mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: case le_cof_ord.refine_1
c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
hc₀ : ℵ₀ ≤ succ c
hc' : (succ c).ord.cof < succ c
f : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)
hf : IsFundamentalSeq f
⊢ (succ c).ord.cof.ord.card ≤ c

pp: case le_cof_ord.refine_2
c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
hc₀ : ℵ₀ ≤ succ c
hc' : (succ c).ord.cof < succ c
f : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)
hf : IsFundamentalSeq f
i : ↑(Iio (succ c).ord.cof.ord)
⊢ (↑(f i) + 1).card ≤ c

Tactic:
·

Goal state after:
case le_cof_ord.refine_1
c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
hc₀ : ℵ₀ ≤ succ c
hc' : (succ c).ord.cof < succ c
f : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)
hf : IsFundamentalSeq f
⊢ (succ c).ord.cof.ord.card ≤ c
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: case le_cof_ord.refine_2
c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
hc₀ : ℵ₀ ≤ succ c
hc' : (succ c).ord.cof < succ c
f : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)
hf : IsFundamentalSeq f
i : ↑(Iio (succ c).ord.cof.ord)
⊢ (↑(f i) + 1).card ≤ c

Tactic:
·

Goal state after:
case le_cof_ord.refine_2
c : Cardinal.{u_1}
hc : ℵ₀ ≤ c
hc₀ : ℵ₀ ≤ succ c
hc' : (succ c).ord.cof < succ c
f : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)
hf : IsFundamentalSeq f
i : ↑(Iio (succ c).ord.cof.ord)
⊢ (↑(f i) + 1).card ≤ c
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case inr.inr
a b : Ordinal.{u_1}
hb✝ : b ≠ 0
hb : IsSuccLimit b
⊢ (a + b).cof = b.cof

Tactic:
·

Goal state after:
case inr.inr
a b : Ordinal.{u_1}
hb✝ : b ≠ 0
hb : IsSuccLimit b
⊢ (a + b).cof = b.cof
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case a
ι : Type u
inst✝ : Infinite ι
c : ι → Cardinal.{v}
h₁ : ∀ (i : ι), 2 ≤ c i
h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι
⊢ (prod fun i ↦ lift.{u, v} (c i)) ≤ lift.{v, u} (2 ^ #ι)

pp: case a
ι : Type u
inst✝ : Infinite ι
c : ι → Cardinal.{v}
h₁ : ∀ (i : ι), 2 ≤ c i
h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι
⊢ lift.{v, u} (2 ^ #ι) ≤ prod fun i ↦ lift.{u, v} (c i)

Tactic:
·

Goal state after:
case a
ι : Type u
inst✝ : Infinite ι
c : ι → Cardinal.{v}
h₁ : ∀ (i : ι), 2 ≤ c i
h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι
⊢ (prod fun i ↦ lift.{u, v} (c i)) ≤ lift.{v, u} (2 ^ #ι)
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case pos
a b : Ordinal.{u_1}
ha : a ≠ 0
hb : IsSuccPrelimit b
hb' : IsMin b
⊢ (a * b).cof = b.cof

pp: case neg
a b : Ordinal.{u_1}
ha : a ≠ 0
hb : IsSuccPrelimit b
hb' : ¬IsMin b
⊢ (a * b).cof = b.cof

Tactic:
·

Goal state after:
case pos
a b : Ordinal.{u_1}
ha : a ≠ 0
hb : IsSuccPrelimit b
hb' : IsMin b
⊢ (a * b).cof = b.cof
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case neg
a b : Ordinal.{u_1}
ha : a ≠ 0
hb : IsSuccPrelimit b
hb' : ¬IsMin b
⊢ (a * b).cof = b.cof

Tactic:
·

Goal state after:
case neg
a b : Ordinal.{u_1}
ha : a ≠ 0
hb : IsSuccPrelimit b
hb' : ¬IsMin b
⊢ (a * b).cof = b.cof
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case pos
o : Ordinal.{u_1}
ho : IsSuccPrelimit o
h : IsMin o
⊢ (preOmega o).cof = o.cof

pp: case neg
o : Ordinal.{u_1}
ho : IsSuccPrelimit o
h : ¬IsMin o
⊢ (preOmega o).cof = o.cof

