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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ord_aleph_eq_enum_card :
ord ∘ aleph = enum_ord {b : ordinal | b.card.ord = b ∧ ω ≤ b} | begin
rw ←eq_enum_ord _ ord_card_unbounded',
use aleph_is_normal.strict_mono,
rw range_eq_iff,
refine ⟨(λ a, ⟨_, _⟩), λ b hb, eq_aleph_of_eq_card_ord hb.1 hb.2⟩,
{ rw card_ord },
{ rw [←ord_aleph_0, ord_le_ord],
exact aleph_0_le_aleph _ }
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 infinite ordinals that are cardinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beth (o : ordinal.{u}) : cardinal.{u} | limit_rec_on o aleph_0 (λ _ x, 2 ^ x) (λ a ha IH, ⨆ b : Iio a, IH b.1 b.2) | def | cardinal.beth | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [] | Beth numbers are defined so that `beth 0 = ℵ₀`, `beth (succ o) = 2 ^ (beth o)`, and when `o` is
a limit ordinal, `beth o` is the supremum of `beth o'` for `o' < o`.
Assuming the generalized continuum hypothesis, which is undecidable in ZFC, `beth o = aleph o` for
every `o`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
beth_zero : beth 0 = aleph_0 | limit_rec_on_zero _ _ _ | theorem | cardinal.beth_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 | |
beth_succ (o : ordinal) : beth (succ o) = 2 ^ beth o | limit_rec_on_succ _ _ _ _ | theorem | cardinal.beth_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 | |
beth_limit {o : ordinal} : o.is_limit → beth o = ⨆ a : Iio o, beth a | limit_rec_on_limit _ _ _ _ | theorem | cardinal.beth_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 | |
beth_strict_mono : strict_mono beth | begin
intros a b,
induction b using ordinal.induction with b IH generalizing a,
intro h,
rcases zero_or_succ_or_limit b with rfl | ⟨c, rfl⟩ | hb,
{ exact (ordinal.not_lt_zero a h).elim },
{ rw lt_succ_iff at h,
rw beth_succ,
apply lt_of_le_of_lt _ (cantor _),
rcases eq_or_lt_of_le h with rfl | h... | theorem | cardinal.beth_strict_mono | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"eq_or_lt_of_le",
"le_csupr",
"ordinal.induction",
"ordinal.not_lt_zero",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
beth_mono : monotone beth | beth_strict_mono.monotone | lemma | cardinal.beth_mono | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
beth_lt {o₁ o₂ : ordinal} : beth o₁ < beth o₂ ↔ o₁ < o₂ | beth_strict_mono.lt_iff_lt | theorem | cardinal.beth_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 | |
beth_le {o₁ o₂ : ordinal} : beth o₁ ≤ beth o₂ ↔ o₁ ≤ o₂ | beth_strict_mono.le_iff_le | theorem | cardinal.beth_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_le_beth (o : ordinal) : aleph o ≤ beth o | begin
apply limit_rec_on o,
{ simp },
{ intros o h,
rw [aleph_succ, beth_succ, succ_le_iff],
exact (cantor _).trans_le (power_le_power_left two_ne_zero h) },
{ intros o ho IH,
rw [aleph_limit ho, beth_limit ho],
exact csupr_mono (bdd_above_of_small _) (λ x, IH x.1 x.2) }
end | theorem | cardinal.aleph_le_beth | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"csupr_mono",
"ordinal",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_beth (o : ordinal) : ℵ₀ ≤ beth o | (aleph_0_le_aleph o).trans $ aleph_le_beth o | theorem | cardinal.aleph_0_le_beth | 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 | |
beth_pos (o : ordinal) : 0 < beth o | aleph_0_pos.trans_le $ aleph_0_le_beth o | theorem | cardinal.beth_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 | |
beth_ne_zero (o : ordinal) : beth o ≠ 0 | (beth_pos o).ne' | theorem | cardinal.beth_ne_zero | 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 | |
beth_normal : is_normal.{u} (λ o, (beth o).ord) | (is_normal_iff_strict_mono_limit _).2 ⟨ord_strict_mono.comp beth_strict_mono, λ o ho a ha,
by { rw [beth_limit ho, ord_le], exact csupr_le' (λ b, ord_le.1 (ha _ b.2)) }⟩ | lemma | cardinal.beth_normal | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"csupr_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_self {c : cardinal} (h : ℵ₀ ≤ c) : c * c = c | begin
refine le_antisymm _
(by simpa only [mul_one] using
mul_le_mul_left' (one_le_aleph_0.trans h) c),
-- the only nontrivial part is `c * c ≤ c`. We prove it inductively.
