mathlib-initiative/mathlib-tactics huggingface dataset:
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
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ (have a := choose ⋯;
    have J := choose ⋯;
    ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
      ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
    ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
      ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)

Tactic:
sub_eq_sub_iff_add_eq_add,

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
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_aJ : ↑(J₁ * Ideal.span {a}) = ↑(J * Ideal.span {a₁})
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
h_J₁_ne_zero : J₁ ≠ 0
h_a_ne_zero : Ideal.span {a} ≠ 0
h_J_ne_zero : J ≠ 0
h_a₁' : spanSingleton R⁰ ((algebraMap R K) a₁) ≠ 0
h_a' : spanSingleton R⁰ ((algebraMap R K) a) ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
⊢ ↑((Associates.mk v.asIdeal).count (Associates.mk (choose ⋯)).factors) +
      ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) +
      ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {choose ⋯})).factors)
Module: Mathlib.Data.ENNReal.Operations

Goal state before:
pp: s : Set ℝ≥0∞
hs : ∀ r ∈ s, r ≠ ∞
⊢ (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)

Tactic:
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
  simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using! (toNNReal_iInf hf)

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
v : HeightOneSpectrum R
⊢ 1 = spanSingleton R⁰ 1⁻¹

Tactic:
inv_one,

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
v : HeightOneSpectrum R
⊢ 1 = spanSingleton R⁰ 1
Module: Mathlib.Data.EReal.Basic

Goal state before:
pp: case pos
x : ℝ≥0∞
h_top : x = ∞
⊢ ⊤.toENNReal = ∞

Tactic:
toENNReal_top

Goal state after:
case pos
x : ℝ≥0∞
h_top : x = ∞
⊢ ∞ = ∞
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
v : HeightOneSpectrum R
hv : ↑v.asIdeal ≠ 0
⊢ ↑v.asIdeal = spanSingleton R⁰ 1⁻¹ * ↑v.asIdeal

Tactic:
inv_one,

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
v : HeightOneSpectrum R
hv : ↑v.asIdeal ≠ 0
⊢ ↑v.asIdeal = spanSingleton R⁰ 1 * ↑v.asIdeal
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
⊢ a < b → b⁻¹ < a⁻¹

Tactic:
h

Goal state after:
a b : ℝ≥0∞
h : a < b
⊢ b⁻¹ < a⁻¹
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
v w : HeightOneSpectrum R
hw : w ≠ v
⊢ ↑w.asIdeal = spanSingleton R⁰ 1⁻¹ * ↑w.asIdeal

Tactic:
inv_one,

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
v w : HeightOneSpectrum R
hw : w ≠ v
⊢ ↑w.asIdeal = spanSingleton R⁰ 1 * ↑w.asIdeal
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
v w : HeightOneSpectrum R
hw : w ≠ v
hw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal
hw_ne_zero : ↑w.asIdeal ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
hw' : Irreducible (Associates.mk w.asIdeal)
⊢ ↑((Associates.mk v.asIdeal).count (Associates.mk w.asIdeal).factors) = 0

Tactic:
natCast_eq_zero,

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
v w : HeightOneSpectrum R
hw : w ≠ v
hw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal
hw_ne_zero : ↑w.asIdeal ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
hw' : Irreducible (Associates.mk w.asIdeal)
⊢ (Associates.mk v.asIdeal).count (Associates.mk w.asIdeal).factors = 0
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ c / b = d / a ↔ a * c = b * d

Tactic:
eq_div_iff ha ha'

Goal state after:
a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ a * (c / b) = d ↔ a * c = b * d
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u_2
m m' : PseudoEMetricSpace α
h : m.toEDist = m'.toEDist
⊢ m = m'

Tactic:
by
  obtain ⟨_, _, _, U, hU⟩ := m; rename EDist α => ed
  obtain ⟨_, _, _, U', hU'⟩ := m'; rename EDist α => ed'
  congr 1
  exact UniformSpace.ext (((show ed = ed' from h) ▸ hU).trans hU'.symm)

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

Goal state before:
pp: case e'_2.h
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
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
v : HeightOneSpectrum R
hv : v ∈ mulSupport fun v ↦ ↑v.asIdeal ^ exps v
⊢ v ∈ {x | (fun v ↦ exps v = 0) x}ᶜ

