mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.RamificationInertia.Ramification

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : Algebra R S
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : IsScalarTower R S T
q : Ideal S
r : Ideal T
inst✝⁵ : q.IsPrime
inst✝⁴ : r.IsPrime
inst✝³ : r.LiesOver q
inst✝² : Algebra (Localization.AtPrime q) (Localization.AtPrime r)
inst✝¹ : Localization.AtPrime.IsLiesOverAlgebra q r
inst✝ : Module.Flat (Localization.AtPrime q) (Localization.AtPrime r)
this : q.LiesOver (under R r)
f : (Localization.AtPrime r ⧸ map (algebraMap R (Localization.AtPrime r)) (under R r)) ≃ₐ[Localization.AtPrime r]
  TensorProduct (Localization.AtPrime q) (Localization.AtPrime r)
    (Localization.AtPrime q ⧸ map (algebraMap R (Localization.AtPrime q)) (under R r)) :=
  (quotientEquivAlgOfEq (Localization.AtPrime r) ⋯).trans
    (Algebra.TensorProduct.quotIdealMapEquivTensorQuot (Localization.AtPrime r)
      (map (algebraMap R (Localization.AtPrime q)) (under R r)))
⊢ (Module.length (Localization.AtPrime r)
        (Localization.AtPrime r ⧸ map (algebraMap R (Localization.AtPrime r)) (under R r))).toNat =
    (Module.length (Localization.AtPrime q)
          (Localization.AtPrime q ⧸ map (algebraMap R (Localization.AtPrime q)) (under R r))).toNat *
      r.ramificationIdx S

Tactic:
ramificationIdx_eq q,

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : Algebra R S
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : IsScalarTower R S T
q : Ideal S
r : Ideal T
inst✝⁵ : q.IsPrime
inst✝⁴ : r.IsPrime
inst✝³ : r.LiesOver q
inst✝² : Algebra (Localization.AtPrime q) (Localization.AtPrime r)
inst✝¹ : Localization.AtPrime.IsLiesOverAlgebra q r
inst✝ : Module.Flat (Localization.AtPrime q) (Localization.AtPrime r)
this : q.LiesOver (under R r)
f : (Localization.AtPrime r ⧸ map (algebraMap R (Localization.AtPrime r)) (under R r)) ≃ₐ[Localization.AtPrime r]
  TensorProduct (Localization.AtPrime q) (Localization.AtPrime r)
    (Localization.AtPrime q ⧸ map (algebraMap R (Localization.AtPrime q)) (under R r)) :=
  (quotientEquivAlgOfEq (Localization.AtPrime r) ⋯).trans
    (Algebra.TensorProduct.quotIdealMapEquivTensorQuot (Localization.AtPrime r)
      (map (algebraMap R (Localization.AtPrime q)) (under R r)))
⊢ (Module.length (Localization.AtPrime r)
        (Localization.AtPrime r ⧸ map (algebraMap R (Localization.AtPrime r)) (under R r))).toNat =
    (Module.length (Localization.AtPrime q)
          (Localization.AtPrime q ⧸ map (algebraMap R (Localization.AtPrime q)) (under R r))).toNat *
      (Module.length (Localization.AtPrime r)
          (Localization.AtPrime r ⧸ map (algebraMap S (Localization.AtPrime r)) q)).toNat
Module: Mathlib.LinearAlgebra.Countable

Goal state before:
pp: M : Type u_1
R : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Countable R
inst✝ : Countable ι
v : ι → M
⊢ Countable ↥(Submodule.span R (Set.range v))

Tactic:
refine Set.countable_coe_iff.mpr (Set.Countable.mono ?_ (Set.countable_range
      (fun c : (ι →₀ R) => c.sum fun i _ => (c i) • v i)))
  exact fun _ h => Finsupp.mem_span_range_iff_exists_finsupp.mp (SetLike.mem_coe.mp h)

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
hf : HasFiniteMulSupport fun v ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
⊢ ¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I

Tactic:
rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
hf : HasFiniteMulSupport fun v ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
⊢ ¬v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors * v.asIdeal ∣
      v.maxPowDividing I * ∏ i ∈ (Finite.toFinset hf).erase v, i.maxPowDividing I
Module: Mathlib.Data.NNReal.Defs

Goal state before:
pp: ι : Sort u_2
s : ι → ℝ≥0
⊢ ↑(⨅ i, s i) = ⨅ i, ↑(s i)

