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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blsub_comp {o o' : ordinal} {f : Π a < o, ordinal}
(hf : ∀ {i j} (hi) (hj), i ≤ j → f i hi ≤ f j hj) {g : Π a < o', ordinal} (hg : blsub o' g = o) :
blsub o' (λ a ha, f (g a ha) (by { rw ←hg, apply lt_blsub })) = blsub o f | @bsup_comp o _ (λ a ha, succ (f a ha)) (λ i j _ _ h, succ_le_succ_iff.2 (hf _ _ h)) g hg | theorem | ordinal.blsub_comp | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.bsup_eq {f} (H : is_normal f) {o : ordinal} (h : is_limit o) :
bsup.{u} o (λ x _, f x) = f o | by { rw [←is_normal.bsup.{u u} H (λ x _, x) h.1, bsup_id_limit h.2] } | theorem | ordinal.is_normal.bsup_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.blsub_eq {f} (H : is_normal f) {o : ordinal} (h : is_limit o) :
blsub.{u} o (λ x _, f x) = f o | by { rw [←H.bsup_eq h, bsup_eq_blsub_of_lt_succ_limit h], exact (λ a _, H.1 a) } | theorem | ordinal.is_normal.blsub_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal_iff_lt_succ_and_bsup_eq {f} :
is_normal f ↔ (∀ a, f a < f (succ a)) ∧ ∀ o, is_limit o → bsup o (λ x _, f x) = f o | ⟨λ h, ⟨h.1, @is_normal.bsup_eq f h⟩, λ ⟨h₁, h₂⟩, ⟨h₁, λ o ho a,
(by {rw ←h₂ o ho, exact bsup_le_iff})⟩⟩ | theorem | ordinal.is_normal_iff_lt_succ_and_bsup_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal_iff_lt_succ_and_blsub_eq {f} :
is_normal f ↔ (∀ a, f a < f (succ a)) ∧ ∀ o, is_limit o → blsub o (λ x _, f x) = f o | begin
rw [is_normal_iff_lt_succ_and_bsup_eq, and.congr_right_iff],
intro h,
split;
intros H o ho;
have := H o ho;
rwa ←bsup_eq_blsub_of_lt_succ_limit ho (λ a _, h a) at *
end | theorem | ordinal.is_normal_iff_lt_succ_and_blsub_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"and.congr_right_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.eq_iff_zero_and_succ {f g : ordinal.{u} → ordinal.{u}} (hf : is_normal f)
(hg : is_normal g) : f = g ↔ f 0 = g 0 ∧ ∀ a, f a = g a → f (succ a) = g (succ a) | ⟨λ h, by simp [h], λ ⟨h₁, h₂⟩, funext (λ a, begin
apply a.limit_rec_on,
assumption',
intros o ho H,
rw [←is_normal.bsup_eq.{u u} hf ho, ←is_normal.bsup_eq.{u u} hg ho],
congr,
ext b hb,
exact H b hb
end)⟩ | theorem | ordinal.is_normal.eq_iff_zero_and_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub₂ (o₁ o₂ : ordinal) (op : Π (a < o₁) (b < o₂), ordinal) : ordinal | lsub (λ x : o₁.out.α × o₂.out.α,
op (typein (<) x.1) (typein_lt_self _) (typein (<) x.2) (typein_lt_self _)) | def | ordinal.blsub₂ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | A two-argument version of `ordinal.blsub`.
