statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
sup_empty {ι} [is_empty ι] (f : ι → ordinal) : sup f = 0 | csupr_of_empty f | theorem | ordinal.sup_empty | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"csupr_of_empty",
"is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_const {ι} [hι : nonempty ι] (o : ordinal) : sup (λ _ : ι, o) = o | csupr_const | theorem | ordinal.sup_const | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"csupr_const",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_unique {ι} [unique ι] (f : ι → ordinal) : sup f = f default | supr_unique | theorem | ordinal.sup_unique | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"supr_unique",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le_of_range_subset {ι ι'} {f : ι → ordinal} {g : ι' → ordinal}
(h : set.range f ⊆ set.range g) : sup.{u (max v w)} f ≤ sup.{v (max u w)} g | sup_le $ λ i, match h (mem_range_self i) with ⟨j, hj⟩ := hj ▸ le_sup _ _ end | theorem | ordinal.sup_le_of_range_subset | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"set.range",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_of_range_eq {ι ι'} {f : ι → ordinal} {g : ι' → ordinal}
(h : set.range f = set.range g) : sup.{u (max v w)} f = sup.{v (max u w)} g | (sup_le_of_range_subset h.le).antisymm (sup_le_of_range_subset.{v u w} h.ge) | theorem | ordinal.sup_eq_of_range_eq | 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 | |
sup_sum {α : Type u} {β : Type v} (f : α ⊕ β → ordinal) : sup.{(max u v) w} f =
max (sup.{u (max v w)} (λ a, f (sum.inl a))) (sup.{v (max u w)} (λ b, f (sum.inr b))) | begin
apply (sup_le_iff.2 _).antisymm (max_le_iff.2 ⟨_, _⟩),
{ rintro (i|i),
{ exact le_max_of_le_left (le_sup _ i) },
{ exact le_max_of_le_right (le_sup _ i) } },
all_goals
{ apply sup_le_of_range_subset.{_ (max u v) w},
rintros i ⟨a, rfl⟩,
apply mem_range_self }
end | theorem | ordinal.sup_sum | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_max_of_le_left",
"le_max_of_le_right",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unbounded_range_of_sup_ge {α β : Type u} (r : α → α → Prop) [is_well_order α r] (f : β → α)
(h : type r ≤ sup.{u u} (typein r ∘ f)) : unbounded r (range f) | (not_bounded_iff _).1 $ λ ⟨x, hx⟩, not_lt_of_le h $ lt_of_le_of_lt
(sup_le $ λ y, le_of_lt $ (typein_lt_typein r).2 $ hx _ $ mem_range_self y)
(typein_lt_type r x) | lemma | ordinal.unbounded_range_of_sup_ge | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"not_lt_of_le",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sup_shrink_equiv {s : set ordinal.{u}} (hs : small.{u} s) (a) (ha : a ∈ s) :
a ≤ sup.{u u} (λ x, ((@equiv_shrink s hs).symm x).val) | by { convert le_sup.{u u} _ ((@equiv_shrink s hs) ⟨a, ha⟩), rw symm_apply_apply } | theorem | ordinal.le_sup_shrink_equiv | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"equiv_shrink"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_Iio (o : ordinal.{u}) : small.{u} (set.Iio o) | let f : o.out.α → set.Iio o := λ x, ⟨typein (<) x, typein_lt_self x⟩ in
let hf : surjective f := λ b, ⟨enum (<) b.val (by { rw type_lt, exact b.prop }),
subtype.ext (typein_enum _ _)⟩ in
small_of_surjective hf | instance | ordinal.small_Iio | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"set.Iio",
"small_of_surjective",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
