mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Order.Archimedean

Goal state before:
pp: case inl
G : Type u_1
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : TopologicalSpace G
inst✝¹ : OrderTopology G
inst✝ : MulArchimedean G
S : Subgroup G
hS : ∀ ε > 1, ∃ g ∈ S, g ∈ Ioo 1 ε
h✝ : Subsingleton G
x : G
⊢ 1 ∈ ↑S

Tactic:
exact one_mem S

Goal state after:
Module: Mathlib.Topology.Instances.Rat

Goal state before:
pp: R : Type u_1
inst✝⁴ : TopologicalSpace R
inst✝³ : MulAction ℚ R
inst✝² : MulAction ℚ≥0 R
inst✝¹ : IsScalarTower ℚ≥0 ℚ R
inst✝ : ContinuousSMul ℚ R
⊢ Continuous[instTopologicalSpaceProd, inst✝⁴] fun p ↦ p.1 • p.2

Tactic:
conv in _ • _ => rw [← NNRat.cast_smul_eq_nnqsmul ℚ]
    fun_prop

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

Goal state before:
pp: case coe
a✝ : ℝ
h : ↑a✝ ≠ ⊥
⊢ ⊤ + ↑a✝ = ⊤

pp: case top
h : ⊤ ≠ ⊥
⊢ ⊤ + ⊤ = ⊤

Tactic:
· exact top_add_coe _

Goal state after:
case top
h : ⊤ ≠ ⊥
⊢ ⊤ + ⊤ = ⊤
Module: Mathlib.Data.EReal.Operations

Goal state before:
pp: x y : EReal
⊢ (x + y).toENNReal ≤ x.toENNReal + y.toENNReal

Tactic:
induction x <;> induction y <;> try {· simp}

Goal state after:
case coe.coe
a✝¹ a✝ : ℝ
⊢ (↑a✝¹ + ↑a✝).toENNReal ≤ (↑a✝¹).toENNReal + (↑a✝).toENNReal
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLattice α
f : Filter β
u : β → α
p : β → Prop
⊢ blimsup u f p = limsup (u ∘ Subtype.val) (comap Subtype.val f)

Tactic:
blimsup_eq_limsup,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : ConditionallyCompleteLattice α
f : Filter β
u : β → α
p : β → Prop
⊢ limsup u (f ⊓ 𝓟 {x | p x}) = limsup (u ∘ Subtype.val) (comap Subtype.val f)
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : ConditionallyCompleteLattice α
u : β → α
inst✝ : Nonempty β
hu : BddBelow (range u)
⊢ liminf u ⊤ = ⨅ i, u i

Tactic:
liminf,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : ConditionallyCompleteLattice α
u : β → α
inst✝ : Nonempty β
hu : BddBelow (range u)
⊢ (map u ⊤).limsInf = ⨅ i, u i
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
f : ℕ → α
k : ℕ
⊢ liminf (f ∘ fun i ↦ i + k) atTop = liminf f atTop

Tactic:
liminf,

Goal state after:
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
f : ℕ → α
k : ℕ
⊢ (map (f ∘ fun i ↦ i + k) atTop).limsInf = liminf f atTop
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
inst✝ : ConditionallyCompleteLattice α
f : ℕ → α
k : ℕ
⊢ (map (f ∘ fun i ↦ i + k) atTop).limsInf = liminf f atTop

Tactic:
liminf,

Goal state after:
α : Type u_1
inst✝ : ConditionallyCompleteLattice α
f : ℕ → α
k : ℕ
⊢ (map (f ∘ fun i ↦ i + k) atTop).limsInf = (map f atTop).limsInf
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : CompleteLattice α
u : β → α
⊢ liminf u ⊤ = ⨅ i, u i

Tactic:
liminf,

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : CompleteLattice α
u : β → α
⊢ (map u ⊤).limsInf = ⨅ i, u i
Module: Mathlib.Order.LiminfLimsup

Goal state before:
pp: case a
ι : Type u_6
R : Type u_7
F : Filter ι
inst✝ : CompleteLattice R
a : ι → R
⊢ limsup a F ≤ sInf ((fun I ↦ sSup (a '' I)) '' F.sets)