refine acc.rec_on (cardinal.lt_wf.apply c) (λ c _,
quotient.induction_on c $ λ α IH ol, _) h,
-- consider the minimal well-or... | theorem | cardinal.mul_eq_self | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"equiv.refl",
"equiv.set.insert",
"equiv.set.prod",
"is_well_order",
"le_of_forall_lt",
"linear_order_of_STO",
"mul_le_mul_left'",
"mul_one",
"order.preimage",
"ordinal",
"prod.lex_def",
"rel_embedding.preimage",
"set.embedding_of_subset"
] | If `α` is an infinite type, then `α × α` and `α` have the same cardinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_eq_max {a b : cardinal} (ha : ℵ₀ ≤ a) (hb : ℵ₀ ≤ b) : a * b = max a b | le_antisymm
(mul_eq_self (ha.trans (le_max_left a b)) ▸
mul_le_mul' (le_max_left _ _) (le_max_right _ _)) $
max_le
(by simpa only [mul_one] using
mul_le_mul_left' (one_le_aleph_0.trans hb) a)
(by simpa only [one_mul] using
mul_le_mul_right' (one_le_aleph_0.trans ha) b) | theorem | cardinal.mul_eq_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_le_mul'",
"mul_le_mul_left'",
"mul_le_mul_right'",
"mul_one",
"one_mul"
] | If `α` and `β` are infinite types, then the cardinality of `α × β` is the maximum
of the cardinalities of `α` and `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_mk_eq_max {α β : Type*} [infinite α] [infinite β] : #α * #β = max (#α) (#β) | mul_eq_max (aleph_0_le_mk α) (aleph_0_le_mk β) | theorem | cardinal.mul_mk_eq_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_mul_aleph (o₁ o₂ : ordinal) : aleph o₁ * aleph o₂ = aleph (max o₁ o₂) | by rw [cardinal.mul_eq_max (aleph_0_le_aleph o₁) (aleph_0_le_aleph o₂), max_aleph_eq] | theorem | cardinal.aleph_mul_aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal.mul_eq_max",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_mul_eq {a : cardinal} (ha : ℵ₀ ≤ a) : ℵ₀ * a = a | (mul_eq_max le_rfl ha).trans (max_eq_right ha) | theorem | cardinal.aleph_0_mul_eq | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_aleph_0_eq {a : cardinal} (ha : ℵ₀ ≤ a) : a * ℵ₀ = a | (mul_eq_max ha le_rfl).trans (max_eq_left ha) | theorem | cardinal.mul_aleph_0_eq | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_mul_mk_eq {α : Type*} [infinite α] : ℵ₀ * #α = #α | aleph_0_mul_eq (aleph_0_le_mk α) | theorem | cardinal.aleph_0_mul_mk_eq | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_aleph_0_eq {α : Type*} [infinite α] : #α * ℵ₀ = #α | mul_aleph_0_eq (aleph_0_le_mk α) | theorem | cardinal.mk_mul_aleph_0_eq | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_mul_aleph (o : ordinal) : ℵ₀ * aleph o = aleph o | aleph_0_mul_eq (aleph_0_le_aleph o) | theorem | cardinal.aleph_0_mul_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_mul_aleph_0 (o : ordinal) : aleph o * ℵ₀ = aleph o | mul_aleph_0_eq (aleph_0_le_aleph o) | theorem | cardinal.aleph_mul_aleph_0 | 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 | |
mul_lt_of_lt {a b c : cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a * b < c | (mul_le_mul' (le_max_left a b) (le_max_right a b)).trans_lt $
(lt_or_le (max a b) ℵ₀).elim
(λ h, (mul_lt_aleph_0 h h).trans_le hc)
(λ h, by { rw mul_eq_self h, exact max_lt h1 h2 }) | theorem | cardinal.mul_lt_of_lt | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_max_of_aleph_0_le_left {a b : cardinal} (h : ℵ₀ ≤ a) : a * b ≤ max a b | begin
convert mul_le_mul' (le_max_left a b) (le_max_right a b),
rw mul_eq_self,
refine h.trans (le_max_left a b)
end | lemma | cardinal.mul_le_max_of_aleph_0_le_left | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_le_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_max_of_aleph_0_le_left {a b : cardinal} (h : ℵ₀ ≤ a) (h' : b ≠ 0) : a * b = max a b | begin
cases le_or_lt ℵ₀ b with hb hb, { exact mul_eq_max h hb },
refine (mul_le_max_of_aleph_0_le_left h).antisymm _,
have : b ≤ a, from hb.le.trans h,
rw [max_eq_left this],
convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') _, rw [mul_one],
