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
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|---|---|---|---|---|---|---|---|---|---|---|
mk_union_of_disjoint {α : Type u} {S T : set α} (H : disjoint S T) :
#(S ∪ T : set α) = #S + #T | quot.sound ⟨equiv.set.union H.le_bot⟩ | theorem | cardinal.mk_union_of_disjoint | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_insert {α : Type u} {s : set α} {a : α} (h : a ∉ s) :
#(insert a s : set α) = #s + 1 | by { rw [← union_singleton, mk_union_of_disjoint, mk_singleton], simpa } | theorem | cardinal.mk_insert | 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_sum_compl {α} (s : set α) : #s + #(sᶜ : set α) = #α | mk_congr (equiv.set.sum_compl s) | lemma | cardinal.mk_sum_compl | 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.sum_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_le_mk_of_subset {α} {s t : set α} (h : s ⊆ t) : #s ≤ #t | ⟨set.embedding_of_subset s t h⟩ | lemma | cardinal.mk_le_mk_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_subtype_mono {p q : α → Prop} (h : ∀ x, p x → q x) : #{x // p x} ≤ #{x // q x} | ⟨embedding_of_subset _ _ h⟩ | lemma | cardinal.mk_subtype_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mk_diff_add_mk (S T : set α) : #S ≤ #(S \ T : set α) + #T | (mk_le_mk_of_subset $ subset_diff_union _ _).trans $ mk_union_le _ _ | lemma | cardinal.le_mk_diff_add_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_diff_add_mk {S T : set α} (h : T ⊆ S) : #(S \ T : set α) + #T = #S | (mk_union_of_disjoint $ by exact disjoint_sdiff_self_left).symm.trans $ by rw diff_union_of_subset h | lemma | cardinal.mk_diff_add_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_sdiff_self_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_union_le_aleph_0 {α} {P Q : set α} : #((P ∪ Q : set α)) ≤ ℵ₀ ↔ #P ≤ ℵ₀ ∧ #Q ≤ ℵ₀ | by simp | lemma | cardinal.mk_union_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_image_eq_lift {α : Type u} {β : Type v} (f : α → β) (s : set α) (h : injective f) :
lift.{u} (#(f '' s)) = lift.{v} (#s) | lift_mk_eq.{v u 0}.mpr ⟨(equiv.set.image f s h).symm⟩ | lemma | cardinal.mk_image_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.set.image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_image_eq_of_inj_on_lift {α : Type u} {β : Type v} (f : α → β) (s : set α)
(h : inj_on f s) : lift.{u} (#(f '' s)) = lift.{v} (#s) | lift_mk_eq.{v u 0}.mpr ⟨(equiv.set.image_of_inj_on f s h).symm⟩ | lemma | cardinal.mk_image_eq_of_inj_on_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.set.image_of_inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_image_eq_of_inj_on {α β : Type u} (f : α → β) (s : set α) (h : inj_on f s) :
#(f '' s) = #s | mk_congr ((equiv.set.image_of_inj_on f s h).symm) | lemma | cardinal.mk_image_eq_of_inj_on | 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_of_inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_subtype_of_equiv {α β : Type u} (p : β → Prop) (e : α ≃ β) :
#{a : α // p (e a)} = #{b : β // p b} | mk_congr (equiv.subtype_equiv_of_subtype e) | lemma | cardinal.mk_subtype_of_equiv | 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.subtype_equiv_of_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_sep (s : set α) (t : α → Prop) : #({ x ∈ s | t x } : set α) = #{ x : s | t x.1 } | mk_congr (equiv.set.sep s t) | lemma | cardinal.mk_sep | 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.sep"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_preimage_of_injective_lift {α : Type u} {β : Type v} (f : α → β) (s : set β)
(h : injective f) : lift.{v} (#(f ⁻¹' s)) ≤ lift.{u} (#s) | begin
rw lift_mk_le.{u v 0}, use subtype.coind (λ x, f x.1) (λ x, x.2),
apply subtype.coind_injective, exact h.comp subtype.val_injective
end | lemma | cardinal.mk_preimage_of_injective_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"
] | [
"subtype.coind",
"subtype.coind_injective",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_preimage_of_subset_range_lift {α : Type u} {β : Type v} (f : α → β) (s : set β)
(h : s ⊆ range f) : lift.{u} (#s) ≤ lift.{v} (#(f ⁻¹' s)) | begin