Tactic:
rw [limsup_eq]
    refine sInf_le_sInf fun x hx => ?_
    rcases (mem_image _ F.sets x).mp hx with ⟨I, ⟨I_mem_F, hI⟩⟩
    filter_upwards [I_mem_F] with i hi
    exact hI ▸ le_sSup (mem_image_of_mem _ hi)

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

Goal state before:
pp: case pos
α : Type u_2
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
f : Filter α
inst✝ : f.NeBot
hc : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
hb : IsBounded (fun x1 x2 ↦ x1 ≤ x2) f
hn : Nontrivial α
⊢ ClusterPt f.limsSup f

Tactic:
by_cases! htop : ∀ x, x ≤ f.limsSup
    · let : OrderTop α := { top := f.limsSup, le_top := htop }
      exact nhds_top_basis.clusterPt_iff_frequently |>.mpr fun a => frequently_lt_of_lt_limsSup hc
    · by_cases! hbot : ∀ x, f.limsSup ≤ x
      · let : OrderBot α := { bot := f.limsSup, bot_le := hbot }
        refine nhds_bot_basis.clusterPt_iff_frequently |>.mpr fun a h => ?_
        exact lt_mem_sets_of_limsSup_lt hb h |>.frequently
      · refine (nhds_basis_Ioo' hbot htop).clusterPt_iff_frequently |>.mpr fun a ⟨hl, hg⟩ => ?_
        exact frequently_lt_of_lt_limsSup hc hl |>.and_eventually <| lt_mem_sets_of_limsSup_lt hb hg

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

Goal state before:
pp: case neg
α : Type u_2
inst✝³ : ConditionallyCompleteLinearOrder α
inst✝² : TopologicalSpace α
inst✝¹ : OrderTopology α
f : Filter α
inst✝ : f.NeBot
hc : IsCobounded (fun x1 x2 ↦ x1 ≤ x2) f
hb : IsBounded (fun x1 x2 ↦ x1 ≤ x2) f
hn : Subsingleton α
⊢ ClusterPt f.limsSup f

Tactic:
· simp_all [ClusterPt, Filter.eq_top_of_neBot]

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

Goal state before:
pp: case pos
ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
h : (Set.Ioo c b).Nonempty
⊢ h.some < b

pp: case neg
ι : Type u_1
α : Type u_7
β : Type u_8
inst✝³ : ConditionallyCompleteLattice α
inst✝² : CompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
u : ι → α → β
c : β
h_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)
h_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop (𝓝 c)
h_anti : ∀ (i : ι), Antitone (u i)
n0 : ι
b : β
hb : b > c
b' : β := if h : (Set.Ioo c b).Nonempty then h.some else c
h : ¬(Set.Ioo c b).Nonempty
⊢ c < b

Tactic:
exacts [h.some_mem.2, hb]

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

Goal state before:
pp: x y : EReal
⊢ (∀ (z : ℝ), x < ↑z → y ≤ ↑z) ↔ y ≤ x

Tactic:
refine ⟨fun h ↦ WithBot.le_of_forall_lt_iff_le.1 ?_, fun h _ x_z ↦ h.trans x_z.le⟩

Goal state after:
x y : EReal
h : ∀ (z : ℝ), x < ↑z → y ≤ ↑z
⊢ ∀ (z : WithTop ℝ), x < ↑z → y ≤ ↑z
Module: Mathlib.Topology.EMetricSpace.Lipschitz

Goal state before:
pp: case h₂
α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < ↑δ
hδ : ↑δ * ↑(3 * K) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : edist x x' < ↑δ
A : s ∩ Metric.eball x ↑δ ∈ 𝓝[s] x
B : t ∩ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x ↑δ
hbt : (a, b).2 ∈ t
hby : (a, b).2 ∈ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2}
⊢ edist (f (x', b)) (f (x', y)) ≤ ε / 2

pp: case h₂
α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < ↑δ
hδ : ↑δ * ↑(3 * K) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : edist x x' < ↑δ
A : s ∩ Metric.eball x ↑δ ∈ 𝓝[s] x
B : t ∩ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x ↑δ
hbt : (a, b).2 ∈ t
hby : (a, b).2 ∈ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2}
⊢ edist (f (x', y)) (f (x, y)) ≤ ↑K * ↑δ