Tactic:
·

Goal state after:
case pos
o : Ordinal.{u_1}
ho : IsSuccPrelimit o
h : IsMin o
⊢ (preOmega o).cof = o.cof
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case neg
o : Ordinal.{u_1}
ho : IsSuccPrelimit o
h : ¬IsMin o
⊢ (preOmega o).cof = o.cof

Tactic:
·

Goal state after:
case neg
o : Ordinal.{u_1}
ho : IsSuccPrelimit o
h : ¬IsMin o
⊢ (preOmega o).cof = o.cof
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case a
ι : Type u
inst✝ : Infinite ι
c : ι → Cardinal.{v}
h₁ : ∀ (i : ι), 2 ≤ c i
h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι
⊢ lift.{v, u} (2 ^ #ι) ≤ prod fun i ↦ lift.{u, v} (c i)

Tactic:
·

Goal state after:
case a
ι : Type u
inst✝ : Infinite ι
c : ι → Cardinal.{v}
h₁ : ∀ (i : ι), 2 ≤ c i
h₂ : ∀ (i : ι), lift.{u, v} (c i) ≤ lift.{v, u} #ι
⊢ lift.{v, u} (2 ^ #ι) ≤ prod fun i ↦ lift.{u, v} (c i)
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: case aleph0_le
κ : Cardinal.{v}
h₁ : ℵ₀ ≤ κ
h₂ : κ ≤ κ.ord.cof
⊢ ℵ₀ ≤ Cardinal.lift.{u, v} κ

pp: case le_cof_ord
κ : Cardinal.{v}
h₁ : ℵ₀ ≤ κ
h₂ : κ ≤ κ.ord.cof
⊢ Cardinal.lift.{u, v} κ ≤ (Cardinal.lift.{u, v} κ).ord.cof

Tactic:
·

Goal state after:
case aleph0_le
κ : Cardinal.{v}
h₁ : ℵ₀ ≤ κ
h₂ : κ ≤ κ.ord.cof
⊢ ℵ₀ ≤ Cardinal.lift.{u, v} κ
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: case le_cof_ord
κ : Cardinal.{v}
h₁ : ℵ₀ ≤ κ
h₂ : κ ≤ κ.ord.cof
⊢ Cardinal.lift.{u, v} κ ≤ (Cardinal.lift.{u, v} κ).ord.cof

Tactic:
·

Goal state after:
case le_cof_ord
κ : Cardinal.{v}
h₁ : ℵ₀ ≤ κ
h₂ : κ ≤ κ.ord.cof
⊢ Cardinal.lift.{u, v} κ ≤ (Cardinal.lift.{u, v} κ).ord.cof
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case inl
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha : #α = 0
⊢ #{ s // Bounded r s } = #α

pp: case inr
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha : #α ≠ 0
⊢ #{ s // Bounded r s } = #α

Tactic:
·

Goal state after:
case inl
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha : #α = 0
⊢ #{ s // Bounded r s } = #α
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case inr.a
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ #{ s // Bounded r s } ≤ #α

pp: case a
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ #α ≤ #{ s // Bounded r s }

Tactic:
·

Goal state after:
case inr.a
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ #{ s // Bounded r s } ≤ #α
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case refine_1
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ {x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a} ⊆ s

pp: case refine_2
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ θ ≤ #↑{x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a}

pp: case refine_3
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ ∀ ⦃x : β⦄ (hx : x ∈ {x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a}), f ⟨x, ⋯⟩ = a

Tactic:
·

Goal state after:
case refine_1
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ {x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a} ⊆ s
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inl
c : Cardinal.{u}
n : ℕ
hc : ℵ₀ ≤ c
⊢ c ^ ↑n ≤ max c ℵ₀

pp: case inr
c : Cardinal.{u}
n : ℕ
hc : c < ℵ₀
⊢ c ^ ↑n ≤ max c ℵ₀

Tactic:
·

Goal state after:
case inl
c : Cardinal.{u}
n : ℕ
hc : ℵ₀ ≤ c
⊢ c ^ ↑n ≤ max c ℵ₀
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case refine_1
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ {x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a} ⊆ s

Tactic:
rintro x ⟨hx, _⟩

Goal state after:
case refine_1
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
x : β
hx : x ∈ s
h✝ : f ⟨x, hx⟩ = a
⊢ x ∈ s
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inr
c : Cardinal.{u}
n : ℕ
hc : c < ℵ₀
⊢ c ^ ↑n ≤ max c ℵ₀

