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chore(SetTheory/Cardinal/Basic): move IsStrongLimit #19723

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chore(SetTheory/Cardinal/Basic): move IsStrongLimit #19723

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move stuff
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35 changes: 33 additions & 2 deletions Mathlib/SetTheory/Cardinal/Basic.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -31,7 +31,9 @@ We define cardinal numbers as a quotient of types under the equivalence relation
* Multiplication `c₁ * c₂` is defined by `Cardinal.mul_def : #α * #β = #(α × β)`.
* The order `c₁ ≤ c₂` is defined by `Cardinal.le_def α β : #α ≤ #β ↔ Nonempty (α ↪ β)`.
* Exponentiation `c₁ ^ c₂` is defined by `Cardinal.power_def α β : #α ^ #β = #(β → α)`.
* `Cardinal.isLimit c` means that `c` is a (weak) limit cardinal: `c ≠ 0 ∧ ∀ x < c, succ x < c`.
* `IsSuccLimit c` means that `c` is a (weak) limit cardinal: `c ≠ 0 ∧ ∀ x < c, succ x < c`.
* `Cardinal.IsStrongLimit c` means that `c` is a strong limit cardinal:
`c ≠ 0 ∧ ∀ x < c, 2 ^ x < c`.
* `Cardinal.aleph0` or `ℵ₀` is the cardinality of `ℕ`. This definition is universe polymorphic:
`Cardinal.aleph0.{u} : Cardinal.{u}` (contrast with `ℕ : Type`, which lives in a specific
universe). In some cases the universe level has to be given explicitly.
Expand Down Expand Up @@ -797,6 +799,8 @@ theorem lift_succ (a) : lift.{v, u} (succ a) = succ (lift.{v, u} a) :=
exact h.not_lt (lt_succ _))
(succ_le_of_lt <| lift_lt.2 <| lt_succ a)

/-! ### Limit cardinals -/

/-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
cardinal by this definition, but `0` isn't.
Deprecated. Use `Order.IsSuccLimit` instead. -/
Expand Down Expand Up @@ -839,6 +843,25 @@ alias isSuccLimit_zero := isSuccPrelimit_zero

end deprecated

/-- A cardinal is a strong limit if it is not zero and it is closed under powersets. Note that `ℵ₀`
is a strong limit by this definition. -/
structure IsStrongLimit (c : Cardinal) : Prop where
ne_zero : c ≠ 0
two_power_lt {x} : x < c → 2 ^ x < c

protected theorem IsStrongLimit.isSuccLimit {c} (H : IsStrongLimit c) : IsSuccLimit c := by
rw [Cardinal.isSuccLimit_iff]
exact ⟨H.ne_zero, isSuccPrelimit_of_succ_lt fun x h ↦
(succ_le_of_lt <| cantor x).trans_lt (H.two_power_lt h)⟩

protected theorem IsStrongLimit.isSuccPrelimit {c} (H : IsStrongLimit c) : IsSuccPrelimit c :=
H.isSuccLimit.isSuccPrelimit

set_option linter.deprecated false in
@[deprecated IsStrongLimit.isSuccLimit (since := "2024-09-17")]
theorem IsStrongLimit.isLimit {c} (H : IsStrongLimit c) : IsLimit c :=
⟨H.ne_zero, H.isSuccPrelimit⟩

/-! ### Indexed cardinal `sum` -/

/-- The indexed sum of cardinals is the cardinality of the
Expand Down Expand Up @@ -1519,6 +1542,14 @@ theorem aleph0_le_of_isSuccLimit {c : Cardinal} (h : IsSuccLimit c) : ℵ₀ ≤
contrapose! h
exact not_isSuccLimit_of_lt_aleph0 h

theorem isStrongLimit_aleph0 : IsStrongLimit ℵ₀ := by
refine ⟨aleph0_ne_zero, fun hx ↦ ?_⟩
obtain ⟨n, rfl⟩ := lt_aleph0.1 hx
exact_mod_cast nat_lt_aleph0 _

theorem IsStrongLimit.aleph0_le {c} (H : IsStrongLimit c) : ℵ₀ ≤ c :=
aleph0_le_of_isSuccLimit H.isSuccLimit

section deprecated

set_option linter.deprecated false in
Expand Down Expand Up @@ -2167,4 +2198,4 @@ end Cardinal

