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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nat_power_aleph_0 {n : ℕ} (hn : 2 ≤ n) : (n ^ aleph_0.{u} : cardinal.{u}) = 𝔠 | nat_power_eq le_rfl hn | lemma | cardinal.nat_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 | |
continuum_power_aleph_0 : continuum.{u} ^ aleph_0.{u} = 𝔠 | by rw [←two_power_aleph_0, ←power_mul, mul_eq_left le_rfl le_rfl aleph_0_ne_zero] | lemma | cardinal.continuum_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 | |
is_unit_iff : is_unit a ↔ a = 1 | begin
refine ⟨λ h, _, by { rintro rfl, exact is_unit_one }⟩,
rcases eq_or_ne a 0 with rfl | ha,
{ exact (not_is_unit_zero h).elim },
rw is_unit_iff_forall_dvd at h,
cases h 1 with t ht,
rw [eq_comm, mul_eq_one_iff'] at ht,
{ exact ht.1 },
all_goals { rwa one_le_iff_ne_zero },
{ rintro rfl,
rw mul_... | lemma | cardinal.is_unit_iff | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"eq_or_ne",
"is_unit",
"is_unit_iff_forall_dvd",
"is_unit_one",
"mul_eq_one_iff'",
"mul_zero",
"not_is_unit_zero",
"zero_ne_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_dvd : ∀ {a b : cardinal}, b ≠ 0 → a ∣ b → a ≤ b | | a _ b0 ⟨b, rfl⟩ := by simpa only [mul_one] using mul_le_mul_left'
(one_le_iff_ne_zero.2 (λ h : b = 0, by simpa only [h, mul_zero] using b0)) a | theorem | cardinal.le_of_dvd | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"cardinal",
"mul_le_mul_left'",
"mul_one",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_le_of_aleph_0_le (ha : a ≠ 0) (h : a ≤ b) (hb : ℵ₀ ≤ b) : a ∣ b | ⟨b, (mul_eq_right hb h ha).symm⟩ | lemma | cardinal.dvd_of_le_of_aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prime_of_aleph_0_le (ha : ℵ₀ ≤ a) : prime a | begin
refine ⟨(aleph_0_pos.trans_le ha).ne', _, λ b c hbc, _⟩,
{ rw is_unit_iff,
exact (one_lt_aleph_0.trans_le ha).ne' },
cases eq_or_ne (b * c) 0 with hz hz,
{ rcases mul_eq_zero.mp hz with rfl | rfl; simp },
wlog h : c ≤ b,
{ cases le_total c b; [skip, rw or_comm]; apply_assumption, assumption',
... | lemma | cardinal.prime_of_aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"eq_or_ne",
"max_def'",
"mul_comm",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_irreducible_of_aleph_0_le (ha : ℵ₀ ≤ a) : ¬irreducible a | begin
rw [irreducible_iff, not_and_distrib],
refine or.inr (λ h, _),
simpa [mul_aleph_0_eq ha, is_unit_iff, (one_lt_aleph_0.trans_le ha).ne', one_lt_aleph_0.ne']
using h a ℵ₀
end | lemma | cardinal.not_irreducible_of_aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"irreducible",
"irreducible_iff",
"not_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_coe_dvd_iff : (n : cardinal) ∣ m ↔ n ∣ m | begin
refine ⟨_, λ ⟨h, ht⟩, ⟨h, by exact_mod_cast ht⟩⟩,
rintro ⟨k, hk⟩,
have : ↑m < ℵ₀ := nat_lt_aleph_0 m,
rw [hk, mul_lt_aleph_0_iff] at this,
rcases this with h | h | ⟨-, hk'⟩,
iterate 2 { simp only [h, mul_zero, zero_mul, nat.cast_eq_zero] at hk, simp [hk] },
lift k to ℕ using hk',
exact ⟨k, by exa... | lemma | cardinal.nat_coe_dvd_iff | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"cardinal",
"lift",
"mul_zero",
"nat.cast_eq_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_is_prime_iff : prime (n : cardinal) ↔ n.prime | begin
simp only [prime, nat.prime_iff],
refine and_congr (by simp) (and_congr _ ⟨λ h b c hbc, _, λ h b c hbc, _⟩),
{ simp only [is_unit_iff, nat.is_unit_iff],
exact_mod_cast iff.rfl },
{ exact_mod_cast h b c (by exact_mod_cast hbc) },
cases lt_or_le (b * c) ℵ₀ with h' h',