small_Iic (o : ordinal.{u}) : small.{u} (set.Iic o) | by { rw ←Iio_succ, apply_instance } | instance | ordinal.small_Iic | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"set.Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_iff_small {s : set ordinal.{u}} : bdd_above s ↔ small.{u} s | ⟨λ ⟨a, h⟩, small_subset $ show s ⊆ Iic a, from λ x hx, h hx,
λ h, ⟨sup.{u u} (λ x, ((@equiv_shrink s h).symm x).val), le_sup_shrink_equiv h⟩⟩ | theorem | ordinal.bdd_above_iff_small | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"bdd_above",
"equiv_shrink",
"small_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_of_small (s : set ordinal.{u}) [h : small.{u} s] : bdd_above s | bdd_above_iff_small.2 h | theorem | ordinal.bdd_above_of_small | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"bdd_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_Sup {s : set ordinal.{u}} (hs : small.{u} s) :
sup.{u u} (λ x, (@equiv_shrink s hs).symm x) = Sup s | let hs' := bdd_above_iff_small.2 hs in
((cSup_le_iff' hs').2 (le_sup_shrink_equiv hs)).antisymm'
(sup_le (λ x, le_cSup hs' (subtype.mem _))) | theorem | ordinal.sup_eq_Sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"antisymm'",
"cSup_le_iff'",
"equiv_shrink",
"le_cSup",
"subtype.mem",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_ord {s : set cardinal.{u}} (hs : bdd_above s) : (Sup s).ord = Sup (ord '' s) | eq_of_forall_ge_iff $ λ a, begin
rw [cSup_le_iff' (bdd_above_iff_small.2 (@small_image _ _ _ s
(cardinal.bdd_above_iff_small.1 hs))), ord_le, cSup_le_iff' hs],
simp [ord_le]
end | theorem | ordinal.Sup_ord | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"bdd_above",
"cSup_le_iff'",
"eq_of_forall_ge_iff",
"small_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_ord {ι} {f : ι → cardinal} (hf : bdd_above (range f)) :
(supr f).ord = ⨆ i, (f i).ord | by { unfold supr, convert Sup_ord hf, rw range_comp } | theorem | ordinal.supr_ord | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"bdd_above",
"cardinal",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le_sup {ι ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop)
[is_well_order ι r] [is_well_order ι' r'] {o} (ho : type r = o) (ho' : type r' = o)
(f : Π a < o, ordinal) : sup (family_of_bfamily' r ho f) ≤ sup (family_of_bfamily' r' ho' f) | sup_le $ λ i, begin
cases typein_surj r' (by { rw [ho', ←ho], exact typein_lt_type r i }) with j hj,
simp_rw [family_of_bfamily', ←hj],
apply le_sup
end | theorem | ordinal.sup_le_sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"ordinal",
"sup_le",
"sup_le_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq_sup {ι ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop) [is_well_order ι r]
[is_well_order ι' r'] {o : ordinal.{u}} (ho : type r = o) (ho' : type r' = o)
(f : Π a < o, ordinal.{max u v}) :
sup (family_of_bfamily' r ho f) = sup (family_of_bfamily' r' ho' f) | sup_eq_of_range_eq.{u u v} (by simp) | theorem | ordinal.sup_eq_sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup (o : ordinal.{u}) (f : Π a < o, ordinal.{max u v}) : ordinal.{max u v} | sup (family_of_bfamily o f) | def | ordinal.bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | The supremum of a family of ordinals indexed by the set of ordinals less than some
`o : ordinal.{u}`. This is a special case of `sup` over the family provided by
`family_of_bfamily`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_eq_bsup {o} (f : Π a < o, ordinal) : sup (family_of_bfamily o f) = bsup o f | rfl | theorem | ordinal.sup_eq_bsup | 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_eq_bsup' {o ι} (r : ι → ι → Prop) [is_well_order ι r] (ho : type r = o) (f) :
