mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: 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}

Tactic:
]

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}
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x✝ y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
x : α
⊢ 0 < ∞

Tactic:
]

Goal state after:
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x✝ y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
x : α
⊢ 0 < ∞
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case refine_2
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n < x
h'n : x ≤ y ^ (n + 1)
⊢ x ≤ y ^ (n + 1)

Tactic:
]

Goal state after:
case refine_2
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n < x
h'n : x ≤ y ^ (n + 1)
⊢ x ≤ y ^ (n + 1)
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
x✝ y✝ : α
h : edist x✝ y✝ < ∞
⊢ edist x✝ y✝ < ∞

Tactic:
]

Goal state after:
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
ε ε₁ ε₂ : ℝ≥0∞
s t : Set α
x✝ y✝ : α
h : edist x✝ y✝ < ∞
⊢ edist x✝ y✝ < ∞
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case neg
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 : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
this : ∃ I_1, ↑I_1 = J⁻¹ * I
⊢ count K v J ≤ count K v I

Tactic:
]

Goal state after:
case neg
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 : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
this : ∃ I_1, ↑I_1 = J⁻¹ * I
⊢ count K v J ≤ count K v I
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Sort u_3
x y : α
dist : α → α → ℝ
dist_self : ∀ (x : α), dist x x = 0
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
⊢ dist y x + dist y x = dist y x + dist y x

Tactic:
]

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

Goal state before:
pp: case ofNat.ofNat
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.ofNat a ≤ Int.ofNat b
⊢ x ^ Int.ofNat a ≤ x ^ Int.ofNat b

Tactic:
simp only [Int.ofNat_eq_natCast, zpow_natCast]

Goal state after:
case ofNat.ofNat
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.ofNat a ≤ Int.ofNat b
⊢ x ^ a ≤ x ^ b
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:
]

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

Goal state before:
pp: case ofNat.ofNat
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.ofNat a ≤ Int.ofNat b
⊢ x ^ Int.ofNat a ≤ x ^ Int.ofNat b

pp: case ofNat.negSucc
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.ofNat a ≤ Int.negSucc b
⊢ x ^ Int.ofNat a ≤ x ^ Int.negSucc b

pp: case negSucc.ofNat
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.negSucc a ≤ Int.ofNat b
⊢ x ^ Int.negSucc a ≤ x ^ Int.ofNat b

pp: case negSucc.negSucc
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.negSucc a ≤ Int.negSucc b
⊢ x ^ Int.negSucc a ≤ x ^ Int.negSucc b

Tactic:
· simp only [Int.ofNat_eq_natCast, zpow_natCast]
    exact pow_right_mono₀ hx (Int.le_of_ofNat_le_ofNat h)

Goal state after:
case ofNat.negSucc
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.ofNat a ≤ Int.negSucc b
⊢ x ^ Int.ofNat a ≤ x ^ Int.negSucc b

case negSucc.ofNat
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.negSucc a ≤ Int.ofNat b
⊢ x ^ Int.negSucc a ≤ x ^ Int.ofNat b

case negSucc.negSucc
x : ℝ≥0∞
hx : 1 ≤ x
a b : ℕ
h : Int.negSucc a ≤ Int.negSucc b
⊢ x ^ Int.negSucc a ≤ x ^ Int.negSucc b
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case neg
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 : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
J' : Ideal R
hJ' : ↑J' = J⁻¹ * I
⊢ 0 ≤ count K v ↑J'

pp: case neg
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 : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
J' : Ideal R
hJ' : ↑J' = J⁻¹ * I
⊢ ↑J' ≠ 0

Tactic:
]

Goal state after:
case neg
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 : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
J' : Ideal R
hJ' : ↑J' = J⁻¹ * I
⊢ 0 ≤ count K v ↑J'

case neg
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 : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
J' : Ideal R
hJ' : ↑J' = J⁻¹ * I
⊢ ↑J' ≠ 0
Module: Mathlib.Topology.EMetricSpace.Defs

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

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : TopologicalSpace β
f : β → α
b : β
⊢ (∀ ε > 0, ∀ᶠ (x : β) in 𝓝 b, edist (f x) (f b) < ε) ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝 b, edist (f x) (f b) < ε

Tactic:
]

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

Goal state before:
pp: x : ℝ≥0∞
x_ne_zero : x ≠ 0
x_ne_top : x ≠ ∞
m n : ℤ
⊢ x ^ m * (x ^ n)⁻¹ = x ^ m * (x ^ n)⁻¹

Tactic:
]

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

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : PseudoEMetricSpace α
inst✝ : TopologicalSpace β
f : β → α
b : β
s : Set β
⊢ (∀ ε > 0, ∀ᶠ (x : β) in 𝓝[s] b, edist (f x) (f b) < ε) ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝[s] b, edist (f x) (f b) < ε