end | lemma | cardinal.mul_eq_max_of_aleph_0_le_left | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_le_mul_left'",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_max_of_aleph_0_le_right {a b : cardinal} (h : ℵ₀ ≤ b) : a * b ≤ max a b | by simpa only [mul_comm, max_comm] using mul_le_max_of_aleph_0_le_left h | lemma | cardinal.mul_le_max_of_aleph_0_le_right | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_max_of_aleph_0_le_right {a b : cardinal} (h' : a ≠ 0) (h : ℵ₀ ≤ b) : a * b = max a b | begin
rw [mul_comm, max_comm],
exact mul_eq_max_of_aleph_0_le_left h h'
end | lemma | cardinal.mul_eq_max_of_aleph_0_le_right | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_max' {a b : cardinal} (h : ℵ₀ ≤ a * b) : a * b = max a b | begin
rcases aleph_0_le_mul_iff.mp h with ⟨ha, hb, ha' | hb'⟩,
{ exact mul_eq_max_of_aleph_0_le_left ha' hb },
{ exact mul_eq_max_of_aleph_0_le_right ha hb' }
end | lemma | cardinal.mul_eq_max' | 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 | |
mul_le_max (a b : cardinal) : a * b ≤ max (max a b) ℵ₀ | begin
rcases eq_or_ne a 0 with rfl | ha0, { simp },
rcases eq_or_ne b 0 with rfl | hb0, { simp },
cases le_or_lt ℵ₀ a with ha ha,
{ rw [mul_eq_max_of_aleph_0_le_left ha hb0],
exact le_max_left _ _ },
{ cases le_or_lt ℵ₀ b with hb hb,
{ rw [mul_comm, mul_eq_max_of_aleph_0_le_left hb ha0, max_comm],
... | theorem | cardinal.mul_le_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"eq_or_ne",
"le_max_of_le_right",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_left {a b : cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) (hb' : b ≠ 0) : a * b = a | by { rw [mul_eq_max_of_aleph_0_le_left ha hb', max_eq_left hb] } | lemma | cardinal.mul_eq_left | 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 | |
mul_eq_right {a b : cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) (ha' : a ≠ 0) : a * b = b | by { rw [mul_comm, mul_eq_left hb ha ha'] } | lemma | cardinal.mul_eq_right | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_left {a b : cardinal} (h : b ≠ 0) : a ≤ b * a | by { convert mul_le_mul_right' (one_le_iff_ne_zero.mpr h) _,
rw [one_mul] } | lemma | cardinal.le_mul_left | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"le_mul_left",
"mul_le_mul_right'",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_right {a b : cardinal} (h : b ≠ 0) : a ≤ a * b | by { rw [mul_comm], exact le_mul_left h } | lemma | cardinal.le_mul_right | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"le_mul_left",
"le_mul_right",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_left_iff {a b : cardinal} : a * b = a ↔ ((max ℵ₀ b ≤ a ∧ b ≠ 0) ∨ b = 1 ∨ a = 0) | begin
rw max_le_iff,
refine ⟨λ h, _, _⟩,
{ cases le_or_lt ℵ₀ a with ha ha,
{ have : a ≠ 0, { rintro rfl, exact ha.not_lt aleph_0_pos },
left, use ha,
{ rw ←not_lt, exact λ hb, ne_of_gt (hb.trans_le (le_mul_left this)) h },
{ rintro rfl, apply this, rw mul_zero at h, exact h.symm }},
righ... | lemma | cardinal.mul_eq_left_iff | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"le_mul_left",
"max_le_iff",
"mul_lt_mul_left",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_self {c : cardinal} (h : ℵ₀ ≤ c) : c + c = c | le_antisymm
(by simpa only [nat.cast_bit0, nat.cast_one, mul_eq_self h, two_mul] using
mul_le_mul_right' ((nat_lt_aleph_0 2).le.trans h) c)
(self_le_add_left c c) | theorem | cardinal.add_eq_self | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"mul_le_mul_right'",
"nat.cast_bit0",
"nat.cast_one",
"two_mul"
] | If `α` is an infinite type, then `α ⊕ α` and `α` have the same cardinality. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_eq_max {a b : cardinal} (ha : ℵ₀ ≤ a) : a + b = max a b | le_antisymm
(add_eq_self (ha.trans (le_max_left a b)) ▸
add_le_add (le_max_left _ _) (le_max_right _ _)) $
max_le (self_le_add_right _ _) (self_le_add_left _ _) | theorem | cardinal.add_eq_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal"
] | If `α` is an infinite type, then the cardinality of `α ⊕ β` is the maximum
of the cardinalities of `α` and `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_eq_max' {a b : cardinal} (ha : ℵ₀ ≤ b) : a + b = max a b | by rw [add_comm, max_comm, add_eq_max ha] | theorem | cardinal.add_eq_max' | 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 | |