rw lift_mk_le.{v u 0},
refine ⟨⟨_, _⟩⟩,
{ rintro ⟨y, hy⟩, rcases classical.subtype_of_exists (h hy) with ⟨x, rfl⟩, exact ⟨x, hy⟩ },
rintro ⟨y, hy⟩ ⟨y', hy'⟩, dsimp,
rcases classical.subtype_of_exists (h hy) with ⟨x, rfl⟩,
rcases classical.subtype_of_exists (h hy') with ⟨x', rfl⟩,
simp, intro hxx', r... | lemma | cardinal.mk_preimage_of_subset_range_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"
] | [
"classical.subtype_of_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_preimage_of_injective_of_subset_range_lift {β : Type v} (f : α → β) (s : set β)
(h : injective f) (h2 : s ⊆ range f) : lift.{v} (#(f ⁻¹' s)) = lift.{u} (#s) | le_antisymm (mk_preimage_of_injective_lift f s h) (mk_preimage_of_subset_range_lift f s h2) | lemma | cardinal.mk_preimage_of_injective_of_subset_range_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_preimage_of_injective (f : α → β) (s : set β) (h : injective f) :
#(f ⁻¹' s) ≤ #s | by { convert mk_preimage_of_injective_lift.{u u} f s h using 1; rw [lift_id] } | lemma | cardinal.mk_preimage_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_preimage_of_subset_range (f : α → β) (s : set β)
(h : s ⊆ range f) : #s ≤ #(f ⁻¹' s) | by { convert mk_preimage_of_subset_range_lift.{u u} f s h using 1; rw [lift_id] } | lemma | cardinal.mk_preimage_of_subset_range | 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_preimage_of_injective_of_subset_range (f : α → β) (s : set β)
(h : injective f) (h2 : s ⊆ range f) : #(f ⁻¹' s) = #s | by { convert mk_preimage_of_injective_of_subset_range_lift.{u u} f s h h2 using 1; rw [lift_id] } | lemma | cardinal.mk_preimage_of_injective_of_subset_range | 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_subset_ge_of_subset_image_lift {α : Type u} {β : Type v} (f : α → β) {s : set α}
{t : set β} (h : t ⊆ f '' s) :
lift.{u} (#t) ≤ lift.{v} (#({ x ∈ s | f x ∈ t } : set α)) | by { rw [image_eq_range] at h, convert mk_preimage_of_subset_range_lift _ _ h using 1,
rw [mk_sep], refl } | lemma | cardinal.mk_subset_ge_of_subset_image_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_subset_ge_of_subset_image (f : α → β) {s : set α} {t : set β} (h : t ⊆ f '' s) :
#t ≤ #({ x ∈ s | f x ∈ t } : set α) | by { rw [image_eq_range] at h, convert mk_preimage_of_subset_range _ _ h using 1,
rw [mk_sep], refl } | lemma | cardinal.mk_subset_ge_of_subset_image | 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 | |
le_mk_iff_exists_subset {c : cardinal} {α : Type u} {s : set α} :
c ≤ #s ↔ ∃ p : set α, p ⊆ s ∧ #p = c | begin
rw [le_mk_iff_exists_set, ←subtype.exists_set_subtype],
apply exists_congr, intro t, rw [mk_image_eq], apply subtype.val_injective
end | theorem | cardinal.le_mk_iff_exists_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"
] | [
"cardinal",
"subtype.val_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_le_iff : (2 : cardinal) ≤ #α ↔ ∃x y : α, x ≠ y | by rw [← nat.cast_two, nat_succ, succ_le_iff, nat.cast_one, one_lt_iff_nontrivial, nontrivial_iff] | lemma | cardinal.two_le_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",
"nat.cast_one",
"nat.cast_two",
"nontrivial_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_le_iff' (x : α) : (2 : cardinal) ≤ #α ↔ ∃y : α, y ≠ x | by rw [two_le_iff, ← nontrivial_iff, nontrivial_iff_exists_ne x] | lemma | cardinal.two_le_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",
"nontrivial_iff",
"nontrivial_iff_exists_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_two_iff : #α = 2 ↔ ∃ x y : α, x ≠ y ∧ ({x, y} : set α) = univ | begin
simp only [← @nat.cast_two cardinal, mk_eq_nat_iff_finset, finset.card_eq_two],
split,
{ rintro ⟨t, ht, x, y, hne, rfl⟩,
exact ⟨x, y, hne, by simpa using ht⟩ },
{ rintro ⟨x, y, hne, h⟩,
exact ⟨{x, y}, by simpa using h, x, y, hne, rfl⟩ }
end | lemma | cardinal.mk_eq_two_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",
"finset.card_eq_two",
"nat.cast_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_two_iff' (x : α) : #α = 2 ↔ ∃! y, y ≠ x | begin
rw [mk_eq_two_iff], split,
{ rintro ⟨a, b, hne, h⟩,
