statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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add_swap_covariant_class_le : covariant_class ordinal.{u} ordinal.{u} (swap (+)) (≤) | ⟨λ c a b h, begin
revert h c, exact (
induction_on a $ λ α₁ r₁ hr₁, induction_on b $ λ α₂ r₂ hr₂ ⟨⟨⟨f, fo⟩, fi⟩⟩ c,
induction_on c $ λ β s hs, by exactI
@rel_embedding.ordinal_type_le _ _ (sum.lex r₁ s) (sum.lex r₂ s) _ _
⟨f.sum_map (embedding.refl _), λ a b, begin
split; intro H,
{ cases a with a a; ... | instance | ordinal.add_swap_covariant_class_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"covariant_class",
"rel_embedding.ordinal_type_le",
"sum.lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_right (a b : ordinal) : a ≤ a + b | by simpa only [add_zero] using add_le_add_left (ordinal.zero_le b) a | theorem | ordinal.le_add_right | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_add_left (a b : ordinal) : a ≤ b + a | by simpa only [zero_add] using add_le_add_right (ordinal.zero_le b) a | theorem | ordinal.le_add_left | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_zero_left : ∀ a : ordinal, max 0 a = a | max_bot_left | lemma | ordinal.max_zero_left | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"max_bot_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_zero_right : ∀ a : ordinal, max a 0 = a | max_bot_right | lemma | ordinal.max_zero_right | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"max_bot_right",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_eq_zero {a b : ordinal} : max a b = 0 ↔ a = 0 ∧ b = 0 | max_eq_bot | lemma | ordinal.max_eq_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"max_eq_bot",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_empty : Inf (∅ : set ordinal) = 0 | dif_neg not_nonempty_empty | theorem | ordinal.Inf_empty | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"Inf_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_le_iff' {a b : ordinal} : a + 1 ≤ b ↔ a < b | ⟨lt_of_lt_of_le (induction_on a $ λ α r _, ⟨⟨⟨⟨λ x, sum.inl x, λ _ _, sum.inl.inj⟩,
λ _ _, sum.lex_inl_inl⟩,
sum.inr punit.star, λ b, sum.rec_on b
(λ x, ⟨λ _, ⟨x, rfl⟩, λ _, sum.lex.sep _ _⟩)
(λ x, sum.lex_inr_inr.trans ⟨false.elim, λ ⟨x, H⟩, sum.inl_ne_inr H⟩)⟩⟩),
induction_on a $ λ α r hr, induction_on b ... | theorem | ordinal.succ_le_iff' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"principal_seg.init",
"rel_embedding.of_monotone",
"sum.inl_ne_inr",
"sum.lex_inl_inl",
"sum.lex_inr_inl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_eq_succ (o : ordinal) : o + 1 = succ o | rfl | theorem | ordinal.add_one_eq_succ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_zero : succ (0 : ordinal) = 1 | zero_add 1 | theorem | ordinal.succ_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_one : succ (1 : ordinal) = 2 | rfl | theorem | ordinal.succ_one | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_succ (o₁ o₂ : ordinal) : o₁ + succ o₂ = succ (o₁ + o₂) | (add_assoc _ _ _).symm | theorem | ordinal.add_succ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_iff_pos {o : ordinal} : 1 ≤ o ↔ 0 < o | by rw [← succ_zero, succ_le_iff] | theorem | ordinal.one_le_iff_pos | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_iff_ne_zero {o : ordinal} : 1 ≤ o ↔ o ≠ 0 | by rw [one_le_iff_pos, ordinal.pos_iff_ne_zero] | theorem | ordinal.one_le_iff_ne_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"ordinal.pos_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_pos (o : ordinal) : 0 < succ o | bot_lt_succ o | theorem | ordinal.succ_pos | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
