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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
nsmul_lt_aleph_0_iff_of_ne_zero {n : ℕ} {a : cardinal} (h : n ≠ 0) : n • a < ℵ₀ ↔ a < ℵ₀ | nsmul_lt_aleph_0_iff.trans $ or_iff_right h | lemma | cardinal.nsmul_lt_aleph_0_iff_of_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"or_iff_right"
] | See also `cardinal.nsmul_lt_aleph_0_iff` for a hypothesis-free version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_lt_aleph_0 {a b : cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a * b < ℵ₀ | match a, b, lt_aleph_0.1 ha, lt_aleph_0.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by rw [← nat.cast_mul]; apply nat_lt_aleph_0
end | theorem | cardinal.mul_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_aleph_0_iff {a b : cardinal} : a * b < ℵ₀ ↔ a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀ | begin
refine ⟨λ h, _, _⟩,
{ by_cases ha : a = 0, { exact or.inl ha },
right, by_cases hb : b = 0, { exact or.inl hb },
right, rw [←ne, ←one_le_iff_ne_zero] at ha hb, split,
{ rw ←mul_one a,
refine (mul_le_mul' le_rfl hb).trans_lt h },
{ rw ←one_mul b,
refine (mul_le_mul' ha le_rfl).trans... | lemma | cardinal.mul_lt_aleph_0_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"le_rfl",
"mul_le_mul'",
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_mul_iff {a b : cardinal} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b) | let h := (@mul_lt_aleph_0_iff a b).not in
by rwa [not_lt, not_or_distrib, not_or_distrib, not_and_distrib, not_lt, not_lt] at h | lemma | cardinal.aleph_0_le_mul_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"not_and_distrib",
"not_or_distrib"
] | See also `cardinal.aleph_0_le_mul_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
aleph_0_le_mul_iff' {a b : cardinal.{u}} : ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ ℵ₀ ≤ b ∨ ℵ₀ ≤ a ∧ b ≠ 0 | begin
have : ∀ {a : cardinal.{u}}, ℵ₀ ≤ a → a ≠ 0, from λ a, ne_bot_of_le_ne_bot aleph_0_ne_zero,
simp only [aleph_0_le_mul_iff, and_or_distrib_left, and_iff_right_of_imp this,
@and.left_comm (a ≠ 0)],
simp only [and.comm, or.comm]
end | lemma | cardinal.aleph_0_le_mul_iff' | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"and_iff_right_of_imp",
"and_or_distrib_left",
"ne_bot_of_le_ne_bot"
] | See also `cardinal.aleph_0_le_mul_iff'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_lt_aleph_0_iff_of_ne_zero {a b : cardinal} (ha : a ≠ 0) (hb : b ≠ 0) :
a * b < ℵ₀ ↔ a < ℵ₀ ∧ b < ℵ₀ | by simp [mul_lt_aleph_0_iff, ha, hb] | lemma | cardinal.mul_lt_aleph_0_iff_of_ne_zero | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
power_lt_aleph_0 {a b : cardinal} (ha : a < ℵ₀) (hb : b < ℵ₀) : a ^ b < ℵ₀ | match a, b, lt_aleph_0.1 ha, lt_aleph_0.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by rw [← nat_cast_pow]; apply nat_lt_aleph_0
end | theorem | cardinal.power_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_iff_unique {α : Type*} :
#α = 1 ↔ subsingleton α ∧ nonempty α | calc #α = 1 ↔ #α ≤ 1 ∧ 1 ≤ #α : le_antisymm_iff
... ↔ subsingleton α ∧ nonempty α :
le_one_iff_subsingleton.and (one_le_iff_ne_zero.trans mk_ne_zero_iff) | lemma | cardinal.eq_one_iff_unique | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infinite_iff {α : Type u} : infinite α ↔ ℵ₀ ≤ #α | by rw [← not_lt, lt_aleph_0_iff_finite, not_finite_iff_infinite] | theorem | cardinal.infinite_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"infinite",
