderedField α
a b c d : α
n : ℤ
hb : 0 < b
⊢ a / b < 1 ↔ 0 < b ∧ a < b ∨ b = 0 ∨ b < 0 ∧ b < a
Theorem: Commute.add_pow_prime_pow_eq
File: Mathlib/Algebra/CharP/Basic.lean

tactic: trans x ^ p ^ n + y ^ p ^ n + ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * (p ^ n).choose k
state_before: R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n : ℕ
⊢ (x + y) ^ p ^ n =
    x ^ p ^ n + y ^ p ^ n + ↑p * ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k / p)

tactic: simp_rw [h.add_pow, ← Nat.Ico_zero_eq_range, Nat.Ico_succ_right, Icc_eq_cons_Ico (zero_le _),
  Finset.sum_cons, Ico_eq_cons_Ioo (pow_pos hp.pos _), Finset.sum_cons, tsub_self, tsub_zero,
  pow_zero, Nat.choose_zero_right, Nat.choose_self, Nat.cast_one, mul_one, one_mul, ← add_assoc]
state_before: R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n : ℕ
⊢ (x + y) ^ p ^ n = x ^ p ^ n + y ^ p ^ n + ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k)

tactic: congr 1
state_before: R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n : ℕ
⊢ x ^ p ^ n + y ^ p ^ n + ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k) =
    x ^ p ^ n + y ^ p ^ n + ↑p * ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k / p)

tactic: simp_rw [Finset.mul_sum, Nat.cast_comm, mul_assoc _ _ (p : R), ← Nat.cast_mul]
state_before: case e_a
R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n : ℕ
⊢ ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k) =
    ↑p * ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k / p)

tactic: refine' Finset.sum_congr rfl fun i hi => _
state_before: case e_a
R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n : ℕ
⊢ ∑ k in Ioo 0 (p ^ n), x ^ k * y ^ (p ^ n - k) * ↑(Nat.choose (p ^ n) k) =
    ∑ x_1 in Ioo 0 (p ^ n), x ^ x_1 * y ^ (p ^ n - x_1) * ↑(Nat.choose (p ^ n) x_1 / p * p)

tactic: rw [mem_Ioo] at hi
state_before: case e_a
R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n i : ℕ
hi : i ∈ Ioo 0 (p ^ n)
⊢ x ^ i * y ^ (p ^ n - i) * ↑(Nat.choose (p ^ n) i) = x ^ i * y ^ (p ^ n - i) * ↑(Nat.choose (p ^ n) i / p * p)

tactic: rw [Nat.div_mul_cancel (hp.dvd_choose_pow hi.1.ne' hi.2.ne)]
state_before: case e_a
R : Type u_1
inst✝ : Semiring R
p : ℕ
x y : R
hp : Nat.Prime p
h : Commute x y
n i : ℕ
hi : 0 < i ∧ i < p ^ n
⊢ x ^ i * y ^ (p ^ n - i) * ↑(Nat.choose (p ^ n) i) = x ^ i * y ^ (p ^ n - i) * ↑(Nat.choose (p ^ n) i / p * p)
Theorem: Order.Ideal.isProper_of_not_mem
File: Mathlib/Order/Ideal.lean

tactic: have := mem_univ p
state_before: P : Type u_1
inst✝ : LE P
I✝ J s t : Ideal P
x y : P
I : Ideal P
p : P
nmem : ¬p ∈ I
hp : ↑I = univ
⊢ False

tactic: rw [← hp] at this
state_before: P : Type u_1
inst✝ : LE P
I✝ J s t : Ideal P
x y : P
I : Ideal P
p : P
nmem : ¬p ∈ I
hp : ↑I = univ
this : p ∈ univ
⊢ False

tactic: exact nmem this
state_before: P : Type u_1
inst✝ : LE P
I✝ J s t : Ideal P
x y : P
I : Ideal P
p : P
nmem : ¬p ∈ I
hp : ↑I = univ
this : p ∈ ↑I
⊢ False
Theorem: orthonormal_empty
File: Mathlib/Analysis/InnerProductSpace/Basic.lean

