mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
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 [zpow_negSucc, Int.ofNat_eq_natCast, zpow_natCast]
    refine (ENNReal.inv_le_one.2 ?_).trans ?_ <;> exact one_le_pow_of_one_le' hx _

Goal state after:
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.Data.ENNReal.Inv

Goal state before:
pp: x : ℝ≥0∞
hx : x ≠ 0
h'x : x ≠ ∞
m n : ℤ
⊢ x ^ (m + n) = x ^ m * x ^ n

Tactic:
by
  lift x to ℝ≥0 using h'x
  replace hx : x ≠ 0 := by simpa only [Ne, coe_eq_zero] using hx
  simp only [← coe_zpow hx, zpow_add₀ hx, coe_mul]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ y ∈ ball x ε ↔ dist x y < ε

Tactic:
dist_comm,

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ y ∈ closedBall x ε ↔ dist x y ≤ ε

Tactic:
dist_comm,

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ y ∈ closedBall x ε ↔ dist y x ≤ ε
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ y ∈ sphere x ε ↔ dist x y = ε

Tactic:
dist_comm,

Goal state after:
α : Type u
inst✝ : PseudoMetricSpace α
x y : α
ε : ℝ
⊢ y ∈ sphere x ε ↔ dist y x = ε
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x r y : ℝ
⊢ dist y x < r ↔ y ∈ Ioo (x - r) (x + r)

Tactic:
dist_comm,

Goal state after:
x r y : ℝ
⊢ dist x y < r ↔ y ∈ Ioo (x - r) (x + r)
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: x r y : ℝ
⊢ dist y x ≤ r ↔ y ∈ Icc (x - r) (x + r)

Tactic:
dist_comm,

Goal state after:
x r y : ℝ
⊢ dist x y ≤ r ↔ y ∈ Icc (x - r) (x + r)
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
s : Finset (HeightOneSpectrum R) := ⋯.toFinset
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : HeightOneSpectrum R → R
ha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ p ∈ s, a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
e : p = ⊥
⊢ I = 0

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
x y : α
⊢ Inseparable x y ↔ dist x y = 0

Tactic:
by
  rw [Metric.inseparable_iff_nndist, dist_nndist, NNReal.coe_eq_zero]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x : α
y₁ y₂ : β
⊢ dist (x, y₁) (x, y₂) = dist y₁ y₂

Tactic:
by
  simp [Prod.dist_eq]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
x₁ x₂ : α
y : β
⊢ dist (x₁, y) (x₂, y) = dist x₁ x₂

Tactic:
by
  simp [Prod.dist_eq]

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

Goal state before:
pp: case h₁
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
hab : a ≠ b
⊢ closure[inst✝³] (Ioc a b) ⊆ Icc a b

pp: case h₂
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure[inst✝³] (Ioc a b)

Tactic:
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc

Goal state after:
case h₂
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
hab : a ≠ b
⊢ Icc a b ⊆ closure[inst✝³] (Ioc a b)
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
⊢ nndist f g < r ↔ ∀ (b : β), nndist (f b) (g b) < r

Tactic:
simp [nndist_pi_def, Finset.sup_lt_iff hr]

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsClosed[inst✝³] (Ico a b)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsClosed[inst✝³] (Ioc a b)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsClosed[inst✝³] (Ioo a b)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : DenselyOrdered α
inst✝¹ : OrderTopology α
inst✝ : FirstCountableTopology α
x y : α
hy : y < x
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x)

Tactic:
have hx : x ∉ Ioo y x := fun h => (lt_irrefl x h.2).elim

Goal state after:
α : Type u_3
inst✝⁴ : LinearOrder α
inst✝³ : TopologicalSpace α
inst✝² : DenselyOrdered α
inst✝¹ : OrderTopology α
inst✝ : FirstCountableTopology α
x y : α
hy : y < x
hx : x ∉ Ioo y x
⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x)
Module: Mathlib.Topology.Order.IsLUB