"not_finite_iff_infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_le_mk (α : Type u) [infinite α] : ℵ₀ ≤ #α | infinite_iff.1 ‹_› | lemma | cardinal.aleph_0_le_mk | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_aleph_0 (α : Type*) [countable α] [infinite α] : #α = ℵ₀ | mk_le_aleph_0.antisymm $ aleph_0_le_mk _ | lemma | cardinal.mk_eq_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"countable",
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
denumerable_iff {α : Type u} : nonempty (denumerable α) ↔ #α = ℵ₀ | ⟨λ ⟨h⟩, mk_congr ((@denumerable.eqv α h).trans equiv.ulift.symm),
λ h, by { cases quotient.exact h with f, exact ⟨denumerable.mk' $ f.trans equiv.ulift⟩ }⟩ | lemma | cardinal.denumerable_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"denumerable",
"denumerable.eqv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_denumerable (α : Type u) [denumerable α] : #α = ℵ₀ | denumerable_iff.1 ⟨‹_›⟩ | lemma | cardinal.mk_denumerable | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"denumerable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_add_aleph_0 : ℵ₀ + ℵ₀ = ℵ₀ | mk_denumerable _ | lemma | cardinal.aleph_0_add_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_mul_aleph_0 : ℵ₀ * ℵ₀ = ℵ₀ | mk_denumerable _ | lemma | cardinal.aleph_0_mul_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_mul_aleph_0 {n : ℕ} (hn : n ≠ 0) : ↑n * ℵ₀ = ℵ₀ | le_antisymm (lift_mk_fin n ▸ mk_le_aleph_0) $ le_mul_of_one_le_left (zero_le _) $
by rwa [← nat.cast_one, nat_cast_le, nat.one_le_iff_ne_zero] | lemma | cardinal.nat_mul_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"le_mul_of_one_le_left",
"nat.cast_one",
"nat.one_le_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_mul_nat {n : ℕ} (hn : n ≠ 0) : ℵ₀ * n = ℵ₀ | by rw [mul_comm, nat_mul_aleph_0 hn] | lemma | cardinal.aleph_0_mul_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_aleph_0 {c₁ c₂ : cardinal} : c₁ + c₂ ≤ ℵ₀ ↔ c₁ ≤ ℵ₀ ∧ c₂ ≤ ℵ₀ | ⟨λ h, ⟨le_self_add.trans h, le_add_self.trans h⟩, λ h, aleph_0_add_aleph_0 ▸ add_le_add h.1 h.2⟩ | lemma | cardinal.add_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_add_nat (n : ℕ) : ℵ₀ + n = ℵ₀ | (add_le_aleph_0.2 ⟨le_rfl, (nat_lt_aleph_0 n).le⟩).antisymm le_self_add | lemma | cardinal.aleph_0_add_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_add_aleph_0 (n : ℕ) : ↑n + ℵ₀ = ℵ₀ | by rw [add_comm, aleph_0_add_nat] | lemma | cardinal.nat_add_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat : zero_hom cardinal ℕ | ⟨λ c, if h : c < aleph_0.{v} then classical.some (lt_aleph_0.1 h) else 0,
begin
have h : 0 < ℵ₀ := nat_lt_aleph_0 0,
rw [dif_pos h, ← cardinal.nat_cast_inj, ← classical.some_spec (lt_aleph_0.1 h), nat.cast_zero],
end⟩ | def | cardinal.to_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"cardinal.nat_cast_inj",
"nat.cast_zero",
"to_nat",
"zero_hom"
] | This function sends finite cardinals to the corresponding natural, and infinite cardinals
to 0. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_nat_apply_of_lt_aleph_0 {c : cardinal} (h : c < ℵ₀) :
c.to_nat = classical.some (lt_aleph_0.1 h) | dif_pos h | lemma | cardinal.to_nat_apply_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_apply_of_aleph_0_le {c : cardinal} (h : ℵ₀ ≤ c) : c.to_nat = 0 | dif_neg h.not_lt | lemma | cardinal.to_nat_apply_of_aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_to_nat_of_lt_aleph_0 {c : cardinal} (h : c < ℵ₀) : ↑c.to_nat = c | by rw [to_nat_apply_of_lt_aleph_0 h, ← classical.some_spec (lt_aleph_0.1 h)] | lemma | cardinal.cast_to_nat_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cast_to_nat_of_aleph_0_le {c : cardinal} (h : ℵ₀ ≤ c) : ↑c.to_nat = (0 : cardinal) | by rw [to_nat_apply_of_aleph_0_le h, nat.cast_zero] | lemma | cardinal.cast_to_nat_of_aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_eq_iff_eq_of_lt_aleph_0 {c d : cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