simp only [eq_univ_iff_forall, mem_insert_iff, mem_singleton_iff] at h,
rcases h x with rfl|rfl,
exacts [⟨b, hne.symm, λ z, (h z).resolve_left⟩, ⟨a, hne, λ z, (h z).resolve_right⟩] },
{ rintro ⟨y, hne, hy⟩,
exact ⟨x, y, hne.symm, eq_univ_of... | lemma | cardinal.mk_eq_two_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 | |
exists_not_mem_of_length_lt {α : Type*} (l : list α) (h : ↑l.length < # α) :
∃ (z : α), z ∉ l | begin
contrapose! h,
calc # α = # (set.univ : set α) : mk_univ.symm
... ≤ # l.to_finset : mk_le_mk_of_subset (λ x _, list.mem_to_finset.mpr (h x))
... = l.to_finset.card : cardinal.mk_coe_finset
... ≤ l.length : cardinal.nat_cast_le.mpr (list.to_finset_card_le l),
end | lemma | cardinal.exists_not_mem_of_length_lt | 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.mk_coe_finset",
"list.to_finset_card_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
three_le {α : Type*} (h : 3 ≤ # α) (x : α) (y : α) :
∃ (z : α), z ≠ x ∧ z ≠ y | begin
have : ↑(3 : ℕ) ≤ # α, simpa using h,
have : ↑(2 : ℕ) < # α, rwa [← succ_le_iff, ← cardinal.nat_succ],
have := exists_not_mem_of_length_lt [x, y] this,
simpa [not_or_distrib] using this,
end | lemma | cardinal.three_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_succ",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerlt (a b : cardinal.{u}) : cardinal.{u} | ⨆ c : Iio b, a ^ c | def | cardinal.powerlt | 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 function `a ^< b`, defined as the supremum of `a ^ c` for `c < b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_powerlt {b c : cardinal.{u}} (a) (h : c < b) : a ^ c ≤ a ^< b | begin
apply @le_csupr _ _ _ (λ y : Iio b, a ^ y) _ ⟨c, h⟩,
rw ←image_eq_range,
exact bdd_above_image.{u u} _ bdd_above_Iio
end | lemma | cardinal.le_powerlt | 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"
] | [
"bdd_above_Iio",
"le_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerlt_le {a b c : cardinal.{u}} : a ^< b ≤ c ↔ ∀ x < b, a ^ x ≤ c | begin
rw [powerlt, csupr_le_iff'],
{ simp },
{ rw ←image_eq_range,
exact bdd_above_image.{u u} _ bdd_above_Iio }
end | lemma | cardinal.powerlt_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"
] | [
"bdd_above_Iio",
"csupr_le_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerlt_le_powerlt_left {a b c : cardinal} (h : b ≤ c) : a ^< b ≤ a ^< c | powerlt_le.2 $ λ x hx, le_powerlt a $ hx.trans_le h | lemma | cardinal.powerlt_le_powerlt_left | 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 | |
powerlt_mono_left (a) : monotone (λ c, a ^< c) | λ b c, powerlt_le_powerlt_left | lemma | cardinal.powerlt_mono_left | 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"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
powerlt_succ {a b : cardinal} (h : a ≠ 0) : a ^< (succ b) = a ^ b | (powerlt_le.2 $ λ c h', power_le_power_left h $ le_of_lt_succ h').antisymm $
le_powerlt a (lt_succ b) | lemma | cardinal.powerlt_succ | 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 | |
powerlt_min {a b c : cardinal} : a ^< min b c = min (a ^< b) (a ^< c) | (powerlt_mono_left a).map_min | lemma | cardinal.powerlt_min | 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 | |
powerlt_max {a b c : cardinal} : a ^< max b c = max (a ^< b) (a ^< c) | (powerlt_mono_left a).map_max | lemma | cardinal.powerlt_max | 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 | |
zero_powerlt {a : cardinal} (h : a ≠ 0) : 0 ^< a = 1 | begin
apply (powerlt_le.2 (λ c hc, zero_power_le _)).antisymm,
rw ←power_zero,
exact le_powerlt 0 (pos_iff_ne_zero.2 h)
end | lemma | cardinal.zero_powerlt | 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 | |
powerlt_zero {a : cardinal} : a ^< 0 = 0 | begin
convert cardinal.supr_of_empty _,
exact subtype.is_empty_of_false (λ x, (cardinal.zero_le _).not_lt),
end | lemma | cardinal.powerlt_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",
"cardinal.supr_of_empty",
"cardinal.zero_le",
"subtype.is_empty_of_false"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positivity_cardinal_pow : expr → tactic strictness | | `(@has_pow.pow _ _ %%inst %%a %%b) := do
strictness_a ← core a,