Goal state before:
pp: case inl.inr
α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
d : α
h : ∀ (b : α), d < b → ∀ᶠ (x : γ) in l, f x < b
hd : IsGLB (Ioi d) d
h✝ : ∃ b, d < b
u : ℕ → α
hu_tt : Tendsto u atTop (𝓝 d)
hu_gt : ∀ (n : ℕ), u n ∈ Ioi d
⊢ ∀ᶠ (x : γ) in l, f x ≤ d

Tactic:
replace h := fun n ↦ h (u n) (by grind)

Goal state after:
case inl.inr
α : Type u_1
γ : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : FirstCountableTopology α
l : Filter γ
inst✝ : CountableInterFilter l
f : γ → α
d : α
hd : IsGLB (Ioi d) d
h✝ : ∃ b, d < b
u : ℕ → α
hu_tt : Tendsto u atTop (𝓝 d)
hu_gt : ∀ (n : ℕ), u n ∈ Ioi d
h : ∀ (n : ℕ), ∀ᶠ (x : γ) in l, f x < u n
⊢ ∀ᶠ (x : γ) in l, f x ≤ d
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 J : FractionalIdeal R⁰ K
⊢ I.divMod 0 J = 0

Tactic:
by
  simp [divMod]

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 J : FractionalIdeal R⁰ K
⊢ I.divMod J 0 = 0

Tactic:
by
  simp [divMod]

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

Goal state before:
pp: case refine_1
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x y : α
h_nonempty : (Ioo y x).Nonempty
Mf : MonotoneOn f (Ioo y x)
h_bdd : BddAbove (f '' Ioo y x)
l : β
hl : l < sSup (f '' Ioo y x)
z : α
lz : l < f z
yz : y < z
zx : z < x
⊢ ∀ᶠ (b : α) in 𝓝[<] x, l < f b

Tactic:
filter_upwards [Ioo_mem_nhdsLT zx] with w hw

Goal state after:
α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x y : α
h_nonempty : (Ioo y x).Nonempty
Mf : MonotoneOn f (Ioo y x)
h_bdd : BddAbove (f '' Ioo y x)
l : β
hl : l < sSup (f '' Ioo y x)
z : α
lz : l < f z
yz : y < z
zx : z < x
w : α
hw : w ∈ Ioo z x
⊢ l < f w
Module: Mathlib.Topology.Order.Monotone

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝⁵ : LinearOrder α
inst✝⁴ : TopologicalSpace α
inst✝³ : OrderTopology α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
x : α
Mf : MonotoneOn f (Iio x)
h_bdd : BddAbove (f '' Iio x)
h : (Iio x).Nonempty
l : β
hl : l < sSup (f '' Iio x)
⊢ ∃ a < x, l < f a

Tactic:
by
      simpa only [mem_image, exists_prop, exists_exists_and_eq_and] using!
        exists_lt_of_lt_csSup (h.image _) hl

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsCompact (Ico a b)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsCompact (Ioc a b)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a b : α
⊢ b ≤ a → IsCompact (Ioo a b)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : ClosedIicTopology α
s : Set β
hs : IsCompact s
ne_s : s.Nonempty
f : β → α
hf : ContinuousOn f s
⊢ ∃ x ∈ s, IsMinOn f s x

Tactic:
by
  rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
  refine ⟨x, hxs, forall_mem_image.1 (fun _ hb => hx <| mem_image_of_mem f ?_)⟩
  rwa [(image_id' s).symm]

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

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

Tactic:
dist_comm,

Goal state after:
E : Type u_5
inst✝ : SeminormedGroup E
a b : E
⊢ dist b a = ‖b⁻¹ * a‖
Module: Mathlib.Analysis.Normed.Group.Basic

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

Tactic:
dist_comm,

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hb : b ∈ s
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty
this : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (toDual b) (toDual a))
⊢ toDual b ∈ ⇑ofDual ⁻¹' s

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hb : b ∈ s
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty
this : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (toDual b) (toDual a))
⊢ toDual b ≤ toDual a