{ rcases mul_lt_aleph_0_iff.mp ... | lemma | cardinal.nat_is_prime_iff | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"cardinal",
"lift",
"mul_comm",
"mul_eq_zero",
"nat.is_unit_iff",
"nat.prime_iff",
"prime",
"zero_dvd_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_iff {a : cardinal} : prime a ↔ ℵ₀ ≤ a ∨ ∃ p : ℕ, a = p ∧ p.prime | begin
cases le_or_lt ℵ₀ a with h h,
{ simp [h] },
lift a to ℕ using id h,
simp [not_le.mpr h]
end | lemma | cardinal.is_prime_iff | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"cardinal",
"lift",
"prime"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_prime_pow_iff {a : cardinal} :
is_prime_pow a ↔ ℵ₀ ≤ a ∨ ∃ n : ℕ, a = n ∧ is_prime_pow n | begin
by_cases h : ℵ₀ ≤ a,
{ simp [h, (prime_of_aleph_0_le h).is_prime_pow] },
lift a to ℕ using not_le.mp h,
simp only [h, nat.cast_inj, exists_eq_left', false_or, is_prime_pow_nat_iff],
rw is_prime_pow_def,
refine ⟨_, λ ⟨p, k, hp, hk, h⟩, ⟨p, k, nat_is_prime_iff.2 hp, by exact_mod_cast and.intro hk h⟩⟩,
... | lemma | cardinal.is_prime_pow_iff | set_theory.cardinal | src/set_theory/cardinal/divisibility.lean | [
"algebra.is_prime_pow",
"set_theory.cardinal.ordinal"
] | [
"cardinal",
"exists_eq_left'",
"is_prime_pow",
"is_prime_pow_def",
"is_prime_pow_nat_iff",
"lift",
"nat.cast_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card (α : Type*) : ℕ | (mk α).to_nat | def | nat.card | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"to_nat"
] | `nat.card α` is the cardinality of `α` as a natural number.
If `α` is infinite, `nat.card α = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_eq_fintype_card [fintype α] : nat.card α = fintype.card α | mk_to_nat_eq_card | lemma | nat.card_eq_fintype_card | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"fintype",
"fintype.card",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_zero_of_infinite [infinite α] : nat.card α = 0 | mk_to_nat_of_infinite | lemma | nat.card_eq_zero_of_infinite | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"infinite",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_of_card_ne_zero (h : nat.card α ≠ 0) : finite α | not_infinite_iff_finite.mp $ h ∘ @nat.card_eq_zero_of_infinite α | lemma | nat.finite_of_card_ne_zero | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"finite",
"nat.card",
"nat.card_eq_zero_of_infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_congr (f : α ≃ β) : nat.card α = nat.card β | cardinal.to_nat_congr f | lemma | nat.card_congr | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal.to_nat_congr",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_of_bijective (f : α → β) (hf : function.bijective f) : nat.card α = nat.card β | card_congr (equiv.of_bijective f hf) | lemma | nat.card_eq_of_bijective | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"equiv.of_bijective",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_of_equiv_fin {α : Type*} {n : ℕ}
(f : α ≃ fin n) : nat.card α = n | by simpa using card_congr f | lemma | nat.card_eq_of_equiv_fin | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_fin_of_card_pos {α : Type*} (h : nat.card α ≠ 0) :
α ≃ fin (nat.card α) | begin
casesI fintype_or_infinite α,
{ simpa using fintype.equiv_fin α },
{ simpa using h },
end | def | nat.equiv_fin_of_card_pos | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"fintype.equiv_fin",
"fintype_or_infinite",
"nat.card"
] | If the cardinality is positive, that means it is a finite type, so there is
an equivalence between `α` and `fin (nat.card α)`. See also `finite.equiv_fin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_of_subsingleton (a : α) [subsingleton α] : nat.card α = 1 | begin
