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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|---|---|---|---|---|---|---|---|---|---|---|
is_well_order_out_lt (o : ordinal) : is_well_order o.out.α (<) | o.out.wo | instance | is_well_order_out_lt | 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 | |
type (r : α → α → Prop) [wo : is_well_order α r] : ordinal | ⟦⟨α, r, wo⟩⟧ | def | ordinal.type | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal"
] | The order type of a well order is an ordinal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
typein (r : α → α → Prop) [is_well_order α r] (a : α) : ordinal | type (subrel r {b | r b a}) | def | ordinal.typein | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal",
"subrel"
] | The order type of an element inside a well order. For the embedding as a principal segment, see
`typein.principal_seg`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
type_def' (w : Well_order) : ⟦w⟧ = type w.r | by { cases w, refl } | theorem | ordinal.type_def' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"Well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_def (r) [wo : is_well_order α r] : (⟦⟨α, r, wo⟩⟧ : ordinal) = type r | rfl | theorem | ordinal.type_def | 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 | |
type_out (o : ordinal) : ordinal.type o.out.r = o | by rw [ordinal.type, Well_order.eta, quotient.out_eq] | lemma | ordinal.type_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"Well_order.eta",
"ordinal",
"ordinal.type",
"quotient.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_eq {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] : type r = type s ↔ nonempty (r ≃r s) | quotient.eq | theorem | ordinal.type_eq | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"quotient.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.rel_iso.ordinal_type_eq {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (h : r ≃r s) : type r = type s | type_eq.2 ⟨h⟩ | theorem | rel_iso.ordinal_type_eq | 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 | |
type_lt (o : ordinal) : type ((<) : o.out.α → o.out.α → Prop) = o | (type_def' _).symm.trans $ quotient.out_eq o | theorem | ordinal.type_lt | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"quotient.out_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_eq_zero_of_empty (r) [is_well_order α r] [is_empty α] : type r = 0 | (rel_iso.rel_iso_of_is_empty r _).ordinal_type_eq | theorem | ordinal.type_eq_zero_of_empty | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_empty",
"is_well_order",
"rel_iso.rel_iso_of_is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_eq_zero_iff_is_empty [is_well_order α r] : type r = 0 ↔ is_empty α | ⟨λ h, let ⟨s⟩ := type_eq.1 h in s.to_equiv.is_empty, @type_eq_zero_of_empty α r _⟩ | theorem | ordinal.type_eq_zero_iff_is_empty | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_empty",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_ne_zero_iff_nonempty [is_well_order α r] : type r ≠ 0 ↔ nonempty α | by simp | theorem | ordinal.type_ne_zero_iff_nonempty | 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 | |
type_ne_zero_of_nonempty (r) [is_well_order α r] [h : nonempty α] : type r ≠ 0 | type_ne_zero_iff_nonempty.2 h | theorem | ordinal.type_ne_zero_of_nonempty | 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 | |
type_pempty : type (@empty_relation pempty) = 0 | rfl | theorem | ordinal.type_pempty | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_empty : type (@empty_relation empty) = 0 | type_eq_zero_of_empty _ | theorem | ordinal.type_empty | 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_eq_one_of_unique (r) [is_well_order α r] [unique α] : type r = 1 | (rel_iso.rel_iso_of_unique_of_irrefl r _).ordinal_type_eq | theorem | ordinal.type_eq_one_of_unique | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"rel_iso.rel_iso_of_unique_of_irrefl",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_eq_one_iff_unique [is_well_order α r] : type r = 1 ↔ nonempty (unique α) | ⟨λ h, let ⟨s⟩ := type_eq.1 h in ⟨s.to_equiv.unique⟩, λ ⟨h⟩, @type_eq_one_of_unique α r _ h⟩ | theorem | ordinal.type_eq_one_iff_unique | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_punit : type (@empty_relation punit) = 1 | rfl | theorem | ordinal.type_punit | 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_unit : type (@empty_relation unit) = 1 | rfl | theorem | ordinal.type_unit | 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 | |