tactic: classical!
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type ?u.2153778
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type ?u.2153827
dec_ι : DecidableEq ι
⊢ Orthonormal 𝕜 fun x => ↑x

tactic: simp [orthonormal_subtype_iff_ite]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type ?u.2153778
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
dec_E : DecidableEq E
ι : Type ?u.2153827
dec_ι : DecidableEq ι
em✝ : (a : Prop) → Decidable a
⊢ Orthonormal 𝕜 fun x => ↑x
Theorem: Filter.image_coe_mem_of_mem_comap
File: Mathlib/Order/Filter/Basic.lean

tactic: simp [h]
state_before: α : Type u
β : Type v
γ : Type w
δ : Type ?u.259357
ι : Sort x
f✝ f₁ f₂ : Filter α
g g₁ g₂ : Filter β
m : α → β
m' : β → γ
s : Set α
t : Set β
f : Filter α
U : Set α
h : U ∈ f
W : Set ↑U
W_in : W ∈ comap Subtype.val f
⊢ range Subtype.val ∈ f
Theorem: Metric.uniformity_edist_aux
File: Mathlib/Topology/MetricSpace/Basic.lean

tactic: simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
state_before: α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
⊢ (⨅ (ε : ℝ) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}) = ⨅ (ε : ℝ≥0∞) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}

tactic: refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩
state_before: α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
⊢ (∀ (i : ℝ≥0∞), i > 0 → {p | ↑(d p.fst p.snd) < i} ∈ ⨅ (ε : ℝ) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}) ∧
    ∀ (i : ℝ), i > 0 → {p | ↑(d p.fst p.snd) < i} ∈ ⨅ (ε : ℝ≥0∞) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}

tactic: rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hε with ⟨ε', ε'0, ε'ε⟩
state_before: case refine_1
α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
ε : ℝ≥0∞
hε : ε > 0
⊢ {p | ↑(d p.fst p.snd) < ε} ∈ ⨅ (ε : ℝ) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}

tactic: refine mem_iInf_of_mem (ε' : ℝ) (mem_iInf_of_mem (ENNReal.coe_pos.1 ε'0) ?_)
state_before: case refine_1.intro.intro
α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
ε : ℝ≥0∞
hε : ε > 0
ε' : ℝ≥0
ε'0 : 0 < ↑ε'
ε'ε : ↑ε' < ε
⊢ {p | ↑(d p.fst p.snd) < ε} ∈ ⨅ (ε : ℝ) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}

tactic: exact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) ε'ε
state_before: case refine_1.intro.intro
α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
ε : ℝ≥0∞
hε : ε > 0
ε' : ℝ≥0
ε'0 : 0 < ↑ε'
ε'ε : ↑ε' < ε
⊢ {p | ↑(d p.fst p.snd) < ε} ∈ 𝓟 {p | ↑(d p.fst p.snd) < ↑ε'}

tactic: lift ε to ℝ≥0 using le_of_lt hε
state_before: case refine_2
α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
ε : ℝ
hε : ε > 0
⊢ {p | ↑(d p.fst p.snd) < ε} ∈ ⨅ (ε : ℝ≥0∞) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}

tactic: refine mem_iInf_of_mem (ε : ℝ≥0∞) (mem_iInf_of_mem (ENNReal.coe_pos.2 hε) ?_)
state_before: case refine_2.intro
α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
ε : ℝ≥0
hε : ↑ε > 0
⊢ {p | ↑(d p.fst p.snd) < ↑ε} ∈ ⨅ (ε : ℝ≥0∞) (_ : ε > 0), 𝓟 {p | ↑(d p.fst p.snd) < ε}