c.to_nat = d.to_nat ↔ c = d | by rw [←nat_cast_inj, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd] | lemma | cardinal.to_nat_eq_iff_eq_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_le_iff_le_of_lt_aleph_0 {c d : cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
c.to_nat ≤ d.to_nat ↔ c ≤ d | by rw [←nat_cast_le, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd] | lemma | cardinal.to_nat_le_iff_le_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_lt_iff_lt_of_lt_aleph_0 {c d : cardinal} (hc : c < ℵ₀) (hd : d < ℵ₀) :
c.to_nat < d.to_nat ↔ c < d | by rw [←nat_cast_lt, cast_to_nat_of_lt_aleph_0 hc, cast_to_nat_of_lt_aleph_0 hd] | lemma | cardinal.to_nat_lt_iff_lt_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_le_of_le_of_lt_aleph_0 {c d : cardinal} (hd : d < ℵ₀) (hcd : c ≤ d) :
c.to_nat ≤ d.to_nat | (to_nat_le_iff_le_of_lt_aleph_0 (hcd.trans_lt hd) hd).mpr hcd | lemma | cardinal.to_nat_le_of_le_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_lt_of_lt_of_lt_aleph_0 {c d : cardinal} (hd : d < ℵ₀) (hcd : c < d) :
c.to_nat < d.to_nat | (to_nat_lt_iff_lt_of_lt_aleph_0 (hcd.trans hd) hd).mpr hcd | lemma | cardinal.to_nat_lt_of_lt_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_cast (n : ℕ) : cardinal.to_nat n = n | begin
rw [to_nat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), ← nat_cast_inj],
exact (classical.some_spec (lt_aleph_0.1 (nat_lt_aleph_0 n))).symm,
end | lemma | cardinal.to_nat_cast | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_right_inverse : function.right_inverse (coe : ℕ → cardinal) to_nat | to_nat_cast | lemma | cardinal.to_nat_right_inverse | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"to_nat"
] | `to_nat` has a right-inverse: coercion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_nat_surjective : surjective to_nat | to_nat_right_inverse.surjective | lemma | cardinal.to_nat_surjective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nat_eq_of_le_nat {c : cardinal} {n : ℕ} (h : c ≤ n) : ∃ m, m ≤ n ∧ c = m | let he := cast_to_nat_of_lt_aleph_0 (h.trans_lt $ nat_lt_aleph_0 n) in
⟨c.to_nat, nat_cast_le.1 (he.trans_le h), he.symm⟩ | lemma | cardinal.exists_nat_eq_of_le_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_nat_of_infinite [h : infinite α] : (#α).to_nat = 0 | dif_neg (infinite_iff.1 h).not_lt | lemma | cardinal.mk_to_nat_of_infinite | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"infinite",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_to_nat : to_nat ℵ₀ = 0 | to_nat_apply_of_aleph_0_le le_rfl | theorem | cardinal.aleph_0_to_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"le_rfl",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_nat_eq_card [fintype α] : (#α).to_nat = fintype.card α | by simp | lemma | cardinal.mk_to_nat_eq_card | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"fintype",
"fintype.card",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_to_nat : to_nat 0 = 0 | by rw [←to_nat_cast 0, nat.cast_zero] | lemma | cardinal.zero_to_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"nat.cast_zero",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_to_nat : to_nat 1 = 1 | by rw [←to_nat_cast 1, nat.cast_one] | lemma | cardinal.one_to_nat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"nat.cast_one",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_eq_iff {c : cardinal} {n : ℕ} (hn : n ≠ 0) : to_nat c = n ↔ c = n | ⟨λ h, (cast_to_nat_of_lt_aleph_0 (lt_of_not_ge (hn ∘ h.symm.trans ∘