out_empty_iff_eq_zero {o : ordinal} : is_empty o.out.α ↔ o = 0 | by rw [←@type_eq_zero_iff_is_empty o.out.α (<), type_lt] | theorem | ordinal.out_empty_iff_eq_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_out_empty (o : ordinal) [h : is_empty o.out.α] : o = 0 | out_empty_iff_eq_zero.1 h | lemma | ordinal.eq_zero_of_out_empty | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_out_zero : is_empty (0 : ordinal).out.α | out_empty_iff_eq_zero.2 rfl | instance | ordinal.is_empty_out_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
out_nonempty_iff_ne_zero {o : ordinal} : nonempty o.out.α ↔ o ≠ 0 | by rw [←@type_ne_zero_iff_nonempty o.out.α (<), type_lt] | theorem | ordinal.out_nonempty_iff_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 | |
ne_zero_of_out_nonempty (o : ordinal) [h : nonempty o.out.α] : o ≠ 0 | out_nonempty_iff_ne_zero.1 h | lemma | ordinal.ne_zero_of_out_nonempty | 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_ne_zero : (1 : ordinal) ≠ 0 | type_ne_zero_of_nonempty _ | lemma | ordinal.one_ne_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"one_ne_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_preimage {α β : Type u} (r : α → α → Prop) [is_well_order α r] (f : β ≃ α) :
type (f ⁻¹'o r) = type r | (rel_iso.preimage f r).ordinal_type_eq | theorem | ordinal.type_preimage | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {C : ordinal → Prop}
(o : ordinal) (H : ∀ α r [is_well_order α r], by exactI C (type r)) : C o | quot.induction_on o $ λ ⟨α, r, wo⟩, @H α r wo | theorem | ordinal.induction_on | 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 | |
type_le_iff {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] :
type r ≤ type s ↔ nonempty (r ≼i s) | iff.rfl | theorem | ordinal.type_le_iff | 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 | |
type_le_iff' {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] : type r ≤ type s ↔ nonempty (r ↪r s) | ⟨λ ⟨f⟩, ⟨f⟩, λ ⟨f⟩, ⟨f.collapse⟩⟩ | theorem | ordinal.type_le_iff' | 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 | |
_root_.initial_seg.ordinal_type_le {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (h : r ≼i s) : type r ≤ type s | ⟨h⟩ | theorem | initial_seg.ordinal_type_le | 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 | |
_root_.rel_embedding.ordinal_type_le {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (h : r ↪r s) : type r ≤ type s | ⟨h.collapse⟩ | theorem | rel_embedding.ordinal_type_le | 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 | |
type_lt_iff {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] :
type r < type s ↔ nonempty (r ≺i s) | iff.rfl | theorem | ordinal.type_lt_iff | 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 | |
_root_.principal_seg.ordinal_type_lt {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (h : r ≺i s) : type r < type s | ⟨h⟩ | theorem | principal_seg.ordinal_type_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 | |
zero_le (o : ordinal) : 0 ≤ o | induction_on o $ λ α r _, by exactI (initial_seg.of_is_empty _ r).ordinal_type_le | theorem | ordinal.zero_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"initial_seg.of_is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_eq_zero : (⊥ : ordinal) = 0 | rfl | lemma | ordinal.bot_eq_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_zero {o : ordinal} : o ≤ 0 ↔ o = 0 | le_bot_iff | theorem | ordinal.le_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"le_bot_iff",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pos_iff_ne_zero {o : ordinal} : 0 < o ↔ o ≠ 0 | bot_lt_iff_ne_bot | theorem | ordinal.pos_iff_ne_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"bot_lt_iff_ne_bot",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_lt_zero (o : ordinal) : ¬ o < 0 | not_lt_bot | theorem | ordinal.not_lt_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"not_lt_bot",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_or_pos : ∀ a : ordinal, a = 0 ∨ 0 < a | eq_bot_or_bot_lt | theorem | ordinal.eq_zero_or_pos | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"eq_bot_or_bot_lt",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_zero.one : ne_zero (1 : ordinal) | ⟨ordinal.one_ne_zero⟩ | instance | ordinal.ne_zero.one | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ne_zero",
"ne_zero.one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
initial_seg_out {α β : ordinal} (h : α ≤ β) :
initial_seg ((<) : α.out.α → α.out.α → Prop) ((<) : β.out.α → β.out.α → Prop) | begin
change α.out.r ≼i β.out.r,
rw [←quotient.out_eq α, ←quotient.out_eq β] at h, revert h,
cases quotient.out α, cases quotient.out β, exact classical.choice
end | def | ordinal.initial_seg_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"initial_seg",