Tactic:
·

Goal state after:
case inr
c : Cardinal.{u}
n : ℕ
hc : c < ℵ₀
⊢ c ^ ↑n ≤ max c ℵ₀
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case refine_2
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ θ ≤ #↑{x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a}

pp: case refine_3
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ ∀ ⦃x : β⦄ (hx : x ∈ {x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a}), f ⟨x, ⋯⟩ = a

Tactic:
·

Goal state after:
case refine_2
β α : Type u
s : Set β
f : ↑s → α
θ : Cardinal.{u}
hθ : θ ≤ #↑s
h₁ : ℵ₀ ≤ θ
h₂ : #α < θ.ord.cof
a : α
ha : θ ≤ #↑(f ⁻¹' {a})
⊢ θ ≤ #↑{x | ∃ (h : x ∈ s), f ⟨x, h⟩ = a}
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case a
c : Cardinal.{u_1}
h : ℵ₀ ≤ c
⊢ c ^< ℵ₀ ≤ c

pp: case a
c : Cardinal.{u_1}
h : ℵ₀ ≤ c
⊢ c ≤ c ^< ℵ₀

Tactic:
·

Goal state after:
case a
c : Cardinal.{u_1}
h : ℵ₀ ≤ c
⊢ c ^< ℵ₀ ≤ c
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inl
c : Cardinal.{u_1}
h : ℵ₀ ≤ c
⊢ c ^< ℵ₀ ≤ max c ℵ₀

pp: case inr
c : Cardinal.{u_1}
h : c < ℵ₀
⊢ c ^< ℵ₀ ≤ max c ℵ₀

Tactic:
·

Goal state after:
case inl
c : Cardinal.{u_1}
h : ℵ₀ ≤ c
⊢ c ^< ℵ₀ ≤ max c ℵ₀
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case a
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ #α ≤ #{ s // Bounded r s }

Tactic:
·

Goal state after:
case a
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ #α ≤ #{ s // Bounded r s }
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case inl
β α : Type u
f : β → α
h : #α < #β
hβ : ℵ₀ ≤ #β
hα : #α < ℵ₀
⊢ ∃ a, succ #α ≤ #↑(f ⁻¹' {a})

pp: case inr
β α : Type u
f : β → α
h : #α < #β
hβ : ℵ₀ ≤ #β
hα : ℵ₀ ≤ #α
⊢ ∃ a, succ #α ≤ #↑(f ⁻¹' {a})

Tactic:
·

Goal state after:
case inl
β α : Type u
f : β → α
h : #α < #β
hβ : ℵ₀ ≤ #β
hα : #α < ℵ₀
⊢ ∃ a, succ #α ≤ #↑(f ⁻¹' {a})
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case a.refine_1
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
x : α
⊢ Bounded r {x}

pp: case a.refine_2
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ Injective fun x ↦ ⟨{x}, ⋯⟩

Tactic:
·

Goal state after:
case a.refine_1
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
x : α
⊢ Bounded r {x}
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case a.refine_2
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ Injective fun x ↦ ⟨{x}, ⋯⟩

Tactic:
·

Goal state after:
case a.refine_2
α : Type u_1
h : (#α).IsStrongPrelimit
r : α → α → Prop
inst✝ : IsWellOrder α r
hr : (#α).ord = type r
ha✝ : #α ≠ 0
h' : (#α).IsStrongLimit
ha : ℵ₀ ≤ #α
⊢ Injective fun x ↦ ⟨{x}, ⋯⟩
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case inr
β α : Type u
f : β → α
h : #α < #β
hβ : ℵ₀ ≤ #β
hα : ℵ₀ ≤ #α
⊢ ∃ a, succ #α ≤ #↑(f ⁻¹' {a})

Tactic:
·

Goal state after:
case inr
β α : Type u
f : β → α
h : #α < #β
hβ : ℵ₀ ≤ #β
hα : ℵ₀ ≤ #α
⊢ ∃ a, succ #α ≤ #↑(f ⁻¹' {a})
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: c : Cardinal.{u}
ι α : Type u
t : ι → Set α
hc : c.IsRegular
hι : #ι < c
h : ∀ (i : ι), #↑(t i) < c
⊢ ∀ (i : ↑(range fun i ↦ t i)), #↑↑i < c

pp: c : Cardinal.{u}
ι α : Type u
t : ι → Set α
hc : c.IsRegular
hι : #ι < c
h : ∀ (i : ι), #↑(t i) < c
⊢ #ι < c.ord.cof