match strictness_a with
| positive p := positive <$> mk_app ``power_pos [b, p]
| _ := failed -- We already know that `0 ≤ x` for all `x : cardinal`
end
| _ := failed | def | tactic.positivity_cardinal_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"
] | [] | Extension for the `positivity` tactic: The cardinal power of a positive cardinal is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof (r : α → α → Prop) : cardinal | Inf {c | ∃ S : set α, (∀ a, ∃ b ∈ S, r a b) ∧ #S = c} | def | order.cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal"
] | Cofinality of a reflexive order `≼`. This is the smallest cardinality
of a subset `S : set α` such that `∀ a, ∃ b ∈ S, a ≼ b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof_nonempty (r : α → α → Prop) [is_refl α r] :
{c | ∃ S : set α, (∀ a, ∃ b ∈ S, r a b) ∧ #S = c}.nonempty | ⟨_, set.univ, λ a, ⟨a, ⟨⟩, refl _⟩, rfl⟩ | theorem | order.cof_nonempty | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | The set in the definition of `order.cof` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof_le (r : α → α → Prop) {S : set α} (h : ∀ a, ∃ b ∈ S, r a b) : cof r ≤ #S | cInf_le' ⟨S, h, rfl⟩ | lemma | order.cof_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cof {r : α → α → Prop} [is_refl α r] (c : cardinal) :
c ≤ cof r ↔ ∀ {S : set α}, (∀ a, ∃ b ∈ S, r a b) → c ≤ #S | begin
rw [cof, le_cInf_iff'' (cof_nonempty r)],
use λ H S h, H _ ⟨S, h, rfl⟩,
rintro H d ⟨S, h, rfl⟩,
exact H h
end | lemma | order.le_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"le_cInf_iff''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_iso.cof_le_lift {α : Type u} {β : Type v} {r : α → α → Prop} {s}
[is_refl β s] (f : r ≃r s) :
cardinal.lift.{max u v} (order.cof r) ≤ cardinal.lift.{max u v} (order.cof s) | begin
rw [order.cof, order.cof, lift_Inf, lift_Inf,
le_cInf_iff'' (nonempty_image_iff.2 (order.cof_nonempty s))],
rintros - ⟨-, ⟨u, H, rfl⟩, rfl⟩,
apply cInf_le',
refine ⟨_, ⟨f.symm '' u, λ a, _, rfl⟩,
lift_mk_eq.{u v (max u v)}.2 ⟨((f.symm).to_equiv.image u).symm⟩⟩,
rcases H (f a) with ⟨b, hb, hb'⟩,
... | theorem | rel_iso.cof_le_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'",
"le_cInf_iff''",
"order.cof",
"order.cof_nonempty",
"rel_iso.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_iso.cof_eq_lift {α : Type u} {β : Type v} {r s}
[is_refl α r] [is_refl β s] (f : r ≃r s) :
cardinal.lift.{max u v} (order.cof r) = cardinal.lift.{max u v} (order.cof s) | (rel_iso.cof_le_lift f).antisymm (rel_iso.cof_le_lift f.symm) | theorem | rel_iso.cof_eq_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"order.cof",
"rel_iso.cof_le_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_iso.cof_le {α β : Type u} {r : α → α → Prop} {s} [is_refl β s] (f : r ≃r s) :
order.cof r ≤ order.cof s | lift_le.1 (rel_iso.cof_le_lift f) | theorem | rel_iso.cof_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"order.cof",
"rel_iso.cof_le_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_iso.cof_eq {α β : Type u} {r s} [is_refl α r] [is_refl β s] (f : r ≃r s) :
order.cof r = order.cof s | lift_inj.1 (rel_iso.cof_eq_lift f) | theorem | rel_iso.cof_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"order.cof",
"rel_iso.cof_eq_lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_order.cof (r : α → α → Prop) : cardinal | order.cof (swap r)ᶜ | def | strict_order.cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"order.cof"
] | Cofinality of a strict order `≺`. This is the smallest cardinality of a set `S : set α` such
that `∀ a, ∃ b ∈ S, ¬ b ≺ a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_order.cof_nonempty (r : α → α → Prop) [is_irrefl α r] :
{c | ∃ S : set α, unbounded r S ∧ #S = c}.nonempty | @order.cof_nonempty α _ (is_refl.swap rᶜ) | theorem | strict_order.cof_nonempty | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_refl.swap",
"order.cof_nonempty"
] | The set in the definition of `order.strict_order.cof` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof (o : ordinal.{u}) : cardinal.{u} | o.lift_on (λ a, strict_order.cof a.r)
begin