add_mk_eq_max {α β : Type*} [infinite α] : #α + #β = max (#α) (#β) | add_eq_max (aleph_0_le_mk α) | theorem | cardinal.add_mk_eq_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mk_eq_max' {α β : Type*} [infinite β] : #α + #β = max (#α) (#β) | add_eq_max' (aleph_0_le_mk β) | theorem | cardinal.add_mk_eq_max' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_max (a b : cardinal) : a + b ≤ max (max a b) ℵ₀ | begin
cases le_or_lt ℵ₀ a with ha ha,
{ rw [add_eq_max ha],
exact le_max_left _ _ },
{ cases le_or_lt ℵ₀ b with hb hb,
{ rw [add_comm, add_eq_max hb, max_comm],
exact le_max_left _ _ },
{ exact le_max_of_le_right (add_lt_aleph_0 ha hb).le } }
end | theorem | cardinal.add_le_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"le_max_of_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_of_le {a b c : cardinal} (hc : ℵ₀ ≤ c) (h1 : a ≤ c) (h2 : b ≤ c) : a + b ≤ c | (add_le_add h1 h2).trans $ le_of_eq $ add_eq_self hc | theorem | cardinal.add_le_of_le | 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 | |
add_lt_of_lt {a b c : cardinal} (hc : ℵ₀ ≤ c) (h1 : a < c) (h2 : b < c) : a + b < c | (add_le_add (le_max_left a b) (le_max_right a b)).trans_lt $
(lt_or_le (max a b) ℵ₀).elim
(λ h, (add_lt_aleph_0 h h).trans_le hc)
(λ h, by rw add_eq_self h; exact max_lt h1 h2) | theorem | cardinal.add_lt_of_lt | 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 | |
eq_of_add_eq_of_aleph_0_le {a b c : cardinal} (h : a + b = c) (ha : a < c) (hc : ℵ₀ ≤ c) :
b = c | begin
apply le_antisymm,
{ rw [← h], apply self_le_add_left },
rw[← not_lt], intro hb,
have : a + b < c := add_lt_of_lt hc ha hb,
simpa [h, lt_irrefl] using this
end | lemma | cardinal.eq_of_add_eq_of_aleph_0_le | 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 | |
add_eq_left {a b : cardinal} (ha : ℵ₀ ≤ a) (hb : b ≤ a) : a + b = a | by { rw [add_eq_max ha, max_eq_left hb] } | lemma | cardinal.add_eq_left | 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 | |
add_eq_right {a b : cardinal} (hb : ℵ₀ ≤ b) (ha : a ≤ b) : a + b = b | by { rw [add_comm, add_eq_left hb ha] } | lemma | cardinal.add_eq_right | 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 | |
add_eq_left_iff {a b : cardinal} : a + b = a ↔ (max ℵ₀ b ≤ a ∨ b = 0) | begin
rw max_le_iff,
refine ⟨λ h, _, _⟩,
{ cases (le_or_lt ℵ₀ a) with ha ha,
{ left, use ha, rw ←not_lt, apply λ hb, ne_of_gt _ h,
exact hb.trans_le (self_le_add_left b a) },
right, rw [←h, add_lt_aleph_0_iff, lt_aleph_0, lt_aleph_0] at ha,
rcases ha with ⟨⟨n, rfl⟩, ⟨m, rfl⟩⟩, norm_cast at h ⊢,
... | lemma | cardinal.add_eq_left_iff | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"max_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_eq_right_iff {a b : cardinal} : a + b = b ↔ (max ℵ₀ a ≤ b ∨ a = 0) | by { rw [add_comm, add_eq_left_iff] } | lemma | cardinal.add_eq_right_iff | 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 | |
add_nat_eq {a : cardinal} (n : ℕ) (ha : ℵ₀ ≤ a) : a + n = a | add_eq_left ha ((nat_lt_aleph_0 _).le.trans ha) | lemma | cardinal.add_nat_eq | 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 | |
add_one_eq {a : cardinal} (ha : ℵ₀ ≤ a) : a + 1 = a | add_eq_left ha (one_le_aleph_0.trans ha) | lemma | cardinal.add_one_eq | 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 | |
mk_add_one_eq {α : Type*} [infinite α] : #α + 1 = #α | add_one_eq (aleph_0_le_mk α) | lemma | cardinal.mk_add_one_eq | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_add_eq_add_left {a b c : cardinal} (h : a + b = a + c) (ha : a < ℵ₀) :
b = c | begin
cases le_or_lt ℵ₀ b with hb hb,
{ have : a < b := ha.trans_le hb,
rw [add_eq_right hb this.le, eq_comm] at h,
rw [eq_of_add_eq_of_aleph_0_le h this hb] },
{ have hc : c < ℵ₀,
{ rw ←not_le, intro hc,
apply lt_irrefl ℵ₀, apply (hc.trans (self_le_add_left _ a)).trans_lt,
rw ←h, apply ad... | lemma | cardinal.eq_of_add_eq_add_left | 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 | |
eq_of_add_eq_add_right {a b c : cardinal} (h : a + b = c + b) (hb : b < ℵ₀) :