Tactic:
h

Goal state after:
case e'_2.h
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
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
v : HeightOneSpectrum R
hv : v ∈ mulSupport fun v ↦ ↑v.asIdeal ^ exps v
h : v ∈ {x | (fun v ↦ exps v = 0) x}
⊢ False
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case ofNat
x : ℝ≥0∞
a✝ : ℕ
⊢ x⁻¹ ^ Int.ofNat a✝ = (x ^ Int.ofNat a✝)⁻¹

Tactic:
simp [ENNReal.inv_pow]

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

Goal state before:
pp: case negSucc
x : ℝ≥0∞
a✝ : ℕ
⊢ x⁻¹ ^ Int.negSucc a✝ = (x ^ Int.negSucc a✝)⁻¹

Tactic:
simp [ENNReal.inv_pow]

Goal state after:
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
⊢ ∞ * ⨅ i, f i = ⨅ i, ∞ * f i

Tactic:
rw [iInf_eq_bot.2, iInf_eq_bot.2, bot_eq_zero, mul_zero] <;>
        exact fun _ _ ↦ ⟨i, by simpa [hi]⟩

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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
a : R
K : Ideal R
e : J * K = Ideal.span {a}
e' : I + K = ⊤
⊢ J * (I + K) = J

Tactic:
e',

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
a : R
K : Ideal R
e : J * K = Ideal.span {a}
e' : I + K = ⊤
⊢ J * ⊤ = J
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
s : Set α
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s

Tactic:
by
  simp only [isOpen_iff_mem_nhds, mem_nhds_iff]

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

Goal state before:
pp: z : ℝ≥0
⊢ nndist z 0 = z

Tactic:
rw [nndist_comm]
  exact NNReal.nndist_zero_eq_val z

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

Goal state before:
pp: α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
s : Set α
a : α
hsa : a ∈ upperBounds s
hsf : ClusterPt a (𝓟 s)
⊢ IsLUB s a

Tactic:
ClusterPt,

Goal state after:
α : Type u_1
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : OrderTopology α
s : Set α
a : α
hsa : a ∈ upperBounds s
hsf : (𝓝 a ⊓ 𝓟 s).NeBot
⊢ IsLUB s a
Module: Mathlib.Topology.Order.IsLUB

Goal state before:
pp: γ : Type u_2
α : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : ClosedIicTopology α
f : γ → α
inst✝ : TopologicalSpace γ
S : Set γ
hS : Dense S
hf : Continuous[inst✝, inst✝³] f
h : BddAbove (range f)
⊢ sSup (range fun s ↦ f ↑s) = sSup (range fun i ↦ f i)

Tactic:
obtain (_ | _) := isEmpty_or_nonempty γ

Goal state after:
case inl
γ : Type u_2
α : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : ClosedIicTopology α
f : γ → α
inst✝ : TopologicalSpace γ
S : Set γ
hS : Dense S
hf : Continuous[inst✝, inst✝³] f
h : BddAbove (range f)
h✝ : IsEmpty γ
⊢ sSup (range fun s ↦ f ↑s) = sSup (range fun i ↦ f i)

case inr
γ : Type u_2
α : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLattice α
inst✝¹ : ClosedIicTopology α
f : γ → α
inst✝ : TopologicalSpace γ
S : Set γ
hS : Dense S
hf : Continuous[inst✝, inst✝³] f
h : BddAbove (range f)
h✝ : Nonempty γ
⊢ sSup (range fun s ↦ f ↑s) = sSup (range fun i ↦ f i)
Module: Mathlib.Topology.MetricSpace.Pseudo.Pi

Goal state before:
pp: β : Type u_2
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
f g : (b : β) → X b
r : ℝ≥0
hr : 0 < r
⊢ ((∃ x, ¬nndist (f x) (g x) < r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r) ↔
    (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r

Tactic:
not_lt,

Goal state after:
β : Type u_2
X : β → Type u_3
inst✝¹ : Fintype β
inst✝ : (b : β) → PseudoMetricSpace (X b)
f g : (b : β) → X b
r : ℝ≥0
hr : 0 < r
⊢ ((∃ x, r ≤ nndist (f x) (g x)) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r) ↔
    (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r
Module: Mathlib.Topology.MetricSpace.Basic

Goal state before:
pp: α : Type u
β : Type v
X✝ : Type u_1
inst✝¹ : PseudoMetricSpace α
γ : Type w
inst✝ : MetricSpace γ
X : Type u_2
m : EMetricSpace X
d : X → X → ℝ≥0∞
hd : d = edist
⊢ PseudoEMetricSpace.toUniformSpace = PseudoEMetricSpace.toUniformSpace