We don't develop a full API for this, since it's only used in a handful of existence results. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_blsub₂ {o₁ o₂ : ordinal} (op : Π (a < o₁) (b < o₂), ordinal) {a b : ordinal}
(ha : a < o₁) (hb : b < o₂) : op a ha b hb < blsub₂ o₁ o₂ op | begin
convert lt_lsub _ (prod.mk (enum (<) a (by rwa type_lt)) (enum (<) b (by rwa type_lt))),
simp only [typein_enum]
end | theorem | ordinal.lt_blsub₂ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mex {ι : Type u} (f : ι → ordinal.{max u v}) : ordinal | Inf (set.range f)ᶜ | def | ordinal.mex | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"set.range"
] | The minimum excluded ordinal in a family of ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mex_not_mem_range {ι : Type u} (f : ι → ordinal.{max u v}) : mex f ∉ set.range f | Inf_mem (nonempty_compl_range f) | theorem | ordinal.mex_not_mem_range | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"Inf_mem",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mex_of_forall {ι : Type u} {f : ι → ordinal.{max u v}} {a : ordinal}
(H : ∀ b < a, ∃ i, f i = b) : a ≤ mex f | by { by_contra' h, exact mex_not_mem_range f (H _ h) } | theorem | ordinal.le_mex_of_forall | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_mex {ι} (f : ι → ordinal) : ∀ i, f i ≠ mex f | by simpa using mex_not_mem_range f | theorem | ordinal.ne_mex | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mex_le_of_ne {ι} {f : ι → ordinal} {a} (ha : ∀ i, f i ≠ a) : mex f ≤ a | cInf_le' (by simp [ha]) | theorem | ordinal.mex_le_of_ne | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_of_lt_mex {ι} {f : ι → ordinal} {a} (ha : a < mex f) : ∃ i, f i = a | by { by_contra' ha', exact ha.not_le (mex_le_of_ne ha') } | theorem | ordinal.exists_of_lt_mex | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mex_le_lsub {ι} (f : ι → ordinal) : mex f ≤ lsub f | cInf_le' (lsub_not_mem_range f) | theorem | ordinal.mex_le_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mex_monotone {α β} {f : α → ordinal} {g : β → ordinal} (h : set.range f ⊆ set.range g) :
mex f ≤ mex g | begin
refine mex_le_of_ne (λ i hi, _),
cases h ⟨i, rfl⟩ with j hj,
rw ←hj at hi,
exact ne_mex g j hi
end | theorem | ordinal.mex_monotone | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mex_lt_ord_succ_mk {ι} (f : ι → ordinal) : mex f < (succ (#ι)).ord | begin
by_contra' h,
apply (lt_succ (#ι)).not_le,
have H := λ a, exists_of_lt_mex ((typein_lt_self a).trans_le h),
let g : (succ (#ι)).ord.out.α → ι := λ a, classical.some (H a),
have hg : injective g := λ a b h', begin
have Hf : ∀ x, f (g x) = typein (<) x := λ a, classical.some_spec (H a),
apply_fun ... | theorem | ordinal.mex_lt_ord_succ_mk | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cardinal.mk_le_of_injective",
"cardinal.mk_ord_out",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bmex (o : ordinal) (f : Π a < o, ordinal) : ordinal | mex (family_of_bfamily o f) | def | ordinal.bmex | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | The minimum excluded ordinal of a family of ordinals indexed by the set of ordinals less than
some `o : ordinal.{u}`. This is a special case of `mex` over the family provided by
`family_of_bfamily`.
This is to `mex` as `bsup` is to `sup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bmex_not_mem_brange {o : ordinal} (f : Π a < o, ordinal) : bmex o f ∉ brange o f | by { rw ←range_family_of_bfamily, apply mex_not_mem_range } | theorem | ordinal.bmex_not_mem_brange | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_bmex_of_forall {o : ordinal} (f : Π a < o, ordinal) {a : ordinal}
(H : ∀ b < a, ∃ i hi, f i hi = b) : a ≤ bmex o f | by { by_contra' h, exact bmex_not_mem_brange f (H _ h) } | theorem | ordinal.le_bmex_of_forall | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bmex {o : ordinal} (f : Π a < o, ordinal) {i} (hi) : f i hi ≠ bmex o f | begin
convert ne_mex _ (enum (<) i (by rwa type_lt)),
rw family_of_bfamily_enum