Tactic:
· exact hby

Goal state after:
case h₂
α : Type u
β : Type v
γ : Type w
inst✝² : PseudoEMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PseudoEMetricSpace γ
f : α × β → γ
s s' : Set α
t : Set β
hs' : s' ⊆ s
hss' : s ⊆ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s'
K : ℝ≥0
ha : ∀ a ∈ s', ContinuousOn (fun y ↦ f (a, y)) t
hb : ∀ b ∈ t, LipschitzOnWith K (fun x ↦ f (x, b)) s
x : α
y : β
hx : x ∈ s
hy : y ∈ t
ε : ℝ≥0∞
ε0 : 0 < ε / 2
δ : ℝ≥0
δpos : 0 < ↑δ
hδ : ↑δ * ↑(3 * K) < ε / 2
x' : α
hx' : x' ∈ s'
hxx' : edist x x' < ↑δ
A : s ∩ Metric.eball x ↑δ ∈ 𝓝[s] x
B : t ∩ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2} ∈ 𝓝[t] y
a : α
b : β
has : (a, b).1 ∈ s
hax : (a, b).1 ∈ Metric.eball x ↑δ
hbt : (a, b).2 ∈ t
hby : (a, b).2 ∈ {b | edist (f (x', b)) (f (x', y)) ≤ ε / 2}
⊢ edist (f (x', y)) (f (x, y)) ≤ ↑K * ↑δ
Module: Mathlib.Topology.Algebra.InfiniteSum.SummationFilter

Goal state before:
pp: β : Type u_2
L : SummationFilter β
⊢ L.support = (map SetLike.coe L.filter).limsInf

Tactic:
refine eq_of_forall_ge_iff fun c ↦ ?_

Goal state after:
β : Type u_2
L : SummationFilter β
c : Set β
⊢ L.support ⊆ c ↔ (map SetLike.coe L.filter).limsInf ⊆ c
Module: Mathlib.Topology.Algebra.InfiniteSum.SummationFilter

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : Preorder β
inst✝³ : LocallyFiniteOrder β
inst✝² : Nonempty β
inst✝¹ : IsDirectedOrder β
inst✝ : IsCodirectedOrder β
⊢ (conditional β).filter.NeBot

Tactic:
rw [conditional_filter]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : Preorder β
inst✝³ : LocallyFiniteOrder β
inst✝² : Nonempty β
inst✝¹ : IsDirectedOrder β
inst✝ : IsCodirectedOrder β
⊢ (Filter.map (fun p ↦ Finset.Icc p.1 p.2) (atBot ×ˢ atTop)).NeBot
Module: Mathlib.Topology.Algebra.InfiniteSum.SummationFilter

Goal state before:
pp: this : ∀ (n : ℕ), Finset.Iic n = Finset.range (n + 1)
⊢ Filter.map (fun x ↦ Finset.range (x + 1)) atTop = Filter.map Finset.range atTop

Tactic:
apply le_antisymm <;>
      rw [← Tendsto]

Goal state after:
case a
this : ∀ (n : ℕ), Finset.Iic n = Finset.range (n + 1)
⊢ Tendsto (fun x ↦ Finset.range (x + 1)) atTop (Filter.map Finset.range atTop)

case a
this : ∀ (n : ℕ), Finset.Iic n = Finset.range (n + 1)
⊢ Tendsto Finset.range atTop (Filter.map (fun x ↦ Finset.range (x + 1)) atTop)
Module: Mathlib.Topology.Algebra.InfiniteSum.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁵ : CommMonoid α
inst✝⁴ : TopologicalSpace α
L : SummationFilter β
inst✝³ : CommMonoid γ
inst✝² : TopologicalSpace γ
G : Type u_4
inst✝¹ : FunLike G α γ
inst✝ : MonoidHomClass G α γ
g : G
hg : IsInducing ⇑g
f : β → α
a : α
⊢ HasProd (⇑g ∘ f) (g a) L ↔ HasProd f a L

Tactic:
by
  simp_rw [HasProd, comp_apply, ← _root_.map_prod]
  exact hg.tendsto_nhds_iff.symm