sup (family_of_bfamily' r ho f) = bsup o f | sup_eq_sup r _ ho _ f | theorem | ordinal.sup_eq_bsup' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_eq_bsup {o} (f : Π a < o, ordinal) : Sup (brange o f) = bsup o f | by { congr, rw range_family_of_bfamily } | theorem | ordinal.Sup_eq_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_sup' {ι} (r : ι → ι → Prop) [is_well_order ι r] (f : ι → ordinal) :
bsup _ (bfamily_of_family' r f) = sup f | by simp only [←sup_eq_bsup' r, enum_typein, family_of_bfamily', bfamily_of_family'] | theorem | ordinal.bsup_eq_sup' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_bsup {ι : Type u} (r r' : ι → ι → Prop) [is_well_order ι r] [is_well_order ι r']
(f : ι → ordinal) : bsup _ (bfamily_of_family' r f) = bsup _ (bfamily_of_family' r' f) | by rw [bsup_eq_sup', bsup_eq_sup'] | theorem | ordinal.bsup_eq_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_sup {ι} (f : ι → ordinal) : bsup _ (bfamily_of_family f) = sup f | bsup_eq_sup' _ f | theorem | ordinal.bsup_eq_sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_congr {o₁ o₂ : ordinal} (f : Π a < o₁, ordinal) (ho : o₁ = o₂) :
bsup o₁ f = bsup o₂ (λ a h, f a (h.trans_eq ho.symm)) | by subst ho | lemma | ordinal.bsup_congr | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_le_iff {o f a} : bsup.{u v} o f ≤ a ↔ ∀ i h, f i h ≤ a | sup_le_iff.trans ⟨λ h i hi, by { rw ←family_of_bfamily_enum o f, exact h _ }, λ h i, h _ _⟩ | theorem | ordinal.bsup_le_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_le {o : ordinal} {f : Π b < o, ordinal} {a} :
(∀ i h, f i h ≤ a) → bsup.{u v} o f ≤ a | bsup_le_iff.2 | theorem | ordinal.bsup_le | 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_bsup {o} (f : Π a < o, ordinal) (i h) : f i h ≤ bsup o f | bsup_le_iff.1 le_rfl _ _ | theorem | ordinal.le_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_bsup {o} (f : Π a < o, ordinal) {a} : a < bsup o f ↔ ∃ i hi, a < f i hi | by simpa only [not_forall, not_le] using not_congr (@bsup_le_iff _ f a) | theorem | ordinal.lt_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_forall",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.bsup {f} (H : is_normal f) {o} :
∀ (g : Π a < o, ordinal) (h : o ≠ 0), f (bsup o g) = bsup o (λ a h, f (g a h)) | induction_on o $ λ α r _ g h, begin
resetI,
haveI := type_ne_zero_iff_nonempty.1 h,
rw [←sup_eq_bsup' r, H.sup, ←sup_eq_bsup' r];
refl
end | theorem | ordinal.is_normal.bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_bsup_of_ne_bsup {o : ordinal} {f : Π a < o, ordinal} :
(∀ i h, f i h ≠ o.bsup f) ↔ ∀ i h, f i h < o.bsup f | ⟨λ hf _ _, lt_of_le_of_ne (le_bsup _ _ _) (hf _ _), λ hf _ _, ne_of_lt (hf _ _)⟩ | theorem | ordinal.lt_bsup_of_ne_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_not_succ_of_ne_bsup {o} {f : Π a < o, ordinal}
(hf : ∀ {i : ordinal} (h : i < o), f i h ≠ o.bsup f) (a) :
a < bsup o f → succ a < bsup o f | by { rw ←sup_eq_bsup at *, exact sup_not_succ_of_ne_sup (λ i, hf _) } | theorem | ordinal.bsup_not_succ_of_ne_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_zero_iff {o} {f : Π a < o, ordinal} : bsup o f = 0 ↔ ∀ i hi, f i hi = 0 | begin
refine ⟨λ h i hi, _, λ h, le_antisymm
(bsup_le (λ i hi, ordinal.le_zero.2 (h i hi))) (ordinal.zero_le _)⟩,
rw [←ordinal.le_zero, ←h],
exact le_bsup f i hi,
end | theorem | ordinal.bsup_eq_zero_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_bsup_of_limit {o : ordinal} {f : Π a < o, ordinal}