to_nat_apply_of_aleph_0_le))).symm.trans (congr_arg coe h),
λ h, (congr_arg to_nat h).trans (to_nat_cast n)⟩ | lemma | cardinal.to_nat_eq_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_eq_one {c : cardinal} : to_nat c = 1 ↔ c = 1 | by rw [to_nat_eq_iff one_ne_zero, nat.cast_one] | lemma | cardinal.to_nat_eq_one | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_one",
"one_ne_zero",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_eq_one_iff_unique {α : Type*} : (#α).to_nat = 1 ↔ subsingleton α ∧ nonempty α | to_nat_eq_one.trans eq_one_iff_unique | lemma | cardinal.to_nat_eq_one_iff_unique | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_lift (c : cardinal.{v}) : (lift.{u v} c).to_nat = c.to_nat | begin
apply nat_cast_injective,
cases lt_or_ge c ℵ₀ with hc hc,
{ rw [cast_to_nat_of_lt_aleph_0, ←lift_nat_cast, cast_to_nat_of_lt_aleph_0 hc],
rwa [lift_lt_aleph_0] },
{ rw [cast_to_nat_of_aleph_0_le, ←lift_nat_cast, cast_to_nat_of_aleph_0_le hc, lift_zero],
rwa [aleph_0_le_lift] },
end | lemma | cardinal.to_nat_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_congr {β : Type v} (e : α ≃ β) : (#α).to_nat = (#β).to_nat | by rw [←to_nat_lift, lift_mk_eq.mpr ⟨e⟩, to_nat_lift] | lemma | cardinal.to_nat_congr | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_mul (x y : cardinal) : (x * y).to_nat = x.to_nat * y.to_nat | begin
rcases eq_or_ne x 0 with rfl | hx1,
{ rw [zero_mul, zero_to_nat, zero_mul] },
rcases eq_or_ne y 0 with rfl | hy1,
{ rw [mul_zero, zero_to_nat, mul_zero] },
cases lt_or_le x ℵ₀ with hx2 hx2,
{ cases lt_or_le y ℵ₀ with hy2 hy2,
{ lift x to ℕ using hx2, lift y to ℕ using hy2,
rw [← nat.cast_mul... | lemma | cardinal.to_nat_mul | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"eq_or_ne",
"lift",
"mul_zero",
"nat.cast_mul",
"to_nat",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_hom : cardinal →*₀ ℕ | { to_fun := to_nat,
map_zero' := zero_to_nat,
map_one' := one_to_nat,
map_mul' := to_nat_mul } | def | cardinal.to_nat_hom | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"to_nat"
] | `cardinal.to_nat` as a `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_nat_finset_prod (s : finset α) (f : α → cardinal) :
to_nat (∏ i in s, f i) = ∏ i in s, to_nat (f i) | map_prod to_nat_hom _ _ | lemma | cardinal.to_nat_finset_prod | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"finset",
"map_prod",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nat_add_of_lt_aleph_0 {a : cardinal.{u}} {b : cardinal.{v}}
(ha : a < ℵ₀) (hb : b < ℵ₀) : ((lift.{v u} a) + (lift.{u v} b)).to_nat = a.to_nat + b.to_nat | begin
apply cardinal.nat_cast_injective,
replace ha : (lift.{v u} a) < ℵ₀ := by rwa lift_lt_aleph_0,
replace hb : (lift.{u v} b) < ℵ₀ := by rwa lift_lt_aleph_0,
rw [nat.cast_add, ←to_nat_lift.{v u} a, ←to_nat_lift.{u v} b, cast_to_nat_of_lt_aleph_0 ha,
cast_to_nat_of_lt_aleph_0 hb, cast_to_nat_of_lt_aleph_0... | lemma | cardinal.to_nat_add_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.nat_cast_injective",
"nat.cast_add",
"to_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat : cardinal →+ part_enat | { to_fun := λ c, if c < ℵ₀ then c.to_nat else ⊤,
map_zero' := by simp [if_pos (zero_lt_one.trans one_lt_aleph_0)],
map_add' := λ x y, begin
by_cases hx : x < ℵ₀,
{ obtain ⟨x0, rfl⟩ := lt_aleph_0.1 hx,
by_cases hy : y < ℵ₀,
{ obtain ⟨y0, rfl⟩ := lt_aleph_0.1 hy,
simp only [add_lt_aleph_0 ... | def | cardinal.to_part_enat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"nat.cast_add",