succ_ne_zero (o : ordinal) : succ o ≠ 0 | ne_of_gt $ succ_pos o | theorem | ordinal.succ_ne_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_one_iff_zero {a : ordinal} : a < 1 ↔ a = 0 | by simpa using @lt_succ_bot_iff _ _ _ a _ _ | theorem | ordinal.lt_one_iff_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_one_iff {a : ordinal} : a ≤ 1 ↔ a = 0 ∨ a = 1 | by simpa using @le_succ_bot_iff _ _ _ a _ | theorem | ordinal.le_one_iff | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_succ (o : ordinal) : card (succ o) = card o + 1 | by simp only [←add_one_eq_succ, card_add, card_one] | theorem | ordinal.card_succ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_succ (n : ℕ) : ↑n.succ = succ (n : ordinal) | rfl | theorem | ordinal.nat_cast_succ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_Iio_one : unique (Iio (1 : ordinal)) | { default := ⟨0, zero_lt_one⟩,
uniq := λ a, subtype.ext $ lt_one_iff_zero.1 a.prop } | instance | ordinal.unique_Iio_one | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"subtype.ext",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_out_one : unique (1 : ordinal).out.α | { default := enum (<) 0 (by simp),
uniq := λ a, begin
rw ←enum_typein (<) a,
unfold default,
congr,
rw ←lt_one_iff_zero,
apply typein_lt_self
end } | instance | ordinal.unique_out_one | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_out_eq (x : (1 : ordinal).out.α) : x = enum (<) 0 (by simp) | unique.eq_default x | theorem | ordinal.one_out_eq | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"unique.eq_default"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_one_out (x : (1 : ordinal).out.α) : typein (<) x = 0 | by rw [one_out_eq x, typein_enum] | theorem | ordinal.typein_one_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_le_typein (r : α → α → Prop) [is_well_order α r] {x x' : α} :
typein r x ≤ typein r x' ↔ ¬r x' x | by rw [←not_lt, typein_lt_typein] | lemma | ordinal.typein_le_typein | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_le_typein' (o : ordinal) {x x' : o.out.α} :
typein (<) x ≤ typein (<) x' ↔ x ≤ x' | by { rw typein_le_typein, exact not_lt } | lemma | ordinal.typein_le_typein' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_le_enum (r : α → α → Prop) [is_well_order α r] {o o' : ordinal}
(ho : o < type r) (ho' : o' < type r) : ¬r (enum r o' ho') (enum r o ho) ↔ o ≤ o' | by rw [←@not_lt _ _ o' o, enum_lt_enum ho'] | lemma | ordinal.enum_le_enum | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_le_enum' (a : ordinal) {o o' : ordinal}
(ho : o < type (<)) (ho' : o' < type (<)) : enum (<) o ho ≤ @enum a.out.α (<) _ o' ho' ↔ o ≤ o' | by rw [←enum_le_enum (<), ←not_lt] | lemma | ordinal.enum_le_enum' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_zero_le {r : α → α → Prop} [is_well_order α r] (h0 : 0 < type r) (a : α) :
¬ r a (enum r 0 h0) | by { rw [←enum_typein r a, enum_le_enum r], apply ordinal.zero_le } | theorem | ordinal.enum_zero_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_zero_le' {o : ordinal} (h0 : 0 < o) (a : o.out.α) :
@enum o.out.α (<) _ 0 (by rwa type_lt) ≤ a | by { rw ←not_lt, apply enum_zero_le } | theorem | ordinal.enum_zero_le' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_enum_succ {o : ordinal} (a : (succ o).out.α) :
a ≤ @enum (succ o).out.α (<) _ o (by { rw type_lt, exact lt_succ o }) | by { rw [←enum_typein (<) a, enum_le_enum', ←lt_succ_iff], apply typein_lt_self } | theorem | ordinal.le_enum_succ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_inj {r : α → α → Prop} [is_well_order α r] {o₁ o₂ : ordinal} (h₁ : o₁ < type r)
(h₂ : o₂ < type r) : enum r o₁ h₁ = enum r o₂ h₂ ↔ o₁ = o₂ | (typein.principal_seg r).subrel_iso.injective.eq_iff.trans subtype.mk_eq_mk | theorem | ordinal.enum_inj | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal",
"subtype.mk_eq_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_iso (r : α → α → Prop) [is_well_order α r] : subrel (<) (< type r) ≃r r | { to_fun := λ x, enum r x.1 x.2,
inv_fun := λ x, ⟨typein r x, typein_lt_type r x⟩,
..(typein.principal_seg r).subrel_iso } | def | ordinal.enum_iso | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"inv_fun",