(hf : ∀ {a a'} (ha : a < o) (ha' : a' < o), a < a' → f a ha < f a' ha')
(ho : ∀ a < o, succ a < o) (i h) : f i h < bsup o f | (hf _ _ $ lt_succ i).trans_le (le_bsup f (succ i) $ ho _ h) | theorem | ordinal.lt_bsup_of_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 | |
bsup_succ_of_mono {o : ordinal} {f : Π a < succ o, ordinal}
(hf : ∀ {i j} hi hj, i ≤ j → f i hi ≤ f j hj) : bsup _ f = f o (lt_succ o) | le_antisymm (bsup_le $ λ i hi, hf _ _ $ le_of_lt_succ hi) (le_bsup _ _ _) | theorem | ordinal.bsup_succ_of_mono | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_zero (f : Π a < (0 : ordinal), ordinal) : bsup 0 f = 0 | bsup_eq_zero_iff.2 (λ i hi, (ordinal.not_lt_zero i hi).elim) | theorem | ordinal.bsup_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.not_lt_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_const {o : ordinal} (ho : o ≠ 0) (a : ordinal) : bsup o (λ _ _, a) = a | le_antisymm (bsup_le (λ _ _, le_rfl)) (le_bsup _ 0 (ordinal.pos_iff_ne_zero.2 ho)) | theorem | ordinal.bsup_const | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_one (f : Π a < (1 : ordinal), ordinal) : bsup 1 f = f 0 zero_lt_one | by simp_rw [←sup_eq_bsup, sup_unique, family_of_bfamily, family_of_bfamily', typein_one_out] | theorem | ordinal.bsup_one | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_le_of_brange_subset {o o'} {f : Π a < o, ordinal} {g : Π a < o', ordinal}
(h : brange o f ⊆ brange o' g) : bsup.{u (max v w)} o f ≤ bsup.{v (max u w)} o' g | bsup_le $ λ i hi, begin
obtain ⟨j, hj, hj'⟩ := h ⟨i, hi, rfl⟩,
rw ←hj',
apply le_bsup
end | theorem | ordinal.bsup_le_of_brange_subset | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_of_brange_eq {o o'} {f : Π a < o, ordinal} {g : Π a < o', ordinal}
(h : brange o f = brange o' g) : bsup.{u (max v w)} o f = bsup.{v (max u w)} o' g | (bsup_le_of_brange_subset h.le).antisymm (bsup_le_of_brange_subset.{v u w} h.ge) | theorem | ordinal.bsup_eq_of_brange_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 | |
lsub {ι} (f : ι → ordinal) : ordinal | sup (succ ∘ f) | def | ordinal.lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | The least strict upper bound of a family of ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_eq_lsub {ι} (f : ι → ordinal) : sup (succ ∘ f) = lsub f | rfl | theorem | ordinal.sup_eq_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_le_iff {ι} {f : ι → ordinal} {a} : lsub f ≤ a ↔ ∀ i, f i < a | by { convert sup_le_iff, simp only [succ_le_iff] } | theorem | ordinal.lsub_le_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_le {ι} {f : ι → ordinal} {a} : (∀ i, f i < a) → lsub f ≤ a | lsub_le_iff.2 | theorem | ordinal.lsub_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_lsub {ι} (f : ι → ordinal) (i) : f i < lsub f | succ_le_iff.1 (le_sup _ i) | theorem | ordinal.lt_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_lsub_iff {ι} {f : ι → ordinal} {a} : a < lsub f ↔ ∃ i, a ≤ f i | by simpa only [not_forall, not_lt, not_le] using not_congr (@lsub_le_iff _ f a) | theorem | ordinal.lt_lsub_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_forall",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le_lsub {ι} (f : ι → ordinal) : sup f ≤ lsub f | sup_le $ λ i, (lt_lsub f i).le | theorem | ordinal.sup_le_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_le_sup_succ {ι} (f : ι → ordinal) : lsub f ≤ succ (sup f) | lsub_le $ λ i, lt_succ_iff.2 (le_sup f i) | theorem | ordinal.lsub_le_sup_succ | 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_eq_lsub_or_sup_succ_eq_lsub {ι} (f : ι → ordinal) :