end | theorem | ordinal.ne_bmex | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bmex_le_of_ne {o : ordinal} {f : Π a < o, ordinal} {a} (ha : ∀ i hi, f i hi ≠ a) :
bmex o f ≤ a | mex_le_of_ne (λ i, ha _ _) | theorem | ordinal.bmex_le_of_ne | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_of_lt_bmex {o : ordinal} {f : Π a < o, ordinal} {a} (ha : a < bmex o f) :
∃ i hi, f i hi = a | begin
cases exists_of_lt_mex ha with i hi,
exact ⟨_, typein_lt_self i, hi⟩
end | theorem | ordinal.exists_of_lt_bmex | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bmex_le_blsub {o : ordinal} (f : Π a < o, ordinal) : bmex o f ≤ blsub o f | mex_le_lsub _ | theorem | ordinal.bmex_le_blsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bmex_monotone {o o' : ordinal} {f : Π a < o, ordinal} {g : Π a < o', ordinal}
(h : brange o f ⊆ brange o' g) : bmex o f ≤ bmex o' g | mex_monotone (by rwa [range_family_of_bfamily, range_family_of_bfamily]) | theorem | ordinal.bmex_monotone | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bmex_lt_ord_succ_card {o : ordinal} (f : Π a < o, ordinal) :
bmex o f < (succ o.card).ord | by { rw ←mk_ordinal_out, exact (mex_lt_ord_succ_mk (family_of_bfamily o f)) } | theorem | ordinal.bmex_lt_ord_succ_card | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_surjective_of_ordinal {α : Type u} (f : α → ordinal.{u}) : ¬ surjective f | λ h, ordinal.lsub_not_mem_range.{u u} f (h _) | lemma | not_surjective_of_ordinal | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_injective_of_ordinal {α : Type u} (f : ordinal.{u} → α) : ¬ injective f | λ h, not_surjective_of_ordinal _ (inv_fun_surjective h) | lemma | not_injective_of_ordinal | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"inv_fun_surjective",
"not_surjective_of_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_surjective_of_ordinal_of_small {α : Type v} [small.{u} α] (f : α → ordinal.{u}) :
¬ surjective f | λ h, not_surjective_of_ordinal _ (h.comp (equiv_shrink _).symm.surjective) | lemma | not_surjective_of_ordinal_of_small | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"equiv_shrink",
"not_surjective_of_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_injective_of_ordinal_of_small {α : Type v} [small.{u} α] (f : ordinal.{u} → α) :
¬ injective f | λ h, not_injective_of_ordinal _ ((equiv_shrink _).injective.comp h) | lemma | not_injective_of_ordinal_of_small | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"equiv_shrink",
"not_injective_of_ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_small_ordinal : ¬ small.{u} ordinal.{max u v} | λ h, @not_injective_of_ordinal_of_small _ h _ (λ a b, ordinal.lift_inj.1) | theorem | not_small_ordinal | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_injective_of_ordinal_of_small"
] | The type of ordinals in universe `u` is not `small.{u}`. This is the type-theoretic analog of
the Burali-Forti paradox. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_ord (S : set ordinal.{u}) : ordinal → ordinal | lt_wf.fix (λ o f, Inf (S ∩ set.Ici (blsub.{u u} o f))) | def | ordinal.enum_ord | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"set.Ici"
] | Enumerator function for an unbounded set of ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_ord_def' (o) :
enum_ord S o = Inf (S ∩ set.Ici (blsub.{u u} o (λ a _, enum_ord S a))) | lt_wf.fix_eq _ _ | theorem | ordinal.enum_ord_def' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"set.Ici"
] | The equation that characterizes `enum_ord` definitionally. This isn't the nicest expression to
work with, so consider using `enum_ord_def` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_ord_def'_nonempty (hS : unbounded (<) S) (a) : (S ∩ set.Ici a).nonempty | let ⟨b, hb, hb'⟩ := hS a in ⟨b, hb, le_of_not_gt hb'⟩ | theorem | ordinal.enum_ord_def'_nonempty | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"set.Ici"
] | The set in `enum_ord_def'` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_ord_mem_aux (hS : unbounded (<) S) (o) :