letI := fintype.of_subsingleton a,
rw [card_eq_fintype_card, fintype.card_of_subsingleton a]
end | lemma | nat.card_of_subsingleton | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"fintype.card_of_subsingleton",
"fintype.of_subsingleton",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_unique [unique α] : nat.card α = 1 | card_of_subsingleton default | lemma | nat.card_unique | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"nat.card",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_one_iff_unique : nat.card α = 1 ↔ subsingleton α ∧ nonempty α | cardinal.to_nat_eq_one_iff_unique | lemma | nat.card_eq_one_iff_unique | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal.to_nat_eq_one_iff_unique",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_two_iff : nat.card α = 2 ↔ ∃ x y : α, x ≠ y ∧ {x, y} = @set.univ α | (to_nat_eq_iff two_ne_zero).trans $ iff.trans (by rw [nat.cast_two]) mk_eq_two_iff | lemma | nat.card_eq_two_iff | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"nat.card",
"nat.cast_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_two_iff' (x : α) : nat.card α = 2 ↔ ∃! y, y ≠ x | (to_nat_eq_iff two_ne_zero).trans $ iff.trans (by rw [nat.cast_two]) (mk_eq_two_iff' x) | lemma | nat.card_eq_two_iff' | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"nat.card",
"nat.cast_two",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_of_is_empty [is_empty α] : nat.card α = 0 | by simp | theorem | nat.card_of_is_empty | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"is_empty",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_prod (α β : Type*) : nat.card (α × β) = nat.card α * nat.card β | by simp only [nat.card, mk_prod, to_nat_mul, to_nat_lift] | lemma | nat.card_prod | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_ulift (α : Type*) : nat.card (ulift α) = nat.card α | card_congr equiv.ulift | lemma | nat.card_ulift | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_plift (α : Type*) : nat.card (plift α) = nat.card α | card_congr equiv.plift | lemma | nat.card_plift | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"equiv.plift",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_pi {β : α → Type*} [fintype α] : nat.card (Π a, β a) = ∏ a, nat.card (β a) | by simp_rw [nat.card, mk_pi, prod_eq_of_fintype, to_nat_lift, to_nat_finset_prod] | lemma | nat.card_pi | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"fintype",
"nat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_fun [finite α] : nat.card (α → β) = nat.card β ^ nat.card α | begin
haveI := fintype.of_finite α,
rw [nat.card_pi, finset.prod_const, finset.card_univ, ←nat.card_eq_fintype_card],
end | lemma | nat.card_fun | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"finite",
"finset.card_univ",
"finset.prod_const",
"fintype.of_finite",
"nat.card",
"nat.card_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_zmod (n : ℕ) : nat.card (zmod n) = n | begin
cases n,
{ exact nat.card_eq_zero_of_infinite },
{ rw [nat.card_eq_fintype_card, zmod.card] },
end | lemma | nat.card_zmod | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"nat.card",
"nat.card_eq_fintype_card",
"nat.card_eq_zero_of_infinite",
"zmod",
"zmod.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card (α : Type*) : part_enat | (mk α).to_part_enat | def | part_enat.card | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"part_enat"
] | `part_enat.card α` is the cardinality of `α` as an extended natural number.