"ordinal",
"quotient.out"
] | Given two ordinals `α ≤ β`, then `initial_seg_out α β` is the initial segment embedding
of `α` to `β`, as map from a model type for `α` to a model type for `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
principal_seg_out {α β : ordinal} (h : α < β) :
principal_seg ((<) : α.out.α → α.out.α → Prop) ((<) : β.out.α → β.out.α → Prop) | begin
change α.out.r ≺i β.out.r,
rw [←quotient.out_eq α, ←quotient.out_eq β] at h, revert h,
cases quotient.out α, cases quotient.out β, exact classical.choice
end | def | ordinal.principal_seg_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"principal_seg",
"quotient.out"
] | Given two ordinals `α < β`, then `principal_seg_out α β` is the principal segment embedding
of `α` to `β`, as map from a model type for `α` to a model type for `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
typein_lt_type (r : α → α → Prop) [is_well_order α r] (a : α) : typein r a < type r | ⟨principal_seg.of_element _ _⟩ | theorem | ordinal.typein_lt_type | 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_lt_self {o : ordinal} (i : o.out.α) : typein (<) i < o | by { simp_rw ←type_lt o, apply typein_lt_type } | theorem | ordinal.typein_lt_self | 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_top {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (f : r ≺i s) :
typein s f.top = type r | eq.symm $ quot.sound ⟨rel_iso.of_surjective
(rel_embedding.cod_restrict _ f f.lt_top)
(λ ⟨a, h⟩, by rcases f.down.1 h with ⟨b, rfl⟩; exact ⟨b, rfl⟩)⟩ | theorem | ordinal.typein_top | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"rel_embedding.cod_restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_apply {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (f : r ≼i s) (a : α) :
ordinal.typein s (f a) = ordinal.typein r a | eq.symm $ quotient.sound ⟨rel_iso.of_surjective
(rel_embedding.cod_restrict _
((subrel.rel_embedding _ _).trans f)
(λ ⟨x, h⟩, by rw [rel_embedding.trans_apply]; exact f.to_rel_embedding.map_rel_iff.2 h))
(λ ⟨y, h⟩, by rcases f.init h with ⟨a, rfl⟩;
exact ⟨⟨a, f.to_rel_embedding.map_rel_iff.1 h⟩, subtype... | theorem | ordinal.typein_apply | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal.typein",
"rel_embedding.cod_restrict",
"rel_embedding.trans_apply",
"subrel.rel_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_lt_typein (r : α → α → Prop) [is_well_order α r]
{a b : α} : typein r a < typein r b ↔ r a b | ⟨λ ⟨f⟩, begin
have : f.top.1 = a,
{ let f' := principal_seg.of_element r a,
let g' := f.trans (principal_seg.of_element r b),
have : g'.top = f'.top, {rw subsingleton.elim f' g'},
exact this },
rw ← this, exact f.top.2
end, λ h, ⟨principal_seg.cod_restrict _
(principal_seg.of_element r a)
(λ x, @t... | theorem | ordinal.typein_lt_typein | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"principal_seg.of_element"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_surj (r : α → α → Prop) [is_well_order α r]
{o} (h : o < type r) : ∃ a, typein r a = o | induction_on o (λ β s _ ⟨f⟩, by exactI ⟨f.top, typein_top _⟩) h | theorem | ordinal.typein_surj | 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_injective (r : α → α → Prop) [is_well_order α r] : injective (typein r) | injective_of_increasing r (<) (typein r) (λ x y, (typein_lt_typein r).2) | lemma | ordinal.typein_injective | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"injective_of_increasing",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_inj (r : α → α → Prop) [is_well_order α r]
{a b} : typein r a = typein r b ↔ a = b | (typein_injective r).eq_iff | theorem | ordinal.typein_inj | 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.principal_seg (r : α → α → Prop) [is_well_order α r] :
r ≺i ((<) : ordinal → ordinal → Prop) | ⟨⟨⟨typein r, typein_injective r⟩, λ a b, typein_lt_typein r⟩, type r,
λ o, ⟨typein_surj r, λ ⟨a, h⟩, h ▸ typein_lt_type r a⟩⟩ | def | ordinal.typein.principal_seg | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal"
] | Principal segment version of the `typein` function, embedding a well order into
ordinals as a principal segment. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
typein.principal_seg_coe (r : α → α → Prop) [is_well_order α r] :
(typein.principal_seg r : α → ordinal) = typein r | rfl | theorem | ordinal.typein.principal_seg_coe | 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 (r : α → α → Prop) [is_well_order α r] (o) (h : o < type r) : α | (typein.principal_seg r).subrel_iso ⟨o, h⟩ | def | ordinal.enum | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order"
] | `enum r o h` is the `o`-th element of `α` ordered by `r`.