Tactic:
]

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

Goal state before:
pp: x : ℝ≥0∞
n : ℤ
⊢ (x ^ n)⁻¹ = (x ^ n)⁻¹

Tactic:
]

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

Goal state before:
pp: x : ℝ≥0∞
hx : 1 ≤ x
m : ℕ
hn : -↑m ≤ 0
⊢ 1 ≤ x ^ ↑m

Tactic:
]

Goal state after:
x : ℝ≥0∞
hx : 1 ≤ x
m : ℕ
hn : -↑m ≤ 0
⊢ 1 ≤ x ^ ↑m
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
p : α → Prop
a : { a // p a }
r : ℝ≥0∞
⊢ eball (↑a) r ∩ {x | p x} = eball (↑a) r ∩ {a | p a}

Tactic:
]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
p : α → Prop
a : { a // p a }
r : ℝ≥0∞
⊢ closedEBall (↑a) r ∩ {x | p x} = closedEBall (↑a) r ∩ {a | p a}

Tactic:
]

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

Goal state before:
pp: 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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I = spanSingleton R⁰ ((algebraMap R K) (Classical.choose ⋯))⁻¹ * ↑(Classical.choose ⋯)
w : HeightOneSpectrum R
⊢ ↑((Associates.mk w.asIdeal).count (Associates.mk (Classical.choose ⋯)).factors) -
      ↑((Associates.mk w.asIdeal).count (Associates.mk (Ideal.span {Classical.choose ⋯})).factors) =
    ↑((Associates.mk w.asIdeal).count (Associates.mk (Classical.choose ⋯)).factors) -
      ↑((Associates.mk w.asIdeal).count (Associates.mk (Ideal.span {Classical.choose ⋯})).factors)

Tactic:
]

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

Goal state before:
pp: case e_edist
α : Type u_3
toDist✝ : Dist α
dist_self✝¹ : ∀ (x : α), dist x x = 0
dist_comm✝¹ : ∀ (x y : α), dist x y = dist y x
dist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
edist✝¹ : α → α → ℝ≥0∞
hed : ∀ (x y : α), edist✝¹ x y = ENNReal.ofReal (dist x y)
toUniformSpace✝¹ : UniformSpace α
hU : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}
toBornology✝¹ : Bornology α
hB : (cobounded α).sets = {s | ∃ C, ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C}
d : Dist α :=
  { toDist := toDist✝, dist_self := dist_self✝¹, dist_comm := dist_comm✝¹, dist_triangle := dist_triangle✝¹,
      edist := edist✝¹, edist_dist := hed, toUniformSpace := toUniformSpace✝¹, uniformity_dist := hU,
      toBornology := toBornology✝¹, cobounded_sets := hB }.toDist
edist✝ : α → α → ℝ≥0∞
toUniformSpace✝ : UniformSpace α
toBornology✝ : Bornology α
dist_self✝ : ∀ (x : α), dist x x = 0
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
hed' : ∀ (x y : α), edist✝ x y = ENNReal.ofReal (dist x y)
hU' : 𝓤 α = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}
hB' : (cobounded α).sets = {s | ∃ C, ∀ x ∈ sᶜ, ∀ y ∈ sᶜ, dist x y ≤ C}
d' : Dist α :=
  { toDist := d, dist_self := dist_self✝, dist_comm := dist_comm✝, dist_triangle := dist_triangle✝, edist := edist✝,
      edist_dist := hed', toUniformSpace := toUniformSpace✝, uniformity_dist := hU', toBornology := toBornology✝,
      cobounded_sets := hB' }.toDist
x y : α
⊢ ENNReal.ofReal (dist x y) = ENNReal.ofReal (dist x y)

Tactic:
]

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

Goal state before:
pp: α✝ : Type u
β : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u
inst : PseudoEMetricSpace α
⊢ (uniformSpaceOfEDist edist ⋯ ⋯ ⋯).toTopologicalSpace ≤ (uniformSpaceOfEDist edist ⋯ ⋯ ⋯).toTopologicalSpace

Tactic:
]

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

Goal state before:
pp: α✝ : Type u
β✝ : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u_2
β : Type u_3
e : TopologicalSpace α
n : TopologicalSpace β
f : α → β
hf : Topology.IsInducing f
m : WeakPseudoEMetricSpace β
hα : PseudoEMetricSpace α := PseudoEMetricSpace.ofEDist (fun x y ↦ edist (f x) (f y)) ⋯ ⋯ ⋯
hβ : PseudoEMetricSpace β := PseudoEMetricSpace.ofEDist edist ⋯ ⋯ ⋯
⊢ (uniformSpaceOfEDist (fun x y ↦ edist (f x) (f y)) ⋯ ⋯ ⋯).toTopologicalSpace ≤ TopologicalSpace.induced f n