Tactic:
·

Goal state after:
c : Cardinal.{u}
ι α : Type u
t : ι → Set α
hc : c.IsRegular
hι : #ι < c
h : ∀ (i : ι), #↑(t i) < c
⊢ ∀ (i : ↑(range fun i ↦ t i)), #↑↑i < c
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case inl
β α : Type u
f : β → α
h : #α < #β
inst✝ : Infinite β
hα : #α < ℵ₀
⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a})

pp: case inr
β α : Type u
f : β → α
h : #α < #β
inst✝ : Infinite β
hα : ℵ₀ ≤ #α
⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a})

Tactic:
·

Goal state after:
case inl
β α : Type u
f : β → α
h : #α < #β
inst✝ : Infinite β
hα : #α < ℵ₀
⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a})
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case inr
β α : Type u
f : β → α
h : #α < #β
inst✝ : Infinite β
hα : ℵ₀ ≤ #α
⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a})

Tactic:
·

Goal state after:
case inr
β α : Type u
f : β → α
h : #α < #β
inst✝ : Infinite β
hα : ℵ₀ ≤ #α
⊢ ∃ a, ℵ₀ ≤ #↑(f ⁻¹' {a})
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: c : Cardinal.{u}
ι α : Type u
t : ι → Set α
hc : c.IsRegular
hι : #ι < c
h : ∀ (i : ι), #↑(t i) < c
⊢ #ι < c.ord.cof

Tactic:
·

Goal state after:
c : Cardinal.{u}
ι α : Type u
t : ι → Set α
hc : c.IsRegular
hι : #ι < c
h : ∀ (i : ι), #↑(t i) < c
⊢ #ι < c.ord.cof
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case inl
β α : Type u
f : β → α
h : #α < #β
inst✝ : Uncountable β
hα : #α < ℵ₀
⊢ ∃ a, ℵ_ 1 ≤ #↑(f ⁻¹' {a})

pp: case inr
β α : Type u
f : β → α
h : #α < #β
inst✝ : Uncountable β
hα : ℵ₀ ≤ #α
⊢ ∃ a, ℵ_ 1 ≤ #↑(f ⁻¹' {a})

Tactic:
·

Goal state after:
case inl
β α : Type u
f : β → α
h : #α < #β
inst✝ : Uncountable β
hα : #α < ℵ₀
⊢ ∃ a, ℵ_ 1 ≤ #↑(f ⁻¹' {a})
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case inl
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α = 0
⊢ #{ s // #↑s < (#α).ord.cof } = #α

pp: case inr
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
⊢ #{ s // #↑s < (#α).ord.cof } = #α

Tactic:
·

Goal state after:
case inl
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α = 0
⊢ #{ s // #↑s < (#α).ord.cof } = #α
Module: Mathlib.SetTheory.Cardinal.Pigeonhole

Goal state before:
pp: case inr
β α : Type u
f : β → α
h : #α < #β
inst✝ : Uncountable β
hα : ℵ₀ ≤ #α
⊢ ∃ a, ℵ_ 1 ≤ #↑(f ⁻¹' {a})

Tactic:
·

Goal state after:
case inr
β α : Type u
f : β → α
h : #α < #β
inst✝ : Uncountable β
hα : ℵ₀ ≤ #α
⊢ ∃ a, ℵ_ 1 ≤ #↑(f ⁻¹' {a})
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case inr.a
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := ⋯
⊢ #{ s // #↑s < (#α).ord.cof } ≤ #α

pp: case a
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := ⋯
⊢ #α ≤ #{ s // #↑s < (#α).ord.cof }

Tactic:
·

Goal state after:
case inr.a
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := ⋯
⊢ #{ s // #↑s < (#α).ord.cof } ≤ #α
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case refine_1
α : Type u
inst✝ : Infinite α
f : α → Bool
x : α × Bool
⊢ (fun x ↦ (x.1, f x.1 ^^ x.2)) ((fun x ↦ (x.1, f x.1 ^^ x.2)) x) = x

pp: case refine_2
α : Type u
inst✝ : Infinite α
f g : α → Bool
h :
  (fun f ↦ Involutive.toPerm (fun x ↦ (x.1, f x.1 ^^ x.2)) ⋯) f =
    (fun f ↦ Involutive.toPerm (fun x ↦ (x.1, f x.1 ^^ x.2)) ⋯) g
⊢ f = g