"part_enat",
"part_enat.add_top",
"part_enat.top_add"
] | This function sends finite cardinals to the corresponding natural, and infinite cardinals
to `⊤`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_part_enat_apply_of_lt_aleph_0 {c : cardinal} (h : c < ℵ₀) : c.to_part_enat = c.to_nat | if_pos h | lemma | cardinal.to_part_enat_apply_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_apply_of_aleph_0_le {c : cardinal} (h : ℵ₀ ≤ c) : c.to_part_enat = ⊤ | if_neg h.not_lt | lemma | cardinal.to_part_enat_apply_of_aleph_0_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_cast (n : ℕ) : cardinal.to_part_enat n = n | by rw [to_part_enat_apply_of_lt_aleph_0 (nat_lt_aleph_0 n), to_nat_cast] | lemma | cardinal.to_part_enat_cast | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.to_part_enat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_part_enat_of_infinite [h : infinite α] : (#α).to_part_enat = ⊤ | to_part_enat_apply_of_aleph_0_le (infinite_iff.1 h) | lemma | cardinal.mk_to_part_enat_of_infinite | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"infinite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aleph_0_to_part_enat : to_part_enat ℵ₀ = ⊤ | to_part_enat_apply_of_aleph_0_le le_rfl | theorem | cardinal.aleph_0_to_part_enat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_eq_top_iff_le_aleph_0 {c : cardinal} :
to_part_enat c = ⊤ ↔ aleph_0 ≤ c | begin
cases lt_or_ge c aleph_0 with hc hc,
simp only [to_part_enat_apply_of_lt_aleph_0 hc, part_enat.coe_ne_top, false_iff, not_le, hc],
simp only [to_part_enat_apply_of_aleph_0_le hc, eq_self_iff_true, true_iff],
exact hc,
end | lemma | cardinal.to_part_enat_eq_top_iff_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"part_enat.coe_ne_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_le_iff_le_of_le_aleph_0 {c c' : cardinal} (h : c ≤ aleph_0) :
to_part_enat c ≤ to_part_enat c' ↔ c ≤ c' | begin
cases lt_or_ge c aleph_0 with hc hc,
rw to_part_enat_apply_of_lt_aleph_0 hc,
cases lt_or_ge c' aleph_0 with hc' hc',
{ rw to_part_enat_apply_of_lt_aleph_0 hc',
rw part_enat.coe_le_coe,
exact to_nat_le_iff_le_of_lt_aleph_0 hc hc', },
{ simp only [to_part_enat_apply_of_aleph_0_le hc',
le_top, ... | lemma | cardinal.to_part_enat_le_iff_le_of_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"le_top",
"part_enat.coe_le_coe",
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_le_iff_le_of_lt_aleph_0 {c c' : cardinal} (hc' : c' < aleph_0) :
to_part_enat c ≤ to_part_enat c' ↔ c ≤ c' | begin
cases lt_or_ge c aleph_0 with hc hc,
{ rw to_part_enat_apply_of_lt_aleph_0 hc,
rw to_part_enat_apply_of_lt_aleph_0 hc',
rw part_enat.coe_le_coe,
exact to_nat_le_iff_le_of_lt_aleph_0 hc hc', },
{ rw to_part_enat_apply_of_aleph_0_le hc,
simp only [top_le_iff, to_part_enat_eq_top_iff_le_aleph_0... | lemma | cardinal.to_part_enat_le_iff_le_of_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"not_iff_not",
"part_enat.coe_le_coe",
"top_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_eq_iff_eq_of_le_aleph_0 {c c' : cardinal}
(hc : c ≤ aleph_0) (hc' : c' ≤ aleph_0) :
to_part_enat c = to_part_enat c' ↔ c = c' | by
rw [le_antisymm_iff, le_antisymm_iff,
to_part_enat_le_iff_le_of_le_aleph_0 hc, to_part_enat_le_iff_le_of_le_aleph_0 hc'] | lemma | cardinal.to_part_enat_eq_iff_eq_of_le_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_mono {c c' : cardinal} (h : c ≤ c') :
to_part_enat c ≤ to_part_enat c' | begin
cases lt_or_ge c aleph_0 with hc hc,
rw to_part_enat_apply_of_lt_aleph_0 hc,
cases lt_or_ge c' aleph_0 with hc' hc',
rw to_part_enat_apply_of_lt_aleph_0 hc',