rintros ⟨α, r, wo₁⟩ ⟨β, s, wo₂⟩ ⟨⟨f, hf⟩⟩,
haveI := wo₁, haveI := wo₂,
apply @rel_iso.cof_eq _ _ _ _ _ _ ,
{ split, exact λ a b, not_iff_not.2 hf },
{ exact ⟨(is_well_order.is_irrefl r).1⟩ },
{ exact ⟨(is_well_order.is_irrefl s).1⟩ }
end | def | ordinal.cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order.is_irrefl",
"rel_iso.cof_eq",
"strict_order.cof"
] | Cofinality of an ordinal. This is the smallest cardinal of a
subset `S` of the ordinal which is unbounded, in the sense
`∀ a, ∃ b ∈ S, a ≤ b`. It is defined for all ordinals, but
`cof 0 = 0` and `cof (succ o) = 1`, so it is only really
interesting on limit ordinals (when it is an infinite cardinal). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof_type (r : α → α → Prop) [is_well_order α r] : (type r).cof = strict_order.cof r | rfl | lemma | ordinal.cof_type | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order",
"strict_order.cof"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cof_type [is_well_order α r] {c} : c ≤ cof (type r) ↔ ∀ S, unbounded r S → c ≤ #S | (le_cInf_iff'' (strict_order.cof_nonempty r)).trans ⟨λ H S h, H _ ⟨S, h, rfl⟩,
by { rintros H d ⟨S, h, rfl⟩, exact H _ h }⟩ | theorem | ordinal.le_cof_type | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order",
"le_cInf_iff''",
"strict_order.cof_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_type_le [is_well_order α r] {S : set α} (h : unbounded r S) : cof (type r) ≤ #S | le_cof_type.1 le_rfl S h | theorem | ordinal.cof_type_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_cof_type [is_well_order α r] {S : set α} : #S < cof (type r) → bounded r S | by simpa using not_imp_not.2 cof_type_le | theorem | ordinal.lt_cof_type | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_eq (r : α → α → Prop) [is_well_order α r] : ∃ S, unbounded r S ∧ #S = cof (type r) | Inf_mem (strict_order.cof_nonempty r) | theorem | ordinal.cof_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"Inf_mem",
"is_well_order",
"strict_order.cof_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_cof_eq (r : α → α → Prop) [is_well_order α r] :
∃ S, unbounded r S ∧ type (subrel r S) = (cof (type r)).ord | let ⟨S, hS, e⟩ := cof_eq r, ⟨s, _, e'⟩ := cardinal.ord_eq S,
T : set α := {a | ∃ aS : a ∈ S, ∀ b : S, s b ⟨_, aS⟩ → r b a} in
begin
resetI, suffices,
{ refine ⟨T, this,
le_antisymm _ (cardinal.ord_le.2 $ cof_type_le this)⟩,
rw [← e, e'],
refine (rel_embedding.of_monotone (λ a : T, (⟨a, let ⟨aS, _⟩... | theorem | ordinal.ord_cof_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.ord_eq",
"is_order_connected.neg_trans",
"is_well_order",
"rel_embedding.of_monotone",
"subrel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_mem_cof {o} : ∃ {ι} (f : ι → ordinal), lsub.{u u} f = o ∧ #ι = o.card | ⟨_, _, lsub_typein o, mk_ordinal_out o⟩ | theorem | ordinal.card_mem_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_lsub_def_nonempty (o) :
{a : cardinal | ∃ {ι} (f : ι → ordinal), lsub.{u u} f = o ∧ #ι = a}.nonempty | ⟨_, card_mem_cof⟩ | theorem | ordinal.cof_lsub_def_nonempty | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"ordinal"
] | The set in the `lsub` characterization of `cof` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof_eq_Inf_lsub (o : ordinal.{u}) :
cof o = Inf {a : cardinal | ∃ {ι : Type u} (f : ι → ordinal), lsub.{u u} f = o ∧ #ι = a} | begin
refine le_antisymm (le_cInf (cof_lsub_def_nonempty o) _) (cInf_le' _),
{ rintros a ⟨ι, f, hf, rfl⟩,
rw ←type_lt o,
refine (cof_type_le (λ a, _)).trans (@mk_le_of_injective _ _
(λ s : (typein ((<) : o.out.α → o.out.α → Prop))⁻¹' (set.range f), classical.some s.prop)
(λ s t hst, let H := con... | theorem | ordinal.cof_eq_Inf_lsub | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'",
"cardinal",
"le_cInf",
"le_of_forall_lt",
"ordinal",
"set.range",
"subtype.coe_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_cof (o) : (cof o).lift = cof o.lift | begin
refine induction_on o _,
introsI α r _,
apply le_antisymm,
{ refine le_cof_type.2 (λ S H, _),
have : (#(ulift.up ⁻¹' S)).lift ≤ #S,
{ rw [← cardinal.lift_umax, ← cardinal.lift_id' (#S)],
