mathlib documentation

set_theory.cardinal.ordinal

Cardinals and ordinals #

Relationships between cardinals and ordinals, properties of cardinals that are proved using ordinals.

Main definitions #

Main Statements #

Tags #

cardinal arithmetic (for infinite cardinals)

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theorem cardinal.ord_is_limit {c : cardinal} (co : ω ≤ c) :
c.ord.is_limit
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noncomputable def cardinal.aleph_idx.initial_seg  :
initial_seg has_lt.lt has_lt.lt

The aleph' index function, which gives the ordinal index of a cardinal. (The aleph' part is because unlike aleph this counts also the finite stages. So aleph_idx n = n, aleph_idx ω = ω, aleph_idx ℵ₁ = ω + 1 and so on.) In this definition, we register additionally that this function is an initial segment, i.e., it is order preserving and its range is an initial segment of the ordinals. For the basic function version, see aleph_idx. For an upgraded version stating that the range is everything, see aleph_idx.rel_iso.

Equations
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noncomputable def cardinal.aleph_idx  :
cardinal → ordinal

The aleph' index function, which gives the ordinal index of a cardinal. (The aleph' part is because unlike aleph this counts also the finite stages. So aleph_idx n = n, aleph_idx ω = ω, aleph_idx ℵ₁ = ω + 1 and so on.) For an upgraded version stating that the range is everything, see aleph_idx.rel_iso.

Equations
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@[simp]
theorem cardinal.aleph_idx.initial_seg_coe  :
⇑cardinal.aleph_idx.initial_seg = cardinal.aleph_idx
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@[simp]
theorem cardinal.aleph_idx_lt {a b : cardinal} :
a.aleph_idx < b.aleph_idx ↔ a < b
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@[simp]
theorem cardinal.aleph_idx_le {a b : cardinal} :
a.aleph_idx ≤ b.aleph_idx ↔ a ≤ b
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theorem cardinal.aleph_idx.init {a : cardinal} {b : ordinal} :
b < a.aleph_idx → (∃ (c : cardinal), c.aleph_idx = b)
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noncomputable def cardinal.aleph_idx.rel_iso  :
has_lt.lt ≃r has_lt.lt

The aleph' index function, which gives the ordinal index of a cardinal. (The aleph' part is because unlike aleph this counts also the finite stages. So aleph_idx n = n, aleph_idx ω = ω, aleph_idx ℵ₁ = ω + 1 and so on.) In this version, we register additionally that this function is an order isomorphism between cardinals and ordinals. For the basic function version, see aleph_idx.

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@[simp]
theorem cardinal.aleph_idx.rel_iso_coe  :
⇑cardinal.aleph_idx.rel_iso = cardinal.aleph_idx
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@[simp]
theorem cardinal.type_cardinal  :
ordinal.type has_lt.lt = ordinal.univ
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@[simp]
theorem cardinal.mk_cardinal  :
# cardinal = cardinal.univ
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noncomputable def cardinal.aleph'.rel_iso  :
has_lt.lt ≃r has_lt.lt

The aleph' function gives the cardinals listed by their ordinal index, and is the inverse of aleph_idx. aleph' n = n, aleph' ω = ω, aleph' (ω + 1) = ℵ₁, etc. In this version, we register additionally that this function is an order isomorphism between ordinals and cardinals. For the basic function version, see aleph'.

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noncomputable def cardinal.aleph'  :
ordinal → cardinal

The aleph' function gives the cardinals listed by their ordinal index, and is the inverse of aleph_idx. aleph' n = n, aleph' ω = ω, aleph' (ω + 1) = ℵ₁, etc.