simp only [part_enat.coe_le_coe],
exact to_nat_le_of_le_of_lt_aleph_0 hc' h,
rw to_part_enat_apply_of_aleph_0_le hc',
exact le_top,
rw [to... | lemma | cardinal.to_part_enat_mono | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"le_top",
"part_enat.coe_le_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_surjective : surjective to_part_enat | λ x, part_enat.cases_on x ⟨ℵ₀, to_part_enat_apply_of_aleph_0_le le_rfl⟩ $
λ n, ⟨n, to_part_enat_cast n⟩ | lemma | cardinal.to_part_enat_surjective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"part_enat.cases_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_lift (c : cardinal.{v}) : (lift.{u v} c).to_part_enat = c.to_part_enat | begin
cases lt_or_ge c ℵ₀ with hc hc,
{ rw [to_part_enat_apply_of_lt_aleph_0 hc, cardinal.to_part_enat_apply_of_lt_aleph_0 _],
simp only [to_nat_lift],
rw [← lift_aleph_0, lift_lt], exact hc },
{ rw [to_part_enat_apply_of_aleph_0_le hc, cardinal.to_part_enat_apply_of_aleph_0_le _],
rw [← lift_aleph_0, l... | lemma | cardinal.to_part_enat_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal.to_part_enat_apply_of_aleph_0_le",
"cardinal.to_part_enat_apply_of_lt_aleph_0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_part_enat_congr {β : Type v} (e : α ≃ β) : (#α).to_part_enat = (#β).to_part_enat | by rw [←to_part_enat_lift, lift_mk_eq.mpr ⟨e⟩, to_part_enat_lift] | lemma | cardinal.to_part_enat_congr | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_to_part_enat_eq_coe_card [fintype α] : (#α).to_part_enat = fintype.card α | by simp | lemma | cardinal.mk_to_part_enat_eq_coe_card | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_int : #ℤ = ℵ₀ | mk_denumerable ℤ | lemma | cardinal.mk_int | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pnat : #ℕ+ = ℵ₀ | mk_denumerable ℕ+ | lemma | cardinal.mk_pnat | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_lt_prod {ι} (f g : ι → cardinal) (H : ∀ i, f i < g i) :
sum f < prod g | lt_of_not_ge $ λ ⟨F⟩, begin
haveI : inhabited (Π (i : ι), (g i).out),
{ refine ⟨λ i, classical.choice $ mk_ne_zero_iff.1 _⟩,
rw mk_out,
exact (H i).ne_bot },
let G := inv_fun F,
have sG : surjective G := inv_fun_surjective F.2,
choose C hc using show ∀ i, ∃ b, ∀ a, G ⟨i, a⟩ i ≠ b,
{ intro i,
sim... | theorem | cardinal.sum_lt_prod | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"cardinal",
"inv_fun",
"inv_fun_surjective",
"not_exists"
] | **König's theorem** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_empty : #empty = 0 | mk_eq_zero _ | theorem | cardinal.mk_empty | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_pempty : #pempty = 0 | mk_eq_zero _ | theorem | cardinal.mk_pempty | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_punit : #punit = 1 | mk_eq_one punit | theorem | cardinal.mk_punit | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_unit : #unit = 1 | mk_punit | theorem | cardinal.mk_unit | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_singleton {α : Type u} (x : α) : #({x} : set α) = 1 | mk_eq_one _ | theorem | cardinal.mk_singleton | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_plift_true : #(plift true) = 1 | mk_eq_one _ | theorem | cardinal.mk_plift_true | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_plift_false : #(plift false) = 0 | mk_eq_zero _ | theorem | cardinal.mk_plift_false | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_vector (α : Type u) (n : ℕ) : #(vector α n) = (#α) ^ℕ n | (mk_congr (equiv.vector_equiv_fin α n)).trans $ by simp | theorem | cardinal.mk_vector | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.vector_equiv_fin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_list_eq_sum_pow (α : Type u) : #(list α) = sum (λ n : ℕ, (#α) ^ℕ n) | calc #(list α) = #(Σ n, vector α n) : mk_congr (equiv.sigma_fiber_equiv list.length).symm