(enum_ord S o) ∈ S ∩ set.Ici (blsub.{u u} o (λ c _, enum_ord S c)) | by { rw enum_ord_def', exact Inf_mem (enum_ord_def'_nonempty hS _) } | theorem | ordinal.enum_ord_mem_aux | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"Inf_mem",
"set.Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_mem (hS : unbounded (<) S) (o) : enum_ord S o ∈ S | (enum_ord_mem_aux hS o).left | theorem | ordinal.enum_ord_mem | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_le_enum_ord (hS : unbounded (<) S) (o) :
blsub.{u u} o (λ c _, enum_ord S c) ≤ enum_ord S o | (enum_ord_mem_aux hS o).right | theorem | ordinal.blsub_le_enum_ord | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_strict_mono (hS : unbounded (<) S) : strict_mono (enum_ord S) | λ _ _ h, (lt_blsub.{u u} _ _ h).trans_le (blsub_le_enum_ord hS _) | theorem | ordinal.enum_ord_strict_mono | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_def (o) :
enum_ord S o = Inf (S ∩ {b | ∀ c, c < o → enum_ord S c < b}) | begin
rw enum_ord_def',
congr, ext,
exact ⟨λ h a hao, (lt_blsub.{u u} _ _ hao).trans_le h, blsub_le⟩
end | theorem | ordinal.enum_ord_def | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | A more workable definition for `enum_ord`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_ord_def_nonempty (hS : unbounded (<) S) {o} :
{x | x ∈ S ∧ ∀ c, c < o → enum_ord S c < x}.nonempty | (⟨_, enum_ord_mem hS o, λ _ b, enum_ord_strict_mono hS b⟩) | lemma | ordinal.enum_ord_def_nonempty | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | The set in `enum_ord_def` is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
enum_ord_range {f : ordinal → ordinal} (hf : strict_mono f) :
enum_ord (range f) = f | funext (λ o, begin
apply ordinal.induction o,
intros a H,
rw enum_ord_def a,
have Hfa : f a ∈ range f ∩ {b | ∀ c, c < a → enum_ord (range f) c < b} :=
⟨mem_range_self a, λ b hb, (by {rw H b hb, exact hf hb})⟩,
refine (cInf_le' Hfa).antisymm ((le_cInf_iff'' ⟨_, Hfa⟩).2 _),
rintros _ ⟨⟨c, rfl⟩, hc : ∀ b <... | theorem | ordinal.enum_ord_range | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"le_cInf_iff''",
"ordinal",
"ordinal.induction",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_univ : enum_ord set.univ = id | by { rw ←range_id, exact enum_ord_range strict_mono_id } | theorem | ordinal.enum_ord_univ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"strict_mono_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_zero : enum_ord S 0 = Inf S | by { rw enum_ord_def, simp [ordinal.not_lt_zero] } | theorem | ordinal.enum_ord_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal.not_lt_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_succ_le {a b} (hS : unbounded (<) S) (ha : a ∈ S) (hb : enum_ord S b < a) :
enum_ord S (succ b) ≤ a | begin
rw enum_ord_def,
exact cInf_le' ⟨ha, λ c hc, ((enum_ord_strict_mono hS).monotone (le_of_lt_succ hc)).trans_lt hb⟩
end | theorem | ordinal.enum_ord_succ_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_le_of_subset {S T : set ordinal} (hS : unbounded (<) S) (hST : S ⊆ T) (a) :
enum_ord T a ≤ enum_ord S a | begin
apply ordinal.induction a,
intros b H,
rw enum_ord_def,
exact cInf_le' ⟨hST (enum_ord_mem hS b), λ c h, (H c h).trans_lt (enum_ord_strict_mono hS h)⟩
end | theorem | ordinal.enum_ord_le_of_subset | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"ordinal",
"ordinal.induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_surjective (hS : unbounded (<) S) : ∀ s ∈ S, ∃ a, enum_ord S a = s | λ s hs, ⟨Sup {a | enum_ord S a ≤ s}, begin
apply le_antisymm,
{ rw enum_ord_def,
refine cInf_le' ⟨hs, λ a ha, _⟩,
have : enum_ord S 0 ≤ s := by { rw enum_ord_zero, exact cInf_le' hs },
rcases exists_lt_of_lt_cSup (by exact ⟨0, this⟩) ha with ⟨b, hb, hab⟩,
exact (enum_ord_strict_mono hS hab).trans_le... | theorem | ordinal.enum_ord_surjective | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"exists_lt_of_lt_cSup",
"le_cSup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_order_iso (hS : unbounded (<) S) : ordinal ≃o S | strict_mono.order_iso_of_surjective (λ o, ⟨_, enum_ord_mem hS o⟩) (enum_ord_strict_mono hS)