Tactic:
by
  simp +instances [replaceEDist_eq]

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
H : (x '' t).Countable
c : β
hc : c ∈ t
d : β
hd : d ∈ t
⊢ x c = x d → c = d

Tactic:
hcd

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
H : (x '' t).Countable
c : β
hc : c ∈ t
d : β
hd : d ∈ t
hcd : x c = x d
⊢ c = d
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: case h
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 := ⋯
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).PairwiseDisjoint fun x ↦ Ioo x (y (f x))

Tactic:
simp only [PairwiseDisjoint, Set.Pairwise, mem_image, onFun, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂]

Goal state after:
case h
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
⊢ ∀ a ∈ t, ∀ a_2 ∈ t, x a ≠ x a_2 → Disjoint (Ioo (x a) (y (f (x a)))) (Ioo (x a_2) (y (f (x a_2))))
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: case h
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
⊢ x c ≠ x d → Disjoint (Ioo (x c) (y (f (x c)))) (Ioo (x d) (y (f (x d))))

Tactic:
hcd

Goal state after:
case h
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
⊢ Disjoint (Ioo (x c) (y (f (x c)))) (Ioo (x d) (y (f (x d))))
Module: Mathlib.Topology.Order.Compact

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : PartialOrder α
h : ∀ {a b : α}, a < b → IsCompact (Icc a b)
a✝ b✝ : α
hab : a✝ ≤ b✝
⊢ a✝ = b✝ → IsCompact (Icc a✝ b✝)

Tactic:
rintro rfl; simp

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

Goal state before:
pp: α : Type u_2
inst✝² : LinearOrder α
inst✝¹ : TopologicalSpace α
inst✝ : ClosedIicTopology α
s : Set α
hs : IsCompact s
ne_s : s.Nonempty
⊢ ∃ x, IsLeast s x

Tactic:
by
  have : Nonempty s := ne_s.to_subtype
  suffices (s ∩ ⋂ x ∈ s, Iic x).Nonempty from
    ⟨this.choose, this.choose_spec.1, mem_iInter₂.mp this.choose_spec.2⟩
  rw [biInter_eq_iInter]
  by_contra H
  rw [not_nonempty_iff_eq_empty] at H
  rcases hs.elim_directed_family_closed (fun x : s => Iic ↑x) (fun x => isClosed_Iic) H
      (Monotone.directed_ge fun _ _ h => Iic_subset_Iic.mpr h) with ⟨x, hx⟩
  exact not_nonempty_iff_eq_empty.mpr hx ⟨x, x.2, le_rfl⟩

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁶ : ConditionallyCompleteLinearOrder α
inst✝⁵ : TopologicalSpace α
inst✝⁴ : OrderTopology α
inst✝³ : TopologicalSpace β
inst✝² : DenselyOrdered α
inst✝¹ : ConditionallyCompleteLinearOrder β
inst✝ : OrderTopology β
f : α → β
a b c : α
h : ContinuousOn f (Icc a b)
hc : c ∈ Icc a b
⊢ sInf (f '' Icc a b) ≤ f c

Tactic:
by
  have := mem_image_of_mem f hc
  rw [h.image_Icc (hc.1.trans hc.2)] at this
  exact this.1

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

Goal state before:
pp: E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ ‖1⁻¹ * a‖ = ‖a‖

Tactic:
inv_one,

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
a : E
⊢ ‖1 * a‖ = ‖a‖
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
⊢ {s | IsPreconnected s} ⊆
    range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪
      (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})

Tactic:
intro s hs
  rcases hs.mem_intervals with (hs | hs | hs | hs | hs | hs | hs | hs | hs | hs) <;> rw [hs] <;>
    simp only [union_insert, union_singleton, mem_insert_iff, mem_union, mem_range, Prod.exists,
      uncurry_apply_pair, exists_apply_eq_apply, true_or, or_true, exists_apply_eq_apply2]

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

Goal state before:
pp: α : Type u
inst✝³ : TopologicalSpace α
inst✝² : ConditionallyCompleteLinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
s : Set α
hs : IsClosed[inst✝³] (s ∩ Icc a b)
ha : a ∈ s
h : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t
hab : a ≤ b
A : Set α := {t | t ∈ Icc a b ∧ Icc a t ⊆ s}
a_mem : a ∈ A
bdd_A : BddAbove A
t₁ : α := sSup A
t₁_mem : t₁ ∈ Icc a b
⊢ t₁ ∈ A