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@[simp]
theorem cardinal.aleph'.rel_iso_coe  :
⇑cardinal.aleph'.rel_iso = cardinal.aleph'
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@[simp]
theorem cardinal.aleph'_lt {o₁ o₂ : ordinal} :
cardinal.aleph' o₁ < cardinal.aleph' o₂ ↔ o₁ < o₂
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@[simp]
theorem cardinal.aleph'_le {o₁ o₂ : ordinal} :
cardinal.aleph' o₁ ≤ cardinal.aleph' o₂ ↔ o₁ ≤ o₂
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@[simp]
theorem cardinal.aleph'_aleph_idx (c : cardinal) :
cardinal.aleph' c.aleph_idx = c
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@[simp]
theorem cardinal.aleph_idx_aleph' (o : ordinal) :
(cardinal.aleph' o).aleph_idx = o
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@[simp]
theorem cardinal.aleph'_zero  :
cardinal.aleph' 0 = 0
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@[simp]
theorem cardinal.aleph'_succ {o : ordinal} :
cardinal.aleph' o.succ = (cardinal.aleph' o).succ
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@[simp]
theorem cardinal.aleph'_nat (n : ℕ) :
cardinal.aleph' ↑n = ↑n
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theorem cardinal.aleph'_le_of_limit {o : ordinal} (l : o.is_limit) {c : cardinal} :
cardinal.aleph' o ≤ c ↔ ∀ (o' : ordinal), o' < o → cardinal.aleph' o' ≤ c
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@[simp]
theorem cardinal.aleph'_omega  :
cardinal.aleph' ω = ω
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@[simp]
noncomputable def cardinal.aleph'_equiv  :
ordinal ≃ cardinal

aleph' and aleph_idx form an equivalence between ordinal and cardinal

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noncomputable def cardinal.aleph (o : ordinal) :
cardinal

The aleph function gives the infinite cardinals listed by their ordinal index. aleph 0 = ω, aleph 1 = succ ω is the first uncountable cardinal, and so on.

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@[simp]
theorem cardinal.aleph_lt {o₁ o₂ : ordinal} :
cardinal.aleph o₁ < cardinal.aleph o₂ ↔ o₁ < o₂
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@[simp]
theorem cardinal.aleph_le {o₁ o₂ : ordinal} :
cardinal.aleph o₁ ≤ cardinal.aleph o₂ ↔ o₁ ≤ o₂
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@[simp]
theorem cardinal.max_aleph_eq (o₁ o₂ : ordinal) :
max (cardinal.aleph o₁) (cardinal.aleph o₂) = cardinal.aleph (max o₁ o₂)
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@[simp]
theorem cardinal.aleph_succ {o : ordinal} :
cardinal.aleph o.succ = (cardinal.aleph o).succ
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@[simp]
theorem cardinal.aleph_zero  :
cardinal.aleph 0 = ω
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theorem cardinal.omega_le_aleph' {o : ordinal} :
ω ≤ cardinal.aleph' o ↔ ω ≤ o
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theorem cardinal.omega_le_aleph (o : ordinal) :
ω ≤ cardinal.aleph o
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theorem cardinal.aleph'_pos {o : ordinal} (ho : 0 < o) :
0 < cardinal.aleph' o
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theorem cardinal.aleph_pos (o : ordinal) :
0 < cardinal.aleph o
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@[protected, instance]
def cardinal.α.nonempty (o : ordinal) :
nonempty (quotient.out (cardinal.aleph o).ord).α
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theorem cardinal.ord_aleph_is_limit (o : ordinal) :
(cardinal.aleph o).ord.is_limit
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@[protected, instance]
def cardinal.α.no_max_order (o : ordinal) :
no_max_order (quotient.out (cardinal.aleph o).ord).α
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theorem cardinal.exists_aleph {c : cardinal} :
ω ≤ c ↔ ∃ (o : ordinal), c = cardinal.aleph o
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theorem cardinal.aleph'_is_normal  :
ordinal.is_normal (cardinal.ord ∘ cardinal.aleph')
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theorem cardinal.aleph_is_normal  :
ordinal.is_normal (cardinal.ord ∘ cardinal.aleph)
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theorem cardinal.succ_omega  :
ω.succ = cardinal.aleph 1
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theorem cardinal.omega_lt_aleph_one  :
ω < cardinal.aleph 1
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theorem cardinal.countable_iff_lt_aleph_one {α : Type u_1} (s : set α) :
s.countable ↔ # ↥s < cardinal.aleph 1
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theorem cardinal.ord_card_unbounded  :
set.unbounded has_lt.lt {b : ordinal | b.card.ord = b}

Ordinals that are cardinals are unbounded.