(λ s, let ⟨a, ha⟩ := enum_ord_surjective hS s s.prop in ⟨a, subtype.eq ha⟩) | def | ordinal.enum_ord_order_iso | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"strict_mono.order_iso_of_surjective"
] | An order isomorphism between an unbounded set of ordinals and the ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_enum_ord (hS : unbounded (<) S) : range (enum_ord S) = S | by { rw range_eq_iff, exact ⟨enum_ord_mem hS, enum_ord_surjective hS⟩ } | theorem | ordinal.range_enum_ord | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_enum_ord (f : ordinal → ordinal) (hS : unbounded (<) S) :
strict_mono f ∧ range f = S ↔ f = enum_ord S | begin
split,
{ rintro ⟨h₁, h₂⟩,
rwa [←lt_wf.eq_strict_mono_iff_eq_range h₁ (enum_ord_strict_mono hS), range_enum_ord hS] },
{ rintro rfl,
exact ⟨enum_ord_strict_mono hS, range_enum_ord hS⟩ }
end | theorem | ordinal.eq_enum_ord | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"strict_mono"
] | A characterization of `enum_ord`: it is the unique strict monotonic function with range `S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_add_nat_cast (m : ℕ) : 1 + (m : ordinal) = succ m | by { rw [←nat.cast_one, ←nat.cast_add, add_comm], refl } | theorem | ordinal.one_add_nat_cast | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_mul (m : ℕ) : ∀ n : ℕ, ((m * n : ℕ) : ordinal) = m * n | | 0 := by simp
| (n+1) := by rw [nat.mul_succ, nat.cast_add, nat_cast_mul, nat.cast_succ, mul_add_one] | theorem | ordinal.nat_cast_mul | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_add_one",
"nat.cast_add",
"nat.cast_succ",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_le {m n : ℕ} : (m : ordinal) ≤ n ↔ m ≤ n | by rw [←cardinal.ord_nat, ←cardinal.ord_nat, cardinal.ord_le_ord, cardinal.nat_cast_le] | theorem | ordinal.nat_cast_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cardinal.nat_cast_le",
"cardinal.ord_le_ord",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_lt {m n : ℕ} : (m : ordinal) < n ↔ m < n | by simp only [lt_iff_le_not_le, nat_cast_le] | theorem | ordinal.nat_cast_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_inj {m n : ℕ} : (m : ordinal) = n ↔ m = n | by simp only [le_antisymm_iff, nat_cast_le] | theorem | ordinal.nat_cast_inj | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_eq_zero {n : ℕ} : (n : ordinal) = 0 ↔ n = 0 | @nat_cast_inj n 0 | theorem | ordinal.nat_cast_eq_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_ne_zero {n : ℕ} : (n : ordinal) ≠ 0 ↔ n ≠ 0 | not_congr nat_cast_eq_zero | theorem | ordinal.nat_cast_ne_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_pos {n : ℕ} : (0 : ordinal) < n ↔ 0 < n | @nat_cast_lt 0 n | theorem | ordinal.nat_cast_pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_sub (m n : ℕ) : ((m - n : ℕ) : ordinal) = m - n | begin
cases le_total m n with h h,
{ rw [tsub_eq_zero_iff_le.2 h, ordinal.sub_eq_zero_iff_le.2 (nat_cast_le.2 h)],
refl },
{ apply (add_left_cancel n).1,
rw [←nat.cast_add, add_tsub_cancel_of_le h, ordinal.add_sub_cancel_of_le (nat_cast_le.2 h)] }
end | theorem | ordinal.nat_cast_sub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"add_tsub_cancel_of_le",
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_div (m n : ℕ) : ((m / n : ℕ) : ordinal) = m / n | begin
rcases eq_or_ne n 0 with rfl | hn,
{ simp },
{ have hn' := nat_cast_ne_zero.2 hn,
apply le_antisymm,
{ rw [le_div hn', ←nat_cast_mul, nat_cast_le, mul_comm],
apply nat.div_mul_le_self },
{ rw [div_le hn', ←add_one_eq_succ, ←nat.cast_succ, ←nat_cast_mul, nat_cast_lt, mul_comm,
←nat.... | theorem | ordinal.nat_cast_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_or_ne",
"mul_comm",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_mod (m n : ℕ) : ((m % n : ℕ) : ordinal) = m % n | by rw [←add_left_cancel, div_add_mod, ←nat_cast_div, ←nat_cast_mul, ←nat.cast_add, nat.div_add_mod] | theorem | ordinal.nat_cast_mod | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"nat.div_add_mod",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_nat_cast : ∀ n : ℕ, lift.{u v} n = n | | 0 := by simp