tactic: exact fun _ => ENNReal.coe_lt_coe.1
state_before: case refine_2.intro
α✝ : Type u
β : Type v
X : Type ?u.156823
ι : Type ?u.156826
inst✝ : PseudoMetricSpace α✝
α : Type u_1
d : α → α → ℝ≥0
ε : ℝ≥0
hε : ↑ε > 0
⊢ {p | ↑(d p.fst p.snd) < ↑ε} ∈ 𝓟 {p | ↑(d p.fst p.snd) < ↑ε}
Theorem: isFiniteMeasure_iff_isFiniteMeasureOnCompacts_of_compactSpace
File: Mathlib/MeasureTheory/Measure/MeasureSpace.lean

tactic: constructor <;> intros
state_before: α : Type u_1
β : Type ?u.3868099
γ : Type ?u.3868102
δ : Type ?u.3868105
ι : Type ?u.3868108
R : Type ?u.3868111
R' : Type ?u.3868114
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : MeasureTheory.Measure α
inst✝ : CompactSpace α
⊢ IsFiniteMeasure μ ↔ IsFiniteMeasureOnCompacts μ

tactic: infer_instance
state_before: case mp
α : Type u_1
β : Type ?u.3868099
γ : Type ?u.3868102
δ : Type ?u.3868105
ι : Type ?u.3868108
R : Type ?u.3868111
R' : Type ?u.3868114
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : MeasureTheory.Measure α
inst✝ : CompactSpace α
a✝ : IsFiniteMeasure μ
⊢ IsFiniteMeasureOnCompacts μ

tactic: exact CompactSpace.isFiniteMeasure
state_before: case mpr
α : Type u_1
β : Type ?u.3868099
γ : Type ?u.3868102
δ : Type ?u.3868105
ι : Type ?u.3868108
R : Type ?u.3868111
R' : Type ?u.3868114
inst✝² : TopologicalSpace α
inst✝¹ : MeasurableSpace α
μ : MeasureTheory.Measure α
inst✝ : CompactSpace α
a✝ : IsFiniteMeasureOnCompacts μ
⊢ IsFiniteMeasure μ
Theorem: IsLeftRegular.ne_zero
File: Mathlib/Algebra/Regular/Basic.lean

tactic: rintro rfl
state_before: R : Type u_1
inst✝¹ : MulZeroClass R
a b : R
inst✝ : Nontrivial R
la : IsLeftRegular a
⊢ a ≠ 0

tactic: rcases exists_pair_ne R with ⟨x, y, xy⟩
state_before: R : Type u_1
inst✝¹ : MulZeroClass R
b : R
inst✝ : Nontrivial R
la : IsLeftRegular 0
⊢ False

tactic: refine' xy (la (_ : 0 * x = 0 * y))
state_before: case intro.intro
R : Type u_1
inst✝¹ : MulZeroClass R
b : R
inst✝ : Nontrivial R
la : IsLeftRegular 0
x y : R
xy : x ≠ y
⊢ False

tactic: rw [zero_mul, zero_mul]
state_before: case intro.intro
R : Type u_1
inst✝¹ : MulZeroClass R
b : R
inst✝ : Nontrivial R
la : IsLeftRegular 0
x y : R
xy : x ≠ y
⊢ 0 * x = 0 * y
Theorem: Filter.Tendsto.fin_insertNth
File: Mathlib/Topology/Constructions.lean

tactic: simpa
state_before: α : Type u
β : Type v
γ : Type ?u.211843
δ : Type ?u.211846
ε : Type ?u.211849
ζ : Type ?u.211852
ι : Type ?u.211855
π✝ : ι → Type ?u.211860
κ : Type ?u.211863
inst✝² : TopologicalSpace α
inst✝¹ : (i : ι) → TopologicalSpace (π✝ i)
f✝ : α → (i : ι) → π✝ i
n : ℕ
π : Fin (n + 1) → Type u_1
inst✝ : (i : Fin (n + 1)) → TopologicalSpace (π i)
i : Fin (n + 1)
f : β → π i
l : Filter β
x : π i
hf : Tendsto f l (𝓝 x)
g : β → (j : Fin n) → π (↑(Fin.succAbove i) j)
y : (j : Fin n) → π (↑(Fin.succAbove i) j)
hg : Tendsto g l (𝓝 y)
j : Fin (n + 1)
⊢ Tendsto (fun i_1 => Fin.insertNth i (f i_1) (g i_1) i) l (𝓝 (Fin.insertNth i x y i))