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theorem cardinal.eq_aleph'_of_eq_card_ord {o : ordinal} (ho : o.card.ord = o) :
∃ (a : ordinal), (cardinal.aleph' a).ord = o
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theorem cardinal.ord_aleph'_eq_enum_card  :
cardinal.ord ∘ cardinal.aleph' = ordinal.enum_ord {b : ordinal | b.card.ord = b}

ord ∘ aleph' enumerates the ordinals that are cardinals.

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theorem cardinal.ord_card_unbounded'  :
set.unbounded has_lt.lt {b : ordinal | b.card.ord = b ∧ ω ≤ b}

Infinite ordinals that are cardinals are unbounded.

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theorem cardinal.eq_aleph_of_eq_card_ord {o : ordinal} (ho : o.card.ord = o) (ho' : ω ≤ o) :
∃ (a : ordinal), (cardinal.aleph a).ord = o
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theorem cardinal.ord_aleph_eq_enum_card  :
cardinal.ord ∘ cardinal.aleph = ordinal.enum_ord {b : ordinal | b.card.ord = b ∧ ω ≤ b}

ord ∘ aleph enumerates the infinite ordinals that are cardinals.

Properties of mul #

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theorem cardinal.mul_eq_self {c : cardinal} (h : ω ≤ c) :
c * c = c

If α is an infinite type, then α × α and α have the same cardinality.

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theorem cardinal.mul_eq_max {a b : cardinal} (ha : ω ≤ a) (hb : ω ≤ b) :
a * b = max a b

If α and β are infinite types, then the cardinality of α × β is the maximum of the cardinalities of α and β.

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@[simp]
theorem cardinal.mul_mk_eq_max {α β : Type u_1} [infinite α] [infinite β] :
(# α) * # β = max (# α) (# β)
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@[simp]
theorem cardinal.aleph_mul_aleph (o₁ o₂ : ordinal) :
(cardinal.aleph o₁) * cardinal.aleph o₂ = cardinal.aleph (max o₁ o₂)
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@[simp]
theorem cardinal.omega_mul_eq {a : cardinal} (ha : ω ≤ a) :
ω * a = a
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@[simp]
theorem cardinal.mul_omega_eq {a : cardinal} (ha : ω ≤ a) :
a * ω = a
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@[simp]
theorem cardinal.omega_mul_mk_eq {α : Type u_1} [infinite α] :
ω * # α = # α
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@[simp]
theorem cardinal.mk_mul_omega_eq {α : Type u_1} [infinite α] :
(# α) * ω = # α
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@[simp]
theorem cardinal.omega_mul_aleph (o : ordinal) :
ω * cardinal.aleph o = cardinal.aleph o
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@[simp]
theorem cardinal.aleph_mul_omega (o : ordinal) :
(cardinal.aleph o) * ω = cardinal.aleph o
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theorem cardinal.mul_lt_of_lt {a b c : cardinal} (hc : ω ≤ c) (h1 : a < c) (h2 : b < c) :
a * b < c
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theorem cardinal.mul_le_max_of_omega_le_left {a b : cardinal} (h : ω ≤ a) :
a * b ≤ max a b
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theorem cardinal.mul_eq_max_of_omega_le_left {a b : cardinal} (h : ω ≤ a) (h' : b ≠ 0) :
a * b = max a b
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theorem cardinal.mul_eq_max' {a b : cardinal} (h : ω ≤ a * b) :
a * b = max a b
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theorem cardinal.mul_le_max (a b : cardinal) :
a * b ≤ max (max a b) ω
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theorem cardinal.mul_eq_left {a b : cardinal} (ha : ω ≤ a) (hb : b ≤ a) (hb' : b ≠ 0) :
a * b = a
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theorem cardinal.mul_eq_right {a b : cardinal} (hb : ω ≤ b) (ha : a ≤ b) (ha' : a ≠ 0) :
a * b = b
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theorem cardinal.le_mul_left {a b : cardinal} (h : b ≠ 0) :
a ≤ b * a
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theorem cardinal.le_mul_right {a b : cardinal} (h : b ≠ 0) :
a ≤ a * b
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theorem cardinal.mul_eq_left_iff {a b : cardinal} :
a * b = a ↔ max ω b ≤ a ∧ b ≠ 0 ∨ b = 1 ∨ a = 0

Properties of add #

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theorem cardinal.add_eq_self {c : cardinal} (h : ω ≤ c) :
c + c = c

If α is an infinite type, then α ⊕ α and α have the same cardinality.