Tactic:
refine ⟨t₁_mem, fun t ht ↦ ?_⟩
    rcases ht.2.eq_or_lt with rfl | h
    · have : closure A ⊆ s ∩ Icc a b := by
        apply (closure_subset_iff hs).2 (fun t ht ↦ ⟨?_, ht.1⟩)
        have : t ∈ Icc a t := ⟨ht.1.1, le_rfl⟩
        exact ht.2 this
      apply this.trans inter_subset_left
      exact csSup_mem_closure ⟨a, a_mem⟩ bdd_A
    · obtain ⟨c, cA, tc⟩ : ∃ c ∈ A, t < c := (lt_csSup_iff bdd_A ⟨a, a_mem⟩).1 h
      apply cA.2
      exact ⟨ht.1, tc.le⟩

Goal state after:
Module: Mathlib.Order.Filter.AtTopBot.Archimedean

Goal state before:
pp: α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
n : ℕ
hn : 1 ≤ n • r
⊢ ∀ᶠ (a : α) in l, b ≤ r * f a

Tactic:
nsmul_eq_mul'

Goal state after:
α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝³ : Semiring R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Archimedean R
hr : 0 < r
hf : Tendsto f l atTop
b : R
n : ℕ
hn : 1 ≤ r * ↑n
⊢ ∀ᶠ (a : α) in l, b ≤ r * f a
Module: Mathlib.Topology.MetricSpace.Bounded

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
f : ℕ → α
hf : CauchySeq f
⊢ Cauchy (map f Filter.cofinite)

Tactic:
Nat.cofinite_eq_atTop

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
f : ℕ → α
hf : CauchySeq f
⊢ Cauchy (map f atTop)
Module: Mathlib.Topology.Instances.Int

Goal state before:
pp: m n : ℤ
hne : m ≠ n
⊢ 0 < |m - n|

Tactic:
abs_pos,

Goal state after:
m n : ℤ
hne : m ≠ n
⊢ m - n ≠ 0
Module: Mathlib.GroupTheory.Archimedean

Goal state before:
pp: G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
H : Subgroup G
hbot : H ≠ ⊥
a : G
h₀ : 1 < a
hd : Disjoint (↑H) (Ioo 1 a)
hex : ∀ g > 1, ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))
this : ∃ n, (↑H ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (a ^ m) (a ^ (m + 1))).Nonempty
x : G
hxH : x ∈ ↑H
hnx : a ^ n < x
hxn : x ≤ a ^ (n + 1)
hxmin : ¬∃ b, IsLeast {g | g ∈ H ∧ 1 < g} b
⊢ False

Tactic:
simp only [IsLeast, not_and, mem_ofPred_eq, mem_lowerBounds, not_exists, not_forall,
    not_le] at hxmin

Goal state after:
G : Type u_1
inst✝³ : CommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedMonoid G
inst✝ : MulArchimedean G
H : Subgroup G
hbot : H ≠ ⊥
a : G
h₀ : 1 < a
hd : Disjoint (↑H) (Ioo 1 a)
hex : ∀ g > 1, ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))
this : ∃ n, (↑H ∩ Ioc (a ^ n) (a ^ (n + 1))).Nonempty
n : ℕ
hmin : ∀ m < n, ¬(↑H ∩ Ioc (a ^ m) (a ^ (m + 1))).Nonempty
x : G
hxH : x ∈ ↑H
hnx : a ^ n < x
hxn : x ≤ a ^ (n + 1)
hxmin : ∀ (x : G), x ∈ H ∧ 1 < x → ∃ x_1, ∃ (_ : x_1 ∈ H ∧ 1 < x_1), x_1 < x
⊢ False
Module: Mathlib.Data.ENNReal.BigOperators

Goal state before:
pp: ι : Type u_1
f : ι → ℝ≥0∞
s : Finset ι
hf : (↑s).Pairwise fun i j ↦ f i ≠ 0 ∨ f j ≠ ∞
⊢ (∏ i ∈ s, f i)⁻¹ = ∏ i ∈ s, (f i)⁻¹

Tactic:
induction s using Finset.cons_induction with
  | empty => simp
  | cons i s hi ih => ?_