a = c | by { rw [add_comm a b, add_comm c b] at h, exact cardinal.eq_of_add_eq_add_left h hb } | lemma | cardinal.eq_of_add_eq_add_right | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"cardinal.eq_of_add_eq_add_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_add_aleph (o₁ o₂ : ordinal) : aleph o₁ + aleph o₂ = aleph (max o₁ o₂) | by rw [cardinal.add_eq_max (aleph_0_le_aleph o₁), max_aleph_eq] | theorem | cardinal.aleph_add_aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal.add_eq_max",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_add_ord {c : cardinal} (hc : ℵ₀ ≤ c) : ordinal.principal (+) c.ord | λ a b ha hb, by { rw [lt_ord, ordinal.card_add] at *, exact add_lt_of_lt hc ha hb } | theorem | cardinal.principal_add_ord | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"ordinal.card_add",
"ordinal.principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_add_aleph (o : ordinal) : ordinal.principal (+) (aleph o).ord | principal_add_ord $ aleph_0_le_aleph o | theorem | cardinal.principal_add_aleph | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ordinal",
"ordinal.principal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_right_inj_of_lt_aleph_0 {α β γ : cardinal} (γ₀ : γ < aleph_0) :
α + γ = β + γ ↔ α = β | ⟨λ h, cardinal.eq_of_add_eq_add_right h γ₀, λ h, congr_fun (congr_arg (+) h) γ⟩ | lemma | cardinal.add_right_inj_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"cardinal.eq_of_add_eq_add_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_nat_inj {α β : cardinal} (n : ℕ) :
α + n = β + n ↔ α = β | add_right_inj_of_lt_aleph_0 (nat_lt_aleph_0 _) | lemma | cardinal.add_nat_inj | 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 | |
add_one_inj {α β : cardinal} :
α + 1 = β + 1 ↔ α = β | add_right_inj_of_lt_aleph_0 one_lt_aleph_0 | lemma | cardinal.add_one_inj | 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 | |
add_le_add_iff_of_lt_aleph_0 {α β γ : cardinal} (γ₀ : γ < cardinal.aleph_0) :
α + γ ≤ β + γ ↔ α ≤ β | begin
refine ⟨λ h, _, λ h, add_le_add_right h γ⟩,
contrapose h,
rw [not_le, lt_iff_le_and_ne, ne] at h ⊢,
exact ⟨add_le_add_right h.1 γ, mt (add_right_inj_of_lt_aleph_0 γ₀).1 h.2⟩,
end | lemma | cardinal.add_le_add_iff_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"cardinal.aleph_0",
"lt_iff_le_and_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_nat_le_add_nat_iff_of_lt_aleph_0 {α β : cardinal} (n : ℕ) :
α + n ≤ β + n ↔ α ≤ β | add_le_add_iff_of_lt_aleph_0 (nat_lt_aleph_0 n) | lemma | cardinal.add_nat_le_add_nat_iff_of_lt_aleph_0 | 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 | |
add_one_le_add_one_iff_of_lt_aleph_0 {α β : cardinal} :
α + 1 ≤ β + 1 ↔ α ≤ β | add_le_add_iff_of_lt_aleph_0 one_lt_aleph_0 | lemma | cardinal.add_one_le_add_one_iff_of_lt_aleph_0 | 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 | |
pow_le {κ μ : cardinal.{u}} (H1 : ℵ₀ ≤ κ) (H2 : μ < ℵ₀) : κ ^ μ ≤ κ | let ⟨n, H3⟩ := lt_aleph_0.1 H2 in
H3.symm ▸ (quotient.induction_on κ (λ α H1, nat.rec_on n
(lt_of_lt_of_le (by { rw [nat.cast_zero, power_zero], exact one_lt_aleph_0 }) H1).le
(λ n ih, trans_rel_left _
(by { rw [nat.cast_succ, power_add, power_one], exact mul_le_mul_right' ih _ })
(mul_eq_self H1))) H1) | theorem | cardinal.pow_le | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"ih",
"mul_le_mul_right'",
"nat.cast_succ",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_eq {κ μ : cardinal.{u}} (H1 : ℵ₀ ≤ κ) (H2 : 1 ≤ μ) (H3 : μ < ℵ₀) : κ ^ μ = κ | (pow_le H1 H3).antisymm $ self_le_power κ H2 | theorem | cardinal.pow_eq | 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 | |
power_self_eq {c : cardinal} (h : ℵ₀ ≤ c) : c ^ c = 2 ^ c | begin
apply ((power_le_power_right $ (cantor c).le).trans _).antisymm,
{ convert power_le_power_right ((nat_lt_aleph_0 2).le.trans h), apply nat.cast_two.symm },
{ rw [←power_mul, mul_eq_self h] }
end | lemma | cardinal.power_self_eq | 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 | |
prod_eq_two_power {ι : Type u} [infinite ι] {c : ι → cardinal.{v}} (h₁ : ∀ i, 2 ≤ c i)
(h₂ : ∀ i, lift.{u} (c i) ≤ lift.{v} (#ι)) :
prod c = 2 ^ lift.{v} (#ι) | begin
rw [← lift_id' (prod c), lift_prod, ← lift_two_power],
apply le_antisymm,
{ refine (prod_le_prod _ _ h₂).trans_eq _,
rw [prod_const, lift_lift, ← lift_power, power_self_eq (aleph_0_le_mk ι), lift_umax.{u v}] },
{ rw [← prod_const', lift_prod],
refine prod_le_prod _ _ (λ i, _),