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theorem cardinal.add_eq_max {a b : cardinal} (ha : ω ≤ a) :
a + b = max a b

If α is an infinite type, then the cardinality of α ⊕ β is the maximum of the cardinalities of α and β.

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theorem cardinal.add_eq_max' {a b : cardinal} (ha : ω ≤ b) :
a + b = max a b
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@[simp]
theorem cardinal.add_mk_eq_max {α β : Type u_1} [infinite α] :
# α + # β = max (# α) (# β)
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@[simp]
theorem cardinal.add_mk_eq_max' {α β : Type u_1} [infinite β] :
# α + # β = max (# α) (# β)
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theorem cardinal.add_le_max (a b : cardinal) :
a + b ≤ max (max a b) ω
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theorem cardinal.add_le_of_le {a b c : cardinal} (hc : ω ≤ c) (h1 : a ≤ c) (h2 : b ≤ c) :
a + b ≤ c
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theorem cardinal.add_lt_of_lt {a b c : cardinal} (hc : ω ≤ c) (h1 : a < c) (h2 : b < c) :
a + b < c
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theorem cardinal.eq_of_add_eq_of_omega_le {a b c : cardinal} (h : a + b = c) (ha : a < c) (hc : ω ≤ c) :
b = c
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theorem cardinal.add_eq_left {a b : cardinal} (ha : ω ≤ a) (hb : b ≤ a) :
a + b = a
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theorem cardinal.add_eq_right {a b : cardinal} (hb : ω ≤ b) (ha : a ≤ b) :
a + b = b
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theorem cardinal.add_eq_left_iff {a b : cardinal} :
a + b = a ↔ max ω b ≤ a ∨ b = 0
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theorem cardinal.add_eq_right_iff {a b : cardinal} :
a + b = b ↔ max ω a ≤ b ∨ a = 0
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theorem cardinal.add_one_eq {a : cardinal} (ha : ω ≤ a) :
a + 1 = a
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@[simp]
theorem cardinal.mk_add_one_eq {α : Type u_1} [infinite α] :
# α + 1 = # α
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@[protected]
theorem cardinal.eq_of_add_eq_add_left {a b c : cardinal} (h : a + b = a + c) (ha : a < ω) :
b = c
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@[protected]
theorem cardinal.eq_of_add_eq_add_right {a b c : cardinal} (h : a + b = c + b) (hb : b < ω) :
a = c
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@[simp]
theorem cardinal.aleph_add_aleph (o₁ o₂ : ordinal) :
cardinal.aleph o₁ + cardinal.aleph o₂ = cardinal.aleph (max o₁ o₂)
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theorem cardinal.principal_add_ord {c : cardinal} (hc : ω ≤ c) :
ordinal.principal has_add.add c.ord
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theorem cardinal.principal_add_aleph (o : ordinal) :
ordinal.principal has_add.add (cardinal.aleph o).ord

Properties about power #

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theorem cardinal.pow_le {κ μ : cardinal} (H1 : ω ≤ κ) (H2 : μ < ω) :
κ ^ μ ≤ κ
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theorem cardinal.pow_eq {κ μ : cardinal} (H1 : ω ≤ κ) (H2 : 1 ≤ μ) (H3 : μ < ω) :
κ ^ μ = κ
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theorem cardinal.power_self_eq {c : cardinal} (h : ω ≤ c) :
c ^ c = 2 ^ c
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theorem cardinal.prod_eq_two_power {ι : Type u} [infinite ι] {c : ι → cardinal} (h₁ : ∀ (i : ι), 2 ≤ c i) (h₂ : ∀ (i : ι), (c i).lift ≤ (# ι).lift) :
cardinal.prod c = 2 ^ (# ι).lift
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theorem cardinal.power_eq_two_power {c₁ c₂ : cardinal} (h₁ : ω ≤ c₁) (h₂ : 2 ≤ c₂) (h₂' : c₂ ≤ c₁) :
c₂ ^ c₁ = 2 ^ c₁
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theorem cardinal.nat_power_eq {c : cardinal} (h : ω ≤ c) {n : ℕ} (hn : 2 ≤ n) :
↑n ^ c = 2 ^ c
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theorem cardinal.power_nat_le {c : cardinal} {n : ℕ} (h : ω ≤ c) :
c ^ n ≤ c
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theorem cardinal.power_nat_eq {c : cardinal} {n : ℕ} (h1 : ω ≤ c) (h2 : 1 ≤ n) :
c ^ n = c
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theorem cardinal.power_nat_le_max {c : cardinal} {n : ℕ} :
c ^ ↑n ≤ max c ω
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@[simp]
theorem cardinal.powerlt_omega {c : cardinal} (h : ω ≤ c) :
c ^< ω = c
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theorem cardinal.powerlt_omega_le (c : cardinal) :
c ^< ω ≤ max c ω

Computing cardinality of various types #

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theorem cardinal.mk_list_eq_mk (α : Type u) [infinite α] :
# (list α) = # α
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theorem cardinal.mk_list_eq_omega (α : Type u) [encodable α] [nonempty α] :
# (list α) = ω
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theorem cardinal.mk_list_eq_max_mk_omega (α : Type u) [nonempty α] :
# (list α) = max (# α) ω
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theorem cardinal.mk_list_le_max (α : Type u) :
# (list α) ≤ max ω (# α)
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theorem cardinal.mk_finset_eq_mk (α : Type u) [infinite α] :
# (finset α) = # α
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theorem cardinal.mk_bounded_set_le_of_infinite (α : Type u) [infinite α] (c : cardinal) :
# {t // # ↥t ≤ c} ≤ # α ^ c
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theorem cardinal.mk_bounded_set_le (α : Type u) (c : cardinal) :
# {t // # ↥t ≤ c} ≤ max (# α) ω ^ c
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theorem cardinal.mk_bounded_subset_le {α : Type u} (s : set α) (c : cardinal) :
# {t // t ⊆ s ∧ # ↥t ≤ c} ≤ max (# ↥s) ω ^ c

Properties of compl #

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theorem cardinal.mk_compl_of_infinite {α : Type u_1} [infinite α] (s : set α) (h2 : # ↥s < # α) :
# ↥sᶜ = # α
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theorem cardinal.mk_compl_finset_of_infinite {α : Type u_1} [infinite α] (s : finset α) :
# ↥(↑s)ᶜ = # α
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theorem cardinal.mk_compl_eq_mk_compl_infinite {α : Type u_1} [infinite α] {s t : set α} (hs : # ↥s < # α) (ht : # ↥t < # α) :
# ↥sᶜ = # ↥tᶜ
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theorem cardinal.mk_compl_eq_mk_compl_finite_lift {α : Type u} {β : Type v} [fintype α] {s : set α} {t : set β} (h1 : (# α).lift = (# β).lift) (h2 : (# ↥s).lift = (# ↥t).lift) :
(# ↥sᶜ).lift = (# ↥tᶜ).lift
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theorem cardinal.mk_compl_eq_mk_compl_finite {α β : Type u} [fintype α] {s : set α} {t : set β} (h1 : # α = # β) (h : # ↥s = # ↥t) :
# ↥sᶜ = # ↥tᶜ
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theorem cardinal.mk_compl_eq_mk_compl_finite_same {α : Type u_1} [fintype α] {s t : set α} (h : # ↥s = # ↥t) :
# ↥sᶜ = # ↥tᶜ

Extending an injection to an equiv #

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theorem cardinal.extend_function {α : Type u_1} {β : Type u_2} {s : set α} (f : ↥s ↪ β) (h : nonempty (↥sᶜ ≃ ↥(set.range ⇑f)ᶜ)) :
∃ (g : α ≃ β), ∀ (x : ↥s), ⇑g ↑x = ⇑f x
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theorem cardinal.extend_function_finite {α : Type u_1} {β : Type u_2} [fintype α] {s : set α} (f : ↥s ↪ β) (h : nonempty (α ≃ β)) :
∃ (g : α ≃ β), ∀ (x : ↥s), ⇑g ↑x = ⇑f x
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theorem cardinal.extend_function_of_lt {α : Type u_1} {β : Type u_2} {s : set α} (f : ↥s ↪ β) (hs : # ↥s < # α) (h : nonempty (α ≃ β)) :
∃ (g : α ≃ β), ∀ (x : ↥s), ⇑g ↑x = ⇑f x