Tactic:
by
  rw [iInf, iInf, coe_sInf, ← Set.range_comp]; rfl

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠
    0
hv : Irreducible (Associates.mk v.asIdeal)
h_dvd :
  (Associates.mk v.asIdeal).count (Associates.mk I).factors ≤
    (Associates.mk v.asIdeal).count
      (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)).factors
h_not_dvd :
  ¬(Associates.mk v.asIdeal).count (Associates.mk I).factors + 1 ≤
      (Associates.mk v.asIdeal).count
        (Associates.mk
            (∏ᶠ (v : HeightOneSpectrum R),
              v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)).factors
⊢ (Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
    (Associates.mk v.asIdeal).count (Associates.mk I).factors

Tactic:
not_le

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠
    0
hv : Irreducible (Associates.mk v.asIdeal)
h_dvd :
  (Associates.mk v.asIdeal).count (Associates.mk I).factors ≤
    (Associates.mk v.asIdeal).count
      (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)).factors
h_not_dvd :
  (Associates.mk v.asIdeal).count
      (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R),
            v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors)).factors <
    (Associates.mk v.asIdeal).count (Associates.mk I).factors + 1
⊢ (Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors =
    (Associates.mk v.asIdeal).count (Associates.mk I).factors
Module: Mathlib.Data.ENNReal.Basic

Goal state before:
pp: n : ℕ
⊢ (↑n).toNNReal = NNReal.mk ↑n ⋯

Tactic:
ENNReal.toNNReal_natCast,

Goal state after:
n : ℕ
⊢ ↑n = NNReal.mk ↑n ⋯
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: ι : Sort u_1
f : ι → ℝ≥0∞
hf : ∀ (i : ι), f i ≠ ∞
⊢ (iInf f).toReal = ⨅ i, (f i).toReal

Tactic:
simp only [ENNReal.toReal, toNNReal_iInf hf, NNReal.coe_iInf]

Goal state after:
Module: Mathlib.Algebra.Order.Module.Field

Goal state before:
pp: 𝕜 : Type u_1
G : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup G
inst✝³ : PartialOrder G
inst✝² : IsOrderedAddMonoid G
inst✝¹ : Module 𝕜 G
a : 𝕜
b₁ b₂ : G
inst✝ : PosSMulStrictMono 𝕜 G
h : a < 0
⊢ a⁻¹ • b₁ < b₂ ↔ a • b₂ < b₁

Tactic:
← smul_lt_smul_iff_of_neg_left h,

Goal state after:
𝕜 : Type u_1
G : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup G
inst✝³ : PartialOrder G
inst✝² : IsOrderedAddMonoid G
inst✝¹ : Module 𝕜 G
a : 𝕜
b₁ b₂ : G
inst✝ : PosSMulStrictMono 𝕜 G
h : a < 0
⊢ a • b₂ < a • a⁻¹ • b₁ ↔ a • b₂ < b₁
Module: Mathlib.Algebra.Order.Module.Field

Goal state before:
pp: 𝕜 : Type u_1
G : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup G
inst✝³ : PartialOrder G
inst✝² : IsOrderedAddMonoid G
inst✝¹ : Module 𝕜 G
a : 𝕜
b₁ b₂ : G
inst✝ : PosSMulStrictMono 𝕜 G
h : a < 0
⊢ b₁ < a⁻¹ • b₂ ↔ b₂ < a • b₁

Tactic:
← smul_lt_smul_iff_of_neg_left h,

Goal state after:
𝕜 : Type u_1
G : Type u_2
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup G
inst✝³ : PartialOrder G
inst✝² : IsOrderedAddMonoid G
inst✝¹ : Module 𝕜 G
a : 𝕜
b₁ b₂ : G
inst✝ : PosSMulStrictMono 𝕜 G
h : a < 0
⊢ a • a⁻¹ • b₂ < a • b₁ ↔ b₂ < a • b₁
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: case inr
ι : Sort u_1
f : ι → ℝ≥0∞
a : ℝ≥0∞
inst✝ : Nonempty ι
ha : a ≠ ∞
h : ¬∃ i, a < f i
⊢ a - ⨆ i, f i = ⨅ i, a - f i

Tactic:
not_exists,

Goal state after:
case inr
ι : Sort u_1
f : ι → ℝ≥0∞
a : ℝ≥0∞
inst✝ : Nonempty ι
ha : a ≠ ∞
h : ∀ (x : ι), ¬a < f x
⊢ a - ⨆ i, f i = ⨅ i, a - f i
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: x : EReal
h : x ≠ ⊥
⊢ ↑x.toReal ≤ x

Tactic:
by_cases h' : x = ⊤
  · simp only [h', le_top]
  · simp only [le_refl, coe_toReal h' h]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a⁻¹ ≤ 1 ↔ 1 ≤ a