exact mk_preimage_of_injective_lift ulift.up _ ulift.up_injective },
refine (cardinal.lift_le.2 $ cof... | theorem | ordinal.lift_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift_id'",
"cardinal.lift_umax",
"lift",
"ulift.up_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_le_card (o) : cof o ≤ card o | by { rw cof_eq_Inf_lsub, exact cInf_le' card_mem_cof } | theorem | ordinal.cof_le_card | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_ord_le (c : cardinal) : c.ord.cof ≤ c | by simpa using cof_le_card c.ord | theorem | ordinal.cof_ord_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_cof_le (o : ordinal.{u}) : o.cof.ord ≤ o | (ord_le_ord.2 (cof_le_card o)).trans (ord_card_le o) | theorem | ordinal.ord_cof_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_lsub_cof (o : ordinal) : ∃ {ι} (f : ι → ordinal), lsub.{u u} f = o ∧ #ι = cof o | by { rw cof_eq_Inf_lsub, exact Inf_mem (cof_lsub_def_nonempty o) } | theorem | ordinal.exists_lsub_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"Inf_mem",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_lsub_le {ι} (f : ι → ordinal) : cof (lsub.{u u} f) ≤ #ι | by { rw cof_eq_Inf_lsub, exact cInf_le' ⟨ι, f, rfl, rfl⟩ } | theorem | ordinal.cof_lsub_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cInf_le'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_lsub_le_lift {ι} (f : ι → ordinal) : cof (lsub f) ≤ cardinal.lift.{v u} (#ι) | begin
rw ←mk_ulift,
convert cof_lsub_le (λ i : ulift ι, f i.down),
exact lsub_eq_of_range_eq.{u (max u v) max u v}
(set.ext (λ x, ⟨λ ⟨i, hi⟩, ⟨ulift.up i, hi⟩, λ ⟨i, hi⟩, ⟨_, hi⟩⟩))
end | theorem | ordinal.cof_lsub_le_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cof_iff_lsub {o : ordinal} {a : cardinal} :
a ≤ cof o ↔ ∀ {ι} (f : ι → ordinal), lsub.{u u} f = o → a ≤ #ι | begin
rw cof_eq_Inf_lsub,
exact (le_cInf_iff'' (cof_lsub_def_nonempty o)).trans ⟨λ H ι f hf, H _ ⟨ι, f, hf, rfl⟩,
λ H b ⟨ι, f, hf, hb⟩, ( by { rw ←hb, exact H _ hf} )⟩
end | theorem | ordinal.le_cof_iff_lsub | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"le_cInf_iff''",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_lt_ord_lift {ι} {f : ι → ordinal} {c : ordinal} (hι : cardinal.lift (#ι) < c.cof)
(hf : ∀ i, f i < c) : lsub.{u v} f < c | lt_of_le_of_ne (lsub_le hf) (λ h, by { subst h, exact (cof_lsub_le_lift f).not_lt hι }) | theorem | ordinal.lsub_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_lt_ord {ι} {f : ι → ordinal} {c : ordinal} (hι : #ι < c.cof) :
(∀ i, f i < c) → lsub.{u u} f < c | lsub_lt_ord_lift (by rwa (#ι).lift_id) | theorem | ordinal.lsub_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_sup_le_lift {ι} {f : ι → ordinal} (H : ∀ i, f i < sup f) : cof (sup f) ≤ (#ι).lift | by { rw ←sup_eq_lsub_iff_lt_sup at H, rw H, exact cof_lsub_le_lift f } | theorem | ordinal.cof_sup_le_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_sup_le {ι} {f : ι → ordinal} (H : ∀ i, f i < sup.{u u} f) : cof (sup.{u u} f) ≤ #ι | by { rw ←(#ι).lift_id, exact cof_sup_le_lift H } | theorem | ordinal.cof_sup_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_lt_ord_lift {ι} {f : ι → ordinal} {c : ordinal} (hι : cardinal.lift (#ι) < c.cof)
(hf : ∀ i, f i < c) : sup.{u v} f < c | (sup_le_lsub.{u v} f).trans_lt (lsub_lt_ord_lift hι hf) | theorem | ordinal.sup_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_lt_ord {ι} {f : ι → ordinal} {c : ordinal} (hι : #ι < c.cof) :
(∀ i, f i < c) → sup.{u u} f < c | sup_lt_ord_lift (by rwa (#ι).lift_id) | theorem | ordinal.sup_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lt_lift {ι} {f : ι → cardinal} {c : cardinal} (hι : cardinal.lift (#ι) < c.ord.cof)
(hf : ∀ i, f i < c) : supr f < c | begin
rw [←ord_lt_ord, supr_ord (cardinal.bdd_above_range _)],
refine sup_lt_ord_lift hι (λ i, _),
rw ord_lt_ord,
apply hf
end | theorem | ordinal.supr_lt_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"cardinal.bdd_above_range",
"cardinal.lift",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_lt {ι} {f : ι → cardinal} {c : cardinal} (hι : #ι < c.ord.cof) :