That is, `enum` maps an initial segment of the ordinals, those
less than the order type of `r`, to the elements of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_type {α β} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s] (f : s ≺i r)
{h : type s < type r} : enum r (type s) h = f.top | (typein.principal_seg r).injective $
((typein.principal_seg r).apply_subrel_iso _).trans (typein_top _).symm | theorem | ordinal.enum_type | 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 | |
enum_typein (r : α → α → Prop) [is_well_order α r] (a : α) :
enum r (typein r a) (typein_lt_type r a) = a | enum_type (principal_seg.of_element r a) | theorem | ordinal.enum_typein | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"principal_seg.of_element"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
typein_enum (r : α → α → Prop) [is_well_order α r]
{o} (h : o < type r) : typein r (enum r o h) = o | let ⟨a, e⟩ := typein_surj r h in
by clear _let_match; subst e; rw enum_typein | theorem | ordinal.typein_enum | 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 | |
enum_lt_enum {r : α → α → Prop} [is_well_order α r]
{o₁ o₂ : ordinal} (h₁ : o₁ < type r) (h₂ : o₂ < type r) :
r (enum r o₁ h₁) (enum r o₂ h₂) ↔ o₁ < o₂ | by rw [← typein_lt_typein r, typein_enum, typein_enum] | theorem | ordinal.enum_lt_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 | |
rel_iso_enum' {α β : Type u} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s]
(f : r ≃r s) (o : ordinal) : ∀(hr : o < type r) (hs : o < type s),
f (enum r o hr) = enum s o hs | begin
refine induction_on o _, rintros γ t wo ⟨g⟩ ⟨h⟩,
resetI, rw [enum_type g, enum_type (principal_seg.lt_equiv g f)], refl
end | lemma | ordinal.rel_iso_enum' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"ordinal",
"principal_seg.lt_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rel_iso_enum {α β : Type u} {r : α → α → Prop} {s : β → β → Prop}
[is_well_order α r] [is_well_order β s]
(f : r ≃r s) (o : ordinal) (hr : o < type r) :
f (enum r o hr) =
enum s o (by {convert hr using 1, apply quotient.sound, exact ⟨f.symm⟩ }) | rel_iso_enum' _ _ _ _ | lemma | ordinal.rel_iso_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 | |
lt_wf : @well_founded ordinal (<) | well_founded_iff_well_founded_subrel.mpr $ λ a, induction_on a $ λ α r wo, by exactI
rel_hom_class.well_founded (typein.principal_seg r).subrel_iso wo.wf | theorem | ordinal.lt_wf | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal",
"rel_hom_class.well_founded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction {p : ordinal.{u} → Prop} (i : ordinal.{u})
(h : ∀ j, (∀ k, k < j → p k) → p j) : p i | lt_wf.induction i h | lemma | ordinal.induction | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | Reformulation of well founded induction on ordinals as a lemma that works with the
`induction` tactic, as in `induction i using ordinal.induction with i IH`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card : ordinal → cardinal | quotient.map Well_order.α $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨e⟩, ⟨e.to_equiv⟩ | def | ordinal.card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal",
"ordinal",
"quotient.map"
] | The cardinal of an ordinal is the cardinality of any type on which a relation with that order
type is defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_type (r : α → α → Prop) [is_well_order α r] : card (type r) = #α | rfl | theorem | ordinal.card_type | 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 {r : α → α → Prop} [wo : is_well_order α r] (x : α) :