| (n+1) := by simp [lift_nat_cast n] | theorem | ordinal.lift_nat_cast | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ord_aleph_0 : ord.{u} ℵ₀ = ω | le_antisymm (ord_le.2 $ le_rfl) $
le_of_forall_lt $ λ o h, begin
rcases ordinal.lt_lift_iff.1 h with ⟨o, rfl, h'⟩,
rw [lt_ord, ←lift_card, lift_lt_aleph_0, ←typein_enum (<) h'],
exact lt_aleph_0_iff_fintype.2 ⟨set.fintype_lt_nat _⟩
end | theorem | cardinal.ord_aleph_0 | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_of_forall_lt",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_one_of_aleph_0_le {c} (h : ℵ₀ ≤ c) : c + 1 = c | begin
rw [add_comm, ←card_ord c, ←card_one, ←card_add, one_add_of_omega_le],
rwa [←ord_aleph_0, ord_le_ord]
end | theorem | cardinal.add_one_of_aleph_0_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_add_of_limit {a b c : ordinal.{u}}
(h : is_limit c) : a < b + c ↔ ∃ c' < c, a < b + c' | by rw [←is_normal.bsup_eq.{u u} (add_is_normal b) h, lt_bsup] | theorem | ordinal.lt_add_of_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_omega {o : ordinal} : o < ω ↔ ∃ n : ℕ, o = n | by simp_rw [←cardinal.ord_aleph_0, cardinal.lt_ord, lt_aleph_0, card_eq_nat] | theorem | ordinal.lt_omega | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cardinal.lt_ord",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_lt_omega (n : ℕ) : ↑n < ω | lt_omega.2 ⟨_, rfl⟩ | theorem | ordinal.nat_lt_omega | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega_pos : 0 < ω | nat_lt_omega 0 | theorem | ordinal.omega_pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_omega : 1 < ω | by simpa only [nat.cast_one] using nat_lt_omega 1 | theorem | ordinal.one_lt_omega | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"nat.cast_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega_is_limit : is_limit ω | ⟨omega_ne_zero, λ o h,
let ⟨n, e⟩ := lt_omega.1 h in
by rw [e]; exact nat_lt_omega (n+1)⟩ | theorem | ordinal.omega_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega_le {o : ordinal} : ω ≤ o ↔ ∀ n : ℕ, ↑n ≤ o | ⟨λ h n, (nat_lt_omega _).le.trans h,
λ H, le_of_forall_lt $ λ a h,
let ⟨n, e⟩ := lt_omega.1 h in
by rw [e, ←succ_le_iff]; exact H (n+1)⟩ | theorem | ordinal.omega_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_of_forall_lt",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_nat_cast : sup nat.cast = ω | (sup_le $ λ n, (nat_lt_omega n).le).antisymm $ omega_le.2 $ le_sup _ | theorem | ordinal.sup_nat_cast | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"nat.cast",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_lt_limit {o} (h : is_limit o) : ∀ n : ℕ, ↑n < o | | 0 := lt_of_le_of_ne (ordinal.zero_le o) h.1.symm
| (n+1) := h.2 _ (nat_lt_limit n) | theorem | ordinal.nat_lt_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
omega_le_of_is_limit {o} (h : is_limit o) : ω ≤ o | omega_le.2 $ λ n, le_of_lt $ nat_lt_limit h n | theorem | ordinal.omega_le_of_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_iff_omega_dvd {a : ordinal} : is_limit a ↔ a ≠ 0 ∧ ω ∣ a | begin
refine ⟨λ l, ⟨l.1, ⟨a / ω, le_antisymm _ (mul_div_le _ _)⟩⟩, λ h, _⟩,
{ refine (limit_le l).2 (λ x hx, le_of_lt _),
rw [←div_lt omega_ne_zero, ←succ_le_iff, le_div omega_ne_zero, mul_succ,
add_le_of_limit omega_is_limit],
intros b hb,
rcases lt_omega.1 hb with ⟨n, rfl⟩,
exact (add_le_add... | theorem | ordinal.is_limit_iff_omega_dvd | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_limit_aux {a b c : ordinal} (ba : b + a = a)
(l : is_limit c)
(IH : ∀ c' < c, (a + b) * succ c' = a * succ c' + b) :
(a + b) * c = a * c | le_antisymm
((mul_le_of_limit l).2 $ λ c' h, begin
apply (mul_le_mul_left' (le_succ c') _).trans,
rw IH _ h,
apply (add_le_add_left _ _).trans,
{ rw ← mul_succ, exact mul_le_mul_left' (succ_le_of_lt $ l.2 _ h) _ },
{ apply_instance },
{ rw ← ba, exact le_add_right _ _ }
end)
(mul_le_mul_ri... | theorem | ordinal.add_mul_limit_aux | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_le_mul_left'",