"is_well_order",
"subrel"
] | A well order `r` is order isomorphic to the set of ordinals smaller than `type r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_iso_out (o : ordinal) : set.Iio o ≃o o.out.α | { to_fun := λ x, enum (<) x.1 $ by { rw type_lt, exact x.2 },
inv_fun := λ x, ⟨typein (<) x, typein_lt_self x⟩,
left_inv := λ ⟨o', h⟩, subtype.ext_val (typein_enum _ _),
right_inv := λ h, enum_typein _ _,
map_rel_iff' := by { rintros ⟨a, _⟩ ⟨b, _⟩, apply enum_le_enum' } } | def | ordinal.enum_iso_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"inv_fun",
"ordinal",
"set.Iio",
"subtype.ext_val"
] | The order isomorphism between ordinals less than `o` and `o.out.α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
out_order_bot_of_pos {o : ordinal} (ho : 0 < o) : order_bot o.out.α | ⟨_, enum_zero_le' ho⟩ | def | ordinal.out_order_bot_of_pos | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"order_bot",
"ordinal"
] | `o.out.α` is an `order_bot` whenever `0 < o`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_zero_eq_bot {o : ordinal} (ho : 0 < o) :
enum (<) 0 (by rwa type_lt) = by { haveI H := out_order_bot_of_pos ho, exact ⊥ } | rfl | theorem | ordinal.enum_zero_eq_bot | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
univ : ordinal.{max (u + 1) v} | lift.{v (u+1)} (@type ordinal (<) _) | def | ordinal.univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | `univ.{u v}` is the order type of the ordinals of `Type u` as a member
of `ordinal.{v}` (when `u < v`). It is an inaccessible cardinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
univ_id : univ.{u (u+1)} = @type ordinal (<) _ | lift_id _ | theorem | ordinal.univ_id | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_univ : lift.{w} univ.{u v} = univ.{u (max v w)} | lift_lift _ | theorem | ordinal.lift_univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
univ_umax : univ.{u (max (u+1) v)} = univ.{u v} | congr_fun lift_umax _ | theorem | ordinal.univ_umax | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift.principal_seg : @principal_seg ordinal.{u} ordinal.{max (u+1) v} (<) (<) | ⟨↑lift.initial_seg.{u (max (u+1) v)}, univ.{u v}, begin
refine λ b, induction_on b _, introsI β s _,
rw [univ, ← lift_umax], split; intro h,
{ rw ← lift_id (type s) at h ⊢,
cases lift_type_lt.1 h with f, cases f with f a hf,
existsi a, revert hf,
apply induction_on a, introsI α r _ hf,
refine lift... | def | ordinal.lift.principal_seg | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"principal_seg",
"rel_embedding.of_monotone",
"rel_iso.of_surjective"
] | Principal segment version of the lift operation on ordinals, embedding `ordinal.{u}` in
`ordinal.{v}` as a principal segment when `u < v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift.principal_seg_coe :
(lift.principal_seg.{u v} : ordinal → ordinal) = lift.{max (u+1) v} | rfl | theorem | ordinal.lift.principal_seg_coe | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift.principal_seg_top : lift.principal_seg.top = univ | rfl | theorem | ordinal.lift.principal_seg_top | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift.principal_seg_top' :
lift.principal_seg.{u (u+1)}.top = @type ordinal (<) _ | by simp only [lift.principal_seg_top, univ_id] | theorem | ordinal.lift.principal_seg_top' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ordinal_out (o : ordinal) : #(o.out.α) = o.card | (ordinal.card_type _).symm.trans $ by rw ordinal.type_lt | theorem | cardinal.mk_ordinal_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"ordinal.card_type",
"ordinal.type_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord (c : cardinal) : ordinal | let F := λ α : Type u, ⨅ r : {r // is_well_order α r}, @type α r.1 r.2 in
quot.lift_on c F
begin
suffices : ∀ {α β}, α ≈ β → F α ≤ F β,