#{y // r y x} = (typein r x).card | rfl | lemma | ordinal.card_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 | |
card_le_card {o₁ o₂ : ordinal} : o₁ ≤ o₂ → card o₁ ≤ card o₂ | induction_on o₁ $ λ α r _, induction_on o₂ $ λ β s _ ⟨⟨⟨f, _⟩, _⟩⟩, ⟨f⟩ | theorem | ordinal.card_le_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 | |
card_zero : card 0 = 0 | rfl | theorem | ordinal.card_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 | |
card_eq_zero {o} : card o = 0 ↔ o = 0 | ⟨induction_on o $ λ α r _ h, begin
haveI := cardinal.mk_eq_zero_iff.1 h,
apply type_eq_zero_of_empty
end, λ e, by simp only [e, card_zero]⟩ | theorem | ordinal.card_eq_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 | |
card_one : card 1 = 1 | rfl | theorem | ordinal.card_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 (o : ordinal.{v}) : ordinal.{max v u} | quotient.lift_on o (λ w, type $ ulift.down ⁻¹'o w.r) $
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨f⟩, quot.sound ⟨(rel_iso.preimage equiv.ulift r).trans $
f.trans (rel_iso.preimage equiv.ulift s).symm⟩ | def | ordinal.lift | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"lift",
"rel_iso.preimage"
] | The universe lift operation for ordinals, which embeds `ordinal.{u}` as
a proper initial segment of `ordinal.{v}` for `v > u`. For the initial segment version,
see `lift.initial_seg`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
type_ulift (r : α → α → Prop) [is_well_order α r] :
type (ulift.down ⁻¹'o r) = lift.{v} (type r) | rfl | theorem | ordinal.type_ulift | 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 | |
_root_.rel_iso.ordinal_lift_type_eq {α : Type u} {β : Type v}
{r : α → α → Prop} {s : β → β → Prop} [is_well_order α r] [is_well_order β s] (f : r ≃r s) :
lift.{v} (type r) = lift.{u} (type s) | ((rel_iso.preimage equiv.ulift r).trans $
f.trans (rel_iso.preimage equiv.ulift s).symm).ordinal_type_eq | theorem | rel_iso.ordinal_lift_type_eq | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_lift_preimage {α : Type u} {β : Type v} (r : α → α → Prop) [is_well_order α r]
(f : β ≃ α) : lift.{u} (type (f ⁻¹'o r)) = lift.{v} (type r) | (rel_iso.preimage f r).ordinal_lift_type_eq | theorem | ordinal.type_lift_preimage | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_umax : lift.{(max u v) u} = lift.{v u} | funext $ λ a, induction_on a $ λ α r _,
quotient.sound ⟨(rel_iso.preimage equiv.ulift r).trans (rel_iso.preimage equiv.ulift r).symm⟩ | theorem | ordinal.lift_umax | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"rel_iso.preimage"
] | `lift.{(max u v) u}` equals `lift.{v u}`. Using `set_option pp.universes true` will make it much
easier to understand what's happening when using this lemma. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_id' (a : ordinal) : lift a = a | induction_on a $ λ α r _, quotient.sound ⟨rel_iso.preimage equiv.ulift r⟩ | theorem | ordinal.lift_id' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"lift",
"ordinal"
] | An ordinal lifted to a lower or equal universe equals itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_id : ∀ a, lift.{u u} a = a | lift_id'.{u u} | theorem | ordinal.lift_id | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | An ordinal lifted to the same universe equals itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_uzero (a : ordinal.{u}) : lift.{0} a = a | lift_id'.{0 u} a | theorem | ordinal.lift_uzero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | An ordinal lifted to the zero universe equals itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_lift (a : ordinal) : lift.{w} (lift.{v} a) = lift.{max v w} a | induction_on a $ λ α r _,