If `α` is infinite, `part_enat.card α = ⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_eq_coe_fintype_card [fintype α] : card α = fintype.card α | mk_to_part_enat_eq_coe_card | lemma | part_enat.card_eq_coe_fintype_card | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_top_of_infinite [infinite α] : card α = ⊤ | mk_to_part_enat_of_infinite | lemma | part_enat.card_eq_top_of_infinite | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_congr {α : Type*} {β : Type*} (f : α ≃ β) :
part_enat.card α = part_enat.card β | cardinal.to_part_enat_congr f | lemma | part_enat.card_congr | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal.to_part_enat_congr",
"part_enat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_ulift (α : Type*) : card (ulift α) = card α | card_congr equiv.ulift | lemma | part_enat.card_ulift | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"equiv.ulift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_plift (α : Type*) : card (plift α) = card α | card_congr equiv.plift | lemma | part_enat.card_plift | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"equiv.plift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_image_of_inj_on {α : Type*} {β : Type*} {f : α → β} {s : set α} (h : set.inj_on f s) :
card (f '' s) = card s | card_congr (equiv.set.image_of_inj_on f s h).symm | lemma | part_enat.card_image_of_inj_on | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"equiv.set.image_of_inj_on",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_image_of_injective {α : Type*} {β : Type*}
(f : α → β) (s : set α) (h : function.injective f) :
card (f '' s) = card s | card_image_of_inj_on (set.inj_on_of_injective h s) | lemma | part_enat.card_image_of_injective | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"set.inj_on_of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cardinal.coe_nat_le_to_part_enat_iff {n : ℕ} {c : cardinal} :
↑n ≤ to_part_enat c ↔ ↑n ≤ c | by rw [← to_part_enat_cast n, to_part_enat_le_iff_le_of_le_aleph_0 (le_of_lt (nat_lt_aleph_0 n))] | lemma | cardinal.coe_nat_le_to_part_enat_iff | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cardinal.to_part_enat_le_coe_nat_iff {c : cardinal} {n : ℕ} :
to_part_enat c ≤ n ↔ c ≤ n | by rw [← to_part_enat_cast n,
to_part_enat_le_iff_le_of_lt_aleph_0 (nat_lt_aleph_0 n)] | lemma | cardinal.to_part_enat_le_coe_nat_iff | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cardinal.coe_nat_eq_to_part_enat_iff {n : ℕ} {c : cardinal} :
↑n = to_part_enat c ↔ ↑n = c | by rw [le_antisymm_iff, le_antisymm_iff,
cardinal.coe_nat_le_to_part_enat_iff, cardinal.to_part_enat_le_coe_nat_iff] | lemma | cardinal.coe_nat_eq_to_part_enat_iff | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cardinal.coe_nat_le_to_part_enat_iff",
"cardinal.to_part_enat_le_coe_nat_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cardinal.to_part_enat_eq_coe_nat_iff {c : cardinal} {n : ℕ} :
to_part_enat c = n ↔ c = n | by rw [eq_comm, cardinal.coe_nat_eq_to_part_enat_iff, eq_comm] | lemma | cardinal.to_part_enat_eq_coe_nat_iff | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cardinal.coe_nat_eq_to_part_enat_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cardinal.coe_nat_lt_coe_iff_lt {n : ℕ} {c : cardinal} :
↑n < to_part_enat c ↔ ↑n < c | by simp only [← not_le, cardinal.to_part_enat_le_coe_nat_iff] | lemma | cardinal.coe_nat_lt_coe_iff_lt | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cardinal.to_part_enat_le_coe_nat_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.cardinal.lt_coe_nat_iff_lt {n : ℕ} {c : cardinal} :
to_part_enat c < n ↔ c < n | by simp only [← not_le, cardinal.coe_nat_le_to_part_enat_iff] | lemma | cardinal.lt_coe_nat_iff_lt | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cardinal.coe_nat_le_to_part_enat_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_zero_iff_empty (α : Type*) : card α = 0 ↔ is_empty α | begin
rw ← cardinal.mk_eq_zero_iff,
conv_rhs { rw ← nat.cast_zero },
rw ← cardinal.to_part_enat_eq_coe_nat_iff,
simp only [part_enat.card, nat.cast_zero]
end | lemma | part_enat.card_eq_zero_iff_empty | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal.mk_eq_zero_iff",