rw [lift_two, ← ... | lemma | cardinal.prod_eq_two_power | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_eq_two_power {c₁ c₂ : cardinal} (h₁ : ℵ₀ ≤ c₁) (h₂ : 2 ≤ c₂) (h₂' : c₂ ≤ c₁) :
c₂ ^ c₁ = 2 ^ c₁ | le_antisymm (power_self_eq h₁ ▸ power_le_power_right h₂') (power_le_power_right h₂) | lemma | cardinal.power_eq_two_power | 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 | |
nat_power_eq {c : cardinal.{u}} (h : ℵ₀ ≤ c) {n : ℕ} (hn : 2 ≤ n) :
(n : cardinal.{u}) ^ c = 2 ^ c | power_eq_two_power h (by assumption_mod_cast) ((nat_lt_aleph_0 n).le.trans h) | lemma | cardinal.nat_power_eq | 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 | |
power_nat_le {c : cardinal.{u}} {n : ℕ} (h : ℵ₀ ≤ c) : c ^ n ≤ c | pow_le h (nat_lt_aleph_0 n) | lemma | cardinal.power_nat_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 | |
power_nat_eq {c : cardinal.{u}} {n : ℕ} (h1 : ℵ₀ ≤ c) (h2 : 1 ≤ n) : c ^ n = c | pow_eq h1 (by exact_mod_cast h2) (nat_lt_aleph_0 n) | lemma | cardinal.power_nat_eq | 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 | |
power_nat_le_max {c : cardinal.{u}} {n : ℕ} : c ^ (n : cardinal.{u}) ≤ max c ℵ₀ | begin
cases le_or_lt ℵ₀ c with hc hc,
{ exact le_max_of_le_left (power_nat_le hc) },
{ exact le_max_of_le_right ((power_lt_aleph_0 hc (nat_lt_aleph_0 _)).le) }
end | lemma | cardinal.power_nat_le_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"le_max_of_le_left",
"le_max_of_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerlt_aleph_0 {c : cardinal} (h : ℵ₀ ≤ c) : c ^< ℵ₀ = c | begin
apply le_antisymm,
{ rw powerlt_le, intro c', rw lt_aleph_0, rintro ⟨n, rfl⟩, apply power_nat_le h },
convert le_powerlt c one_lt_aleph_0, rw power_one
end | lemma | cardinal.powerlt_aleph_0 | 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 | |
powerlt_aleph_0_le (c : cardinal) : c ^< ℵ₀ ≤ max c ℵ₀ | begin
cases le_or_lt ℵ₀ c,
{ rw powerlt_aleph_0 h, apply le_max_left },
rw powerlt_le,
exact λ c' hc', (power_lt_aleph_0 h hc').le.trans (le_max_right _ _)
end | lemma | cardinal.powerlt_aleph_0_le | 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 | |
mk_list_eq_mk (α : Type u) [infinite α] : #(list α) = #α | have H1 : ℵ₀ ≤ #α := aleph_0_le_mk α,
eq.symm $ le_antisymm ⟨⟨λ x, [x], λ x y H, (list.cons.inj H).1⟩⟩ $
calc #(list α)
= sum (λ n : ℕ, #α ^ (n : cardinal.{u})) : mk_list_eq_sum_pow α
... ≤ sum (λ n : ℕ, #α) : sum_le_sum _ _ $ λ n, pow_le H1 $ nat_lt_aleph_0 n
... = #α : by simp [H1] | theorem | cardinal.mk_list_eq_mk | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_list_eq_aleph_0 (α : Type u) [countable α] [nonempty α] : #(list α) = ℵ₀ | mk_le_aleph_0.antisymm (aleph_0_le_mk _) | theorem | cardinal.mk_list_eq_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_list_eq_max_mk_aleph_0 (α : Type u) [nonempty α] : #(list α) = max (#α) ℵ₀ | begin
casesI finite_or_infinite α,
{ rw [mk_list_eq_aleph_0, eq_comm, max_eq_right],
exact mk_le_aleph_0 },
{ rw [mk_list_eq_mk, eq_comm, max_eq_left],
exact aleph_0_le_mk α }
end | theorem | cardinal.mk_list_eq_max_mk_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finite_or_infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_list_le_max (α : Type u) : #(list α) ≤ max ℵ₀ (#α) | begin
casesI finite_or_infinite α,
{ exact mk_le_aleph_0.trans (le_max_left _ _) },
{ rw mk_list_eq_mk,
apply le_max_right }
end | theorem | cardinal.mk_list_le_max | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finite_or_infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finset_of_infinite (α : Type u) [infinite α] : #(finset α) = #α | eq.symm $ le_antisymm (mk_le_of_injective (λ x y, finset.singleton_inj.1)) $
calc #(finset α) ≤ #(list α) : mk_le_of_surjective list.to_finset_surjective
... = #α : mk_list_eq_mk α | theorem | cardinal.mk_finset_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finset",
"infinite",
"list.to_finset_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_lift_of_infinite (α : Type u) (β : Type v) [infinite α] [has_zero β]
[nontrivial β] : #(α →₀ β) = max (lift.{v} (#α)) (lift.{u} (#β)) | begin
apply le_antisymm,
{ calc #(α →₀ β) ≤ # (finset (α × β)) : mk_le_of_injective (finsupp.graph_injective α β)
... = #(α × β) : mk_finset_of_infinite _
... = max (lift.{v} (#α)) (lift.{u} (#β)) :