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans $
(rel_iso.preimage equiv.ulift _).trans (rel_iso.preimage equiv.ulift _).symm⟩ | theorem | ordinal.lift_lift | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"ordinal",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_type_le {α : Type u} {β : Type v} {r s} [is_well_order α r] [is_well_order β s] :
lift.{max v w} (type r) ≤ lift.{max u w} (type s) ↔ nonempty (r ≼i s) | ⟨λ ⟨f⟩, ⟨(initial_seg.of_iso (rel_iso.preimage equiv.ulift r).symm).trans $
f.trans (initial_seg.of_iso (rel_iso.preimage equiv.ulift s))⟩,
λ ⟨f⟩, ⟨(initial_seg.of_iso (rel_iso.preimage equiv.ulift r)).trans $
f.trans (initial_seg.of_iso (rel_iso.preimage equiv.ulift s).symm)⟩⟩ | theorem | ordinal.lift_type_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"initial_seg.of_iso",
"is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_type_eq {α : Type u} {β : Type v} {r s} [is_well_order α r] [is_well_order β s] :
lift.{max v w} (type r) = lift.{max u w} (type s) ↔ nonempty (r ≃r s) | quotient.eq.trans
⟨λ ⟨f⟩, ⟨(rel_iso.preimage equiv.ulift r).symm.trans $
f.trans (rel_iso.preimage equiv.ulift s)⟩,
λ ⟨f⟩, ⟨(rel_iso.preimage equiv.ulift r).trans $
f.trans (rel_iso.preimage equiv.ulift s).symm⟩⟩ | theorem | ordinal.lift_type_eq | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_type_lt {α : Type u} {β : Type v} {r s} [is_well_order α r] [is_well_order β s] :
lift.{max v w} (type r) < lift.{max u w} (type s) ↔ nonempty (r ≺i s) | by haveI := @rel_embedding.is_well_order _ _ (@equiv.ulift.{max v w} α ⁻¹'o r)
r (rel_iso.preimage equiv.ulift.{max v w} r) _;
haveI := @rel_embedding.is_well_order _ _ (@equiv.ulift.{max u w} β ⁻¹'o s)
s (rel_iso.preimage equiv.ulift.{max u w} s) _; exact
⟨λ ⟨f⟩, ⟨(f.equiv_lt (rel_iso.preimage equiv.ulift... | theorem | ordinal.lift_type_lt | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"equiv.ulift",
"initial_seg.of_iso",
"is_well_order",
"rel_embedding.is_well_order",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_le {a b : ordinal} : lift.{u v} a ≤ lift b ↔ a ≤ b | induction_on a $ λ α r _, induction_on b $ λ β s _, by { rw ← lift_umax, exactI lift_type_le } | theorem | ordinal.lift_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_inj {a b : ordinal} : lift a = lift b ↔ a = b | by simp only [le_antisymm_iff, lift_le] | theorem | ordinal.lift_inj | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_lt {a b : ordinal} : lift a < lift b ↔ a < b | by simp only [lt_iff_le_not_le, lift_le] | theorem | ordinal.lift_lt | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_zero : lift 0 = 0 | type_eq_zero_of_empty _ | theorem | ordinal.lift_zero | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_one : lift 1 = 1 | type_eq_one_of_unique _ | theorem | ordinal.lift_one | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_card (a) : (card a).lift = card (lift a) | induction_on a $ λ α r _, rfl | theorem | ordinal.lift_card | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_down' {a : cardinal.{u}} {b : ordinal.{max u v}}
(h : card b ≤ a.lift) : ∃ a', lift a' = b | let ⟨c, e⟩ := cardinal.lift_down h in
cardinal.induction_on c (λ α, induction_on b $ λ β s _ e', begin
resetI,
rw [card_type, ← cardinal.lift_id'.{(max u v) u} (#β),
← cardinal.lift_umax.{u v}, lift_mk_eq.{u (max u v) (max u v)}] at e',
cases e' with f,
have g := rel_iso.preimage f s,
haveI := (g : ⇑f ⁻... | theorem | ordinal.lift_down' | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"cardinal.induction_on",
"cardinal.lift_down",
"is_well_order",
"lift",