Tactic:
by rw [inv_le_iff_inv_le, inv_one]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ ∞ / a = if a = ∞ then 0 else ∞

Tactic:
simp [div_eq_mul_inv, top_mul']

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
hc : c < a * b
⊢ ∃ b' < b, c < a * b'

Tactic:
simp_rw [mul_comm a] at hc ⊢

Goal state after:
a b c : ℝ≥0∞
hc : c < b * a
⊢ ∃ b' < b, c < b' * a
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
h : a ≠ 0
⊢ ∃ n, (↑n)⁻¹ < a⁻¹⁻¹

Tactic:
by simp only [ENNReal.inv_lt_inv, ENNReal.exists_nat_gt (inv_ne_top.2 h)]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z = edist y z

Tactic:
apply le_antisymm

Goal state after:
case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z ≤ edist y z

case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist y z ≤ edist x z
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: case inr
α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
α : Type u_3
dist : α → α → ℝ
dist_comm : ∀ (x y : α), dist x y = dist y x
dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
s : Set α
hs : s ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
t : Set α
ht : t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
x : α
hx : x ∈ s
⊢ s ∪ t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}

Tactic:
rcases t.eq_empty_or_nonempty with rfl | ⟨y, hy⟩

Goal state after:
case inr.inl
α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
α : Type u_3
dist : α → α → ℝ
dist_comm : ∀ (x y : α), dist x y = dist y x
dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
s : Set α
hs : s ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
x : α
hx : x ∈ s
ht : ∅ ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
⊢ s ∪ ∅ ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}

case inr.inr
α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
α : Type u_3
dist : α → α → ℝ
dist_comm : ∀ (x y : α), dist x y = dist y x
dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
s : Set α
hs : s ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
t : Set α
ht : t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
x : α
hx : x ∈ s
y : α
hy : y ∈ t
⊢ s ∪ t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
s : Set α
inst✝ : PseudoEMetricSpace β
f : α → β
a : α
b : β
⊢ (∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → edist x a < δ → f x ∈ univ ∧ edist (f x) b < ε) ↔
    ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → edist x a < δ → edist (f x) b < ε

Tactic:
simp only [mem_univ, true_and]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
m m' : WeakPseudoEMetricSpace α
h : m.toEDist = m'.toEDist
⊢ m = m'

Tactic:
by
  cases m; cases m'; congr

Goal state after:
Module: Mathlib.Topology.MetricSpace.Defs

Goal state before:
pp: case inl
γ : Type w
inst✝ : MetricSpace γ
x : γ
r : ℝ
hr✝ : r ≤ 0
hr : r < 0
⊢ (closedBall x r).Subsingleton

Tactic:
rw [closedBall_eq_empty.2 hr]

Goal state after:
case inl
γ : Type w
inst✝ : MetricSpace γ
x : γ
r : ℝ
hr✝ : r ≤ 0
hr : r < 0
⊢ ∅.Subsingleton
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)
x y : E
⊢ dist x y = dist 1 (x⁻¹ * y)

Tactic:
apply le_antisymm

Goal state after:
case a
𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)
x y : E
⊢ dist x y ≤ dist 1 (x⁻¹ * y)

case a
𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)
x y : E
⊢ dist 1 (x⁻¹ * y) ≤ dist x y
Module: Mathlib.Analysis.Normed.Group.Defs

Goal state before:
pp: 𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y
x y : E
⊢ dist x y = dist 1 (x⁻¹ * y)

Tactic:
apply le_antisymm

Goal state after:
case a
𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y
x y : E
⊢ dist x y ≤ dist 1 (x⁻¹ * y)

case a
𝓕 : Type u_1
α : Type u_2
ι : Type u_3
κ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝² : Norm E
inst✝¹ : Group E
inst✝ : PseudoMetricSpace E
h₁ : ∀ (x : E), ‖x‖ = dist 1 x
h₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y
x y : E
⊢ dist 1 (x⁻¹ * y) ≤ dist x y
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ
⊢ ¬0 ≤ ε ↔ ε < 0

Tactic:
not_le

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ
⊢ ε < 0 ↔ ε < 0
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
ι : Type u_2
inst✝ : PseudoMetricSpace α
f : Filter ι
x₀ : α
p : α × ι → Prop
⊢ (∀ᶠ (x : α × ι) in 𝓝 x₀ ×ˢ f, p x) ↔
    ∃ ε > 0, ∃ pa, (∀ᶠ (i : ι) in f, pa i) ∧ ∀ ⦃x : α⦄, dist x x₀ < ε → ∀ ⦃i : ι⦄, pa i → p (x, i)