Tactic:
]

Goal state after:
α✝ : Type u
β✝ : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u_2
β : Type u_3
e : TopologicalSpace α
n : TopologicalSpace β
f : α → β
hf : Topology.IsInducing f
m : WeakPseudoEMetricSpace β
hα : PseudoEMetricSpace α := PseudoEMetricSpace.ofEDist (fun x y ↦ edist (f x) (f y)) ⋯ ⋯ ⋯
hβ : PseudoEMetricSpace β := PseudoEMetricSpace.ofEDist edist ⋯ ⋯ ⋯
⊢ (uniformSpaceOfEDist (fun x y ↦ edist (f x) (f y)) ⋯ ⋯ ⋯).toTopologicalSpace ≤ TopologicalSpace.induced f n
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α✝ : Type u
β✝ : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u_2
β : Type u_3
e : TopologicalSpace α
n : TopologicalSpace β
f : α → β
hf : Topology.IsInducing f
m : WeakPseudoEMetricSpace β
hα : PseudoEMetricSpace α := PseudoEMetricSpace.ofEDist (fun x y ↦ edist (f x) (f y)) ⋯ ⋯ ⋯
hβ : PseudoEMetricSpace β := PseudoEMetricSpace.ofEDist edist ⋯ ⋯ ⋯
s : Set β
hs : ∀ x ∈ s, ∃ ε > 0, Metric.eball x ε ⊆ s
⊢ ∀ x ∈ f ⁻¹' s, ∃ ε > 0, Metric.eball x ε ⊆ f ⁻¹' s

Tactic:
]

Goal state after:
α✝ : Type u
β✝ : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u_2
β : Type u_3
e : TopologicalSpace α
n : TopologicalSpace β
f : α → β
hf : Topology.IsInducing f
m : WeakPseudoEMetricSpace β
hα : PseudoEMetricSpace α := PseudoEMetricSpace.ofEDist (fun x y ↦ edist (f x) (f y)) ⋯ ⋯ ⋯
hβ : PseudoEMetricSpace β := PseudoEMetricSpace.ofEDist edist ⋯ ⋯ ⋯
s : Set β
hs : ∀ x ∈ s, ∃ ε > 0, Metric.eball x ε ⊆ s
⊢ ∀ x ∈ f ⁻¹' s, ∃ ε > 0, Metric.eball x ε ⊆ f ⁻¹' s
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case inr.inl.inl
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
i : ι
hi : f i = 0
⊢ 0 = 0

pp: case inr.inl.inl
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
i : ι
hi : f i = 0
⊢ ∀ (b : ℝ≥0∞), ⊥ < b → ∃ i, ∞ * f i < b

pp: case inr.inl.inl
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
i : ι
hi : f i = 0
⊢ ∀ (b : ℝ≥0∞), ⊥ < b → ∃ i, f i < b

Tactic:
]

Goal state after:
case inr.inl.inl
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
i : ι
hi : f i = 0
⊢ ∀ (b : ℝ≥0∞), ⊥ < b → ∃ i, ∞ * f i < b

case inr.inl.inl
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
i : ι
hi : f i = 0
⊢ ∀ (b : ℝ≥0∞), ⊥ < b → ∃ i, f i < b
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: 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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  v ∈
    {v |
      ¬↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
            ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
          0}
hv_irred : Irreducible v.asIdeal
h_notMem : ¬(v.asIdeal ∣ J ∨ v.asIdeal ∣ Ideal.span {a})
⊢ False

Tactic:
]

Goal state after:
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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  v ∈
    {v |
      ¬↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
            ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
          0}
hv_irred : Irreducible v.asIdeal
h_notMem : ¬(v.asIdeal ∣ J ∨ v.asIdeal ∣ Ideal.span {a})
⊢ False
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α✝ : Type u
β✝ : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u_2
β : Type u_3
e : TopologicalSpace α
n : TopologicalSpace β
f : α → β
hf : Topology.IsInducing f
m : WeakPseudoEMetricSpace β
x : α
r : ℝ≥0∞
u : Set β
hu : IsOpen[n] u
uy : Subtype.val ⁻¹' u = Metric.eball (f x) ∞ ↓∩ Metric.eball (f x) r
⊢ IsOpen[instTopologicalSpaceSubtype] (Metric.eball x ∞ ↓∩ Metric.eball x r)

Tactic:
]

Goal state after:
α✝ : Type u
β✝ : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α✝
γ : Type w
inst✝ : EMetricSpace γ
α : Type u_2
β : Type u_3
e : TopologicalSpace α
n : TopologicalSpace β
f : α → β
hf : Topology.IsInducing f
m : WeakPseudoEMetricSpace β
x : α
r : ℝ≥0∞
u : Set β
hu : IsOpen[n] u
uy : Subtype.val ⁻¹' u = Metric.eball (f x) ∞ ↓∩ Metric.eball (f x) r
⊢ IsOpen[instTopologicalSpaceSubtype] (Metric.eball x ∞ ↓∩ Metric.eball x r)
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case inr.inl.inr
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
hf : ¬∃ i, f i = 0
⊢ ∀ (i : ι), ∞ * f i = ∞