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Order

Goal state before:
pp: case refine_1
ι : Type u_4
α : Type u_5
L : SummationFilter ι
inst✝⁶ : CommMonoid α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedMonoid α
inst✝³ : TopologicalSpace α
inst✝² : OrderClosedTopology α
f : ι → α
inst✝¹ : L.LeAtTop
inst✝ : L.NeBot
hf : ∀ (i : ι), 1 ≤ f i
hf' : HasProd f 1 L
⊢ f = 1

pp: case refine_2
ι : Type u_4
α : Type u_5
L : SummationFilter ι
inst✝⁶ : CommMonoid α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedMonoid α
inst✝³ : TopologicalSpace α
inst✝² : OrderClosedTopology α
f : ι → α
inst✝¹ : L.LeAtTop
inst✝ : L.NeBot
hf : ∀ (i : ι), 1 ≤ f i
⊢ f = 1 → HasProd f 1 L

Tactic:
· ext i
    exact (hf i).antisymm' (le_hasProd hf' _ fun j _ ↦ hf j)

Goal state after:
case refine_2
ι : Type u_4
α : Type u_5
L : SummationFilter ι
inst✝⁶ : CommMonoid α
inst✝⁵ : PartialOrder α
inst✝⁴ : IsOrderedMonoid α
inst✝³ : TopologicalSpace α
inst✝² : OrderClosedTopology α
f : ι → α
inst✝¹ : L.LeAtTop
inst✝ : L.NeBot
hf : ∀ (i : ι), 1 ≤ f i
⊢ f = 1 → HasProd f 1 L
Module: Mathlib.Topology.Algebra.InfiniteSum.Group

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : CommGroup G
inst✝ : IsTopologicalGroup G
f : α → G
⊢ Tendsto (fun s ↦ ∏' (a : { x // x ∉ s }), f ↑a) atTop (𝓝 1)

Tactic:
by_cases H : Multipliable f

Goal state after:
case pos
α : Type u_1
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : CommGroup G
inst✝ : IsTopologicalGroup G
f : α → G
H : Multipliable f
⊢ Tendsto (fun s ↦ ∏' (a : { x // x ∉ s }), f ↑a) atTop (𝓝 1)

case neg
α : Type u_1
G : Type u_4
inst✝² : TopologicalSpace G
inst✝¹ : CommGroup G
inst✝ : IsTopologicalGroup G
f : α → G
H : ¬Multipliable f
⊢ Tendsto (fun s ↦ ∏' (a : { x // x ∉ s }), f ↑a) atTop (𝓝 1)
Module: Mathlib.Topology.Algebra.InfiniteSum.NatInt

Goal state before:
pp: M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x ↦ ↑x) v'
u2 : Finset ℤ := image (fun x ↦ -↑x) v'
x : ℤ
hx : x ∈ u
⊢ x ∈ u1 ∪ u2

Tactic:
simp only [u1, u2, mem_union, mem_image]

Goal state after:
M : Type u_1
inst✝² : CommMonoid M
inst✝¹ : TopologicalSpace M
m : M
inst✝ : ContinuousMul M
f : ℤ → M
hf : HasProd f m
u : Finset ℤ
v' : Finset ℕ
hv' : image Int.natAbs u ⊆ v'
u1 : Finset ℤ := image (fun x ↦ ↑x) v'
u2 : Finset ℤ := image (fun x ↦ -↑x) v'
x : ℤ
hx : x ∈ u
⊢ (∃ a ∈ v', ↑a = x) ∨ ∃ a ∈ v', -↑a = x
Module: Mathlib.Topology.Metrizable.Uniformity

Goal state before:
pp: X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun x y ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length = m → d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
⊢ l.length ∈ upperBounds s

Tactic:
intro m

Goal state after:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun x y ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length = m → d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
m : ℕ
⊢ m ∈ s → m ≤ l.length
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: α : Type u_1
f : α → ℝ≥0
l : Filter α
⊢ (∀ (x : ℝ≥0), ∀ᶠ (a : α) in l, ↑x < ↑(f a)) ↔ Tendsto f l atTop