sup f = lsub f ∨ succ (sup f) = lsub f | begin
cases eq_or_lt_of_le (sup_le_lsub f),
{ exact or.inl h },
{ exact or.inr ((succ_le_of_lt h).antisymm (lsub_le_sup_succ f)) }
end | theorem | ordinal.sup_eq_lsub_or_sup_succ_eq_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_or_lt_of_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_succ_le_lsub {ι} (f : ι → ordinal) : succ (sup f) ≤ lsub f ↔ ∃ i, f i = sup f | begin
refine ⟨λ h, _, _⟩,
{ by_contra' hf,
exact (succ_le_iff.1 h).ne ((sup_le_lsub f).antisymm
(lsub_le (ne_sup_iff_lt_sup.1 hf))) },
rintro ⟨_, hf⟩,
rw [succ_le_iff, ←hf],
exact lt_lsub _ _
end | theorem | ordinal.sup_succ_le_lsub | 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_succ_eq_lsub {ι} (f : ι → ordinal) : succ (sup f) = lsub f ↔ ∃ i, f i = sup f | (lsub_le_sup_succ f).le_iff_eq.symm.trans (sup_succ_le_lsub f) | theorem | ordinal.sup_succ_eq_lsub | 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_eq_lsub_iff_succ {ι} (f : ι → ordinal) :
sup f = lsub f ↔ ∀ a < lsub f, succ a < lsub f | begin
refine ⟨λ h, _, λ hf, le_antisymm (sup_le_lsub f) (lsub_le (λ i, _))⟩,
{ rw ←h,
exact λ a, sup_not_succ_of_ne_sup (λ i, (lsub_le_iff.1 (le_of_eq h.symm) i).ne) },
by_contra' hle,
have heq := (sup_succ_eq_lsub f).2 ⟨i, le_antisymm (le_sup _ _) hle⟩,
have := hf _ (by { rw ←heq, exact lt_succ (sup f) }... | theorem | ordinal.sup_eq_lsub_iff_succ | 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_eq_lsub_iff_lt_sup {ι} (f : ι → ordinal) : sup f = lsub f ↔ ∀ i, f i < sup f | ⟨λ h i, (by { rw h, apply lt_lsub }), λ h, le_antisymm (sup_le_lsub f) (lsub_le h)⟩ | theorem | ordinal.sup_eq_lsub_iff_lt_sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_empty {ι} [h : is_empty ι] (f : ι → ordinal) : lsub f = 0 | by { rw [←ordinal.le_zero, lsub_le_iff], exact h.elim } | lemma | ordinal.lsub_empty | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_pos {ι} [h : nonempty ι] (f : ι → ordinal) : 0 < lsub f | h.elim $ λ i, (ordinal.zero_le _).trans_lt (lt_lsub f i) | lemma | ordinal.lsub_pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_eq_zero_iff {ι} {f : ι → ordinal} : lsub f = 0 ↔ is_empty ι | begin
refine ⟨λ h, ⟨λ i, _⟩, λ h, @lsub_empty _ h _⟩,
have := @lsub_pos _ ⟨i⟩ f,
rw h at this,
exact this.false
end | theorem | ordinal.lsub_eq_zero_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_empty",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_const {ι} [hι : nonempty ι] (o : ordinal) : lsub (λ _ : ι, o) = succ o | sup_const (succ o) | theorem | ordinal.lsub_const | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_unique {ι} [hι : unique ι] (f : ι → ordinal) : lsub f = succ (f default) | sup_unique _ | theorem | ordinal.lsub_unique | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_le_of_range_subset {ι ι'} {f : ι → ordinal} {g : ι' → ordinal}
(h : set.range f ⊆ set.range g) : lsub.{u (max v w)} f ≤ lsub.{v (max u w)} g | sup_le_of_range_subset (by convert set.image_subset _ h; apply set.range_comp) | theorem | ordinal.lsub_le_of_range_subset | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"set.image_subset",
"set.range",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_eq_of_range_eq {ι ι'} {f : ι → ordinal} {g : ι' → ordinal}
(h : set.range f = set.range g) : lsub.{u (max v w)} f = lsub.{v (max u w)} g | (lsub_le_of_range_subset h.le).antisymm (lsub_le_of_range_subset.{v u w} h.ge) | theorem | ordinal.lsub_eq_of_range_eq | 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 | |