Tactic:
refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_

Goal state after:
α : Type u
ι : Type u_2
inst✝ : PseudoMetricSpace α
f : Filter ι
x₀ : α
p : α × ι → Prop
⊢ (∃ i, (0 < i.1 ∧ i.2 ∈ f) ∧ ∀ ⦃x : α × ι⦄, x ∈ ball x₀ i.1 ×ˢ id i.2 → p x) ↔
    ∃ ε > 0, ∃ pa, (∀ᶠ (i : ι) in f, pa i) ∧ ∀ ⦃x : α⦄, dist x x₀ < ε → ∀ ⦃i : ι⦄, pa i → p (x, i)
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
s : Set α
inst✝ : PseudoMetricSpace β
f : α → β
a : α
b : β
⊢ (∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → f x ∈ univ ∧ dist (f x) b < ε) ↔
    ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) b < ε

Tactic:
simp only [mem_univ, true_and]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u_3
inst✝ : PseudoMetricSpace α
r : ℝ
hpos : 0 < r
hr : Pairwise fun x1 x2 ↦ r ≤ dist x1 x2
⊢ ⊥.toTopologicalSpace = ⊥

Tactic:
UniformSpace.toTopologicalSpace_bot

Goal state after:
α : Type u_3
inst✝ : PseudoMetricSpace α
r : ℝ
hpos : 0 < r
hr : Pairwise fun x1 x2 ↦ r ≤ dist x1 x2
⊢ ⊥ = ⊥
Module: Mathlib.Topology.Order.DenselyOrdered

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
inst✝ : DiscreteTopology α
⊢ Subsingleton α

Tactic:
suffices ∀ a b : α, b ≤ a from ⟨fun a b ↦ le_antisymm (this b a) (this a b)⟩

Goal state after:
α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
inst✝ : DiscreteTopology α
⊢ ∀ (a b : α), b ≤ a
Module: Mathlib.Topology.Order.IsLUB

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
t : Set α
x : α
inst✝ : (𝓝 x).IsCountablyGenerated
htx : IsLUB t x
ht : t.Nonempty
⊢ ∃ u, Monotone u ∧ (∀ (n : ℕ), u n ≤ x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t

Tactic:
by_cases h : x ∈ t
  · exact ⟨fun _ => x, monotone_const, fun n => le_rfl, tendsto_const_nhds, fun _ => h⟩
  · rcases htx.exists_seq_strictMono_tendsto_of_notMem h ht with ⟨u, hu⟩
    exact ⟨u, hu.1.monotone, fun n => (hu.2.1 n).le, hu.2.2⟩

Goal state after:
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := x '' t
c : β
hc : c ∈ t
d : β
hd : d ∈ t
hcd : x c ≠ x d
H : c < d
a : α
xca : x c < a
ayc : a < y c
xda : x d < a
ayd : a < y (f (x d))
⊢ False

Tactic:
have : x d ≤ y c := (xda.trans ayc).le

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : LinearOrder β
s : Set α
f : α → β
inst✝ : SecondCountableTopology α
hf : MonotoneOn f s
a✝ : Nontrivial α
t : Set β := {c | ∃ x y, x ∈ s ∧ y ∈ s ∧ x < y ∧ f x = c ∧ f y = c}
x y : β → α
hxs : ∀ c ∈ t, x c ∈ s
hys : ∀ c ∈ t, y c ∈ s
hxy : ∀ c ∈ t, x c < y c
hfx : ∀ c ∈ t, f (x c) = c
hfy : ∀ c ∈ t, f (y c) = c
u : Set α := x '' t
c : β
hc : c ∈ t
d : β
hd : d ∈ t
hcd : x c ≠ x d
H : c < d
a : α
xca : x c < a
ayc : a < y c
xda : x d < a
ayd : a < y (f (x d))
this : x d ≤ y c
⊢ False
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: X : Type u
α : Type v
inst✝⁴ : TopologicalSpace X
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderClosedTopology α
s : Set X
hs : IsPreconnected s
a : X
l : Filter X
ha : a ∈ s
inst✝ : l.NeBot
hl : l ≤ 𝓟 s
f g : X → α
hf : ContinuousOn f s
hg : ContinuousOn g s
ha' : f a ≤ g a
he : g ≤ᶠ[l] f
⊢ ∃ x ∈ s, f x = g x