"cardinal.to_part_enat_eq_coe_nat_iff",
"is_empty",
"nat.cast_zero",
"part_enat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_one_iff_subsingleton (α : Type*) : card α ≤ 1 ↔ subsingleton α | begin
rw ← le_one_iff_subsingleton,
conv_rhs { rw ← nat.cast_one},
rw ← cardinal.to_part_enat_le_coe_nat_iff,
simp only [part_enat.card, nat.cast_one]
end | lemma | part_enat.card_le_one_iff_subsingleton | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal.to_part_enat_le_coe_nat_iff",
"nat.cast_one",
"part_enat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_card_iff_nontrivial (α : Type*) : 1 < card α ↔ nontrivial α | begin
rw ← one_lt_iff_nontrivial,
conv_rhs { rw ← nat.cast_one},
rw ← cardinal.coe_nat_lt_coe_iff_lt,
simp only [part_enat.card, nat.cast_one]
end | lemma | part_enat.one_lt_card_iff_nontrivial | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"cardinal.coe_nat_lt_coe_iff_lt",
"nat.cast_one",
"nontrivial",
"part_enat.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_finite_of_card {α : Type*} {n : ℕ} (hα : part_enat.card α = n) :
finite α | begin
apply or.resolve_right (finite_or_infinite α),
intro h, resetI,
apply part_enat.coe_ne_top n,
rw ← hα,
exact part_enat.card_eq_top_of_infinite,
end | lemma | part_enat.is_finite_of_card | set_theory.cardinal | src/set_theory/cardinal/finite.lean | [
"data.zmod.defs",
"set_theory.cardinal.basic"
] | [
"finite",
"finite_or_infinite",
"part_enat.card",
"part_enat.card_eq_top_of_infinite",
"part_enat.coe_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_is_limit {c} (co : ℵ₀ ≤ c) : (ord c).is_limit | begin
refine ⟨λ h, aleph_0_ne_zero _, λ a, lt_imp_lt_of_le_imp_le (λ h, _)⟩,
{ rw [←ordinal.le_zero, ord_le] at h,
simpa only [card_zero, nonpos_iff_eq_zero] using co.trans h },
{ rw ord_le at h ⊢,
rwa [←@add_one_of_aleph_0_le (card a), ←card_succ],
rw [←ord_le, ←le_succ_of_is_limit, ord_le],
{ ex... | theorem | cardinal.ord_is_limit | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"lt_imp_lt_of_le_imp_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_idx.initial_seg : @initial_seg cardinal ordinal (<) (<) | @rel_embedding.collapse cardinal ordinal (<) (<) _ cardinal.ord.order_embedding.lt_embedding | def | cardinal.aleph_idx.initial_seg | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"initial_seg",
"ordinal",
"rel_embedding.collapse"
] | 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 seg... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_idx : cardinal → ordinal | aleph_idx.initial_seg | def | cardinal.aleph_idx | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"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_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_idx.initial_seg_coe :
(aleph_idx.initial_seg : cardinal → ordinal) = aleph_idx | rfl | theorem | cardinal.aleph_idx.initial_seg_coe | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_idx_lt {a b} : aleph_idx a < aleph_idx b ↔ a < b | aleph_idx.initial_seg.to_rel_embedding.map_rel_iff | theorem | cardinal.aleph_idx_lt | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_idx_le {a b} : aleph_idx a ≤ aleph_idx b ↔ a ≤ b | by rw [← not_lt, ← not_lt, aleph_idx_lt] | theorem | cardinal.aleph_idx_le | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_idx.init {a b} : b < aleph_idx a → ∃ c, aleph_idx c = b | aleph_idx.initial_seg.init | theorem | cardinal.aleph_idx.init | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_idx.rel_iso : @rel_iso cardinal.{u} ordinal.{u} (<) (<) | @rel_iso.of_surjective cardinal.{u} ordinal.{u} (<) (<) aleph_idx.initial_seg.{u} $
(initial_seg.eq_or_principal aleph_idx.initial_seg.{u}).resolve_right $
λ ⟨o, e⟩, begin
have : ∀ c, aleph_idx c < o := λ c, (e _).2 ⟨_, rfl⟩,
refine ordinal.induction_on o _ this, introsI α r _ h,
let s := ⨆ a, inv_fun aleph_idx (... | def | cardinal.aleph_idx.rel_iso | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"initial_seg.eq_or_principal",
"inv_fun",
"le_csupr",
"left_inverse_inv_fun",
"ordinal.enum",
"ordinal.induction_on",
"ordinal.typein",
"rel_iso",
"rel_iso.of_surjective"
] | 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 ℵ₀ = ω`,
`aleph_idx ℵ₁ = ω + 1` and so on.)