Goal state after:
case cons
ι : Type u_1
f : ι → ℝ≥0∞
s✝ : Finset ι
i : ι
s : Finset ι
hi : i ∉ s
ih : ((↑s).Pairwise fun i j ↦ f i ≠ 0 ∨ f j ≠ ∞) → (∏ i ∈ s, f i)⁻¹ = ∏ i ∈ s, (f i)⁻¹
hf : (↑(Finset.cons i s hi)).Pairwise fun i j ↦ f i ≠ 0 ∨ f j ≠ ∞
⊢ (∏ i ∈ Finset.cons i s hi, f i)⁻¹ = ∏ i ∈ Finset.cons i s hi, (f i)⁻¹
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: α : Type u_1
f : Filter α
inst✝¹ : LinearOrder α
inst✝ : f.NeBot
t : α
ht : ∀ (a : α), (∀ᶠ (x : α) in f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) t a
t' : α
ht' : t' < t
ev : ∀ᶠ (x : α) in f, ¬(fun x ↦ t' ≤ x) x
a✝ : α
⊢ ¬t' ≤ a✝ → a✝ ≤ t'

Tactic:
h

Goal state after:
α : Type u_1
f : Filter α
inst✝¹ : LinearOrder α
inst✝ : f.NeBot
t : α
ht : ∀ (a : α), (∀ᶠ (x : α) in f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) t a
t' : α
ht' : t' < t
ev : ∀ᶠ (x : α) in f, ¬(fun x ↦ t' ≤ x) x
a✝ : α
h : ¬t' ≤ a✝
⊢ a✝ ≤ t'
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: α : Type u_5
f : Filter α
R : Type u_6
inst✝³ : Preorder R
inst✝² : Add R
inst✝¹ : AddLeftMono R
inst✝ : AddRightMono R
u v : α → R
u_bdd_le : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
v_bdd_le : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (u + v)

Tactic:
by
  obtain ⟨U, hU⟩ := u_bdd_le
  obtain ⟨V, hV⟩ := v_bdd_le
  use U + V
  simp only [eventually_map, Pi.add_apply] at hU hV ⊢
  filter_upwards [hU, hV] with a hu hv using add_le_add hu hv

Goal state after:
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: R : Type u_5
S : Type u_6
F : Filter R
inst✝² : LinearOrder R
inst✝¹ : LinearOrder S
f : R → S
f_decr : Antitone f
inst✝ : F.NeBot
cobdd : IsCobounded (fun x1 x2 ↦ x2 ≤ x1) F
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f

Tactic:
obtain ⟨l, hl⟩ := IsCobounded.frequently_le cobdd

Goal state after:
R : Type u_5
S : Type u_6
F : Filter R
inst✝² : LinearOrder R
inst✝¹ : LinearOrder S
f : R → S
f_decr : Antitone f
inst✝ : F.NeBot
cobdd : IsCobounded (fun x1 x2 ↦ x2 ≤ x1) F
l : R
hl : ∃ᶠ (x : R) in F, x ≤ l
⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) F f
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
inst✝¹ : CompleteDistribLattice α
f : Filter β
u : β → α
s : Set β
inst✝ : DecidablePred fun x ↦ x ∈ s
v : β → α
a✝ : β
⊢ a✝ ∈ s → s.piecewise u v a✝ = u a✝

Tactic:
h

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
inst✝¹ : CompleteDistribLattice α
f : Filter β
u : β → α
s : Set β
inst✝ : DecidablePred fun x ↦ x ∈ s
v : β → α
a✝ : β
h : a✝ ∈ s
⊢ s.piecewise u v a✝ = u a✝
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : CompleteDistribLattice α
f : Filter β
u : β → α
s : Set β
inst✝ : DecidablePred fun x ↦ x ∈ s
v : β → α
a✝ : β
⊢ a✝ ∉ s → s.piecewise u v a✝ = v a✝

Tactic:
h

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
inst✝¹ : CompleteDistribLattice α
f : Filter β
u : β → α
s : Set β
inst✝ : DecidablePred fun x ↦ x ∈ s
v : β → α
a✝ : β
h : a✝ ∉ s
⊢ s.piecewise u v a✝ = v a✝
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: case symm
x✝ y✝ : EReal
h : (x✝ * y✝).toReal = x✝.toReal * y✝.toReal
⊢ (y✝ * x✝).toReal = y✝.toReal * x✝.toReal

Tactic:
rwa [mul_comm, EReal.mul_comm]

Goal state after:
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: a b : EReal
⊢ a * b = 0 → a = 0 ∨ b = 0

Tactic:
h

Goal state after:
a b : EReal
h : a * b = 0
⊢ a = 0 ∨ b = 0