Tactic:
by simp_all

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

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

Tactic:
by
  simpa [-tsub_le_iff_right, tsub_le_iff_left, two_mul, add_assoc]
    using norm_mul₃_le' (a := (u / v)) (b := v) (c := v)

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

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hb : b ∈ s
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty
this : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (toDual b) (toDual a))
x : αᵒᵈ
hx : x ∈ ⇑ofDual ⁻¹' (s ∩ Ioc a b)
⊢ (⇑ofDual ⁻¹' s ∩ Ioc x (toDual a)).Nonempty

Tactic:
← Equiv.image_symm_eq_preimage,

Goal state after:
α : Type u
inst✝² : TopologicalSpace α
inst✝¹ : ConditionallyCompleteLinearOrder α
inst✝ : OrderTopology α
a b : α
s : Set α
hs : IsClosed[inst✝²] (s ∩ Icc a b)
hb : b ∈ s
hab : a ≤ b
hgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty
this : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (toDual b) (toDual a))
x : αᵒᵈ
hx : x ∈ ⇑ofDual.symm '' (s ∩ Ioc a b)
⊢ (⇑ofDual ⁻¹' s ∩ Ioc x (toDual a)).Nonempty
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: case inl
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
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
t' : α
ht' : t' ∈ Icc a t
h't' : t' ≤ t₁
⊢ t' ∈ s

pp: case inr
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
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
t' : α
ht' : t' ∈ Icc a t
h't' : t₁ < t'
⊢ t' ∈ s

Tactic:
· exact ht₁ ⟨ht'.1, h't'⟩

Goal state after:
case inr
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
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
t' : α
ht' : t' ∈ Icc a t
h't' : t₁ < t'
⊢ t' ∈ s
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
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
this : t ∈ A
⊢ False

Tactic:
have : t ≤ t₁ := le_csSup bdd_A this

Goal state after:
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
ht₁ : Icc a t₁ ⊆ s
t₁a : a ≤ t₁
t₁b : t₁ ≤ b
t₁b' : t₁ < b
m : α
t₁m : m > t₁
H : Ioo t₁ m ⊆ s
t : α
hat : t₁ < t
ht : t < min m b
this✝ : t ∈ A
this : t ≤ t₁
⊢ False
Module: Mathlib.Topology.Order.IntermediateValue

Goal state before:
pp: α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
a : α
f : α → δ
hf : ContinuousOn f (Iic a)
hbot : Tendsto f atBot atBot
⊢ Iio (f a) ⊆ f '' Iio a

Tactic:
intro y hy
  have := intermediate_value_Iic hf hbot (mem_Iic_of_Iio hy)
  grind

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

Goal state before:
pp: case inl
α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
s : Set α
hs : s.OrdConnected
f : α → δ
hf : ContinuousOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
hab : f a ≤ f b
⊢ SurjOn f s [[f a, f b]]

Tactic:
simp [hf.surjOn_Icc, *]

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

Goal state before:
pp: case inr
α : Type u
inst✝⁶ : TopologicalSpace α
inst✝⁵ : ConditionallyCompleteLinearOrder α
inst✝⁴ : OrderTopology α
inst✝³ : DenselyOrdered α
δ : Type u_1
inst✝² : LinearOrder δ
inst✝¹ : TopologicalSpace δ
inst✝ : OrderClosedTopology δ
s : Set α
hs : s.OrdConnected
f : α → δ
hf : ContinuousOn f s
a b : α
ha : a ∈ s
hb : b ∈ s
hab : f b ≤ f a
⊢ SurjOn f s [[f a, f b]]

Tactic:
simp [hf.surjOn_Icc, *]

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

Goal state before:
pp: α : Type u
inst✝ : PseudoMetricSpace α
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅
⊢ ⋃ i, (fun i ↦ ∅) i = ∅ ∧
    ∀ (i : ℕ),
      Bornology.IsBounded ((fun i ↦ ∅) i) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun i ↦ ∅) i)