In this version, we register additionally that this function is an order isomorp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_idx.rel_iso_coe :
(aleph_idx.rel_iso : cardinal → ordinal) = aleph_idx | rfl | theorem | cardinal.aleph_idx.rel_iso_coe | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_cardinal : @type cardinal (<) _ = ordinal.univ.{u (u+1)} | by rw ordinal.univ_id; exact quotient.sound ⟨aleph_idx.rel_iso⟩ | theorem | cardinal.type_cardinal | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal.univ_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_cardinal : #cardinal = univ.{u (u+1)} | by simpa only [card_type, card_univ] using congr_arg card type_cardinal | theorem | cardinal.mk_cardinal | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'.rel_iso | cardinal.aleph_idx.rel_iso.symm | def | cardinal.aleph'.rel_iso | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | 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) = succ ℵ₀`, etc.
In this version, we register additionally that this function is an order isomorphism
between ordinals and cardinals.
For the basic functi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph' : ordinal → cardinal | aleph'.rel_iso | def | cardinal.aleph' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal"
] | 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) = succ ℵ₀`, etc. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph'.rel_iso_coe :
(aleph'.rel_iso : ordinal → cardinal) = aleph' | rfl | theorem | cardinal.aleph'.rel_iso_coe | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_lt {o₁ o₂ : ordinal} : aleph' o₁ < aleph' o₂ ↔ o₁ < o₂ | aleph'.rel_iso.map_rel_iff | theorem | cardinal.aleph'_lt | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_le {o₁ o₂ : ordinal} : aleph' o₁ ≤ aleph' o₂ ↔ o₁ ≤ o₂ | le_iff_le_iff_lt_iff_lt.2 aleph'_lt | theorem | cardinal.aleph'_le | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_aleph_idx (c : cardinal) : aleph' c.aleph_idx = c | cardinal.aleph_idx.rel_iso.to_equiv.symm_apply_apply c | theorem | cardinal.aleph'_aleph_idx | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_idx_aleph' (o : ordinal) : (aleph' o).aleph_idx = o | cardinal.aleph_idx.rel_iso.to_equiv.apply_symm_apply o | theorem | cardinal.aleph_idx_aleph' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_zero : aleph' 0 = 0 | by { rw [← nonpos_iff_eq_zero, ← aleph'_aleph_idx 0, aleph'_le], apply ordinal.zero_le } | theorem | cardinal.aleph'_zero | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_succ {o : ordinal} : aleph' (succ o) = succ (aleph' o) | begin
apply (succ_le_of_lt $ aleph'_lt.2 $ lt_succ o).antisymm' (cardinal.aleph_idx_le.1 $ _),
rw [aleph_idx_aleph', succ_le_iff, ← aleph'_lt, aleph'_aleph_idx],
apply lt_succ
end | theorem | cardinal.aleph'_succ | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"antisymm'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_nat : ∀ n : ℕ, aleph' n = n | | 0 := aleph'_zero
| (n+1) := show aleph' (succ n) = n.succ,
by rw [aleph'_succ, aleph'_nat, nat_succ] | theorem | cardinal.aleph'_nat | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_le_of_limit {o : ordinal} (l : o.is_limit) {c} :
aleph' o ≤ c ↔ ∀ o' < o, aleph' o' ≤ c | ⟨λ h o' h', (aleph'_le.2 $ h'.le).trans h,
λ h, begin
rw [←aleph'_aleph_idx c, aleph'_le, limit_le l],
intros x h',
rw [←aleph'_le, aleph'_aleph_idx],
exact h _ h'
end⟩ | theorem | cardinal.aleph'_le_of_limit | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_limit {o : ordinal} (ho : o.is_limit) : aleph' o = ⨆ a : Iio o, aleph' a | begin
refine le_antisymm _ (csupr_le' (λ i, aleph'_le.2 (le_of_lt i.2))),
rw aleph'_le_of_limit ho,
exact λ a ha, le_csupr (bdd_above_of_small _) (⟨a, ha⟩ : Iio o)