... = sum (λ n : ℕ, (#α) ^ℕ n) : by simp | theorem | cardinal.mk_list_eq_sum_pow | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.sigma_fiber_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_quot_le {α : Type u} {r : α → α → Prop} : #(quot r) ≤ #α | mk_le_of_surjective quot.exists_rep | theorem | cardinal.mk_quot_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_quotient_le {α : Type u} {s : setoid α} : #(quotient s) ≤ #α | mk_quot_le | theorem | cardinal.mk_quotient_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_subtype_le_of_subset {α : Type u} {p q : α → Prop} (h : ∀ ⦃x⦄, p x → q x) :
#(subtype p) ≤ #(subtype q) | ⟨embedding.subtype_map (embedding.refl α) h⟩ | theorem | cardinal.mk_subtype_le_of_subset | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_emptyc (α : Type u) : #(∅ : set α) = 0 | mk_eq_zero _ | theorem | cardinal.mk_emptyc | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_emptyc_iff {α : Type u} {s : set α} : #s = 0 ↔ s = ∅ | begin
split,
{ intro h,
rw mk_eq_zero_iff at h,
exact eq_empty_iff_forall_not_mem.2 (λ x hx, h.elim' ⟨x, hx⟩) },
{ rintro rfl, exact mk_emptyc _ }
end | lemma | cardinal.mk_emptyc_iff | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_univ {α : Type u} : #(@univ α) = #α | mk_congr (equiv.set.univ α) | theorem | cardinal.mk_univ | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_image_le {α β : Type u} {f : α → β} {s : set α} : #(f '' s) ≤ #s | mk_le_of_surjective surjective_onto_image | theorem | cardinal.mk_image_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_image_le_lift {α : Type u} {β : Type v} {f : α → β} {s : set α} :
lift.{u} (#(f '' s)) ≤ lift.{v} (#s) | lift_mk_le.{v u 0}.mpr ⟨embedding.of_surjective _ surjective_onto_image⟩ | theorem | cardinal.mk_image_le_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_range_le {α β : Type u} {f : α → β} : #(range f) ≤ #α | mk_le_of_surjective surjective_onto_range | theorem | cardinal.mk_range_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_range_le_lift {α : Type u} {β : Type v} {f : α → β} :
lift.{u} (#(range f)) ≤ lift.{v} (#α) | lift_mk_le.{v u 0}.mpr ⟨embedding.of_surjective _ surjective_onto_range⟩ | theorem | cardinal.mk_range_le_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_range_eq (f : α → β) (h : injective f) : #(range f) = #α | mk_congr ((equiv.of_injective f h).symm) | lemma | cardinal.mk_range_eq | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_range_eq_of_injective {α : Type u} {β : Type v} {f : α → β} (hf : injective f) :
lift.{u} (#(range f)) = lift.{v} (#α) | lift_mk_eq'.mpr ⟨(equiv.of_injective f hf).symm⟩ | lemma | cardinal.mk_range_eq_of_injective | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_range_eq_lift {α : Type u} {β : Type v} {f : α → β} (hf : injective f) :
lift.{max u w} (# (range f)) = lift.{max v w} (# α) | lift_mk_eq.mpr ⟨(equiv.of_injective f hf).symm⟩ | lemma | cardinal.mk_range_eq_lift | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.of_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_image_eq {α β : Type u} {f : α → β} {s : set α} (hf : injective f) :
#(f '' s) = #s | mk_congr ((equiv.set.image f s hf).symm) | theorem | cardinal.mk_image_eq | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"equiv.set.image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_Union_le_sum_mk {α ι : Type u} {f : ι → set α} : #(⋃ i, f i) ≤ sum (λ i, #(f i)) | calc #(⋃ i, f i) ≤ #(Σ i, f i) : mk_le_of_surjective (set.sigma_to_Union_surjective f)