from λ α β h, (this h).antisymm (this (setoid.symm h)),
rintros α β ⟨f⟩,
refine le_cinfi_iff'.2 (λ i, _),
haveI := @rel_embedding.is_well_order _ _ (f ⁻¹'o i.1) _ ↑(rel_iso.p... | def | cardinal.ord | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cinfi_le'",
"is_well_order",
"ordinal",
"rel_embedding.is_well_order",
"rel_iso.preimage"
] | The ordinal corresponding to a cardinal `c` is the least ordinal
whose cardinal is `c`. For the order-embedding version, see `ord.order_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ord_eq_Inf (α : Type u) : ord (#α) = ⨅ r : {r // is_well_order α r}, @type α r.1 r.2 | rfl | lemma | cardinal.ord_eq_Inf | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_eq (α) : ∃ (r : α → α → Prop) [wo : is_well_order α r], ord (#α) = @type α r wo | let ⟨r, wo⟩ := infi_mem (λ r : {r // is_well_order α r}, @type α r.1 r.2) in ⟨r.1, r.2, wo.symm⟩ | theorem | cardinal.ord_eq | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"infi_mem",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_le_type (r : α → α → Prop) [h : is_well_order α r] : ord (#α) ≤ type r | cinfi_le' _ (subtype.mk r h) | theorem | cardinal.ord_le_type | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cinfi_le'",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_le {c o} : ord c ≤ o ↔ c ≤ o.card | induction_on c $ λ α, ordinal.induction_on o $ λ β s _,
let ⟨r, _, e⟩ := ord_eq α in begin
resetI, simp only [card_type], split; intro h,
{ rw e at h, exact let ⟨f⟩ := h in ⟨f.to_embedding⟩ },
{ cases h with f,
have g := rel_embedding.preimage f s,
haveI := rel_embedding.is_well_order g,
exact le_tran... | theorem | cardinal.ord_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal.induction_on",
"rel_embedding.is_well_order",
"rel_embedding.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_ord_card : galois_connection ord card | λ _ _, ord_le | theorem | cardinal.gc_ord_card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_ord {c o} : o < ord c ↔ o.card < c | gc_ord_card.lt_iff_lt | theorem | cardinal.lt_ord | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_ord (c) : (ord c).card = c | quotient.induction_on c $ λ α,
let ⟨r, _, e⟩ := ord_eq α in by simp only [mk_def, e, card_type] | theorem | cardinal.card_ord | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gci_ord_card : galois_coinsertion ord card | gc_ord_card.to_galois_coinsertion $ λ c, c.card_ord.le | def | cardinal.gci_ord_card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"galois_coinsertion"
] | Galois coinsertion between `cardinal.ord` and `ordinal.card`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ord_card_le (o : ordinal) : o.card.ord ≤ o | gc_ord_card.l_u_le _ | theorem | cardinal.ord_card_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_ord_succ_card (o : ordinal) : o < (succ o.card).ord | lt_ord.2 $ lt_succ _ | lemma | cardinal.lt_ord_succ_card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_strict_mono : strict_mono ord | gci_ord_card.strict_mono_l | theorem | cardinal.ord_strict_mono | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_mono : monotone ord | gc_ord_card.monotone_l | theorem | cardinal.ord_mono | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_le_ord {c₁ c₂} : ord c₁ ≤ ord c₂ ↔ c₁ ≤ c₂ | gci_ord_card.l_le_l_iff | theorem | cardinal.ord_le_ord | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_lt_ord {c₁ c₂} : ord c₁ < ord c₂ ↔ c₁ < c₂ | ord_strict_mono.lt_iff_lt | theorem | cardinal.ord_lt_ord | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_zero : ord 0 = 0 | gc_ord_card.l_bot | theorem | cardinal.ord_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_nat (n : ℕ) : ord n = n | (ord_le.2 (card_nat n).ge).antisymm begin
induction n with n IH,
{ apply ordinal.zero_le },
{ exact succ_le_of_lt (IH.trans_lt $ ord_lt_ord.2 $ nat_cast_lt.2 (nat.lt_succ_self n)) }