end | theorem | cardinal.aleph'_limit | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"csupr_le'",
"le_csupr",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_omega : aleph' ω = ℵ₀ | eq_of_forall_ge_iff $ λ c, begin
simp only [aleph'_le_of_limit omega_is_limit, lt_omega, exists_imp_distrib, aleph_0_le],
exact forall_swap.trans (forall_congr $ λ n, by simp only [forall_eq, aleph'_nat]),
end | theorem | cardinal.aleph'_omega | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"eq_of_forall_ge_iff",
"exists_imp_distrib",
"forall_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_equiv : ordinal ≃ cardinal | ⟨aleph', aleph_idx, aleph_idx_aleph', aleph'_aleph_idx⟩ | def | cardinal.aleph'_equiv | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal"
] | `aleph'` and `aleph_idx` form an equivalence between `ordinal` and `cardinal` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph (o : ordinal) : cardinal | aleph' (ω + o) | def | cardinal.aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal"
] | The `aleph` function gives the infinite cardinals listed by their
ordinal index. `aleph 0 = ℵ₀`, `aleph 1 = succ ℵ₀` is the first
uncountable cardinal, and so on. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_lt {o₁ o₂ : ordinal} : aleph o₁ < aleph o₂ ↔ o₁ < o₂ | aleph'_lt.trans (add_lt_add_iff_left _) | theorem | cardinal.aleph_lt | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_le {o₁ o₂ : ordinal} : aleph o₁ ≤ aleph o₂ ↔ o₁ ≤ o₂ | le_iff_le_iff_lt_iff_lt.2 aleph_lt | theorem | cardinal.aleph_le | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_aleph_eq (o₁ o₂ : ordinal) : max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) | begin
cases le_total (aleph o₁) (aleph o₂) with h h,
{ rw [max_eq_right h, max_eq_right (aleph_le.1 h)] },
{ rw [max_eq_left h, max_eq_left (aleph_le.1 h)] }
end | theorem | cardinal.max_aleph_eq | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_succ {o : ordinal} : aleph (succ o) = succ (aleph o) | by rw [aleph, add_succ, aleph'_succ, aleph] | theorem | cardinal.aleph_succ | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_zero : aleph 0 = ℵ₀ | by rw [aleph, add_zero, aleph'_omega] | theorem | cardinal.aleph_zero | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_limit {o : ordinal} (ho : o.is_limit) : aleph o = ⨆ a : Iio o, aleph a | begin
apply le_antisymm _ (csupr_le' _),
{ rw [aleph, aleph'_limit (ho.add _)],
refine csupr_mono' (bdd_above_of_small _) _,
rintro ⟨i, hi⟩,
cases lt_or_le i ω,
{ rcases lt_omega.1 h with ⟨n, rfl⟩,
use ⟨0, ho.pos⟩,
simpa using (nat_lt_aleph_0 n).le },
{ exact ⟨⟨_, (sub_lt_of_le h).2 ... | theorem | cardinal.aleph_limit | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"csupr_le'",
"csupr_mono'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_aleph' {o : ordinal} : ℵ₀ ≤ aleph' o ↔ ω ≤ o | by rw [← aleph'_omega, aleph'_le] | theorem | cardinal.aleph_0_le_aleph' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_aleph (o : ordinal) : ℵ₀ ≤ aleph o | by { rw [aleph, aleph_0_le_aleph'], apply ordinal.le_add_right } | theorem | cardinal.aleph_0_le_aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal",
"ordinal.le_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_pos {o : ordinal} (ho : 0 < o) : 0 < aleph' o | by rwa [←aleph'_zero, aleph'_lt] | theorem | cardinal.aleph'_pos | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_pos (o : ordinal) : 0 < aleph o | aleph_0_pos.trans_le (aleph_0_le_aleph o) | theorem | cardinal.aleph_pos | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_to_nat (o : ordinal) : (aleph o).to_nat = 0 | to_nat_apply_of_aleph_0_le $ aleph_0_le_aleph o | theorem | cardinal.aleph_to_nat | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_to_part_enat (o : ordinal) : (aleph o).to_part_enat = ⊤ | to_part_enat_apply_of_aleph_0_le $ aleph_0_le_aleph o | theorem | cardinal.aleph_to_part_enat | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_out_aleph (o : ordinal) : nonempty (aleph o).ord.out.α | begin