"mul_le_mul_right'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_succ {a b : ordinal} (c) (ba : b + a = a) :
(a + b) * succ c = a * succ c + b | begin
apply limit_rec_on c,
{ simp only [succ_zero, mul_one] },
{ intros c IH,
rw [mul_succ, IH, ← add_assoc, add_assoc _ b, ba, ← mul_succ] },
{ intros c l IH,
have := add_mul_limit_aux ba l IH,
rw [mul_succ, add_mul_limit_aux ba l IH, mul_succ, add_assoc] }
end | theorem | ordinal.add_mul_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_mul_limit {a b c : ordinal} (ba : b + a = a)
(l : is_limit c) : (a + b) * c = a * c | add_mul_limit_aux ba l (λ c' _, add_mul_succ c' ba) | theorem | ordinal.add_mul_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_le_of_forall_add_lt {a b c : ordinal} (hb : 0 < b) (h : ∀ d < b, a + d < c) :
a + b ≤ c | begin
have H : a + (c - a) = c := ordinal.add_sub_cancel_of_le (by {rw ←add_zero a, exact (h _ hb).le}),
rw ←H,
apply add_le_add_left _ a,
by_contra' hb,
exact (h _ hb).ne H
end | theorem | ordinal.add_le_of_forall_add_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.apply_omega {f : ordinal.{u} → ordinal.{u}} (hf : is_normal f) :
sup.{0 u} (f ∘ nat.cast) = f ω | by rw [←sup_nat_cast, is_normal.sup.{0 u u} hf] | theorem | ordinal.is_normal.apply_omega | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"nat.cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_add_nat (o : ordinal) : sup (λ n : ℕ, o + n) = o + ω | (add_is_normal o).apply_omega | theorem | ordinal.sup_add_nat | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_mul_nat (o : ordinal) : sup (λ n : ℕ, o * n) = o * ω | begin
rcases eq_zero_or_pos o with rfl | ho,
{ rw zero_mul, exact sup_eq_zero_iff.2 (λ n, zero_mul n) },
{ exact (mul_is_normal ho).apply_omega }
end | theorem | ordinal.sup_mul_nat | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank (h : acc r a) : ordinal.{u} | acc.rec_on h $ λ a h ih, ordinal.sup.{u u} $ λ b : {b // r b a}, order.succ $ ih b b.2 | def | acc.rank | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ih",
"order.succ"
] | The rank of an element `a` accessible under a relation `r` is defined inductively as the
smallest ordinal greater than the ranks of all elements below it (i.e. elements `b` such that
`r b a`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rank_eq (h : acc r a) :
h.rank = ordinal.sup.{u u} (λ b : {b // r b a}, order.succ (h.inv b.2).rank) | by { change (acc.intro a $ λ _, h.inv).rank = _, refl } | lemma | acc.rank_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"order.succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank_lt_of_rel (hb : acc r b) (h : r a b) : (hb.inv h).rank < hb.rank | (order.lt_succ _).trans_le $ by { rw hb.rank_eq, refine le_trans _ (ordinal.le_sup _ ⟨a, h⟩), refl } | lemma | acc.rank_lt_of_rel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"order.lt_succ",
"ordinal.le_sup"
] | if `r a b` then the rank of `a` is less than the rank of `b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rank (a : α) : ordinal.{u} | (hwf.apply a).rank | def | well_founded.rank | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | The rank of an element `a` under a well-founded relation `r` is defined inductively as the
smallest ordinal greater than the ranks of all elements below it (i.e. elements `b` such that
`r b a`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rank_eq : hwf.rank a = ordinal.sup.{u u} (λ b : {b // r b a}, order.succ $ hwf.rank b) | by { rw [rank, acc.rank_eq], refl } | lemma | well_founded.rank_eq | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"acc.rank_eq",
"order.succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank_lt_of_rel (h : r a b) : hwf.rank a < hwf.rank b | acc.rank_lt_of_rel _ h | lemma | well_founded.rank_lt_of_rel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"acc.rank_lt_of_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank_strict_mono [preorder α] [well_founded_lt α] :