tactic: simpa using tendsto_pi_nhds.1 hg
state_before: α : Type u
β : Type v
γ : Type ?u.211843
δ : Type ?u.211846
ε : Type ?u.211849
ζ : Type ?u.211852
ι : Type ?u.211855
π✝ : ι → Type ?u.211860
κ : Type ?u.211863
inst✝² : TopologicalSpace α
inst✝¹ : (i : ι) → TopologicalSpace (π✝ i)
f✝ : α → (i : ι) → π✝ i
n : ℕ
π : Fin (n + 1) → Type u_1
inst✝ : (i : Fin (n + 1)) → TopologicalSpace (π i)
i : Fin (n + 1)
f : β → π i
l : Filter β
x : π i
hf : Tendsto f l (𝓝 x)
g : β → (j : Fin n) → π (↑(Fin.succAbove i) j)
y : (j : Fin n) → π (↑(Fin.succAbove i) j)
hg : Tendsto g l (𝓝 y)
j : Fin (n + 1)
⊢ ∀ (j : Fin n),
    Tendsto (fun i_1 => Fin.insertNth i (f i_1) (g i_1) (↑(Fin.succAbove i) j)) l
      (𝓝 (Fin.insertNth i x y (↑(Fin.succAbove i) j)))
Theorem: Option.map_map₂_antidistrib_right
File: Mathlib/Data/Option/NAry.lean

tactic: cases a <;> cases b <;> simp [h_antidistrib]
state_before: α : Type u_4
β : Type u_5
γ : Type u_3
f : α → β → γ
a : Option α
b : Option β
c : Option γ
δ : Type u_1
α' : Type u_2
g : γ → δ
f' : β → α' → δ
g' : α → α'
h_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' b (g' a)
⊢ Option.map g (map₂ f a b) = map₂ f' b (Option.map g' a)
Theorem: div_le_div''
File: Mathlib/Algebra/Order/Group/Defs.lean

tactic: rw [div_eq_mul_inv, div_eq_mul_inv, mul_comm b, mul_inv_le_inv_mul_iff, mul_comm]
state_before: α : Type u
inst✝² : CommGroup α
inst✝¹ : Preorder α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α
hab : a ≤ b
hcd : c ≤ d
⊢ a / d ≤ b / c

tactic: exact mul_le_mul' hab hcd
state_before: α : Type u
inst✝² : CommGroup α
inst✝¹ : Preorder α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α
hab : a ≤ b
hcd : c ≤ d
⊢ a * c ≤ b * d
Theorem: Ordinal.mul_mod
File: Mathlib/SetTheory/Ordinal/Arithmetic.lean

tactic: simpa using mul_add_mod_self x y 0
state_before: α : Type ?u.267174
β : Type ?u.267177
γ : Type ?u.267180
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
x y : Ordinal
⊢ x * y % x = 0
Theorem: Submonoid.LocalizationMap.mk'_eq_of_eq'
File: Mathlib/GroupTheory/MonoidLocalization.lean

tactic: simpa only [mul_comm] using H
state_before: M : Type u_1
inst✝² : CommMonoid M
S : Submonoid M
N : Type u_2
inst✝¹ : CommMonoid N
P : Type ?u.908945
inst✝ : CommMonoid P
f : LocalizationMap S N
a₁ b₁ : M
a₂ b₂ : { x // x ∈ S }
H : b₁ * ↑a₂ = a₁ * ↑b₂
⊢ ↑a₂ * b₁ = ↑b₂ * a₁
Theorem: EuclideanGeometry.Sphere.secondInter_mem
File: Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean

tactic: simp_rw [mem_sphere, Sphere.secondInter_dist]
state_before: V : Type u_2
P : Type u_1
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p : P
v : V
⊢ secondInter s p v ∈ s ↔ p ∈ s
Theorem: Continuous.exists_forall_le
File: Mathlib/Topology/Algebra/Order/Compact.lean