rw [out_nonempty_iff_ne_zero, ←ord_zero],
exact λ h, (ord_injective h).not_gt (aleph_pos o)
end | instance | cardinal.nonempty_out_aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_aleph_is_limit (o : ordinal) : (aleph o).ord.is_limit | ord_is_limit $ aleph_0_le_aleph _ | theorem | cardinal.ord_aleph_is_limit | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_aleph {c : cardinal} : ℵ₀ ≤ c ↔ ∃ o, c = aleph o | ⟨λ h, ⟨aleph_idx c - ω,
by { rw [aleph, ordinal.add_sub_cancel_of_le, aleph'_aleph_idx],
rwa [← aleph_0_le_aleph', aleph'_aleph_idx] }⟩,
λ ⟨o, e⟩, e.symm ▸ aleph_0_le_aleph _⟩ | theorem | cardinal.exists_aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph'_is_normal : is_normal (ord ∘ aleph') | ⟨λ o, ord_lt_ord.2 $ aleph'_lt.2 $ lt_succ o,
λ o l a, by simp only [ord_le, aleph'_le_of_limit l]⟩ | theorem | cardinal.aleph'_is_normal | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_is_normal : is_normal (ord ∘ aleph) | aleph'_is_normal.trans $ add_is_normal ω | theorem | cardinal.aleph_is_normal | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_aleph_0 : succ ℵ₀ = aleph 1 | by rw [←aleph_zero, ←aleph_succ, ordinal.succ_zero] | theorem | cardinal.succ_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal.succ_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_lt_aleph_one : ℵ₀ < aleph 1 | by { rw ←succ_aleph_0, apply lt_succ } | lemma | cardinal.aleph_0_lt_aleph_one | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_iff_lt_aleph_one {α : Type*} (s : set α) : s.countable ↔ #s < aleph 1 | by rw [←succ_aleph_0, lt_succ_iff, le_aleph_0_iff_set_countable] | lemma | cardinal.countable_iff_lt_aleph_one | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_card_unbounded : unbounded (<) {b : ordinal | b.card.ord = b} | unbounded_lt_iff.2 $ λ a, ⟨_, ⟨(by { dsimp, rw card_ord }), (lt_ord_succ_card a).le⟩⟩ | theorem | cardinal.ord_card_unbounded | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | Ordinals that are cardinals are unbounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_aleph'_of_eq_card_ord {o : ordinal} (ho : o.card.ord = o) : ∃ a, (aleph' a).ord = o | ⟨cardinal.aleph_idx.rel_iso o.card, by simpa using ho⟩ | theorem | cardinal.eq_aleph'_of_eq_card_ord | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_aleph'_eq_enum_card : ord ∘ aleph' = enum_ord {b : ordinal | b.card.ord = b} | begin
rw [←eq_enum_ord _ ord_card_unbounded, range_eq_iff],
exact ⟨aleph'_is_normal.strict_mono, ⟨(λ a, (by { dsimp, rw card_ord })),
λ b hb, eq_aleph'_of_eq_card_ord hb⟩⟩
end | theorem | cardinal.ord_aleph'_eq_enum_card | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | `ord ∘ aleph'` enumerates the ordinals that are cardinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ord_card_unbounded' : unbounded (<) {b : ordinal | b.card.ord = b ∧ ω ≤ b} | (unbounded_lt_inter_le ω).2 ord_card_unbounded | theorem | cardinal.ord_card_unbounded' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal"
] | Infinite ordinals that are cardinals are unbounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_aleph_of_eq_card_ord {o : ordinal} (ho : o.card.ord = o) (ho' : ω ≤ o) :
∃ a, (aleph a).ord = o | begin
cases eq_aleph'_of_eq_card_ord ho with a ha,
use a - ω,
unfold aleph,
rwa ordinal.add_sub_cancel_of_le,
rwa [←aleph_0_le_aleph', ←ord_le_ord, ha, ord_aleph_0]
end | theorem | cardinal.eq_aleph_of_eq_card_ord | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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