strict_mono (rank $ @is_well_founded.wf α (<) _) | λ _ _, rank_lt_of_rel _ | lemma | well_founded.rank_strict_mono | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"strict_mono",
"well_founded_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rank_strict_anti [preorder α] [well_founded_gt α] :
strict_anti (rank $ @is_well_founded.wf α (>) _) | λ _ _, rank_lt_of_rel $ @is_well_founded.wf α (>) _ | lemma | well_founded.rank_strict_anti | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"strict_anti",
"well_founded_gt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_embedding_to_cardinal : nonempty (σ ↪ cardinal.{u}) | (embedding.total _ _).resolve_left $ λ ⟨⟨f, hf⟩⟩,
let g : σ → cardinal.{u} := inv_fun f in
let ⟨x, (hx : g x = 2 ^ sum g)⟩ := inv_fun_surjective hf (2 ^ sum g) in
have g x ≤ sum g, from le_sum.{u u} g x,
not_le_of_gt (by rw hx; exact cantor _) this | theorem | nonempty_embedding_to_cardinal | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"inv_fun",
"inv_fun_surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_to_cardinal : σ ↪ cardinal.{u} | classical.choice nonempty_embedding_to_cardinal | def | embedding_to_cardinal | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"nonempty_embedding_to_cardinal"
] | An embedding of any type to the set of cardinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
well_ordering_rel : σ → σ → Prop | embedding_to_cardinal ⁻¹'o (<) | def | well_ordering_rel | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"embedding_to_cardinal"
] | Any type can be endowed with a well order, obtained by pulling back the well order over
cardinals by some embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
well_ordering_rel.is_well_order : is_well_order σ well_ordering_rel | (rel_embedding.preimage _ _).is_well_order | instance | well_ordering_rel.is_well_order | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order",
"rel_embedding.preimage",
"well_ordering_rel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_well_order.subtype_nonempty : nonempty {r // is_well_order σ r} | ⟨⟨well_ordering_rel, infer_instance⟩⟩ | instance | is_well_order.subtype_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 | |
Well_order : Type (u+1) | (α : Type u)
(r : α → α → Prop)
(wo : is_well_order α r) | structure | Well_order | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order"
] | Bundled structure registering a well order on a type. Ordinals will be defined as a quotient
of this type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eta (o : Well_order) : mk o.α o.r o.wo = o | by { cases o, refl } | lemma | Well_order.eta | 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 | |
ordinal.is_equivalent : setoid Well_order | { r := λ ⟨α, r, wo⟩ ⟨β, s, wo'⟩, nonempty (r ≃r s),
iseqv := ⟨λ ⟨α, r, _⟩, ⟨rel_iso.refl _⟩,
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨e⟩, ⟨e.symm⟩,
λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨e₁⟩ ⟨e₂⟩, ⟨e₁.trans e₂⟩⟩ } | instance | ordinal.is_equivalent | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"Well_order"
] | Equivalence relation on well orders on arbitrary types in universe `u`, given by order
isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ordinal : Type (u + 1) | quotient ordinal.is_equivalent | def | ordinal | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"ordinal.is_equivalent"
] | `ordinal.{u}` is the type of well orders in `Type u`, up to order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_well_founded_out (o : ordinal) : has_well_founded o.out.α | ⟨o.out.r, o.out.wo.wf⟩ | instance | has_well_founded_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 | |
linear_order_out (o : ordinal) : linear_order o.out.α | is_well_order.linear_order o.out.r | instance | linear_order_out | set_theory.ordinal | src/set_theory/ordinal/basic.lean | [
"data.sum.order",
"order.initial_seg",
"set_theory.cardinal.basic"
] | [
"is_well_order.linear_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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