lsub_sum {α : Type u} {β : Type v} (f : α ⊕ β → ordinal) : lsub.{(max u v) w} f =
max (lsub.{u (max v w)} (λ a, f (sum.inl a))) (lsub.{v (max u w)} (λ b, f (sum.inr b))) | sup_sum _ | theorem | ordinal.lsub_sum | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_not_mem_range {ι} (f : ι → ordinal) : lsub f ∉ set.range f | λ ⟨i, h⟩, h.not_lt (lt_lsub f i) | theorem | ordinal.lsub_not_mem_range | 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 | |
nonempty_compl_range {ι : Type u} (f : ι → ordinal.{max u v}) : (set.range f)ᶜ.nonempty | ⟨_, lsub_not_mem_range f⟩ | theorem | ordinal.nonempty_compl_range | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_typein (o : ordinal) :
lsub.{u u} (typein ((<) : o.out.α → o.out.α → Prop)) = o | (lsub_le.{u u} typein_lt_self).antisymm begin
by_contra' h,
nth_rewrite 0 ←type_lt o at h,
simpa [typein_enum] using lt_lsub.{u u} (typein (<)) (enum (<) _ h)
end | theorem | ordinal.lsub_typein | 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_typein_limit {o : ordinal} (ho : ∀ a, a < o → succ a < o) :
sup.{u u} (typein ((<) : o.out.α → o.out.α → Prop)) = o | by rw (sup_eq_lsub_iff_succ.{u u} (typein (<))).2; rwa lsub_typein o | theorem | ordinal.sup_typein_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 | |
sup_typein_succ {o : ordinal} :
sup.{u u} (typein ((<) : (succ o).out.α → (succ o).out.α → Prop)) = o | begin
cases sup_eq_lsub_or_sup_succ_eq_lsub.{u u}
(typein ((<) : (succ o).out.α → (succ o).out.α → Prop)) with h h,
{ rw sup_eq_lsub_iff_succ at h,
simp only [lsub_typein] at h,
exact (h o (lt_succ o)).false.elim },
rw [←succ_eq_succ_iff, h],
apply lsub_typein
end | theorem | ordinal.sup_typein_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub (o : ordinal.{u}) (f : Π a < o, ordinal.{max u v}) : ordinal.{max u v} | o.bsup (λ a ha, succ (f a ha)) | def | ordinal.blsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | The least strict upper bound of a family of ordinals indexed by the set of ordinals less than
some `o : ordinal.{u}`.
This is to `lsub` as `bsup` is to `sup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bsup_eq_blsub (o : ordinal) (f : Π a < o, ordinal) :
bsup o (λ a ha, succ (f a ha)) = blsub o f | rfl | theorem | ordinal.bsup_eq_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 | |
lsub_eq_blsub' {ι} (r : ι → ι → Prop) [is_well_order ι r] {o} (ho : type r = o) (f) :
lsub (family_of_bfamily' r ho f) = blsub o f | sup_eq_bsup' r ho (λ a ha, succ (f a ha)) | theorem | ordinal.lsub_eq_blsub' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_eq_lsub {ι ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop)
[is_well_order ι r] [is_well_order ι' r'] {o} (ho : type r = o) (ho' : type r' = o)
(f : Π a < o, ordinal) : lsub (family_of_bfamily' r ho f) = lsub (family_of_bfamily' r' ho' f) | by rw [lsub_eq_blsub', lsub_eq_blsub'] | theorem | ordinal.lsub_eq_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lsub_eq_blsub {o} (f : Π a < o, ordinal) :
lsub (family_of_bfamily o f) = blsub o f | lsub_eq_blsub' _ _ _ | theorem | ordinal.lsub_eq_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 | |
blsub_eq_lsub' {ι} (r : ι → ι → Prop) [is_well_order ι r] (f : ι → ordinal) :
blsub _ (bfamily_of_family' r f) = lsub f | bsup_eq_sup' r (succ ∘ f) | theorem | ordinal.blsub_eq_lsub' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_eq_blsub {ι : Type u} (r r' : ι → ι → Prop) [is_well_order ι r] [is_well_order ι r']