Tactic:
]

Goal state after:
case inr.inl.inr
ι : Sort u_1
f : ι → ℝ≥0∞
hinfty : ∞ = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0
h₀ : ∞ = 0 → Nonempty ι
ha₀ : ∞ ≠ 0
hf : ¬∃ i, f i = 0
⊢ ∀ (i : ι), ∞ * f i = ∞
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: 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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  v ∈
    {v |
      ¬↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
            ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
          0}
hv_irred : Irreducible v.asIdeal
h_notMem :
  (Associates.mk v.asIdeal).count (Associates.mk J).factors = 0 ∧
    (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors = 0
⊢ False

Tactic:
]

Goal state after:
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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  v ∈
    {v |
      ¬↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
            ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
          0}
hv_irred : Irreducible v.asIdeal
h_notMem :
  (Associates.mk v.asIdeal).count (Associates.mk J).factors = 0 ∧
    (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors = 0
⊢ False
Module: Mathlib.Topology.MetricSpace.Defs

Goal state before:
pp: γ : Type w
inst✝ : MetricSpace γ
x y : γ
⊢ x = y ↔ x = y

Tactic:
]

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
⊢ ∅.Subsingleton

Tactic:
]

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

Goal state before:
pp: case inr
γ : Type w
inst✝ : MetricSpace γ
x : γ
hr : 0 ≤ 0
⊢ {x}.Subsingleton

Tactic:
]

Goal state after:
case inr
γ : Type w
inst✝ : MetricSpace γ
x : γ
hr : 0 ≤ 0
⊢ {x}.Subsingleton
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ι : Sort u_1
κ : Sort u_2
f : ι → ℝ≥0∞
a : ℝ≥0∞
g : κ → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
hg : ∃ j, g j ≠ ∞
ha : ∀ (i : ι) (j : κ), a ≤ f i * g j
⊢ a ≤ (⨅ i, f ↑i) * ⨅ j, g j

Tactic:
]

Goal state after:
ι : Sort u_1
κ : Sort u_2
f : ι → ℝ≥0∞
a : ℝ≥0∞
g : κ → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
hg : ∃ j, g j ≠ ∞
ha : ∀ (i : ι) (j : κ), a ≤ f i * g j
⊢ a ≤ (⨅ i, f ↑i) * ⨅ j, g j
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: 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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv : ¬0 = 0
hv_irred : Irreducible v.asIdeal
h_notMem :
  (Associates.mk v.asIdeal).count (Associates.mk J).factors = 0 ∧
    (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors = 0
⊢ False

Tactic:
]

Goal state after:
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
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
ha_ne_zero : Ideal.span {a} ≠ 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv : ¬0 = 0
hv_irred : Irreducible v.asIdeal
h_notMem :
  (Associates.mk v.asIdeal).count (Associates.mk J).factors = 0 ∧
    (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors = 0
⊢ False
Module: Mathlib.Topology.MetricSpace.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
γ : Type w
inst✝ : MetricSpace γ
x✝² x✝¹ x✝ : Empty
⊢ 0 ≤ 0

Tactic:
]

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

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝¹ : PseudoMetricSpace α
γ : Type w
inst✝ : MetricSpace γ
x✝² x✝¹ x✝ : PUnit.{u + 1}
⊢ 0 ≤ 0

Tactic:
]

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

Goal state before:
pp: ι : Sort u_1
κ : Sort u_2
f : ι → ℝ≥0∞
a : ℝ≥0∞
g : κ → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
ha : ∀ (i : ι) (j : κ), a ≤ f i * g j
this✝ : Nonempty { a // f a ≠ ∞ }
this : Nonempty κ
hg : ⨅ j, g j ≠ ∞
⊢ ∀ (i : { i // f i ≠ ∞ }), a ≤ f ↑i * ⨅ j, g j

Tactic:
]

Goal state after:
ι : Sort u_1
κ : Sort u_2
f : ι → ℝ≥0∞
a : ℝ≥0∞
g : κ → ℝ≥0∞
hf : ∃ i, f i ≠ ∞
ha : ∀ (i : ι) (j : κ), a ≤ f i * g j
this✝ : Nonempty { a // f a ≠ ∞ }
this : Nonempty κ
hg : ⨅ j, g j ≠ ∞
⊢ ∀ (i : { i // f i ≠ ∞ }), a ≤ f ↑i * ⨅ j, g j
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:
]

Goal state after:
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)
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:
]

Goal state after:
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)