Tactic:
·

Goal state after:
case refine_1
α : Type u
inst✝ : Infinite α
f : α → Bool
x : α × Bool
⊢ (fun x ↦ (x.1, f x.1 ^^ x.2)) ((fun x ↦ (x.1, f x.1 ^^ x.2)) x) = x
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case a
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := ⋯
⊢ #α ≤ #{ s // #↑s < (#α).ord.cof }

Tactic:
·

Goal state after:
case a
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := ⋯
⊢ #α ≤ #{ s // #↑s < (#α).ord.cof }
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case refine_2
α : Type u
inst✝ : Infinite α
f g : α → Bool
h :
  (fun f ↦ Involutive.toPerm (fun x ↦ (x.1, f x.1 ^^ x.2)) ⋯) f =
    (fun f ↦ Involutive.toPerm (fun x ↦ (x.1, f x.1 ^^ x.2)) ⋯) g
⊢ f = g

Tactic:
·

Goal state after:
case refine_2
α : Type u
inst✝ : Infinite α
f g : α → Bool
h :
  (fun f ↦ Involutive.toPerm (fun x ↦ (x.1, f x.1 ^^ x.2)) ⋯) f =
    (fun f ↦ Involutive.toPerm (fun x ↦ (x.1, f x.1 ^^ x.2)) ⋯) g
⊢ f = g
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case a.refine_1
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := linearOrderOfSTO r
x : α
⊢ #↑{x} < (#α).ord.cof

pp: case a.refine_2
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := linearOrderOfSTO r
⊢ Injective fun x ↦ ⟨{x}, ⋯⟩

Tactic:
·

Goal state after:
case a.refine_1
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := linearOrderOfSTO r
x : α
⊢ #↑{x} < (#α).ord.cof
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
a : Ordinal.{max u v}
ha : a < c.ord
⊢ ℵ₀ < c

pp: ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
a : Ordinal.{max u v}
ha : a < c.ord
⊢ lift.{v, u} #ι < c

Tactic:
·

Goal state after:
ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
a : Ordinal.{max u v}
ha : a < c.ord
⊢ ℵ₀ < c
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
a : Ordinal.{max u v}
ha : a < c.ord
⊢ lift.{v, u} #ι < c

Tactic:
·

Goal state after:
ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
a : Ordinal.{max u v}
ha : a < c.ord
⊢ lift.{v, u} #ι < c
Module: Mathlib.SetTheory.Cardinal.Cofinality.Ordinal

Goal state before:
pp: case a.refine_2
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := linearOrderOfSTO r
⊢ Injective fun x ↦ ⟨{x}, ⋯⟩

Tactic:
·

Goal state after:
case a.refine_2
α : Type u_1
h : (#α).IsStrongPrelimit
ha : #α ≠ 0
h' : (#α).IsStrongLimit
r : α → α → Prop
wo : IsWellOrder α r
hr : (#α).ord = type r
this : LinearOrder α := linearOrderOfSTO r
⊢ Injective fun x ↦ ⟨{x}, ⋯⟩
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inl
α : Type u
inst✝ : Nonempty α
h✝ : Finite α
⊢ #(List α) = max ℵ₀ #α

pp: case inr
α : Type u
inst✝ : Nonempty α
h✝ : Infinite α
⊢ #(List α) = max ℵ₀ #α

Tactic:
·

Goal state after:
case inl
α : Type u
inst✝ : Nonempty α
h✝ : Finite α
⊢ #(List α) = max ℵ₀ #α
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inr
α : Type u
inst✝ : Nonempty α
h✝ : Infinite α
⊢ #(List α) = max ℵ₀ #α