"rel_iso.preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_down {a : ordinal.{u}} {b : ordinal.{max u v}}
(h : b ≤ lift a) : ∃ a', lift a' = b | @lift_down' (card a) _ (by rw lift_card; exact card_le_card h) | theorem | ordinal.lift_down | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_lift_iff {a : ordinal.{u}} {b : ordinal.{max u v}} :
b ≤ lift a ↔ ∃ a', lift a' = b ∧ a' ≤ a | ⟨λ h, let ⟨a', e⟩ := lift_down h in ⟨a', e, lift_le.1 $ e.symm ▸ h⟩,
λ ⟨a', e, h⟩, e ▸ lift_le.2 h⟩ | theorem | ordinal.le_lift_iff | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_lift_iff {a : ordinal.{u}} {b : ordinal.{max u v}} :
b < lift a ↔ ∃ a', lift a' = b ∧ a' < a | ⟨λ h, let ⟨a', e⟩ := lift_down (le_of_lt h) in
⟨a', e, lift_lt.1 $ e.symm ▸ h⟩,
λ ⟨a', e, h⟩, e ▸ lift_lt.2 h⟩ | theorem | ordinal.lt_lift_iff | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift.initial_seg : @initial_seg ordinal.{u} ordinal.{max u v} (<) (<) | ⟨⟨⟨lift.{v}, λ a b, lift_inj.1⟩, λ a b, lift_lt⟩,
λ a b h, lift_down (le_of_lt h)⟩ | def | ordinal.lift.initial_seg | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"initial_seg"
] | Initial segment version of the lift operation on ordinals, embedding `ordinal.{u}` in
`ordinal.{v}` as an initial segment when `u ≤ v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift.initial_seg_coe : (lift.initial_seg : ordinal → ordinal) = lift | rfl | theorem | ordinal.lift.initial_seg_coe | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega : ordinal.{u} | lift $ @type ℕ (<) _ | def | ordinal.omega | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | `ω` is the first infinite ordinal, defined as the order type of `ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
type_nat_lt : @type ℕ (<) _ = ω | (lift_id _).symm | theorem | ordinal.type_nat_lt | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [] | Note that the presence of this lemma makes `simp [omega]` form a loop. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
card_omega : card ω = ℵ₀ | rfl | theorem | ordinal.card_omega | 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_omega : lift ω = ω | lift_lift _ | theorem | ordinal.lift_omega | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_add (o₁ o₂ : ordinal) : card (o₁ + o₂) = card o₁ + card o₂ | induction_on o₁ $ λ α r _, induction_on o₂ $ λ β s _, rfl | theorem | ordinal.card_add | 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 | |
type_sum_lex {α β : Type u} (r : α → α → Prop) (s : β → β → Prop)
[is_well_order α r] [is_well_order β s] : type (sum.lex r s) = type r + type s | rfl | theorem | ordinal.type_sum_lex | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"sum.lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_nat (n : ℕ) : card.{u} n = n | by induction n; [refl, simp only [card_add, card_one, nat.cast_succ, *]] | theorem | ordinal.card_nat | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"nat.cast_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_covariant_class_le : covariant_class ordinal.{u} ordinal.{u} (+) (≤) | ⟨λ c a b h, begin
revert h c, exact (
induction_on a $ λ α₁ r₁ _, induction_on b $ λ α₂ r₂ _ ⟨⟨⟨f, fo⟩, fi⟩⟩ c,
induction_on c $ λ β s _,
⟨⟨⟨(embedding.refl _).sum_map f,
λ a b, match a, b with
| sum.inl a, sum.inl b := sum.lex_inl_inl.trans sum.lex_inl_inl.symm
| sum.inl a, sum.inr b := by appl... | instance | ordinal.add_covariant_class_le | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"covariant_class",
"iff_of_false",
"iff_of_true",
"sum.lex_inr_inl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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