end | theorem | cardinal.ord_nat | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_one : ord 1 = 1 | by simpa using ord_nat 1 | theorem | cardinal.ord_one | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_ord (c) : (ord c).lift = ord (lift c) | begin
refine le_antisymm (le_of_forall_lt (λ a ha, _)) _,
{ rcases ordinal.lt_lift_iff.1 ha with ⟨a, rfl, h⟩,
rwa [lt_ord, ← lift_card, lift_lt, ← lt_ord, ← ordinal.lift_lt] },
{ rw [ord_le, ← lift_card, card_ord] }
end | theorem | cardinal.lift_ord | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"le_of_forall_lt",
"lift",
"ordinal.lift_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ord_out (c : cardinal) : #c.ord.out.α = c | by simp | lemma | cardinal.mk_ord_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_typein_lt (r : α → α → Prop) [is_well_order α r] (x : α)
(h : ord (#α) = type r) : card (typein r x) < #α | by { rw [←lt_ord, h], apply typein_lt_type } | lemma | cardinal.card_typein_lt | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_typein_out_lt (c : cardinal) (x : c.ord.out.α) : card (typein (<) x) < c | by { rw ←lt_ord, apply typein_lt_self } | lemma | cardinal.card_typein_out_lt | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_injective : injective ord | by { intros c c' h, rw [←card_ord c, ←card_ord c', h] } | lemma | cardinal.ord_injective | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord.order_embedding : cardinal ↪o ordinal | rel_embedding.order_embedding_of_lt_embedding
(rel_embedding.of_monotone cardinal.ord $ λ a b, cardinal.ord_lt_ord.2) | def | cardinal.ord.order_embedding | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cardinal.ord",
"ordinal",
"rel_embedding.of_monotone",
"rel_embedding.order_embedding_of_lt_embedding"
] | The ordinal corresponding to a cardinal `c` is the least ordinal
whose cardinal is `c`. This is the order-embedding version. For the regular function, see `ord`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ord.order_embedding_coe :
(ord.order_embedding : cardinal → ordinal) = ord | rfl | theorem | cardinal.ord.order_embedding_coe | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
univ | lift.{v (u+1)} (#ordinal) | def | cardinal.univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | The cardinal `univ` is the cardinality of ordinal `univ`, or
equivalently the cardinal of `ordinal.{u}`, or `cardinal.{u}`,
as an element of `cardinal.{v}` (when `u < v`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
univ_id : univ.{u (u+1)} = #ordinal | lift_id _ | theorem | cardinal.univ_id | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt_univ (c : cardinal) : lift.{(u+1) u} c < univ.{u (u+1)} | by simpa only [lift.principal_seg_coe, lift_ord, lift_succ, ord_le, succ_le_iff] using le_of_lt
(lift.principal_seg.{u (u+1)}.lt_top (succ c).ord) | theorem | cardinal.lift_lt_univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt_univ' (c : cardinal) : lift.{(max (u+1) v) u} c < univ.{u v} | by simpa only [lift_lift, lift_univ, univ_umax] using
lift_lt.{_ (max (u+1) v)}.2 (lift_lt_univ c) | theorem | cardinal.lift_lt_univ' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_univ : ord univ.{u v} = ordinal.univ.{u v} | le_antisymm (ord_card_le _) $ le_of_forall_lt $ λ o h,
lt_ord.2 begin
rcases lift.principal_seg.{u v}.down.1
(by simpa only [lift.principal_seg_coe] using h) with ⟨o', rfl⟩,
simp only [lift.principal_seg_coe], rw [← lift_card],
apply lift_lt_univ'
end | theorem | cardinal.ord_univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"le_of_forall_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_univ {c} : c < univ.{u (u+1)} ↔ ∃ c', c = lift.{(u+1) u} c' | ⟨λ h, begin
have := ord_lt_ord.2 h,
rw ord_univ at this,
cases lift.principal_seg.{u (u+1)}.down.1
(by simpa only [lift.principal_seg_top]) with o e,
have := card_ord c,
rw [← e, lift.principal_seg_coe, ← lift_card] at this,