Tactic:
·

Goal state after:
case inr
α : Type u
inst✝ : Nonempty α
h✝ : Infinite α
⊢ #(List α) = max ℵ₀ #α
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inl
α : Type u
h✝ : Finite α
⊢ #(List α) ≤ max ℵ₀ #α

pp: case inr
α : Type u
h✝ : Infinite α
⊢ #(List α) ≤ max ℵ₀ #α

Tactic:
·

Goal state after:
case inl
α : Type u
h✝ : Finite α
⊢ #(List α) ≤ max ℵ₀ #α
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case inr
α : Type u
h✝ : Infinite α
⊢ #(List α) ≤ max ℵ₀ #α

Tactic:
·

Goal state after:
case inr
α : Type u
h✝ : Infinite α
⊢ #(List α) ≤ max ℵ₀ #α
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case mk.f
α : Type u
inst✝ : Infinite α
β : Type u
⊢ (fun α β ↦ β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β → { t // #↑t ≤ #β }

pp: case mk.hf
α : Type u
inst✝ : Infinite α
β : Type u
⊢ Surjective ?mk.f

Tactic:
·

Goal state after:
case mk.f
α : Type u
inst✝ : Infinite α
β : Type u
⊢ (fun α β ↦ β → α) (α ⊕ ULift.{u, 0} (Fin 1)) β → { t // #↑t ≤ #β }
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case h.mp
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
⊢ x ∈
      ↑((fun f ↦ ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y ↦
          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) →
    x ∈ ↑⟨s, ⋯⟩

pp: case h.mpr
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
⊢ x ∈ ↑⟨s, ⋯⟩ →
    x ∈
      ↑((fun f ↦ ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y ↦
          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 })

Tactic:
·

Goal state after:
case h.mp
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
⊢ x ∈
      ↑((fun f ↦ ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y ↦
          if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) →
    x ∈ ↑⟨s, ⋯⟩
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case pos
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
y : β
h : (if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) = Sum.inl x
h' : ∃ z, g z = y
⊢ x ∈ ↑⟨s, ⋯⟩

pp: case neg
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
y : β
h : (if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) = Sum.inl x
h' : ¬∃ z, g z = y
⊢ x ∈ ↑⟨s, ⋯⟩

Tactic:
·

Goal state after:
case pos
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
y : β
h : (if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) = Sum.inl x
h' : ∃ z, g z = y
⊢ x ∈ ↑⟨s, ⋯⟩
Module: Mathlib.SetTheory.Cardinal.Regular

Goal state before:
pp: case limit.ha
ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
hω : ℵ₀ < c.ord.cof
b : Ordinal.{max u v}
hb : IsSuccLimit b
H : ∀ o' < b, o' < c.ord → derivFamily f o' < c.ord
hb' : b < c.ord
⊢ lift.{max u v, max (u + 1) (v + 1)} #↑(Iio b) < (Ordinal.lift.{max (u + 1) (v + 1), max u v} c.ord).cof

pp: case limit.hf
ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
hω : ℵ₀ < c.ord.cof
b : Ordinal.{max u v}
hb : IsSuccLimit b
H : ∀ o' < b, o' < c.ord → derivFamily f o' < c.ord
hb' : b < c.ord
⊢ ∀ (i : ↑(Iio b)), derivFamily f ↑i < c.ord

Tactic:
·

Goal state after:
case limit.ha
ι : Type u
f : ι → Ordinal.{max u v} → Ordinal.{max u v}
c : Cardinal.{max u v}
hc : c.IsRegular
hι : lift.{v, u} #ι < c
hc' : c ≠ ℵ₀
hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord
hω : ℵ₀ < c.ord.cof
b : Ordinal.{max u v}
hb : IsSuccLimit b
H : ∀ o' < b, o' < c.ord → derivFamily f o' < c.ord
hb' : b < c.ord
⊢ lift.{max u v, max (u + 1) (v + 1)} #↑(Iio b) < (Ordinal.lift.{max (u + 1) (v + 1), max u v} c.ord).cof
Module: Mathlib.SetTheory.Cardinal.Arithmetic

Goal state before:
pp: case neg
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
y : β
h : (if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) = Sum.inl x
h' : ¬∃ z, g z = y
⊢ x ∈ ↑⟨s, ⋯⟩

Tactic:
·

Goal state after:
case neg
α : Type u
inst✝ : Infinite α
β : Type u
s : Set α
g : ↑s ↪ β
x : α
y : β
h : (if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) = Sum.inl x
h' : ¬∃ z, g z = y
⊢ x ∈ ↑⟨s, ⋯⟩