Tactic:
atTop_basis_Ioi.tendsto_right_iff

Goal state after:
α : Type u_1
f : α → ℝ≥0
l : Filter α
⊢ (∀ (x : ℝ≥0), ∀ᶠ (a : α) in l, ↑x < ↑(f a)) ↔ ∀ (i : ℝ≥0), True → ∀ᶠ (x : α) in l, f x ∈ Ioi i
Module: Mathlib.Topology.UniformSpace.AbstractCompletion

Goal state before:
pp: α : Type uα
inst✝² : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝¹ : UniformSpace β
f : α → β
inst✝ : T2Space β
hf : UniformContinuous f
a : α
⊢ pkg.extend f (pkg.coe a) = f a

Tactic:
pkg.extend_def hf

Goal state after:
α : Type uα
inst✝² : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝¹ : UniformSpace β
f : α → β
inst✝ : T2Space β
hf : UniformContinuous f
a : α
⊢ ⋯.extend f (pkg.coe a) = f a
Module: Mathlib.Topology.UniformSpace.AbstractCompletion

Goal state before:
pp: case pos
α : Type uα
inst✝² : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝¹ : UniformSpace β
f : α → β
inst✝ : CompleteSpace β
hf : UniformContinuous f
⊢ UniformContinuous (pkg.extend f)

Tactic:
pkg.extend_def hf

Goal state after:
case pos
α : Type uα
inst✝² : UniformSpace α
pkg : AbstractCompletion.{vα, uα} α
β : Type uβ
inst✝¹ : UniformSpace β
f : α → β
inst✝ : CompleteSpace β
hf : UniformContinuous f
⊢ UniformContinuous (⋯.extend f)
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: case mp
β : Type u_4
inst✝¹ : Nonempty β
inst✝ : SemilatticeSup β
f : β → ℝ≥0∞
hf : Antitone f
h : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, f n ≤ ε
ε : ℝ≥0∞
hε : 0 < ε
⊢ ∃ n, f n < ε

Tactic:
obtain ⟨n, hn⟩ := h (min 1 (ε / 2))
      (lt_min_iff.mpr ⟨zero_lt_one, (ENNReal.div_pos_iff.mpr ⟨hε.ne', by finiteness⟩)⟩)
    · refine ⟨n, hn.trans_lt ?_⟩
      by_cases hε_top : ε = ∞
      · simp [hε_top]
      refine (min_le_right _ _).trans_lt ?_
      rw [ENNReal.div_lt_iff (Or.inr hε.ne') (Or.inr hε_top)]
      conv_lhs => rw [← mul_one ε]
      gcongr; simp

Goal state after:
Module: Mathlib.Topology.UniformSpace.Completion

Goal state before:
pp: α : Type u
inst✝ : UniformSpace α
f g : CauchyFilter α
a b : α
ha : Tendsto id (↑f) (𝓝 a)
hb : Tendsto id (↑g) (𝓝 b)
⊢ Inseparable a b ↔ ↑f ×ˢ ↑g ≤ 𝓤 α

Tactic:
(ha.comp tendsto_fst).inseparable_iff_uniformity (hb.comp tendsto_snd)

Goal state after:
α : Type u
inst✝ : UniformSpace α
f g : CauchyFilter α
a b : α
ha : Tendsto id (↑f) (𝓝 a)
hb : Tendsto id (↑g) (𝓝 b)
⊢ Tendsto (fun x ↦ ((id ∘ Prod.fst) x, (id ∘ Prod.snd) x)) (↑f ×ˢ ↑g) (𝓤 α) ↔ ↑f ×ˢ ↑g ≤ 𝓤 α
Module: Mathlib.Analysis.Normed.Group.Continuity

Goal state before:
pp: E : Type u_4
F : Type u_5
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
j : E →* F
b : F
hb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range
f : ℕ → ℝ
b_pos : ∀ (n : ℕ), 0 < f n
v : ℕ → F
sum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)
hv₀ : ‖(v 0)⁻¹ * b‖ < f 0
hv_pos : ∀ (n : ℕ), 0 < n → ‖v n‖ < f n
g : ℕ → E
hg : ∀ (n : ℕ), j (g n) = v n
⊢ ‖(j (g 0))⁻¹ * b‖ < f 0