exact ⟨_, this.symm⟩
end, λ ⟨c', e⟩, e.symm ▸ lift_lt_univ _⟩ | theorem | cardinal.lt_univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_univ' {c} : c < univ.{u v} ↔ ∃ c', c = lift.{(max (u+1) v) u} c' | ⟨λ h, let ⟨a, e, h'⟩ := lt_lift_iff.1 h in begin
rw [← univ_id] at h',
rcases lt_univ.{u}.1 h' with ⟨c', rfl⟩,
exact ⟨c', by simp only [e.symm, lift_lift]⟩
end, λ ⟨c', e⟩, e.symm ▸ lift_lt_univ' _⟩ | theorem | cardinal.lt_univ' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_iff_lift_mk_lt_univ {α : Type u} :
small.{v} α ↔ cardinal.lift (#α) < univ.{v (max u (v + 1))} | begin
rw lt_univ',
split,
{ rintro ⟨β, e⟩,
exact ⟨#β, lift_mk_eq.{u _ (v + 1)}.2 e⟩ },
{ rintro ⟨c, hc⟩,
exact ⟨⟨c.out, lift_mk_eq.{u _ (v + 1)}.1 (hc.trans (congr rfl c.mk_out.symm))⟩⟩ }
end | theorem | cardinal.small_iff_lift_mk_lt_univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal.lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_univ : card univ = cardinal.univ | rfl | theorem | ordinal.card_univ | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_le_card {o} {n : ℕ} : (n : cardinal) ≤ card o ↔ (n : ordinal) ≤ o | by rw [← cardinal.ord_le, cardinal.ord_nat] | theorem | ordinal.nat_le_card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal",
"cardinal.ord_le",
"cardinal.ord_nat",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_lt_card {o} {n : ℕ} : (n : cardinal) < card o ↔ (n : ordinal) < o | by { rw [←succ_le_iff, ←succ_le_iff, ←nat_succ, nat_le_card], refl } | theorem | ordinal.nat_lt_card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_lt_nat {o} {n : ℕ} : card o < n ↔ o < n | lt_iff_lt_of_le_iff_le nat_le_card | theorem | ordinal.card_lt_nat | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_le_nat {o} {n : ℕ} : card o ≤ n ↔ o ≤ n | le_iff_le_iff_lt_iff_lt.2 nat_lt_card | theorem | ordinal.card_le_nat | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_eq_nat {o} {n : ℕ} : card o = n ↔ o = n | by simp only [le_antisymm_iff, card_le_nat, nat_le_card] | theorem | ordinal.card_eq_nat | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_fintype (r : α → α → Prop) [is_well_order α r] [fintype α] :
type r = fintype.card α | by rw [←card_eq_nat, card_type, mk_fintype] | theorem | ordinal.type_fintype | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"fintype",
"fintype.card",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_fin (n : ℕ) : @type (fin n) (<) _ = n | by simp | theorem | ordinal.type_fin | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF_rec (b : ordinal)
{C : ordinal → Sort*} (H0 : C 0) (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : ∀ o, C o | | o :=
if ho : o = 0 then by rwa ho else
let hwf := mod_opow_log_lt_self b ho in H o ho (CNF_rec (o % b ^ log b o))
using_well_founded {dec_tac := `[assumption]} | def | ordinal.CNF_rec | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | Inducts on the base `b` expansion of an ordinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CNF_rec_zero {C : ordinal → Sort*} (b : ordinal)
(H0 : C 0) (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) : @CNF_rec b C H0 H 0 = H0 | by { rw [CNF_rec, dif_pos rfl], refl } | theorem | ordinal.CNF_rec_zero | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF_rec_pos (b : ordinal) {o : ordinal} {C : ordinal → Sort*} (ho : o ≠ 0)
(H0 : C 0) (H : ∀ o, o ≠ 0 → C (o % b ^ log b o) → C o) :
@CNF_rec b C H0 H o = H o ho (@CNF_rec b C H0 H _) | by rw [CNF_rec, dif_neg ho] | theorem | ordinal.CNF_rec_pos | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF (b o : ordinal) : list (ordinal × ordinal) | CNF_rec b [] (λ o ho IH, (log b o, o / b ^ log b o) :: IH) o | def | ordinal.CNF | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | The Cantor normal form of an ordinal `o` is the list of coefficients and exponents in the
base-`b` expansion of `o`.