(∀ i, f i < c) → supr f < c | supr_lt_lift (by rwa (#ι).lift_id) | theorem | ordinal.supr_lt | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_lt_ord_lift {ι} {f : ι → ordinal → ordinal} {c} (hc : ℵ₀ < cof c)
(hc' : (#ι).lift < cof c) (hf : ∀ i (b < c), f i b < c) {a} (ha : a < c) :
nfp_family.{u v} f a < c | begin
refine sup_lt_ord_lift ((cardinal.lift_le.2 (mk_list_le_max ι)).trans_lt _) (λ l, _),
{ rw lift_max,
apply max_lt _ hc',
rwa cardinal.lift_aleph_0 },
{ induction l with i l H,
{ exact ha },
{ exact hf _ _ H } }
end | theorem | ordinal.nfp_family_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.lift_aleph_0",
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_family_lt_ord {ι} {f : ι → ordinal → ordinal} {c} (hc : ℵ₀ < cof c)
(hc' : #ι < cof c) (hf : ∀ i (b < c), f i b < c) {a} : a < c → nfp_family.{u u} f a < c | nfp_family_lt_ord_lift hc (by rwa (#ι).lift_id) hf | theorem | ordinal.nfp_family_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_bfamily_lt_ord_lift {o : ordinal} {f : Π a < o, ordinal → ordinal} {c} (hc : ℵ₀ < cof c)
(hc' : o.card.lift < cof c) (hf : ∀ i hi (b < c), f i hi b < c) {a} :
a < c → nfp_bfamily.{u v} o f a < c | nfp_family_lt_ord_lift hc (by rwa mk_ordinal_out) (λ i, hf _ _) | theorem | ordinal.nfp_bfamily_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_bfamily_lt_ord {o : ordinal} {f : Π a < o, ordinal → ordinal} {c} (hc : ℵ₀ < cof c)
(hc' : o.card < cof c) (hf : ∀ i hi (b < c), f i hi b < c) {a} :
a < c → nfp_bfamily.{u u} o f a < c | nfp_bfamily_lt_ord_lift hc (by rwa o.card.lift_id) hf | theorem | ordinal.nfp_bfamily_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nfp_lt_ord {f : ordinal → ordinal} {c} (hc : ℵ₀ < cof c) (hf : ∀ i < c, f i < c) {a} :
a < c → nfp f a < c | nfp_family_lt_ord_lift hc (by simpa using cardinal.one_lt_aleph_0.trans hc) (λ _, hf) | theorem | ordinal.nfp_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_blsub_cof (o : ordinal) : ∃ (f : Π a < (cof o).ord, ordinal), blsub.{u u} _ f = o | begin
rcases exists_lsub_cof o with ⟨ι, f, hf, hι⟩,
rcases cardinal.ord_eq ι with ⟨r, hr, hι'⟩,
rw ←@blsub_eq_lsub' ι r hr at hf,
rw [←hι, hι'],
exact ⟨_, hf⟩
end | theorem | ordinal.exists_blsub_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.ord_eq",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_cof_iff_blsub {b : ordinal} {a : cardinal} :
a ≤ cof b ↔ ∀ {o} (f : Π a < o, ordinal), blsub.{u u} o f = b → a ≤ o.card | le_cof_iff_lsub.trans ⟨λ H o f hf, by simpa using H _ hf, λ H ι f hf, begin
rcases cardinal.ord_eq ι with ⟨r, hr, hι'⟩,
rw ←@blsub_eq_lsub' ι r hr at hf,
simpa using H _ hf
end⟩ | theorem | ordinal.le_cof_iff_blsub | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal",
"cardinal.ord_eq",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_blsub_le_lift {o} (f : Π a < o, ordinal) :
cof (blsub o f) ≤ cardinal.lift.{v u} (o.card) | by { convert cof_lsub_le_lift _, exact (mk_ordinal_out o).symm } | theorem | ordinal.cof_blsub_le_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_blsub_le {o} (f : Π a < o, ordinal) : cof (blsub.{u u} o f) ≤ o.card | by { rw ←(o.card).lift_id, exact cof_blsub_le_lift f } | theorem | ordinal.cof_blsub_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_lt_ord_lift {o : ordinal} {f : Π a < o, ordinal} {c : ordinal}
(ho : o.card.lift < c.cof) (hf : ∀ i hi, f i hi < c) : blsub.{u v} o f < c | lt_of_le_of_ne (blsub_le hf) (λ h, ho.not_le
(by simpa [←supr_ord, hf, h] using cof_blsub_le_lift.{u} f)) | theorem | ordinal.blsub_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_lt_ord {o : ordinal} {f : Π a < o, ordinal} {c : ordinal} (ho : o.card < c.cof)
(hf : ∀ i hi, f i hi < c) : blsub.{u u} o f < c | blsub_lt_ord_lift (by rwa (o.card).lift_id) hf | theorem | ordinal.blsub_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_bsup_le_lift {o : ordinal} {f : Π a < o, ordinal} (H : ∀ i h, f i h < bsup o f) :
cof (bsup o f) ≤ o.card.lift | by { rw ←bsup_eq_blsub_iff_lt_bsup at H, rw H, exact cof_blsub_le_lift f } | theorem | ordinal.cof_bsup_le_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_bsup_le {o : ordinal} {f : Π a < o, ordinal} :