Tactic:
by simpa [hg 0] using hv₀

Goal state after:
Module: Mathlib.Order.Interval.Set.ProjIcc

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a b : α
h : a ≤ b
f : α → β
ha : ∀ x < a, f x = f a
hb : ∀ (x : α), b < x → f x = f b
x : α
⊢ IccExtend h (f ∘ Subtype.val) x = f x

Tactic:
rcases lt_or_ge x a with hxa | hax

Goal state after:
case inl
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a b : α
h : a ≤ b
f : α → β
ha : ∀ x < a, f x = f a
hb : ∀ (x : α), b < x → f x = f b
x : α
hxa : x < a
⊢ IccExtend h (f ∘ Subtype.val) x = f x

case inr
α : Type u_1
β : Type u_2
inst✝ : LinearOrder α
a b : α
h : a ≤ b
f : α → β
ha : ∀ x < a, f x = f a
hb : ∀ (x : α), b < x → f x = f b
x : α
hax : a ≤ x
⊢ IccExtend h (f ∘ Subtype.val) x = f x
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: case inr.refine_2
s : Set ℝ
h : Bornology.IsBounded s
hne : s.Nonempty
⊢ sSup s - sInf s ≤ (ediam s).toReal

Tactic:
← Metric.diam,

Goal state after:
case inr.refine_2
s : Set ℝ
h : Bornology.IsBounded s
hne : s.Nonempty
⊢ sSup s - sInf s ≤ Metric.diam s
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: s : Set ℝ
h : Bornology.IsBounded s
⊢ (ediam s).toReal = sSup s - sInf s

Tactic:
Real.ediam_eq h,

Goal state after:
s : Set ℝ
h : Bornology.IsBounded s
⊢ (ENNReal.ofReal (sSup s - sInf s)).toReal = sSup s - sInf s
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: ι : Type u_4
f : Filter ι
u : ι → ℝ≥0∞
inst✝ : f.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ∞
le_b : ∀ᶠ (i : ι) in f, u i ≤ b
liminf_le : liminf u f ≤ b
i : ι
i_le_b : u i ≤ b
⊢ (u i).toReal = b.truncateToReal (u i)

Tactic:
simp only [truncateToReal_eq_toReal b_ne_top i_le_b]

Goal state after:
Module: Mathlib.Topology.Instances.ENNReal.Lemmas

Goal state before:
pp: ι : Type u_4
f : Filter ι
u : ι → ℝ≥0∞
inst✝ : f.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ∞
le_b : ∀ᶠ (i : ι) in f, u i ≤ b
i : ι
i_le_b : u i ≤ b
⊢ (u i).toReal = b.truncateToReal (u i)

Tactic:
simp only [truncateToReal_eq_toReal b_ne_top i_le_b]

Goal state after:
Module: Mathlib.Analysis.Normed.Order.Lattice

Goal state before:
pp: α : Type u_1
inst✝³ : NormedAddCommGroup α
inst✝² : Lattice α
inst✝¹ : HasSolidNorm α
inst✝ : IsOrderedAddMonoid α
a b c d : α
⊢ |a ⊓ b - c ⊓ b| + |c ⊓ b - c ⊓ d| ≤ |a - c| + |b - d|

Tactic:
gcongr ?_ + ?_

Goal state after:
case h₁
α : Type u_1
inst✝³ : NormedAddCommGroup α
inst✝² : Lattice α
inst✝¹ : HasSolidNorm α
inst✝ : IsOrderedAddMonoid α
a b c d : α
⊢ |a ⊓ b - c ⊓ b| ≤ |a - c|

case h₂
α : Type u_1
inst✝³ : NormedAddCommGroup α
inst✝² : Lattice α
inst✝¹ : HasSolidNorm α
inst✝ : IsOrderedAddMonoid α
a b c d : α
⊢ |c ⊓ b - c ⊓ d| ≤ |b - d|
Module: Mathlib.Analysis.Normed.Order.Lattice