Tactic:
by simp_all

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

Goal state before:
pp: α : Type u_1
R : Type u_2
l : Filter α
r : R
inst✝³ : AddCommGroup R
inst✝² : LinearOrder R
inst✝¹ : IsOrderedAddMonoid R
inst✝ : Archimedean R
f : α → ℤ
hr : 0 < r
hf : Tendsto (fun x ↦ -f x) l atTop
⊢ Tendsto (fun x ↦ -f x • r) l atTop

Tactic:
exact hf.atTop_zsmul_const hr

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

Goal state before:
pp: z : ℤ
c : ℝ≥0
⊢ |z| ≤ ↑⌊c⌋₊ ↔ ‖z‖₊ ≤ c

Tactic:
rw [Int.abs_eq_natAbs, Int.ofNat_le, Nat.le_floor_iff zero_le, NNReal.natCast_natAbs z]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology α
inst✝ : OrderTopology β
e : α ≃o β
⊢ Continuous[inst✝³, inst✝²] ⇑e

Tactic:
‹OrderTopology β›.topology_eq_generate_intervals,

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : OrderTopology α
inst✝ : OrderTopology β
e : α ≃o β
⊢ Continuous[inst✝³, Preorder.topology β] ⇑e
Module: Mathlib.Order.Filter.IsBounded

Goal state before:
pp: case empty
α : Type u_5
f : Filter α
R : Type u_6
κ : Type u_7
inst✝ : AddCommMonoid R
r : R → R → Prop
hr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)
hr₀ : r 0 0
u : κ → α → R
h : ∀ k ∈ ∅, IsBoundedUnder r f (u k)
⊢ IsBoundedUnder r f (∑ k ∈ ∅, u k)

pp: case cons
α : Type u_5
f : Filter α
R : Type u_6
κ : Type u_7
inst✝ : AddCommMonoid R
r : R → R → Prop
hr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)
hr₀ : r 0 0
u : κ → α → R
a✝¹ : κ
s✝ : Finset κ
h✝ : a✝¹ ∉ s✝
a✝ : (∀ k ∈ s✝, IsBoundedUnder r f (u k)) → IsBoundedUnder r f (∑ k ∈ s✝, u k)
h : ∀ k ∈ Finset.cons a✝¹ s✝ h✝, IsBoundedUnder r f (u k)
⊢ IsBoundedUnder r f (∑ k ∈ Finset.cons a✝¹ s✝ h✝, u k)

Tactic:
case empty =>
    rw [Finset.sum_empty]
    exact ⟨0, by simp_all only [eventually_map, Pi.zero_apply, eventually_true]⟩

Goal state after:
case cons
α : Type u_5
f : Filter α
R : Type u_6
κ : Type u_7
inst✝ : AddCommMonoid R
r : R → R → Prop
hr : ∀ (v₁ v₂ : α → R), IsBoundedUnder r f v₁ → IsBoundedUnder r f v₂ → IsBoundedUnder r f (v₁ + v₂)
hr₀ : r 0 0
u : κ → α → R
a✝¹ : κ
s✝ : Finset κ
h✝ : a✝¹ ∉ s✝
a✝ : (∀ k ∈ s✝, IsBoundedUnder r f (u k)) → IsBoundedUnder r f (∑ k ∈ s✝, u k)
h : ∀ k ∈ Finset.cons a✝¹ s✝ h✝, IsBoundedUnder r f (u k)
⊢ IsBoundedUnder r f (∑ k ∈ Finset.cons a✝¹ s✝ h✝, u k)
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x : ℝ
x✝ : EReal
h : ↑x + x✝ ≤ ↑x + ⊥
⊢ x✝ ≤ ⊥