-- end Tactic

set_option linter.style.longFile 2200
set_option linter.style.longFile 2400
42 changes: 4 additions & 38 deletions Mathlib/SetTheory/Cardinal/Cofinality.lean
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Expand Up @@ -17,8 +17,6 @@ This file contains the definition of cofinality of an ordinal number and regular
* `Ordinal.cof o` is the cofinality of the ordinal `o`.
If `o` is the order type of the relation `<` on `α`, then `o.cof` is the smallest cardinality of a
subset `s` of α that is *cofinal* in `α`, i.e. `∀ x : α, ∃ y ∈ s, ¬ y < x`.
* `Cardinal.IsStrongLimit c` means that `c` is a strong limit cardinal:
`c ≠ 0 ∧ ∀ x < c, 2 ^ x < c`.
* `Cardinal.IsRegular c` means that `c` is a regular cardinal: `ℵ₀ ≤ c ∧ c.ord.cof = c`.
* `Cardinal.IsInaccessible c` means that `c` is strongly inaccessible:
`ℵ₀ < c ∧ IsRegular c ∧ IsStrongLimit c`.
Expand Down Expand Up @@ -797,43 +795,11 @@ namespace Cardinal

open Ordinal

/-- A cardinal is a strong limit if it is not zero and it is
closed under powersets. Note that `ℵ₀` is a strong limit by this definition. -/
def IsStrongLimit (c : Cardinal) : Prop :=
c ≠ 0 ∧ ∀ x < c, (2^x) < c

theorem IsStrongLimit.ne_zero {c} (h : IsStrongLimit c) : c ≠ 0 :=
h.1

theorem IsStrongLimit.two_power_lt {x c} (h : IsStrongLimit c) : x < c → (2^x) < c :=
h.2 x

theorem isStrongLimit_aleph0 : IsStrongLimit ℵ₀ :=
⟨aleph0_ne_zero, fun x hx => by
rcases lt_aleph0.1 hx with ⟨n, rfl⟩
exact mod_cast nat_lt_aleph0 (2 ^ n)⟩

protected theorem IsStrongLimit.isSuccLimit {c} (H : IsStrongLimit c) : IsSuccLimit c := by
rw [Cardinal.isSuccLimit_iff]
exact ⟨H.ne_zero, isSuccPrelimit_of_succ_lt fun x h =>
(succ_le_of_lt <| cantor x).trans_lt (H.two_power_lt h)⟩

protected theorem IsStrongLimit.isSuccPrelimit {c} (H : IsStrongLimit c) : IsSuccPrelimit c :=
H.isSuccLimit.isSuccPrelimit

theorem IsStrongLimit.aleph0_le {c} (H : IsStrongLimit c) : ℵ₀ ≤ c :=
aleph0_le_of_isSuccLimit H.isSuccLimit

set_option linter.deprecated false in
@[deprecated IsStrongLimit.isSuccLimit (since := "2024-09-17")]
theorem IsStrongLimit.isLimit {c} (H : IsStrongLimit c) : IsLimit c :=
⟨H.ne_zero, H.isSuccPrelimit⟩

theorem isStrongLimit_beth {o : Ordinal} (H : IsSuccPrelimit o) : IsStrongLimit (ℶ_ o) := by
rcases eq_or_ne o 0 with (rfl | h)
· rw [beth_zero]
exact isStrongLimit_aleph0
· refine ⟨beth_ne_zero o, fun a ha => ?_⟩
· refine ⟨beth_ne_zero o, fun ha => ?_⟩
rw [beth_limit ⟨h, isSuccPrelimit_iff_succ_lt.1 H⟩] at ha
rcases exists_lt_of_lt_ciSup' ha with ⟨⟨i, hi⟩, ha⟩
have := power_le_power_left two_ne_zero ha.le
Expand All @@ -849,7 +815,7 @@ theorem mk_bounded_subset {α : Type*} (h : ∀ x < #α, (2^x) < #α) {r : α
constructor
rintro ⟨s, hs⟩
exact (not_unbounded_iff s).2 hs (unbounded_of_isEmpty s)
have h' : IsStrongLimit #α := ⟨ha, h⟩
have h' : IsStrongLimit #α := ⟨ha, @h⟩
have ha := h'.aleph0_le
apply le_antisymm
· have : { s : Set α | Bounded r s } = ⋃ i, 𝒫{ j | r j i } := setOf_exists _
Expand All @@ -874,7 +840,7 @@ theorem mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, (2^x) < #α) :
#{ s : Set α // #s < cof (#α).ord } = #α := by
rcases eq_or_ne #α 0 with (ha | ha)
· simp [ha]
have h' : IsStrongLimit #α := ⟨ha, h⟩
have h' : IsStrongLimit #α := ⟨ha, @h⟩
rcases ord_eq α with ⟨r, wo, hr⟩
haveI := wo
apply le_antisymm
Expand Down Expand Up @@ -1182,7 +1148,7 @@ def IsInaccessible (c : Cardinal) :=

theorem IsInaccessible.mk {c} (h₁ : ℵ₀ < c) (h₂ : c ≤ c.ord.cof) (h₃ : ∀ x < c, (2^x) < c) :
IsInaccessible c :=
⟨h₁, ⟨h₁.le, h₂⟩, (aleph0_pos.trans h₁).ne', h₃⟩
⟨h₁, ⟨h₁.le, h₂⟩, (aleph0_pos.trans h₁).ne', @h₃⟩

-- Lean's foundations prove the existence of ℵ₀ many inaccessible cardinals
theorem univ_inaccessible : IsInaccessible univ.{u, v} :=
Expand Down
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