Goal state before:
pp: α : Type u_1
inst✝³ : NormedAddCommGroup α
inst✝² : Lattice α
inst✝¹ : HasSolidNorm α
inst✝ : IsOrderedAddMonoid α
a b c d : α
⊢ |a ⊔ b - c ⊔ b| + |c ⊔ b - c ⊔ d| ≤ |a - c| + |b - d|

Tactic:
gcongr ?_ + ?_

Goal state after:
case h₁
α : Type u_1
inst✝³ : NormedAddCommGroup α
inst✝² : Lattice α
inst✝¹ : HasSolidNorm α
inst✝ : IsOrderedAddMonoid α
a b c d : α
⊢ |a ⊔ b - c ⊔ b| ≤ |a - c|

case h₂
α : Type u_1
inst✝³ : NormedAddCommGroup α
inst✝² : Lattice α
inst✝¹ : HasSolidNorm α
inst✝ : IsOrderedAddMonoid α
a b c d : α
⊢ |c ⊔ b - c ⊔ d| ≤ |b - d|
Module: Mathlib.Algebra.Order.GroupWithZero.Finset

Goal state before:
pp: ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : MulPosReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
⊢ s.sup' hs f * a = s.sup' hs fun i ↦ f i * a

Tactic:
by_cases! h : 0 = a

Goal state after:
case pos
ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : MulPosReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
h : 0 = a
⊢ s.sup' hs f * a = s.sup' hs fun i ↦ f i * a

case neg
ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : MulPosReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
h : 0 ≠ a
⊢ s.sup' hs f * a = s.sup' hs fun i ↦ f i * a
Module: Mathlib.Algebra.Order.GroupWithZero.Finset

Goal state before:
pp: ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : PosMulReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
⊢ a * s.sup' hs f = s.sup' hs fun i ↦ a * f i

Tactic:
by_cases! h : 0 = a

Goal state after:
case pos
ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : PosMulReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
h : 0 = a
⊢ a * s.sup' hs f = s.sup' hs fun i ↦ a * f i

case neg
ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : PosMulReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
h : 0 ≠ a
⊢ a * s.sup' hs f = s.sup' hs fun i ↦ a * f i
Module: Mathlib.Algebra.Order.GroupWithZero.Finset

Goal state before:
pp: ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : MulPosReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
⊢ s.sup' hs f / a = s.sup' hs fun i ↦ f i / a

Tactic:
by_cases! h : 0 = a

Goal state after:
case pos
ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : MulPosReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
h : 0 = a
⊢ s.sup' hs f / a = s.sup' hs fun i ↦ f i / a

case neg
ι : Type u_1
G₀ : Type u_3
inst✝² : GroupWithZero G₀
inst✝¹ : SemilatticeSup G₀
a : G₀
inst✝ : MulPosReflectLT G₀
ha : 0 ≤ a
f : ι → G₀
s : Finset ι
hs : s.Nonempty
h : 0 ≠ a
⊢ s.sup' hs f / a = s.sup' hs fun i ↦ f i / a
Module: Mathlib.Analysis.Normed.Group.Uniform

Goal state before:
pp: E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
n : ℕ
hn : n ≥ N
⊢ ∀ n ≥ N + 1, u n / v n = 1

Tactic:
intro m

Goal state after:
E : Type u_2
inst✝ : SeminormedCommGroup E
u v : ℕ → E
N : ℕ
huv : ∀ n ≥ N, u n = v n
hv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k
d : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k
n : ℕ
hn : n ≥ N
m : ℕ
⊢ m ≥ N + 1 → u m / v m = 1
Module: Mathlib.Algebra.Order.AddTorsor

Goal state before:
pp: G : Type u_1
P : Type u_2
inst✝³ : Preorder G
inst✝² : Preorder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
a b : G
c d : P
h₁ : a < b
h₂ : c ≤ d
hbad : b • d ≤ a • d
⊢ False

Tactic:
have h : b ≤ a := IsOrderedCancelSMul.le_of_smul_le_smul_right b a d hbad

Goal state after:
G : Type u_1
P : Type u_2
inst✝³ : Preorder G
inst✝² : Preorder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
a b : G
c d : P
h₁ : a < b
h₂ : c ≤ d
hbad : b • d ≤ a • d
h : b ≤ a
⊢ False