tactic: inhabit β
state_before: α : Type u_2
β : Type u_1
γ : Type ?u.49088
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : Nonempty β
f : β → α
hf : Continuous f
hlim : Tendsto f (cocompact β) atTop
⊢ ∃ x, ∀ (y : β), f x ≤ f y

tactic: exact hf.exists_forall_le' default (hlim.eventually <| eventually_ge_atTop _)
state_before: α : Type u_2
β : Type u_1
γ : Type ?u.49088
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : Nonempty β
f : β → α
hf : Continuous f
hlim : Tendsto f (cocompact β) atTop
inhabited_h : Inhabited β
⊢ ∃ x, ∀ (y : β), f x ≤ f y
Theorem: LinearIsometryEquiv.piLpCongrLeft_symm
File: Mathlib/Analysis/NormedSpace/PiLp.lean

tactic: simp only [LinearIsometryEquiv.piLpCongrLeft, LinearIsometryEquiv.symm,
  LinearIsometryEquiv.coe_mk]
state_before: p : ℝ≥0∞
𝕜 : Type u_4
𝕜' : Type ?u.384596
ι : Type u_1
α : ι → Type ?u.384604
β : ι → Type ?u.384609
inst✝⁸ : Fintype ι
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedField 𝕜'
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (β i)
c : 𝕜
x y : PiLp p β
x' y' : (i : ι) → β i
i : ι
ι' : Type u_2
inst✝² : Fintype ι'
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
e : ι ≃ ι'
z : PiLp p fun x => E
⊢ ↑(LinearIsometryEquiv.symm (LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e)) z =
    ↑(LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e.symm) z

tactic: unfold PiLp
state_before: p : ℝ≥0∞
𝕜 : Type u_4
𝕜' : Type ?u.384596
ι : Type u_1
α : ι → Type ?u.384604
β : ι → Type ?u.384609
inst✝⁸ : Fintype ι
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedField 𝕜'
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (β i)
c : 𝕜
x y : PiLp p β
x' y' : (i : ι) → β i
i : ι
ι' : Type u_2
inst✝² : Fintype ι'
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
e : ι ≃ ι'
z : PiLp p fun x => E
⊢ ↑(LinearEquiv.symm (LinearEquiv.piCongrLeft' 𝕜 (fun x => E) e)) z =
    ↑(LinearEquiv.piCongrLeft' 𝕜 (fun x => E) e.symm) z

tactic: ext
state_before: p : ℝ≥0∞
𝕜 : Type u_4
𝕜' : Type ?u.384596
ι : Type u_1
α : ι → Type ?u.384604
β : ι → Type ?u.384609
inst✝⁸ : Fintype ι
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedField 𝕜'
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (β i)
c : 𝕜
x y : PiLp p β
x' y' : (i : ι) → β i
i : ι
ι' : Type u_2
inst✝² : Fintype ι'
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
e : ι ≃ ι'
z : PiLp p fun x => E
⊢ ↑(LinearEquiv.symm (LinearEquiv.piCongrLeft' 𝕜 (fun x => E) e)) z =
    ↑(LinearEquiv.piCongrLeft' 𝕜 (fun x => E) e.symm) z

tactic: simp only [LinearEquiv.piCongrLeft'_symm_apply, eq_rec_constant,
  LinearEquiv.piCongrLeft'_apply, Equiv.symm_symm_apply]
state_before: case h
p : ℝ≥0∞
𝕜 : Type u_4
𝕜' : Type ?u.384596
ι : Type u_1
α : ι → Type ?u.384604
β : ι → Type ?u.384609
inst✝⁸ : Fintype ι
inst✝⁷ : Fact (1 ≤ p)
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedField 𝕜'
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (β i)
c : 𝕜
x y : PiLp p β
x' y' : (i : ι) → β i
i : ι
ι' : Type u_2
inst✝² : Fintype ι'
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
e : ι ≃ ι'
z : PiLp p fun x => E
x✝ : ι
⊢ ↑(LinearEquiv.symm (LinearEquiv.piCongrLeft' 𝕜 (fun x => E) e)) z x✝ =
    ↑(LinearEquiv.piCongrLeft' 𝕜 (fun x => E) e.symm) z x✝
Theorem: FormalMultilinearSeries.nnnorm_changeOriginSeriesTerm_apply_le
File: Mathlib/Analysis/Analytic/Basic.lean