by rw [mk_prod, mul_eq_max_of_aleph_0_le_left]; simp },
{ apply max_le;
rw [←lift_id (# (α →₀ β))... | lemma | cardinal.mk_finsupp_lift_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"exists_ne",
"finset",
"finsupp.graph_injective",
"finsupp.single_injective",
"finsupp.single_left_injective",
"infinite",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_of_infinite (α β : Type u) [infinite α] [has_zero β]
[nontrivial β] : #(α →₀ β) = max (#α) (#β) | by simp | lemma | cardinal.mk_finsupp_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite",
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_lift_of_infinite' (α : Type u) (β : Type v) [nonempty α]
[has_zero β] [infinite β] : #(α →₀ β) = max (lift.{v} (#α)) (lift.{u} (#β)) | begin
casesI fintype_or_infinite α,
{ rw mk_finsupp_lift_of_fintype,
have : ℵ₀ ≤ (#β).lift := aleph_0_le_lift.2 (aleph_0_le_mk β),
rw [max_eq_right (le_trans _ this), power_nat_eq this],
exacts [fintype.card_pos, lift_le_aleph_0.2 (lt_aleph_0_of_finite _).le] },
{ apply mk_finsupp_lift_of_infinite },
... | lemma | cardinal.mk_finsupp_lift_of_infinite' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"fintype.card_pos",
"fintype_or_infinite",
"infinite",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_of_infinite' (α β : Type u) [nonempty α] [has_zero β] [infinite β] :
#(α →₀ β) = max (#α) (#β) | by simp | lemma | cardinal.mk_finsupp_of_infinite' | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_finsupp_nat (α : Type u) [nonempty α] : #(α →₀ ℕ) = max (#α) ℵ₀ | by simp | lemma | cardinal.mk_finsupp_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 | |
mk_multiset_of_nonempty (α : Type u) [nonempty α] : #(multiset α) = max (#α) ℵ₀ | multiset.to_finsupp.to_equiv.cardinal_eq.trans (mk_finsupp_nat α) | lemma | cardinal.mk_multiset_of_nonempty | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_multiset_of_infinite (α : Type u) [infinite α] : #(multiset α) = #α | by simp | lemma | cardinal.mk_multiset_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_multiset_of_is_empty (α : Type u) [is_empty α] : #(multiset α) = 1 | multiset.to_finsupp.to_equiv.cardinal_eq.trans (by simp) | lemma | cardinal.mk_multiset_of_is_empty | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"is_empty",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_multiset_of_countable (α : Type u) [countable α] [nonempty α] : #(multiset α) = ℵ₀ | multiset.to_finsupp.to_equiv.cardinal_eq.trans (by simp) | lemma | cardinal.mk_multiset_of_countable | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"countable",
"multiset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bounded_set_le_of_infinite (α : Type u) [infinite α] (c : cardinal) :
#{t : set α // #t ≤ c} ≤ #α ^ c | begin
refine le_trans _ (by rw [←add_one_eq (aleph_0_le_mk α)]),
induction c using cardinal.induction_on with β,
fapply mk_le_of_surjective,
{ intro f, use sum.inl ⁻¹' range f,
refine le_trans (mk_preimage_of_injective _ _ (λ x y, sum.inl.inj)) _,
apply mk_range_le },
rintro ⟨s, ⟨g⟩⟩,
use λ y, if h ... | lemma | cardinal.mk_bounded_set_le_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"cardinal",
"cardinal.induction_on",
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bounded_set_le (α : Type u) (c : cardinal) :
#{t : set α // #t ≤ c} ≤ max (#α) ℵ₀ ^ c | begin
transitivity #{t : set (ulift.{u} ℕ ⊕ α) // #t ≤ c},
{ refine ⟨embedding.subtype_map _ _⟩, apply embedding.image,
use sum.inr, apply sum.inr.inj, intros s hs, exact mk_image_le.trans hs },
apply (mk_bounded_set_le_of_infinite (ulift.{u} ℕ ⊕ α) c).trans,
rw [max_comm, ←add_eq_max]; refl
end | lemma | cardinal.mk_bounded_set_le | 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 | |
mk_bounded_subset_le {α : Type u} (s : set α) (c : cardinal.{u}) :
#{t : set α // t ⊆ s ∧ #t ≤ c} ≤ max (#s) ℵ₀ ^ c | begin
refine le_trans _ (mk_bounded_set_le s c),
refine ⟨embedding.cod_restrict _ _ _⟩,
use λ t, coe ⁻¹' t.1,
{ rintros ⟨t, ht1, ht2⟩ ⟨t', h1t', h2t'⟩ h, apply subtype.eq, dsimp only at h ⊢,
refine (preimage_eq_preimage' _ _).1 h; rw [subtype.range_coe]; assumption },