(f : ι → ordinal) : blsub _ (bfamily_of_family' r f) = blsub _ (bfamily_of_family' r' f) | by rw [blsub_eq_lsub', blsub_eq_lsub'] | theorem | ordinal.blsub_eq_blsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_eq_lsub {ι} (f : ι → ordinal) : blsub _ (bfamily_of_family f) = lsub f | blsub_eq_lsub' _ _ | theorem | ordinal.blsub_eq_lsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_congr {o₁ o₂ : ordinal} (f : Π a < o₁, ordinal) (ho : o₁ = o₂) :
blsub o₁ f = blsub o₂ (λ a h, f a (h.trans_eq ho.symm)) | by subst ho | lemma | ordinal.blsub_congr | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_le_iff {o f a} : blsub o f ≤ a ↔ ∀ i h, f i h < a | by { convert bsup_le_iff, simp [succ_le_iff] } | theorem | ordinal.blsub_le_iff | 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 {o : ordinal} {f : Π b < o, ordinal} {a} : (∀ i h, f i h < a) → blsub o f ≤ a | blsub_le_iff.2 | theorem | ordinal.blsub_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_blsub {o} (f : Π a < o, ordinal) (i h) : f i h < blsub o f | blsub_le_iff.1 le_rfl _ _ | theorem | ordinal.lt_blsub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_blsub_iff {o f a} : a < blsub o f ↔ ∃ i hi, a ≤ f i hi | by simpa only [not_forall, not_lt, not_le] using not_congr (@blsub_le_iff _ f a) | theorem | ordinal.lt_blsub_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_le_blsub {o} (f : Π a < o, ordinal) : bsup o f ≤ blsub o f | bsup_le (λ i h, (lt_blsub f i h).le) | theorem | ordinal.bsup_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 | |
blsub_le_bsup_succ {o} (f : Π a < o, ordinal) : blsub o f ≤ succ (bsup o f) | blsub_le (λ i h, lt_succ_iff.2 (le_bsup f i h)) | theorem | ordinal.blsub_le_bsup_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_blsub_or_succ_bsup_eq_blsub {o} (f : Π a < o, ordinal) :
bsup o f = blsub o f ∨ succ (bsup o f) = blsub o f | by { rw [←sup_eq_bsup, ←lsub_eq_blsub], exact sup_eq_lsub_or_sup_succ_eq_lsub _ } | theorem | ordinal.bsup_eq_blsub_or_succ_bsup_eq_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 | |
bsup_succ_le_blsub {o} (f : Π a < o, ordinal) :
succ (bsup o f) ≤ blsub o f ↔ ∃ i hi, f i hi = bsup o f | begin
refine ⟨λ h, _, _⟩,
{ by_contra' hf,
exact ne_of_lt (succ_le_iff.1 h) (le_antisymm (bsup_le_blsub f)
(blsub_le (lt_bsup_of_ne_bsup.1 hf))) },
rintro ⟨_, _, hf⟩,
rw [succ_le_iff, ←hf],
exact lt_blsub _ _ _
end | theorem | ordinal.bsup_succ_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 | |
bsup_succ_eq_blsub {o} (f : Π a < o, ordinal) :
succ (bsup o f) = blsub o f ↔ ∃ i hi, f i hi = bsup o f | (blsub_le_bsup_succ f).le_iff_eq.symm.trans (bsup_succ_le_blsub f) | theorem | ordinal.bsup_succ_eq_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 | |
bsup_eq_blsub_iff_succ {o} (f : Π a < o, ordinal) :
bsup o f = blsub o f ↔ ∀ a < blsub o f, succ a < blsub o f | by { rw [←sup_eq_bsup, ←lsub_eq_blsub], apply sup_eq_lsub_iff_succ } | theorem | ordinal.bsup_eq_blsub_iff_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_blsub_iff_lt_bsup {o} (f : Π a < o, ordinal) :
bsup o f = blsub o f ↔ ∀ i hi, f i hi < bsup o f | ⟨λ h i, (by { rw h, apply lt_blsub }), λ h, le_antisymm (bsup_le_blsub f) (blsub_le h)⟩ | theorem | ordinal.bsup_eq_blsub_iff_lt_bsup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_eq_blsub_of_lt_succ_limit {o} (ho : is_limit o) {f : Π a < o, ordinal}
(hf : ∀ a ha, f a ha < f (succ a) (ho.2 a ha)) : bsup o f = blsub o f | begin
rw bsup_eq_blsub_iff_lt_bsup,
exact λ i hi, (hf i hi).trans_le (le_bsup f _ _)