... = sum (λ i, #(f i)) : mk_sigma _ | theorem | cardinal.mk_Union_le_sum_mk | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"set.sigma_to_Union_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_Union_eq_sum_mk {α ι : Type u} {f : ι → set α} (h : ∀i j, i ≠ j → disjoint (f i) (f j)) :
#(⋃ i, f i) = sum (λ i, #(f i)) | calc #(⋃ i, f i) = #(Σ i, f i) : mk_congr (set.Union_eq_sigma_of_disjoint h)
... = sum (λi, #(f i)) : mk_sigma _ | theorem | cardinal.mk_Union_eq_sum_mk | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"disjoint",
"set.Union_eq_sigma_of_disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_Union_le {α ι : Type u} (f : ι → set α) : #(⋃ i, f i) ≤ #ι * ⨆ i, #(f i) | mk_Union_le_sum_mk.trans (sum_le_supr _) | lemma | cardinal.mk_Union_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_sUnion_le {α : Type u} (A : set (set α)) : #(⋃₀ A) ≤ #A * ⨆ s : A, #s | by { rw sUnion_eq_Union, apply mk_Union_le } | lemma | cardinal.mk_sUnion_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_bUnion_le {ι α : Type u} (A : ι → set α) (s : set ι) :
#(⋃ x ∈ s, A x) ≤ #s * ⨆ x : s, #(A x.1) | by { rw bUnion_eq_Union, apply mk_Union_le } | lemma | cardinal.mk_bUnion_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset_card_lt_aleph_0 (s : finset α) : #(↑s : set α) < ℵ₀ | lt_aleph_0_of_finite _ | lemma | cardinal.finset_card_lt_aleph_0 | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_set_eq_nat_iff_finset {α} {s : set α} {n : ℕ} :
#s = n ↔ ∃ t : finset α, (t : set α) = s ∧ t.card = n | begin
split,
{ intro h,
lift s to finset α using lt_aleph_0_iff_set_finite.1 (h.symm ▸ nat_lt_aleph_0 n),
simpa using h },
{ rintro ⟨t, rfl, rfl⟩,
exact mk_coe_finset }
end | theorem | cardinal.mk_set_eq_nat_iff_finset | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finset",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_nat_iff_finset {n : ℕ} : #α = n ↔ ∃ t : finset α, (t : set α) = univ ∧ t.card = n | by rw [← mk_univ, mk_set_eq_nat_iff_finset] | theorem | cardinal.mk_eq_nat_iff_finset | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_nat_iff_fintype {n : ℕ} : #α = n ↔ ∃ (h : fintype α), @fintype.card α h = n | begin
rw [mk_eq_nat_iff_finset],
split,
{ rintro ⟨t, ht, hn⟩,
exact ⟨⟨t, eq_univ_iff_forall.1 ht⟩, hn⟩ },
{ rintro ⟨⟨t, ht⟩, hn⟩,
exact ⟨t, eq_univ_iff_forall.2 ht, hn⟩ }
end | theorem | cardinal.mk_eq_nat_iff_fintype | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [
"fintype",
"fintype.card"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_union_add_mk_inter {α : Type u} {S T : set α} :
#(S ∪ T : set α) + #(S ∩ T : set α) = #S + #T | quot.sound ⟨equiv.set.union_sum_inter S T⟩ | theorem | cardinal.mk_union_add_mk_inter | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_union_le {α : Type u} (S T : set α) : #(S ∪ T : set α) ≤ #S + #T | @mk_union_add_mk_inter α S T ▸ self_le_add_right (#(S ∪ T : set α)) (#(S ∩ T : set α)) | lemma | cardinal.mk_union_le | set_theory.cardinal | src/set_theory/cardinal/basic.lean | [
"data.fintype.big_operators",
"data.finsupp.defs",
"data.nat.part_enat",
"data.set.countable",
"logic.small.basic",
"order.conditionally_complete_lattice.basic",
"order.succ_pred.limit",
"set_theory.cardinal.schroeder_bernstein",
"tactic.positivity"
] | [] | The cardinality of a union is at most the sum of the cardinalities
of the two sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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