rintro ⟨t, h1t, h2t⟩, exact (mk_preim... | lemma | cardinal.mk_bounded_subset_le | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"subtype.range_coe",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_compl_of_infinite {α : Type*} [infinite α] (s : set α) (h2 : #s < #α) :
#(sᶜ : set α) = #α | by { refine eq_of_add_eq_of_aleph_0_le _ h2 (aleph_0_le_mk α), exact mk_sum_compl s } | lemma | cardinal.mk_compl_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_compl_finset_of_infinite {α : Type*} [infinite α] (s : finset α) :
#((↑s)ᶜ : set α) = #α | by { apply mk_compl_of_infinite, exact (finset_card_lt_aleph_0 s).trans_le (aleph_0_le_mk α) } | lemma | cardinal.mk_compl_finset_of_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finset",
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_compl_eq_mk_compl_infinite {α : Type*} [infinite α] {s t : set α} (hs : #s < #α)
(ht : #t < #α) : #(sᶜ : set α) = #(tᶜ : set α) | by { rw [mk_compl_of_infinite s hs, mk_compl_of_infinite t ht] } | lemma | cardinal.mk_compl_eq_mk_compl_infinite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_compl_eq_mk_compl_finite_lift {α : Type u} {β : Type v} [finite α]
{s : set α} {t : set β} (h1 : lift.{max v w} (#α) = lift.{max u w} (#β))
(h2 : lift.{max v w} (#s) = lift.{max u w} (#t)) :
lift.{max v w} (#(sᶜ : set α)) = lift.{max u w} (#(tᶜ : set β)) | begin
casesI nonempty_fintype α,
rcases lift_mk_eq.1 h1 with ⟨e⟩, letI : fintype β := fintype.of_equiv α e,
replace h1 : fintype.card α = fintype.card β := (fintype.of_equiv_card _).symm,
classical,
lift s to finset α using s.to_finite,
lift t to finset β using t.to_finite,
simp only [finset.coe_sort_coe,... | lemma | cardinal.mk_compl_eq_mk_compl_finite_lift | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finite",
"finset",
"finset.card_compl",
"finset.coe_compl",
"finset.coe_sort_coe",
"fintype",
"fintype.card",
"fintype.of_equiv",
"fintype.of_equiv_card",
"lift",
"nat.cast_inj",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_compl_eq_mk_compl_finite {α β : Type u} [finite α] {s : set α} {t : set β}
(h1 : #α = #β) (h : #s = #t) : #(sᶜ : set α) = #(tᶜ : set β) | by { rw ← lift_inj, apply mk_compl_eq_mk_compl_finite_lift; rwa [lift_inj] } | lemma | cardinal.mk_compl_eq_mk_compl_finite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_compl_eq_mk_compl_finite_same {α : Type*} [finite α] {s t : set α}
(h : #s = #t) : #(sᶜ : set α) = #(tᶜ : set α) | mk_compl_eq_mk_compl_finite rfl h | lemma | cardinal.mk_compl_eq_mk_compl_finite_same | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_function {α β : Type*} {s : set α} (f : s ↪ β)
(h : nonempty ((sᶜ : set α) ≃ ((range f)ᶜ : set β))) :
∃ (g : α ≃ β), ∀ x : s, g x = f x | begin
intros, have := h, cases this with g,
let h : α ≃ β := (set.sum_compl (s : set α)).symm.trans
((sum_congr (equiv.of_injective f f.2) g).trans
(set.sum_compl (range f))),
refine ⟨h, _⟩, rintro ⟨x, hx⟩, simp [set.sum_compl_symm_apply_of_mem, hx]
end | theorem | cardinal.extend_function | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_function_finite {α β : Type*} [finite α] {s : set α} (f : s ↪ β)
(h : nonempty (α ≃ β)) : ∃ (g : α ≃ β), ∀ x : s, g x = f x | begin
apply extend_function f,
cases id h with g,
rw [← lift_mk_eq] at h,
rw [←lift_mk_eq, mk_compl_eq_mk_compl_finite_lift h],
rw [mk_range_eq_lift], exact f.2
end | theorem | cardinal.extend_function_finite | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_function_of_lt {α β : Type*} {s : set α} (f : s ↪ β) (hs : #s < #α)
(h : nonempty (α ≃ β)) : ∃ (g : α ≃ β), ∀ x : s, g x = f x | begin
casesI fintype_or_infinite α,
{ exact extend_function_finite f h },
{ apply extend_function f, cases id h with g, haveI := infinite.of_injective _ g.injective,
rw [← lift_mk_eq'] at h ⊢,
rwa [mk_compl_of_infinite s hs, mk_compl_of_infinite],
rwa [← lift_lt, mk_range_eq_of_injective f.injective, ... | theorem | cardinal.extend_function_of_lt | set_theory.cardinal | src/set_theory/cardinal/ordinal.lean | [
"data.finsupp.multiset",
"order.bounded",
"set_theory.ordinal.principal",
"tactic.linarith"
] | [
"fintype_or_infinite",
"infinite.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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