end | theorem | ordinal.bsup_eq_blsub_of_lt_succ_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 | |
blsub_succ_of_mono {o : ordinal} {f : Π a < succ o, ordinal}
(hf : ∀ {i j} hi hj, i ≤ j → f i hi ≤ f j hj) : blsub _ f = succ (f o (lt_succ o)) | bsup_succ_of_mono $ λ i j hi hj h, succ_le_succ (hf hi hj h) | theorem | ordinal.blsub_succ_of_mono | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_eq_zero_iff {o} {f : Π a < o, ordinal} : blsub o f = 0 ↔ o = 0 | by { rw [←lsub_eq_blsub, lsub_eq_zero_iff], exact out_empty_iff_eq_zero } | theorem | ordinal.blsub_eq_zero_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_zero (f : Π a < (0 : ordinal), ordinal) : blsub 0 f = 0 | by rwa blsub_eq_zero_iff | lemma | ordinal.blsub_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 | |
blsub_pos {o : ordinal} (ho : 0 < o) (f : Π a < o, ordinal) : 0 < blsub o f | (ordinal.zero_le _).trans_lt (lt_blsub f 0 ho) | lemma | ordinal.blsub_pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_type (r : α → α → Prop) [is_well_order α r] (f) :
blsub (type r) f = lsub (λ a, f (typein r a) (typein_lt_type _ _)) | eq_of_forall_ge_iff $ λ o,
by rw [blsub_le_iff, lsub_le_iff]; exact
⟨λ H b, H _ _, λ H i h, by simpa only [typein_enum] using H (enum r i h)⟩ | theorem | ordinal.blsub_type | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_of_forall_ge_iff",
"is_well_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_const {o : ordinal} (ho : o ≠ 0) (a : ordinal) : blsub.{u v} o (λ _ _, a) = succ a | bsup_const.{u v} ho (succ a) | theorem | ordinal.blsub_const | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_one (f : Π a < (1 : ordinal), ordinal) : blsub 1 f = succ (f 0 zero_lt_one) | bsup_one _ | theorem | ordinal.blsub_one | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_id : ∀ o, blsub.{u u} o (λ x _, x) = o | lsub_typein | theorem | ordinal.blsub_id | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bsup_id_limit {o : ordinal} : (∀ a < o, succ a < o) → bsup.{u u} o (λ x _, x) = o | sup_typein_limit | theorem | ordinal.bsup_id_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 | |
bsup_id_succ (o) : bsup.{u u} (succ o) (λ x _, x) = o | sup_typein_succ | theorem | ordinal.bsup_id_succ | 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_of_brange_subset {o o'} {f : Π a < o, ordinal} {g : Π a < o', ordinal}
(h : brange o f ⊆ brange o' g) : blsub.{u (max v w)} o f ≤ blsub.{v (max u w)} o' g | bsup_le_of_brange_subset $ λ a ⟨b, hb, hb'⟩, begin
obtain ⟨c, hc, hc'⟩ := h ⟨b, hb, rfl⟩,
simp_rw ←hc' at hb',
exact ⟨c, hc, hb'⟩
end | theorem | ordinal.blsub_le_of_brange_subset | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
blsub_eq_of_brange_eq {o o'} {f : Π a < o, ordinal} {g : Π a < o', ordinal}
(h : {o | ∃ i hi, f i hi = o} = {o | ∃ i hi, g i hi = o}) :
blsub.{u (max v w)} o f = blsub.{v (max u w)} o' g | (blsub_le_of_brange_subset h.le).antisymm (blsub_le_of_brange_subset.{v u w} h.ge) | theorem | ordinal.blsub_eq_of_brange_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 | |
bsup_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) :
bsup o' (λ a ha, f (g a ha) (by { rw ←hg, apply lt_blsub })) = bsup o f | begin
apply le_antisymm;
refine bsup_le (λ i hi, _),
{ apply le_bsup },
{ rw [←hg, lt_blsub_iff] at hi,
rcases hi with ⟨j, hj, hj'⟩,
exact (hf _ _ hj').trans (le_bsup _ _ _) }
end | theorem | ordinal.bsup_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 |
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