tactic: rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
state_before: 𝕜 : Type u_3
E : Type u_2
F : Type u_1
G : Type ?u.1230559
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
p : FormalMultilinearSeries 𝕜 E F
x✝ y✝ : E
r R : ℝ≥0
k l : ℕ
s : Finset (Fin (k + l))
hs : Finset.card s = l
x y : E
⊢ ‖↑(↑(changeOriginSeriesTerm p k l s hs) fun x_1 => x) fun x => y‖₊ ≤ ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k

tactic: apply ContinuousMultilinearMap.le_of_op_nnnorm_le
state_before: 𝕜 : Type u_3
E : Type u_2
F : Type u_1
G : Type ?u.1230559
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
p : FormalMultilinearSeries 𝕜 E F
x✝ y✝ : E
r R : ℝ≥0
k l : ℕ
s : Finset (Fin (k + l))
hs : Finset.card s = l
x y : E
⊢ ‖↑(↑(changeOriginSeriesTerm p k l s hs) fun x_1 => x) fun x => y‖₊ ≤
    (‖changeOriginSeriesTerm p k l s hs‖₊ * ∏ _i : Fin l, ‖x‖₊) * ∏ _i : Fin k, ‖y‖₊

tactic: apply ContinuousMultilinearMap.le_op_nnnorm
state_before: case h
𝕜 : Type u_3
E : Type u_2
F : Type u_1
G : Type ?u.1230559
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
p : FormalMultilinearSeries 𝕜 E F
x✝ y✝ : E
r R : ℝ≥0
k l : ℕ
s : Finset (Fin (k + l))
hs : Finset.card s = l
x y : E
⊢ ‖↑(changeOriginSeriesTerm p k l s hs) fun x_1 => x‖₊ ≤ ‖changeOriginSeriesTerm p k l s hs‖₊ * ∏ _i : Fin l, ‖x‖₊
Theorem: Ordinal.nfpBFamily_fp
File: Mathlib/SetTheory/Ordinal/FixedPoint.lean

tactic: rw [← familyOfBFamily_enum o f]
state_before: o : Ordinal
f : (b : Ordinal) → b < o → Ordinal → Ordinal
i : Ordinal
hi : i < o
H : IsNormal (f i hi)
a : Ordinal
⊢ f i hi (nfpBFamily o f a) = nfpBFamily o f a

tactic: apply nfpFamily_fp
state_before: o : Ordinal
f : (b : Ordinal) → b < o → Ordinal → Ordinal
i : Ordinal
hi : i < o
H : IsNormal (f i hi)
a : Ordinal
⊢ familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)) (nfpBFamily o f a) =
    nfpBFamily o f a

tactic: rw [familyOfBFamily_enum]
state_before: case H
o : Ordinal
f : (b : Ordinal) → b < o → Ordinal → Ordinal
i : Ordinal
hi : i < o
H : IsNormal (f i hi)
a : Ordinal
⊢ IsNormal (familyOfBFamily o f (enum (fun x x_1 => x < x_1) i (_ : i < type fun x x_1 => x < x_1)))

tactic: exact H
state_before: case H
o : Ordinal
f : (b : Ordinal) → b < o → Ordinal → Ordinal
i : Ordinal
hi : i < o
H : IsNormal (f i hi)
a : Ordinal
⊢ IsNormal (f i ?H.hi)

case H.hi
o : Ordinal
f : (b : Ordinal) → b < o → Ordinal → Ordinal
i : Ordinal
hi : i < o
H : IsNormal (f i hi)
a : Ordinal
⊢ i < o