Tactic:
by simpa using h

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

Goal state before:
pp: case h_option
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
p : HeightOneSpectrum R
ι : Type u_4
inst✝¹ : Finite ι
α✝ : Type u_4
inst✝ : Fintype α✝
ih : ∀ (I : α✝ → Ideal R), emultiplicity p.asIdeal (⨆ i, I i) = ⨅ i, emultiplicity p.asIdeal (I i)
I : Option α✝ → Ideal R
⊢ min (emultiplicity p.asIdeal (I none)) (⨅ i, emultiplicity p.asIdeal (I (some i))) =
    ⨅ i, emultiplicity p.asIdeal (I i)

Tactic:
iInf_option

Goal state after:
case h_option
R : Type u_3
inst✝³ : CommRing R
inst✝² : IsDedekindDomain R
p : HeightOneSpectrum R
ι : Type u_4
inst✝¹ : Finite ι
α✝ : Type u_4
inst✝ : Fintype α✝
ih : ∀ (I : α✝ → Ideal R), emultiplicity p.asIdeal (⨆ i, I i) = ⨅ i, emultiplicity p.asIdeal (I i)
I : Option α✝ → Ideal R
⊢ min (emultiplicity p.asIdeal (I none)) (⨅ i, emultiplicity p.asIdeal (I (some i))) =
    min (emultiplicity p.asIdeal (I none)) (⨅ b, emultiplicity p.asIdeal (I (some b)))
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
inst✝ : CompleteLattice α
f : Filter β
p q : β → Prop
u : β → α
h : ∀ᶠ (x : β) in f, u x ≠ ⊥ → (p x ↔ q x)
a : α
b : β
hb : u b ≠ ⊥ → (p b ↔ q b)
hu : u b = ⊥
⊢ p b → u b ≤ a ↔ q b → u b ≤ a

Tactic:
simp [hu]

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

Goal state before:
pp: ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : BoundedLENhdsClass α
inst✝ : BoundedLENhdsClass β
f : Filter ι
u : ι → α
a✝ : α
x : α × β
a : α
ha : ∀ᶠ (x : α) in 𝓝 x.1, (fun x1 x2 ↦ x1 ≤ x2) x a
⊢ IsBounded (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x)

Tactic:
obtain ⟨b, hb⟩ := isBounded_le_nhds x.2

Goal state after:
ι : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
S : Type u_5
π : ι → Type u_6
inst✝⁵ : Preorder α
inst✝⁴ : Preorder β
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : BoundedLENhdsClass α
inst✝ : BoundedLENhdsClass β
f : Filter ι
u : ι → α
a✝ : α
x : α × β
a : α
ha : ∀ᶠ (x : α) in 𝓝 x.1, (fun x1 x2 ↦ x1 ≤ x2) x a
b : β
hb : ∀ᶠ (x : β) in 𝓝 x.2, (fun x1 x2 ↦ x1 ≤ x2) x b
⊢ IsBounded (fun x1 x2 ↦ x1 ≤ x2) (𝓝 x)
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
ι : Type u_4
p : ι → Prop
s : ι → Set α
x : α
h : x ∉ {x | {n | p n ∧ x ∈ s n}.Infinite}
⊢ {x_1 | p x_1 ∧ ¬s x_1 ⊆ {x}ᶜ}.Finite

Tactic:
by simpa using h

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

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝ : Preorder β
f : β → Finset α
h : Monotone f
h' : ∀ (x : α), ∃ n, x ∈ f n
⊢ Tendsto f atTop atTop

Tactic:
simp only [atTop_finset_eq_iInf, tendsto_iInf, tendsto_principal]

Goal state after:
α : Type u_3
β : Type u_4
inst✝ : Preorder β
f : β → Finset α
h : Monotone f
h' : ∀ (x : α), ∃ n, x ∈ f n
⊢ ∀ (i : α), ∀ᶠ (a : β) in atTop, f a ∈ Ici {i}
Module: Mathlib.Topology.Algebra.InfiniteSum.SummationFilter

Goal state before:
pp: β : Type u_2
γ : Type u_3
L : SummationFilter β
f : γ ↪ β
⊢ (L.comap f).support = ⇑f ⁻¹' L.support

Tactic:
by
  simp [support]

Goal state after: