statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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id_of_le_cof (h : o ≤ o.cof.ord) : is_fundamental_sequence o o (λ a _, a) | ⟨h, λ _ _ _ _, id, blsub_id o⟩ | theorem | ordinal.is_fundamental_sequence.id_of_le_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero {f : Π b < (0 : ordinal), ordinal} :
is_fundamental_sequence 0 0 f | ⟨by rw [cof_zero, ord_zero], λ i j hi, (ordinal.not_lt_zero i hi).elim, blsub_zero f⟩ | theorem | ordinal.is_fundamental_sequence.zero | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal",
"ordinal.not_lt_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ : is_fundamental_sequence (succ o) 1 (λ _ _, o) | begin
refine ⟨_, λ i j hi hj h, _, blsub_const ordinal.one_ne_zero o⟩,
{ rw [cof_succ, ord_one] },
{ rw lt_one_iff_zero at hi hj,
rw [hi, hj] at h,
exact h.false.elim }
end | theorem | ordinal.is_fundamental_sequence.succ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal.one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone (hf : is_fundamental_sequence a o f) {i j : ordinal} (hi : i < o)
(hj : j < o) (hij : i ≤ j) : f i hi ≤ f j hj | begin
rcases lt_or_eq_of_le hij with hij | rfl,
{ exact (hf.2.1 hi hj hij).le },
{ refl }
end | theorem | ordinal.is_fundamental_sequence.monotone | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"monotone",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {a o o' : ordinal.{u}} {f : Π b < o, ordinal.{u}}
(hf : is_fundamental_sequence a o f) {g : Π b < o', ordinal.{u}}
(hg : is_fundamental_sequence o o' g) :
is_fundamental_sequence a o' (λ i hi, f (g i hi) (by { rw ←hg.2.2, apply lt_blsub })) | begin
refine ⟨_, λ i j _ _ h, hf.2.1 _ _ (hg.2.1 _ _ h), _⟩,
{ rw hf.cof_eq,
exact hg.1.trans (ord_cof_le o) },
{ rw @blsub_comp.{u u u} o _ f (@is_fundamental_sequence.monotone _ _ f hf),
exact hf.2.2 }
end | theorem | ordinal.is_fundamental_sequence.trans | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_fundamental_sequence (a : ordinal.{u}) :
∃ f, is_fundamental_sequence a a.cof.ord f | begin
rsuffices ⟨o, f, hf⟩ : ∃ o f, is_fundamental_sequence a o f,
{ exact ⟨_, hf.ord_cof⟩ },
rcases exists_lsub_cof a with ⟨ι, f, hf, hι⟩,
rcases ord_eq ι with ⟨r, wo, hr⟩,
haveI := wo,
let r' := subrel r {i | ∀ j, r j i → f j < f i},
let hrr' : r' ↪r r := subrel.rel_embedding _ _,
haveI := hrr'.is_wel... | theorem | ordinal.exists_fundamental_sequence | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"subrel",
"subrel.rel_embedding"
] | Every ordinal has a fundamental sequence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof_cof (a : ordinal.{u}) : cof (cof a).ord = cof a | begin
cases exists_fundamental_sequence a with f hf,
cases exists_fundamental_sequence a.cof.ord with g hg,
exact ord_injective ((hf.trans hg).cof_eq.symm)
end | theorem | ordinal.cof_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.is_fundamental_sequence {f : ordinal.{u} → ordinal.{u}}
(hf : is_normal f) {a o} (ha : is_limit a) {g} (hg : is_fundamental_sequence a o g) :
is_fundamental_sequence (f a) o (λ b hb, f (g b hb)) | begin
refine ⟨_, λ i j _ _ h, hf.strict_mono (hg.2.1 _ _ h), _⟩,
{ rcases exists_lsub_cof (f a) with ⟨ι, f', hf', hι⟩,
rw [←hg.cof_eq, ord_le_ord, ←hι],
suffices : lsub.{u u} (λ i, (Inf {b : ordinal | f' i ≤ f b})) = a,
{ rw ←this,
apply cof_lsub_le },
have H : ∀ i, ∃ b < a, f' i ≤ f b := λ i,... | theorem | ordinal.is_normal.is_fundamental_sequence | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'",
"le_cInf_iff''",
"le_of_forall_lt",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.cof_eq {f} (hf : is_normal f) {a} (ha : is_limit a) : cof (f a) = cof a | let ⟨g, hg⟩ := exists_fundamental_sequence a in
ord_injective (hf.is_fundamental_sequence ha hg).cof_eq | theorem | ordinal.is_normal.cof_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.cof_le {f} (hf : is_normal f) (a) : cof a ≤ cof (f a) | begin
rcases zero_or_succ_or_limit a with rfl | ⟨b, rfl⟩ | ha,
{ rw cof_zero,
exact zero_le _ },
{ rw [cof_succ, cardinal.one_le_iff_ne_zero, cof_ne_zero, ←ordinal.pos_iff_ne_zero],
exact (ordinal.zero_le (f b)).trans_lt (hf.1 b) },
{ rw hf.cof_eq ha }
end | theorem | ordinal.is_normal.cof_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.one_le_iff_ne_zero",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_add (a b : ordinal) : b ≠ 0 → cof (a + b) = cof b | λ h, begin
rcases zero_or_succ_or_limit b with rfl | ⟨c, rfl⟩ | hb,
{ contradiction },
{ rw [add_succ, cof_succ, cof_succ] },
{ exact (add_is_normal a).cof_eq hb }
end | theorem | ordinal.cof_add | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_cof {o} : ℵ₀ ≤ cof o ↔ is_limit o | begin
rcases zero_or_succ_or_limit o with rfl|⟨o,rfl⟩|l,
{ simp [not_zero_is_limit, cardinal.aleph_0_ne_zero] },
{ simp [not_succ_is_limit, cardinal.one_lt_aleph_0] },
{ simp [l], refine le_of_not_lt (λ h, _),
cases cardinal.lt_aleph_0.1 h with n e,
have := cof_cof o,
rw [e, ord_nat] at this,
ca... | theorem | ordinal.aleph_0_le_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.aleph_0_ne_zero",
"cardinal.one_lt_aleph_0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_cof {o : ordinal} (ho : o.is_limit) : (aleph' o).ord.cof = o.cof | aleph'_is_normal.cof_eq ho | theorem | ordinal.aleph'_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_cof {o : ordinal} (ho : o.is_limit) : (aleph o).ord.cof = o.cof | aleph_is_normal.cof_eq ho | theorem | ordinal.aleph_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_omega : cof ω = ℵ₀ | (aleph_0_le_cof.2 omega_is_limit).antisymm' $ by { rw ←card_omega, apply cof_le_card } | theorem | ordinal.cof_omega | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"antisymm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_eq' (r : α → α → Prop) [is_well_order α r] (h : is_limit (type r)) :
∃ S : set α, (∀ a, ∃ b ∈ S, r a b) ∧ #S = cof (type r) | let ⟨S, H, e⟩ := cof_eq r in
⟨S, λ a,
let a' := enum r _ (h.2 _ (typein_lt_type r a)) in
let ⟨b, h, ab⟩ := H a' in
⟨b, h, (is_order_connected.conn a b a' $ (typein_lt_typein r).1
(by { rw typein_enum, exact lt_succ (typein _ _) })).resolve_right ab⟩,
e⟩ | theorem | ordinal.cof_eq' | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_univ : cof univ.{u v} = cardinal.univ | le_antisymm (cof_le_card _) begin
refine le_of_forall_lt (λ c h, _),
rcases lt_univ'.1 h with ⟨c, rfl⟩,
rcases @cof_eq ordinal.{u} (<) _ with ⟨S, H, Se⟩,
rw [univ, ← lift_cof, ← cardinal.lift_lift, cardinal.lift_lt, ← Se],
refine lt_of_not_ge (λ h, _),
cases cardinal.lift_down h with a e,
refine quotient.... | theorem | ordinal.cof_univ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift_down",
"cardinal.lift_lift",
"cardinal.lift_lt",
"cardinal.univ",
"le_of_forall_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unbounded_of_unbounded_sUnion (r : α → α → Prop) [wo : is_well_order α r] {s : set (set α)}
(h₁ : unbounded r $ ⋃₀ s) (h₂ : #s < strict_order.cof r) : ∃ x ∈ s, unbounded r x | begin
by_contra' h,
simp_rw not_unbounded_iff at h,
let f : s → α := λ x : s, wo.wf.sup x (h x.1 x.2),
refine h₂.not_le (le_trans (cInf_le' ⟨range f, λ x, _, rfl⟩) mk_range_le),
rcases h₁ x with ⟨y, ⟨c, hc, hy⟩, hxy⟩,
exact ⟨f ⟨c, hc⟩, mem_range_self _, λ hxz, hxy (trans (wo.wf.lt_sup _ hy) hxz)⟩
end | theorem | ordinal.unbounded_of_unbounded_sUnion | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'",
"is_well_order",
"strict_order.cof"
] | If the union of s is unbounded and s is smaller than the cofinality,
then s has an unbounded member | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unbounded_of_unbounded_Union {α β : Type u} (r : α → α → Prop) [wo : is_well_order α r]
(s : β → set α)
(h₁ : unbounded r $ ⋃ x, s x) (h₂ : #β < strict_order.cof r) : ∃ x : β, unbounded r (s x) | begin
rw ←sUnion_range at h₁,
rcases unbounded_of_unbounded_sUnion r h₁ (mk_range_le.trans_lt h₂) with ⟨_, ⟨x, rfl⟩, u⟩,
exact ⟨x, u⟩
end | theorem | ordinal.unbounded_of_unbounded_Union | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order",
"strict_order.cof"
] | If the union of s is unbounded and s is smaller than the cofinality,
then s has an unbounded member | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infinite_pigeonhole {β α : Type u} (f : β → α) (h₁ : ℵ₀ ≤ #β)
(h₂ : #α < (#β).ord.cof) : ∃ a : α, #(f ⁻¹' {a}) = #β | begin
have : ∃ a, #β ≤ #(f ⁻¹' {a}),
{ by_contra' h,
apply mk_univ.not_lt,
rw [←preimage_univ, ←Union_of_singleton, preimage_Union],
exact mk_Union_le_sum_mk.trans_lt ((sum_le_supr _).trans_lt $ mul_lt_of_lt h₁
(h₂.trans_le $ cof_ord_le _) (supr_lt h₂ h)) },
cases this with x h,
refine ⟨x, h.a... | theorem | ordinal.infinite_pigeonhole | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | The infinite pigeonhole principle | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infinite_pigeonhole_card {β α : Type u} (f : β → α) (θ : cardinal) (hθ : θ ≤ #β)
(h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof) : ∃ a : α, θ ≤ #(f ⁻¹' {a}) | begin
rcases le_mk_iff_exists_set.1 hθ with ⟨s, rfl⟩,
cases infinite_pigeonhole (f ∘ subtype.val : s → α) h₁ h₂ with a ha,
use a, rw [←ha, @preimage_comp _ _ _ subtype.val f],
exact mk_preimage_of_injective _ _ subtype.val_injective
end | theorem | ordinal.infinite_pigeonhole_card | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"subtype.val_injective"
] | Pigeonhole principle for a cardinality below the cardinality of the domain | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
infinite_pigeonhole_set {β α : Type u} {s : set β} (f : s → α) (θ : cardinal)
(hθ : θ ≤ #s) (h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof) :
∃ (a : α) (t : set β) (h : t ⊆ s), θ ≤ #t ∧ ∀ {{x}} (hx : x ∈ t), f ⟨x, h hx⟩ = a | begin
cases infinite_pigeonhole_card f θ hθ h₁ h₂ with a ha,
refine ⟨a, {x | ∃ h, f ⟨x, h⟩ = a}, _, _, _⟩,
{ rintro x ⟨hx, hx'⟩, exact hx },
{ refine ha.trans (ge_of_eq $ quotient.sound ⟨equiv.trans _
(equiv.subtype_subtype_equiv_subtype_exists _ _).symm⟩),
simp only [coe_eq_subtype, mem_singleton_iff... | theorem | ordinal.infinite_pigeonhole_set | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"equiv.subtype_subtype_equiv_subtype_exists",
"ge_of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strong_limit (c : cardinal) : Prop | c ≠ 0 ∧ ∀ x < c, 2 ^ x < c | def | cardinal.is_strong_limit | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_strong_limit.ne_zero {c} (h : is_strong_limit c) : c ≠ 0 | h.1 | theorem | cardinal.is_strong_limit.ne_zero | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strong_limit.two_power_lt {x c} (h : is_strong_limit c) : x < c → 2 ^ x < c | h.2 x | theorem | cardinal.is_strong_limit.two_power_lt | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strong_limit_aleph_0 : is_strong_limit ℵ₀ | ⟨aleph_0_ne_zero, λ x hx, begin
rcases lt_aleph_0.1 hx with ⟨n, rfl⟩,
exact_mod_cast nat_lt_aleph_0 (pow 2 n)
end⟩ | theorem | cardinal.is_strong_limit_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strong_limit.is_succ_limit {c} (H : is_strong_limit c) : is_succ_limit c | is_succ_limit_of_succ_lt $ λ x h, (succ_le_of_lt $ cantor x).trans_lt (H.two_power_lt h) | theorem | cardinal.is_strong_limit.is_succ_limit | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strong_limit.is_limit {c} (H : is_strong_limit c) : is_limit c | ⟨H.ne_zero, H.is_succ_limit⟩ | theorem | cardinal.is_strong_limit.is_limit | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_strong_limit_beth {o : ordinal} (H : is_succ_limit o) : is_strong_limit (beth o) | begin
rcases eq_or_ne o 0 with rfl | h,
{ rw beth_zero,
exact is_strong_limit_aleph_0 },
{ refine ⟨beth_ne_zero o, λ a ha, _⟩,
rw beth_limit ⟨h, is_succ_limit_iff_succ_lt.1 H⟩ at ha,
rcases exists_lt_of_lt_csupr' ha with ⟨⟨i, hi⟩, ha⟩,
have := power_le_power_left two_ne_zero ha.le,
rw ←beth_su... | theorem | cardinal.is_strong_limit_beth | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"eq_or_ne",
"exists_lt_of_lt_csupr'",
"ordinal",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bounded_subset {α : Type*} (h : ∀ x < #α, 2 ^ x < #α) {r : α → α → Prop}
[is_well_order α r] (hr : (#α).ord = type r) : #{s : set α // bounded r s} = #α | begin
rcases eq_or_ne (#α) 0 with ha | ha,
{ rw ha,
haveI := mk_eq_zero_iff.1 ha,
rw mk_eq_zero_iff,
split,
rintro ⟨s, hs⟩,
exact (not_unbounded_iff s).2 hs (unbounded_of_is_empty s) },
have h' : is_strong_limit (#α) := ⟨ha, h⟩,
have ha := h'.is_limit.aleph_0_le,
apply le_antisymm,
{ hav... | theorem | cardinal.mk_bounded_subset | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"csupr_le'",
"eq_or_ne",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, 2 ^ x < #α) :
#{s : set α // #s < cof (#α).ord} = #α | begin
rcases eq_or_ne (#α) 0 with ha | ha,
{ rw ha,
simp [λ s, (cardinal.zero_le s).not_lt] },
have h' : is_strong_limit (#α) := ⟨ha, h⟩,
rcases ord_eq α with ⟨r, wo, hr⟩,
haveI := wo,
apply le_antisymm,
{ nth_rewrite_rhs 0 ←mk_bounded_subset h hr,
apply mk_le_mk_of_subset (λ s hs, _),
rw hr a... | theorem | cardinal.mk_subset_mk_lt_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.zero_le",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular (c : cardinal) : Prop | ℵ₀ ≤ c ∧ c ≤ c.ord.cof | def | cardinal.is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"is_regular"
] | A cardinal is regular if it is infinite and it equals its own cofinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_regular.aleph_0_le {c : cardinal} (H : c.is_regular) : ℵ₀ ≤ c | H.1 | lemma | cardinal.is_regular.aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.cof_eq {c : cardinal} (H : c.is_regular) : c.ord.cof = c | (cof_ord_le c).antisymm H.2 | lemma | cardinal.is_regular.cof_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.pos {c : cardinal} (H : c.is_regular) : 0 < c | aleph_0_pos.trans_le H.1 | lemma | cardinal.is_regular.pos | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular.ord_pos {c : cardinal} (H : c.is_regular) : 0 < c.ord | by { rw cardinal.lt_ord, exact H.pos } | lemma | cardinal.is_regular.ord_pos | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"cardinal.lt_ord"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_cof {o : ordinal} (h : o.is_limit) : is_regular o.cof | ⟨aleph_0_le_cof.2 h, (cof_cof o).ge⟩ | theorem | cardinal.is_regular_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_aleph_0 : is_regular ℵ₀ | ⟨le_rfl, by simp⟩ | theorem | cardinal.is_regular_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_succ {c : cardinal.{u}} (h : ℵ₀ ≤ c) : is_regular (succ c) | ⟨h.trans (le_succ c), succ_le_of_lt begin
cases quotient.exists_rep (@succ cardinal _ _ c) with α αe, simp at αe,
rcases ord_eq α with ⟨r, wo, re⟩, resetI,
have := ord_is_limit (h.trans (le_succ _)),
rw [← αe, re] at this ⊢,
rcases cof_eq' r this with ⟨S, H, Se⟩,
rw [← Se],
apply lt_imp_lt_of_le_imp_le (λ... | theorem | cardinal.is_regular_succ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"is_regular",
"lt_imp_lt_of_le_imp_le",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_aleph_one : is_regular (aleph 1) | by { rw ←succ_aleph_0, exact is_regular_succ le_rfl } | theorem | cardinal.is_regular_aleph_one | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_aleph'_succ {o : ordinal} (h : ω ≤ o) : is_regular (aleph' (succ o)) | by { rw aleph'_succ, exact is_regular_succ (aleph_0_le_aleph'.2 h) } | theorem | cardinal.is_regular_aleph'_succ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_regular_aleph_succ (o : ordinal) : is_regular (aleph (succ o)) | by { rw aleph_succ, exact is_regular_succ (aleph_0_le_aleph o) } | theorem | cardinal.is_regular_aleph_succ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infinite_pigeonhole_card_lt {β α : Type u} (f : β → α)
(w : #α < #β) (w' : ℵ₀ ≤ #α) :
∃ a : α, #α < #(f ⁻¹' {a}) | begin
simp_rw [← succ_le_iff],
exact ordinal.infinite_pigeonhole_card f (succ (#α)) (succ_le_of_lt w)
(w'.trans (lt_succ _).le)
((lt_succ _).trans_le (is_regular_succ w').2.ge),
end | theorem | cardinal.infinite_pigeonhole_card_lt | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal.infinite_pigeonhole_card"
] | A function whose codomain's cardinality is infinite but strictly smaller than its domain's
has a fiber with cardinality strictly great than the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_infinite_fiber {β α : Type*} (f : β → α)
(w : #α < #β) (w' : _root_.infinite α) :
∃ a : α, _root_.infinite (f ⁻¹' {a}) | begin
simp_rw [cardinal.infinite_iff] at ⊢ w',
cases infinite_pigeonhole_card_lt f w w' with a ha,
exact ⟨a, w'.trans ha.le⟩,
end | theorem | cardinal.exists_infinite_fiber | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.infinite_iff"
] | A function whose codomain's cardinality is infinite but strictly smaller than its domain's
has an infinite fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_range_of_union_finset_eq_top
{α β : Type*} [infinite β] (f : α → finset β) (w : (⋃ a, (f a : set β)) = ⊤) :
#β ≤ #(range f) | begin
have k : _root_.infinite (range f),
{ rw infinite_coe_iff,
apply mt (union_finset_finite_of_range_finite f),
rw w,
exact infinite_univ, },
by_contradiction h,
simp only [not_le] at h,
let u : Π b, ∃ a, b ∈ f a := λ b, by simpa using (w.ge : _) (set.mem_univ b),
let u' : β → range f := λ b,... | lemma | cardinal.le_range_of_union_finset_eq_top | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"by_contradiction",
"finset",
"infinite",
"infinite.of_injective",
"set.mem_univ"
] | If an infinite type `β` can be expressed as a union of finite sets,
then the cardinality of the collection of those finite sets
must be at least the cardinality of `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lsub_lt_ord_lift_of_is_regular {ι} {f : ι → ordinal} {c} (hc : is_regular c)
(hι : cardinal.lift (#ι) < c) : (∀ i, f i < c.ord) → ordinal.lsub f < c.ord | lsub_lt_ord_lift (by rwa hc.cof_eq) | theorem | cardinal.lsub_lt_ord_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift",
"is_regular",
"ordinal",
"ordinal.lsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_lt_ord_of_is_regular {ι} {f : ι → ordinal} {c} (hc : is_regular c) (hι : #ι < c) :
(∀ i, f i < c.ord) → ordinal.lsub f < c.ord | lsub_lt_ord (by rwa hc.cof_eq) | theorem | cardinal.lsub_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal",
"ordinal.lsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_lt_ord_lift_of_is_regular {ι} {f : ι → ordinal} {c} (hc : is_regular c)
(hι : cardinal.lift (#ι) < c) : (∀ i, f i < c.ord) → ordinal.sup f < c.ord | sup_lt_ord_lift (by rwa hc.cof_eq) | theorem | cardinal.sup_lt_ord_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift",
"is_regular",
"ordinal",
"ordinal.sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_lt_ord_of_is_regular {ι} {f : ι → ordinal} {c} (hc : is_regular c) (hι : #ι < c) :
(∀ i, f i < c.ord) → ordinal.sup f < c.ord | sup_lt_ord (by rwa hc.cof_eq) | theorem | cardinal.sup_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal",
"ordinal.sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_lt_ord_lift_of_is_regular {o : ordinal} {f : Π a < o, ordinal} {c} (hc : is_regular c)
(ho : cardinal.lift o.card < c) : (∀ i hi, f i hi < c.ord) → ordinal.blsub o f < c.ord | blsub_lt_ord_lift (by rwa hc.cof_eq) | theorem | cardinal.blsub_lt_ord_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift",
"is_regular",
"ordinal",
"ordinal.blsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_lt_ord_of_is_regular {o : ordinal} {f : Π a < o, ordinal} {c} (hc : is_regular c)
(ho : o.card < c) : (∀ i hi, f i hi < c.ord) → ordinal.blsub o f < c.ord | blsub_lt_ord (by rwa hc.cof_eq) | theorem | cardinal.blsub_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal",
"ordinal.blsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_lt_ord_lift_of_is_regular {o : ordinal} {f : Π a < o, ordinal} {c} (hc : is_regular c)
(hι : cardinal.lift o.card < c) : (∀ i hi, f i hi < c.ord) → ordinal.bsup o f < c.ord | bsup_lt_ord_lift (by rwa hc.cof_eq) | theorem | cardinal.bsup_lt_ord_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift",
"is_regular",
"ordinal",
"ordinal.bsup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_lt_ord_of_is_regular {o : ordinal} {f : Π a < o, ordinal} {c} (hc : is_regular c)
(hι : o.card < c) : (∀ i hi, f i hi < c.ord) → ordinal.bsup o f < c.ord | bsup_lt_ord (by rwa hc.cof_eq) | theorem | cardinal.bsup_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal",
"ordinal.bsup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lt_lift_of_is_regular {ι} {f : ι → cardinal} {c} (hc : is_regular c)
(hι : cardinal.lift (#ι) < c) : (∀ i, f i < c) → supr f < c | supr_lt_lift (by rwa hc.cof_eq) | theorem | cardinal.supr_lt_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"cardinal.lift",
"is_regular",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lt_of_is_regular {ι} {f : ι → cardinal} {c} (hc : is_regular c) (hι : #ι < c) :
(∀ i, f i < c) → supr f < c | supr_lt (by rwa hc.cof_eq) | theorem | cardinal.supr_lt_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"is_regular",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_lt_lift_of_is_regular {ι : Type u} {f : ι → cardinal} {c : cardinal} (hc : is_regular c)
(hι : cardinal.lift.{v u} (#ι) < c) (hf : ∀ i, f i < c) : sum f < c | (sum_le_supr_lift _).trans_lt $ mul_lt_of_lt hc.1 hι (supr_lt_lift_of_is_regular hc hι hf) | theorem | cardinal.sum_lt_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_lt_of_is_regular {ι : Type u} {f : ι → cardinal} {c : cardinal} (hc : is_regular c)
(hι : #ι < c) : (∀ i, f i < c) → sum f < c | sum_lt_lift_of_is_regular.{u u} hc (by rwa lift_id) | theorem | cardinal.sum_lt_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"is_regular"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_lt_ord_lift_of_is_regular {ι} {f : ι → ordinal → ordinal} {c} (hc : is_regular c)
(hι : (#ι).lift < c) (hc' : c ≠ ℵ₀) (hf : ∀ i (b < c.ord), f i b < c.ord) {a} (ha : a < c.ord) :
nfp_family.{u v} f a < c.ord | by { apply nfp_family_lt_ord_lift _ _ hf ha; rwa hc.cof_eq, exact lt_of_le_of_ne hc.1 hc'.symm } | theorem | cardinal.nfp_family_lt_ord_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_lt_ord_of_is_regular {ι} {f : ι → ordinal → ordinal} {c} (hc : is_regular c)
(hι : #ι < c) (hc' : c ≠ ℵ₀) {a} (hf : ∀ i (b < c.ord), f i b < c.ord) :
a < c.ord → nfp_family.{u u} f a < c.ord | nfp_family_lt_ord_lift_of_is_regular hc (by rwa lift_id) hc' hf | theorem | cardinal.nfp_family_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_bfamily_lt_ord_lift_of_is_regular {o : ordinal} {f : Π a < o, ordinal → ordinal} {c}
(hc : is_regular c) (ho : o.card.lift < c) (hc' : c ≠ ℵ₀)
(hf : ∀ i hi (b < c.ord), f i hi b < c.ord) {a} : a < c.ord → nfp_bfamily.{u v} o f a < c.ord | nfp_family_lt_ord_lift_of_is_regular hc (by rwa mk_ordinal_out) hc' (λ i, hf _ _) | theorem | cardinal.nfp_bfamily_lt_ord_lift_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_bfamily_lt_ord_of_is_regular {o : ordinal} {f : Π a < o, ordinal → ordinal} {c}
(hc : is_regular c) (ho : o.card < c) (hc' : c ≠ ℵ₀) (hf : ∀ i hi (b < c.ord), f i hi b < c.ord)
{a} : a < c.ord → nfp_bfamily.{u u} o f a < c.ord | nfp_bfamily_lt_ord_lift_of_is_regular hc (by rwa lift_id) hc' hf | theorem | cardinal.nfp_bfamily_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_lt_ord_of_is_regular {f : ordinal → ordinal} {c} (hc : is_regular c) (hc' : c ≠ ℵ₀)
(hf : ∀ i < c.ord, f i < c.ord) {a} : (a < c.ord) → nfp f a < c.ord | nfp_lt_ord (by { rw hc.cof_eq, exact lt_of_le_of_ne hc.1 hc'.symm }) hf | theorem | cardinal.nfp_lt_ord_of_is_regular | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_family_lt_ord_lift {ι} {f : ι → ordinal → ordinal} {c} (hc : is_regular c)
(hι : (#ι).lift < c) (hc' : c ≠ ℵ₀) (hf : ∀ i (b < c.ord), f i b < c.ord) {a} :
a < c.ord → deriv_family.{u v} f a < c.ord | begin
have hω : ℵ₀ < c.ord.cof,
{ rw hc.cof_eq, exact lt_of_le_of_ne hc.1 hc'.symm },
apply a.limit_rec_on,
{ rw deriv_family_zero,
exact nfp_family_lt_ord_lift hω (by rwa hc.cof_eq) hf },
{ intros b hb hb',
rw deriv_family_succ,
exact nfp_family_lt_ord_lift hω (by rwa hc.cof_eq) hf
((ord_is... | theorem | cardinal.deriv_family_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_family_lt_ord {ι} {f : ι → ordinal → ordinal} {c} (hc : is_regular c)
(hι : #ι < c) (hc' : c ≠ ℵ₀) (hf : ∀ i (b < c.ord), f i b < c.ord) {a} :
a < c.ord → deriv_family.{u u} f a < c.ord | deriv_family_lt_ord_lift hc (by rwa lift_id) hc' hf | theorem | cardinal.deriv_family_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_bfamily_lt_ord_lift {o : ordinal} {f : Π a < o, ordinal → ordinal} {c}
(hc : is_regular c) (hι : o.card.lift < c) (hc' : c ≠ ℵ₀)
(hf : ∀ i hi (b < c.ord), f i hi b < c.ord) {a} :
a < c.ord → deriv_bfamily.{u v} o f a < c.ord | deriv_family_lt_ord_lift hc (by rwa mk_ordinal_out) hc' (λ i, hf _ _) | theorem | cardinal.deriv_bfamily_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_bfamily_lt_ord {o : ordinal} {f : Π a < o, ordinal → ordinal} {c} (hc : is_regular c)
(hι : o.card < c) (hc' : c ≠ ℵ₀) (hf : ∀ i hi (b < c.ord), f i hi b < c.ord)
{a} : a < c.ord → deriv_bfamily.{u u} o f a < c.ord | deriv_bfamily_lt_ord_lift hc (by rwa lift_id) hc' hf | theorem | cardinal.deriv_bfamily_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_lt_ord {f : ordinal.{u} → ordinal} {c} (hc : is_regular c) (hc' : c ≠ ℵ₀)
(hf : ∀ i < c.ord, f i < c.ord) {a} : a < c.ord → deriv f a < c.ord | deriv_family_lt_ord_lift hc
(by simpa using cardinal.one_lt_aleph_0.trans (lt_of_le_of_ne hc.1 hc'.symm))
hc' (λ _, hf) | theorem | cardinal.deriv_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"deriv",
"is_regular",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_inaccessible (c : cardinal) | ℵ₀ < c ∧ is_regular c ∧ is_strong_limit c | def | cardinal.is_inaccessible | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"is_regular"
] | A cardinal is inaccessible if it is an uncountable regular strong limit cardinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_inaccessible.mk {c} (h₁ : ℵ₀ < c) (h₂ : c ≤ c.ord.cof) (h₃ : ∀ x < c, 2 ^ x < c) :
is_inaccessible c | ⟨h₁, ⟨h₁.le, h₂⟩, (aleph_0_pos.trans h₁).ne', h₃⟩ | theorem | cardinal.is_inaccessible.mk | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
univ_inaccessible : is_inaccessible (univ.{u v}) | is_inaccessible.mk
(by simpa using lift_lt_univ' ℵ₀)
(by simp)
(λ c h, begin
rcases lt_univ'.1 h with ⟨c, rfl⟩,
rw ← lift_two_power.{u (max (u+1) v)},
apply lift_lt_univ'
end) | theorem | cardinal.univ_inaccessible | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_power_cof {c : cardinal.{u}} : ℵ₀ ≤ c → c < c ^ cof c.ord | quotient.induction_on c $ λ α h, begin
rcases ord_eq α with ⟨r, wo, re⟩, resetI,
have := ord_is_limit h,
rw [mk_def, re] at this ⊢,
rcases cof_eq' r this with ⟨S, H, Se⟩,
have := sum_lt_prod (λ a:S, #{x // r x a}) (λ _, #α) (λ i, _),
{ simp only [cardinal.prod_const, cardinal.lift_id, ← Se, ← mk_sigma, powe... | theorem | cardinal.lt_power_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift_id",
"cardinal.prod_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_cof_power {a b : cardinal} (ha : ℵ₀ ≤ a) (b1 : 1 < b) :
a < cof (b ^ a).ord | begin
have b0 : b ≠ 0 := (zero_lt_one.trans b1).ne',
apply lt_imp_lt_of_le_imp_le (power_le_power_left $ power_ne_zero a b0),
rw [←power_mul, mul_eq_self ha],
exact lt_power_cof (ha.trans $ (cantor' _ b1).le),
end | theorem | cardinal.lt_cof_power | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"lt_imp_lt_of_le_imp_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum : cardinal.{u} | 2 ^ aleph_0.{u} | def | cardinal.continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | Cardinality of continuum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
two_power_aleph_0 : 2 ^ aleph_0.{u} = continuum.{u} | rfl | lemma | cardinal.two_power_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_continuum : lift.{v} 𝔠 = 𝔠 | by rw [←two_power_aleph_0, lift_two_power, lift_aleph_0, two_power_aleph_0] | lemma | cardinal.lift_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_le_lift {c : cardinal.{u}} : 𝔠 ≤ lift.{v} c ↔ 𝔠 ≤ c | by rw [←lift_continuum, lift_le] | lemma | cardinal.continuum_le_lift | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_le_continuum {c : cardinal.{u}} : lift.{v} c ≤ 𝔠 ↔ c ≤ 𝔠 | by rw [←lift_continuum, lift_le] | lemma | cardinal.lift_le_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_lt_lift {c : cardinal.{u}} : 𝔠 < lift.{v} c ↔ 𝔠 < c | by rw [←lift_continuum, lift_lt] | lemma | cardinal.continuum_lt_lift | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt_continuum {c : cardinal.{u}} : lift.{v} c < 𝔠 ↔ c < 𝔠 | by rw [←lift_continuum, lift_lt] | lemma | cardinal.lift_lt_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_lt_continuum : ℵ₀ < 𝔠 | cantor ℵ₀ | lemma | cardinal.aleph_0_lt_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_continuum : ℵ₀ ≤ 𝔠 | aleph_0_lt_continuum.le | lemma | cardinal.aleph_0_le_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
beth_one : beth 1 = 𝔠 | by simpa using beth_succ 0 | lemma | cardinal.beth_one | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_lt_continuum (n : ℕ) : ↑n < 𝔠 | (nat_lt_aleph_0 n).trans aleph_0_lt_continuum | lemma | cardinal.nat_lt_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_set_nat : #(set ℕ) = 𝔠 | by simp | lemma | cardinal.mk_set_nat | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_pos : 0 < 𝔠 | nat_lt_continuum 0 | lemma | cardinal.continuum_pos | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_ne_zero : 𝔠 ≠ 0 | continuum_pos.ne' | lemma | cardinal.continuum_ne_zero | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_one_le_continuum : aleph 1 ≤ 𝔠 | by { rw ←succ_aleph_0, exact order.succ_le_of_lt aleph_0_lt_continuum } | lemma | cardinal.aleph_one_le_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [
"order.succ_le_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_to_nat : continuum.to_nat = 0 | to_nat_apply_of_aleph_0_le aleph_0_le_continuum | theorem | cardinal.continuum_to_nat | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_to_part_enat : continuum.to_part_enat = ⊤ | to_part_enat_apply_of_aleph_0_le aleph_0_le_continuum | theorem | cardinal.continuum_to_part_enat | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_add_continuum : ℵ₀ + 𝔠 = 𝔠 | add_eq_right aleph_0_le_continuum aleph_0_le_continuum | lemma | cardinal.aleph_0_add_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_add_aleph_0 : 𝔠 + ℵ₀ = 𝔠 | (add_comm _ _).trans aleph_0_add_continuum | lemma | cardinal.continuum_add_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_add_self : 𝔠 + 𝔠 = 𝔠 | add_eq_right aleph_0_le_continuum le_rfl | lemma | cardinal.continuum_add_self | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_add_continuum (n : ℕ) : ↑n + 𝔠 = 𝔠 | add_eq_right aleph_0_le_continuum (nat_lt_continuum n).le | lemma | cardinal.nat_add_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_add_nat (n : ℕ) : 𝔠 + n = 𝔠 | (add_comm _ _).trans (nat_add_continuum n) | lemma | cardinal.continuum_add_nat | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_mul_self : 𝔠 * 𝔠 = 𝔠 | mul_eq_left aleph_0_le_continuum le_rfl continuum_ne_zero | lemma | cardinal.continuum_mul_self | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_mul_aleph_0 : 𝔠 * ℵ₀ = 𝔠 | mul_eq_left aleph_0_le_continuum aleph_0_le_continuum aleph_0_ne_zero | lemma | cardinal.continuum_mul_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_mul_continuum : ℵ₀ * 𝔠 = 𝔠 | (mul_comm _ _).trans continuum_mul_aleph_0 | lemma | cardinal.aleph_0_mul_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_mul_continuum {n : ℕ} (hn : n ≠ 0) : ↑n * 𝔠 = 𝔠 | mul_eq_right aleph_0_le_continuum (nat_lt_continuum n).le (nat.cast_ne_zero.2 hn) | lemma | cardinal.nat_mul_continuum | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuum_mul_nat {n : ℕ} (hn : n ≠ 0) : 𝔠 * n = 𝔠 | (mul_comm _ _).trans (nat_mul_continuum hn) | lemma | cardinal.continuum_mul_nat | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_power_aleph_0 : aleph_0.{u} ^ aleph_0.{u} = 𝔠 | power_self_eq le_rfl | lemma | cardinal.aleph_0_power_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/continuum.lean | [
"set_theory.cardinal.ordinal"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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