We special-case `CNF 0 o = CNF 1 o = [(0, o)]` for `o ≠ 0`.
`CNF b (b ^ u₁ * v₁ + b ^ u₂ * v₂) = [(u₁, v₁), (u₂, v₂)]` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CNF_zero (b : ordinal) : CNF b 0 = [] | CNF_rec_zero b _ _ | theorem | ordinal.CNF_zero | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF_ne_zero {b o : ordinal} (ho : o ≠ 0) :
CNF b o = (log b o, o / b ^ log b o) :: CNF b (o % b ^ log b o) | CNF_rec_pos b ho _ _ | theorem | ordinal.CNF_ne_zero | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | Recursive definition for the Cantor normal form. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_CNF {o : ordinal} (ho : o ≠ 0) : CNF 0 o = [⟨0, o⟩] | by simp [CNF_ne_zero ho] | theorem | ordinal.zero_CNF | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_CNF {o : ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩] | by simp [CNF_ne_zero ho] | theorem | ordinal.one_CNF | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF_of_le_one {b o : ordinal} (hb : b ≤ 1) (ho : o ≠ 0) : CNF b o = [⟨0, o⟩] | begin
rcases le_one_iff.1 hb with rfl | rfl,
{ exact zero_CNF ho },
{ exact one_CNF ho }
end | theorem | ordinal.CNF_of_le_one | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF_of_lt {b o : ordinal} (ho : o ≠ 0) (hb : o < b) : CNF b o = [⟨0, o⟩] | by simp [CNF_ne_zero ho, log_eq_zero hb] | theorem | ordinal.CNF_of_lt | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
CNF_foldr (b o : ordinal) : (CNF b o).foldr (λ p r, b ^ p.1 * p.2 + r) 0 = o | CNF_rec b (by { rw CNF_zero, refl })
(λ o ho IH, by rw [CNF_ne_zero ho, foldr_cons, IH, div_add_mod]) o | theorem | ordinal.CNF_foldr | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | Evaluating the Cantor normal form of an ordinal returns the ordinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CNF_fst_le_log {b o : ordinal.{u}} {x : ordinal × ordinal} :
x ∈ CNF b o → x.1 ≤ log b o | begin
refine CNF_rec b _ (λ o ho H, _) o,
{ simp },
{ rw [CNF_ne_zero ho, mem_cons_iff],
rintro (rfl | h),
{ exact le_rfl },
{ exact (H h).trans (log_mono_right _ (mod_opow_log_lt_self b ho).le) } }
end | theorem | ordinal.CNF_fst_le_log | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"le_rfl",
"mem_cons_iff",
"ordinal"
] | Every exponent in the Cantor normal form `CNF b o` is less or equal to `log b o`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CNF_fst_le {b o : ordinal.{u}} {x : ordinal × ordinal} (h : x ∈ CNF b o) : x.1 ≤ o | (CNF_fst_le_log h).trans $ log_le_self _ _ | theorem | ordinal.CNF_fst_le | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | Every exponent in the Cantor normal form `CNF b o` is less or equal to `o`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CNF_lt_snd {b o : ordinal.{u}} {x : ordinal × ordinal} : x ∈ CNF b o → 0 < x.2 | begin
refine CNF_rec b _ (λ o ho IH, _) o,
{ simp },
{ rw CNF_ne_zero ho,
rintro (rfl | h),
{ exact div_opow_log_pos b ho },
{ exact IH h } }
end | theorem | ordinal.CNF_lt_snd | set_theory.ordinal | src/set_theory/ordinal/cantor_normal_form.lean | [
"set_theory.ordinal.arithmetic",
"set_theory.ordinal.exponential"
] | [
"ordinal"
] | Every coefficient in a Cantor normal form is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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