(∀ i h, f i h < bsup.{u u} o f) → cof (bsup.{u u} o f) ≤ o.card | by { rw ←(o.card).lift_id, exact cof_bsup_le_lift } | theorem | ordinal.cof_bsup_le | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_lt_ord_lift {o : ordinal} {f : Π a < o, ordinal} {c : ordinal}
(ho : o.card.lift < c.cof) (hf : ∀ i hi, f i hi < c) : bsup.{u v} o f < c | (bsup_le_blsub f).trans_lt (blsub_lt_ord_lift ho hf) | theorem | ordinal.bsup_lt_ord_lift | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_lt_ord {o : ordinal} {f : Π a < o, ordinal} {c : ordinal} (ho : o.card < c.cof) :
(∀ i hi, f i hi < c) → bsup.{u u} o f < c | bsup_lt_ord_lift (by rwa (o.card).lift_id) | theorem | ordinal.bsup_lt_ord | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_zero : cof 0 = 0 | (cof_le_card 0).antisymm (cardinal.zero_le _) | theorem | ordinal.cof_zero | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_eq_zero {o} : cof o = 0 ↔ o = 0 | ⟨induction_on o $ λ α r _ z, by exactI
let ⟨S, hl, e⟩ := cof_eq r in type_eq_zero_iff_is_empty.2 $
⟨λ a, let ⟨b, h, _⟩ := hl a in
(mk_eq_zero_iff.1 (e.trans z)).elim' ⟨_, h⟩⟩,
λ e, by simp [e]⟩ | theorem | ordinal.cof_eq_zero | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_ne_zero {o} : cof o ≠ 0 ↔ o ≠ 0 | cof_eq_zero.not | theorem | ordinal.cof_ne_zero | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_succ (o) : cof (succ o) = 1 | begin
apply le_antisymm,
{ refine induction_on o (λ α r _, _),
change cof (type _) ≤ _,
rw [← (_ : #_ = 1)], apply cof_type_le,
{ refine λ a, ⟨sum.inr punit.star, set.mem_singleton _, _⟩,
rcases a with a|⟨⟨⟨⟩⟩⟩; simp [empty_relation] },
{ rw [cardinal.mk_fintype, set.card_singleton], simp } },... | theorem | ordinal.cof_succ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"cardinal.mk_fintype",
"cardinal.succ_zero",
"cardinal.zero_le",
"lt_iff_le_and_ne",
"set.card_singleton",
"set.mem_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cof_eq_one_iff_is_succ {o} : cof.{u} o = 1 ↔ ∃ a, o = succ a | ⟨induction_on o $ λ α r _ z, begin
resetI,
rcases cof_eq r with ⟨S, hl, e⟩, rw z at e,
cases mk_ne_zero_iff.1 (by rw e; exact one_ne_zero) with a,
refine ⟨typein r a, eq.symm $ quotient.sound
⟨rel_iso.of_surjective (rel_embedding.of_monotone _
(λ x y, _)) (λ x, _)⟩⟩,
{ apply sum.rec; [exact subtype.... | theorem | ordinal.cof_eq_one_iff_is_succ | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"one_ne_zero",
"order.preimage",
"rel_embedding.of_monotone",
"subrel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_fundamental_sequence (a o : ordinal.{u}) (f : Π b < o, ordinal.{u}) : Prop | o ≤ a.cof.ord ∧ (∀ {i j} (hi hj), i < j → f i hi < f j hj) ∧ blsub.{u u} o f = a | def | ordinal.is_fundamental_sequence | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | A fundamental sequence for `a` is an increasing sequence of length `o = cof a` that converges at
`a`. We provide `o` explicitly in order to avoid type rewrites. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cof_eq (hf : is_fundamental_sequence a o f) : a.cof.ord = o | hf.1.antisymm' $ by { rw ←hf.2.2, exact (ord_le_ord.2 (cof_blsub_le f)).trans (ord_card_le o) } | theorem | ordinal.is_fundamental_sequence.cof_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono (hf : is_fundamental_sequence a o f) {i j} :
∀ hi hj, i < j → f i hi < f j hj | hf.2.1 | theorem | ordinal.is_fundamental_sequence.strict_mono | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_eq (hf : is_fundamental_sequence a o f) : blsub.{u u} o f = a | hf.2.2 | theorem | ordinal.is_fundamental_sequence.blsub_eq | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_cof (hf : is_fundamental_sequence a o f) :
is_fundamental_sequence a a.cof.ord (λ i hi, f i (hi.trans_le (by rw hf.cof_eq))) | by { have H := hf.cof_eq, subst H, exact hf } | theorem | ordinal.is_fundamental_sequence.ord_cof | set_theory.cardinal | src/set_theory/cardinal/cofinality.lean | [
"set_theory.cardinal.ordinal",
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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