This section proves inequalities for bit0 and bit1, enabling simp to solve inequalities for numeral cardinals. The complexity of the resulting algorithm is not good, as in some cases simp reduces an inequality to a disjunction of two situations, depending on whether a cardinal is finite or infinite. Since the evaluation of the branches is not lazy, this is bad. It is good enough for practical situations, though.

For specific numbers, these inequalities could also be deduced from the corresponding inequalities of natural numbers using norm_cast:

example : (37 : cardinal) < 42 :=
by { norm_cast, norm_num }
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theorem cardinal.bit0_ne_zero (a : cardinal) :
¬bit0 a = 0 ↔ ¬a = 0
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@[simp]
theorem cardinal.bit1_ne_zero (a : cardinal) :
¬bit1 a = 0
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@[simp]
theorem cardinal.zero_lt_bit0 (a : cardinal) :
0 < bit0 a ↔ 0 < a
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@[simp]
theorem cardinal.zero_lt_bit1 (a : cardinal) :
0 < bit1 a
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@[simp]
theorem cardinal.one_le_bit0 (a : cardinal) :
1 ≤ bit0 a ↔ 0 < a
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@[simp]
theorem cardinal.one_le_bit1 (a : cardinal) :
1 ≤ bit1 a
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theorem cardinal.bit0_eq_self {c : cardinal} (h : ω ≤ c) :
bit0 c = c
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@[simp]
theorem cardinal.bit0_lt_omega {c : cardinal} :
bit0 c < ω ↔ c < ω
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@[simp]
theorem cardinal.omega_le_bit0 {c : cardinal} :
ω ≤ bit0 c ↔ ω ≤ c
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@[simp]
theorem cardinal.bit1_eq_self_iff {c : cardinal} :
bit1 c = c ↔ ω ≤ c
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@[simp]
theorem cardinal.bit1_lt_omega {c : cardinal} :
bit1 c < ω ↔ c < ω
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@[simp]
theorem cardinal.omega_le_bit1 {c : cardinal} :
ω ≤ bit1 c ↔ ω ≤ c
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@[simp]
theorem cardinal.bit0_le_bit0 {a b : cardinal} :
bit0 a ≤ bit0 b ↔ a ≤ b
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@[simp]
theorem cardinal.bit0_le_bit1 {a b : cardinal} :
bit0 a ≤ bit1 b ↔ a ≤ b
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@[simp]
theorem cardinal.bit1_le_bit1 {a b : cardinal} :
bit1 a ≤ bit1 b ↔ a ≤ b
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@[simp]
theorem cardinal.bit1_le_bit0 {a b : cardinal} :
bit1 a ≤ bit0 b ↔ a < b ∨ a ≤ b ∧ ω ≤ a
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@[simp]
theorem cardinal.bit0_lt_bit0 {a b : cardinal} :
bit0 a < bit0 b ↔ a < b
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@[simp]
theorem cardinal.bit1_lt_bit0 {a b : cardinal} :
bit1 a < bit0 b ↔ a < b
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@[simp]
theorem cardinal.bit1_lt_bit1 {a b : cardinal} :
bit1 a < bit1 b ↔ a < b
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@[simp]
theorem cardinal.bit0_lt_bit1 {a b : cardinal} :
bit0 a < bit1 b ↔ a < b ∨ a ≤ b ∧ a < ω
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theorem cardinal.one_lt_two  :
1 < 2
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@[simp]
theorem cardinal.one_lt_bit0 {a : cardinal} :
1 < bit0 a ↔ 0 < a
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@[simp]
theorem cardinal.one_lt_bit1 (a : cardinal) :
1 < bit1 a ↔ 0 < a
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@[simp]
theorem cardinal.one_le_one  :
1 ≤ 1