Tactic:
by
  rw [continuousOn_iff_continuous_domRestrict] at hf hg
  obtain ⟨b, h⟩ :=
    @intermediate_value_univ₂_eventually₁ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) ⟨a, ha⟩ _
      (comap_coe_neBot_of_le_principal hl) _ _ hf hg ha' (he.comap _)
  exact ⟨b, b.prop, h⟩

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case refine_3
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ ((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ⋯).ker =
    ⊥

Tactic:
Submodule.mapQ,

Goal state after:
case refine_3
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
⊢ ((Submodule.comap (↑I).subtype ↑J).liftQ
        ((Submodule.comap (↑I').subtype ↑J').mkQ ∘ₗ LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯)
        ⋯).ker =
    ⊥
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: case inr.refine_2
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
K : Set β
hK : IsCompact K
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
h0K : K.Nonempty
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z > f x y
h : {x} ×ˢ K ⊆ ↿f ⁻¹' Iio z
u : Set γ
v : Set β
hu : IsOpen[inst✝] u
left✝ : IsOpen[inst✝¹] v
hxu : {x} ⊆ u
hKv : K ⊆ v
huv : u ×ˢ v ⊆ ↿f ⁻¹' Iio z
⊢ ∀ᶠ (b : γ) in 𝓝 x, sSup (f b '' K) < z

Tactic:
refine eventually_of_mem (hu.mem_nhds (singleton_subset_iff.mp hxu)) fun x' hx' => ?_

Goal state after:
case inr.refine_2
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : ConditionallyCompleteLinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : γ → β → α
K : Set β
hK : IsCompact K
hf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f
h0K : K.Nonempty
x : γ
y : β
hyK : y ∈ K
h2y : sSup (f x '' K) = f x y
hy : ∀ y_1 ∈ K, f x y_1 ≤ f x y
this : (∀ a' < f x y, ∀ᶠ (b : γ) in 𝓝 x, a' < f b y) ∧ ∀ a' > f x y, ∀ᶠ (b : γ) in 𝓝 x, f b y < a'
z : α
hz : z > f x y
h : {x} ×ˢ K ⊆ ↿f ⁻¹' Iio z
u : Set γ
v : Set β
hu : IsOpen[inst✝] u
left✝ : IsOpen[inst✝¹] v
hxu : {x} ⊆ u
hKv : K ⊆ v
huv : u ×ˢ v ⊆ ↿f ⁻¹' Iio z
x' : γ
hx' : x' ∈ u
⊢ sSup (f x' '' K) < z
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖v‖ ≤ ‖u‖ + ‖u / v‖

Tactic:
rw [norm_div_rev]

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖v‖ ≤ ‖u‖ + ‖v / u‖
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
u v : E
⊢ ‖v‖ = ‖u⁻¹ * (u * v)‖

Tactic:
rw [← mul_assoc, inv_mul_cancel, one_mul]

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case refine_4
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H✝ : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
H : Submodule.map ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I = ↑(spanSingleton R⁰ (I'.divMod I J')) * ↑I
⊢ ((Submodule.comap (↑I).subtype ↑J).mapQ (Submodule.comap (↑I').subtype ↑J')
        (LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯) ⋯).range =
    ⊤

Tactic:
Submodule.mapQ,

Goal state after:
case refine_4
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J I' J' : FractionalIdeal R⁰ K
H✝ : I * J' = I' * J
h : J ≤ I
h' : J' ≤ I'
hJ' : J' ≠ 0
hI : I ≠ 0
this : J' ⊓ spanSingleton R⁰ (I'.divMod I J') * I = spanSingleton R⁰ (I'.divMod I J') * J
H : Submodule.map ((Algebra.lsmul R R K) (I'.divMod I J')) ↑I = ↑(spanSingleton R⁰ (I'.divMod I J')) * ↑I
⊢ ((Submodule.comap (↑I).subtype ↑J).liftQ
        ((Submodule.comap (↑I').subtype ↑J').mkQ ∘ₗ LinearMap.restrict ((Algebra.lsmul R R K) (I'.divMod I J')) ⋯)
        ⋯).range =
    ⊤
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_8
inst✝ : SeminormedCommGroup E
u v : E
⊢ ‖v * u‖ - ‖v / u‖ ≤ 2 * ‖u‖

Tactic:
exact norm_mul_sub_norm_div_le_two_mul _ _

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Basic

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedCommGroup E
a b : E
r : ℝ
c : E
⊢ a * c ∈ ball (b * c) r ↔ a ∈ ball b r

Tactic:
simp only [mem_ball, dist_eq_norm_inv_mul, mul_comm _ (b * c), mul_comm a⁻¹ b]
  simp

Goal state after: