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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principal_add_omega_opow (o : ordinal) : principal (+) (omega ^ o) | principal_add_iff_add_left_eq_self.2 (λ a, add_omega_opow) | theorem | ordinal.principal_add_omega_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_add_iff_zero_or_omega_opow {o : ordinal} :
principal (+) o ↔ o = 0 ∨ ∃ a, o = omega ^ a | begin
rcases eq_or_ne o 0 with rfl | ho,
{ simp only [principal_zero, or.inl] },
{ rw [principal_add_iff_add_left_eq_self],
simp only [ho, false_or],
refine ⟨λ H, ⟨_, ((lt_or_eq_of_le (opow_log_le_self _ ho))
.resolve_left $ λ h, _).symm⟩, λ ⟨b, e⟩, e.symm ▸ λ a, add_omega_opow⟩,
have := H _ h... | theorem | ordinal.principal_add_iff_zero_or_omega_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"eq_or_ne",
"mul_add_one",
"mul_zero",
"nat.cast_succ",
"nat.cast_zero",
"not_lt_of_le",
"ordinal"
] | The main characterization theorem for additive principal ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opow_principal_add_of_principal_add {a} (ha : principal (+) a) (b : ordinal) :
principal (+) (a ^ b) | begin
rcases principal_add_iff_zero_or_omega_opow.1 ha with rfl | ⟨c, rfl⟩,
{ rcases eq_or_ne b 0 with rfl | hb,
{ rw opow_zero, exact principal_add_one },
{ rwa zero_opow hb } },
{ rw ←opow_mul, exact principal_add_omega_opow _ }
end | theorem | ordinal.opow_principal_add_of_principal_add | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_absorp {a b c : ordinal} (h₁ : a < omega ^ b) (h₂ : omega ^ b ≤ c) : a + c = c | by rw [← ordinal.add_sub_cancel_of_le h₂, ← add_assoc, add_omega_opow h₁] | theorem | ordinal.add_absorp | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_principal_add_is_principal_add (a : ordinal.{u}) {b : ordinal.{u}} (hb₁ : b ≠ 1)
(hb : principal (+) b) : principal (+) (a * b) | begin
rcases eq_zero_or_pos a with rfl | ha,
{ rw zero_mul,
exact principal_zero },
{ rcases eq_zero_or_pos b with rfl | hb₁',
{ rw mul_zero,
exact principal_zero },
{ rw [← succ_le_iff, succ_zero] at hb₁',
intros c d hc hd,
rw lt_mul_of_limit (principal_add_is_limit (lt_of_le_of_ne ... | theorem | ordinal.mul_principal_add_is_principal_add | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_one : principal (*) 1 | by { rw principal_one_iff, exact zero_mul _ } | theorem | ordinal.principal_mul_one | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_two : principal (*) 2 | λ a b ha hb, begin
have h₂ : succ (1 : ordinal) = 2 := rfl,
rw [←h₂, lt_succ_iff] at *,
convert mul_le_mul' ha hb,
exact (mul_one 1).symm
end | theorem | ordinal.principal_mul_two | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_le_mul'",
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_of_le_two {o : ordinal} (ho : o ≤ 2) : principal (*) o | begin
rcases lt_or_eq_of_le ho with ho | rfl,
{ have h₂ : succ (1 : ordinal) = 2 := rfl,
rw [←h₂, lt_succ_iff] at ho,
rcases lt_or_eq_of_le ho with ho | rfl,
{ rw lt_one_iff_zero.1 ho,
exact principal_zero },
{ exact principal_mul_one } },
{ exact principal_mul_two }
end | theorem | ordinal.principal_mul_of_le_two | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_add_of_principal_mul {o : ordinal} (ho : principal (*) o) (ho₂ : o ≠ 2) :
principal (+) o | begin
cases lt_or_gt_of_ne ho₂ with ho₁ ho₂,
{ change o < succ 1 at ho₁,
rw lt_succ_iff at ho₁,
exact principal_add_of_le_one ho₁ },
{ refine λ a b hao hbo, lt_of_le_of_lt _ (ho (max_lt hao hbo) ho₂),
rw mul_two,
exact add_le_add (le_max_left a b) (le_max_right a b) }
end | theorem | ordinal.principal_add_of_principal_mul | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_two",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_is_limit {o : ordinal.{u}} (ho₂ : 2 < o) (ho : principal (*) o) :
o.is_limit | principal_add_is_limit
((lt_succ 1).trans ho₂)
(principal_add_of_principal_mul ho (ne_of_gt ho₂)) | theorem | ordinal.principal_mul_is_limit | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_iff_mul_left_eq {o : ordinal} :
principal (*) o ↔ ∀ a, 0 < a → a < o → a * o = o | begin
refine ⟨λ h a ha₀ hao, _, λ h a b hao hbo, _⟩,
{ cases le_or_gt o 2 with ho ho,
{ convert one_mul o,
apply le_antisymm,
{ have : a < succ 1 := hao.trans_le ho,
rwa lt_succ_iff at this },
{ rwa [←succ_le_iff, succ_zero] at ha₀ } },
{ exact op_eq_self_of_principal hao (mul_is_n... | theorem | ordinal.principal_mul_iff_mul_left_eq | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"eq_or_ne",
"one_mul",
"ordinal",
"strict_mono",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_omega : principal (*) omega | λ a b ha hb, match a, b, lt_omega.1 ha, lt_omega.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by { rw [← nat_cast_mul], apply nat_lt_omega }
end | theorem | ordinal.principal_mul_omega | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_omega {a : ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega | principal_mul_iff_mul_left_eq.1 (principal_mul_omega) a a0 ha | theorem | ordinal.mul_omega | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_omega_opow {a b c : ordinal}
(c0 : 0 < c) (ha : a < omega ^ c) (hb : b < omega) : a * b < omega ^ c | begin
rcases zero_or_succ_or_limit c with rfl|⟨c,rfl⟩|l,
{ exact (lt_irrefl _).elim c0 },
{ rw opow_succ at ha,
rcases ((mul_is_normal $ opow_pos _ omega_pos).limit_lt
omega_is_limit).1 ha with ⟨n, hn, an⟩,
apply (mul_le_mul_right' (le_of_lt an) _).trans_lt,
rw [opow_succ, mul_assoc, mul_lt_mul_... | theorem | ordinal.mul_lt_omega_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_assoc",
"mul_le_mul'",
"mul_le_mul_right'",
"mul_lt_mul_iff_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_omega_opow_opow {a b : ordinal} (a0 : 0 < a) (h : a < omega ^ omega ^ b) :
a * omega ^ omega ^ b = omega ^ omega ^ b | begin
by_cases b0 : b = 0, {rw [b0, opow_zero, opow_one] at h ⊢, exact mul_omega a0 h},
refine le_antisymm _
(by simpa only [one_mul] using mul_le_mul_right' (one_le_iff_pos.2 a0) (omega ^ omega ^ b)),
rcases (lt_opow_of_limit omega_ne_zero (opow_is_limit_left omega_is_limit b0)).1 h
with ⟨x, xb, ax⟩,
a... | theorem | ordinal.mul_omega_opow_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_le_mul_right'",
"one_mul",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_omega_opow_opow (o : ordinal) : principal (*) (omega ^ omega ^ o) | principal_mul_iff_mul_left_eq.2 (λ a, mul_omega_opow_opow) | theorem | ordinal.principal_mul_omega_opow_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_add_of_principal_mul_opow {o b : ordinal} (hb : 1 < b)
(ho : principal (*) (b ^ o)) : principal (+) o | λ x y hx hy, begin
have := ho ((opow_lt_opow_iff_right hb).2 hx) ((opow_lt_opow_iff_right hb).2 hy),
rwa [←opow_add, opow_lt_opow_iff_right hb] at this
end | theorem | ordinal.principal_add_of_principal_mul_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_mul_iff_le_two_or_omega_opow_opow {o : ordinal} :
principal (*) o ↔ o ≤ 2 ∨ ∃ a, o = omega ^ omega ^ a | begin
refine ⟨λ ho, _, _⟩,
{ cases le_or_lt o 2 with ho₂ ho₂,
{ exact or.inl ho₂ },
rcases principal_add_iff_zero_or_omega_opow.1 (principal_add_of_principal_mul ho ho₂.ne')
with rfl | ⟨a, rfl⟩,
{ exact (ordinal.not_lt_zero 2 ho₂).elim },
rcases principal_add_iff_zero_or_omega_opow.1
(pr... | theorem | ordinal.principal_mul_iff_le_two_or_omega_opow_opow | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal",
"ordinal.not_lt_zero"
] | The main characterization theorem for multiplicative principal ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_omega_dvd {a : ordinal}
(a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b | | _ ⟨b, rfl⟩ := by rw [← mul_assoc, mul_omega a0 ha] | theorem | ordinal.mul_omega_dvd | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_assoc",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_opow_log_succ {a b : ordinal.{u}} (ha : a ≠ 0) (hb : principal (*) b) (hb₂ : 2 < b) :
a * b = b ^ succ (log b a) | begin
apply le_antisymm,
{ have hbl := principal_mul_is_limit hb₂ hb,
rw [←is_normal.bsup_eq.{u u} (mul_is_normal (ordinal.pos_iff_ne_zero.2 ha)) hbl, bsup_le_iff],
intros c hcb,
have hb₁ : 1 < b := (lt_succ 1).trans hb₂,
have hbo₀ : b ^ b.log a ≠ 0 := ordinal.pos_iff_ne_zero.1 (opow_pos _ (zero_lt_... | theorem | ordinal.mul_eq_opow_log_succ | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"mul_assoc",
"mul_le_mul_left'",
"mul_le_mul_right'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_opow_omega : principal (^) omega | λ a b ha hb, match a, b, lt_omega.1 ha, lt_omega.1 hb with
| _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ := by { simp_rw ←nat_cast_opow, apply nat_lt_omega }
end | theorem | ordinal.principal_opow_omega | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opow_omega {a : ordinal} (a1 : 1 < a) (h : a < omega) : a ^ omega = omega | le_antisymm
((opow_le_of_limit (one_le_iff_ne_zero.1 $ le_of_lt a1) omega_is_limit).2
(λ b hb, (principal_opow_omega h hb).le))
(right_le_opow _ a1) | theorem | ordinal.opow_omega | set_theory.ordinal | src/set_theory/ordinal/principal.lean | [
"set_theory.ordinal.fixed_point"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_singleton_iff : is_open ({a} : set ordinal) ↔ ¬ is_limit a | begin
refine ⟨λ h ha, _, λ ha, _⟩,
{ obtain ⟨b, c, hbc, hbc'⟩ := (mem_nhds_iff_exists_Ioo_subset'
⟨0, ordinal.pos_iff_ne_zero.2 ha.1⟩ ⟨_, lt_succ a⟩).1 (h.mem_nhds rfl),
have hba := ha.2 b hbc.1,
exact hba.ne (hbc' ⟨lt_succ b, hba.trans hbc.2⟩) },
{ rcases zero_or_succ_or_limit a with rfl | ⟨b, hb⟩ ... | theorem | ordinal.is_open_singleton_iff | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_open",
"is_open_Ioo",
"is_open_gt'",
"mem_nhds_iff_exists_Ioo_subset'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff : is_open s ↔ ∀ o ∈ s, is_limit o → ∃ a < o, set.Ioo a o ⊆ s | begin
classical,
refine ⟨_, λ h, _⟩,
{ rw is_open_iff_generate_intervals,
intros h o hos ho,
have ho₀ := ordinal.pos_iff_ne_zero.2 ho.1,
induction h with t ht t u ht hu ht' hu' t ht H,
{ rcases ht with ⟨a, rfl | rfl⟩,
{ exact ⟨a, hos, λ b hb, hb.1⟩ },
{ exact ⟨0, ho₀, λ b hb, hb.2.tran... | theorem | ordinal.is_open_iff | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_open",
"is_open_Ioo",
"is_open_Union",
"is_open_iff_generate_intervals",
"ordinal",
"set.Ioo",
"set.mem_singleton",
"set.mem_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_sup : a ∈ closure s ↔ ∃ {ι : Type u} [nonempty ι] (f : ι → ordinal),
(∀ i, f i ∈ s) ∧ sup.{u u} f = a | begin
refine mem_closure_iff.trans ⟨λ h, _, _⟩,
{ by_cases has : a ∈ s,
{ exact ⟨punit, by apply_instance, λ _, a, λ _, has, sup_const a⟩ },
{ have H := λ b (hba : b < a), h _ (@is_open_Ioo _ _ _ _ b (a + 1)) ⟨hba, lt_succ a⟩,
let f : a.out.α → ordinal := λ i, classical.some (H (typein (<) i) (typein_... | theorem | ordinal.mem_closure_iff_sup | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"closure",
"eq_or_lt_of_le",
"is_open_Ioo",
"mem_nhds_iff_exists_Ioo_subset'",
"ordinal",
"set.Ioo",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_iff_sup (hs : is_closed s) :
a ∈ s ↔ ∃ {ι : Type u} (hι : nonempty ι) (f : ι → ordinal),
(∀ i, f i ∈ s) ∧ sup.{u u} f = a | by rw [←mem_closure_iff_sup, hs.closure_eq] | theorem | ordinal.mem_closed_iff_sup | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_closed",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff_bsup :
a ∈ closure s ↔ ∃ {o : ordinal} (ho : o ≠ 0) (f : Π a < o, ordinal),
(∀ i hi, f i hi ∈ s) ∧ bsup.{u u} o f = a | mem_closure_iff_sup.trans ⟨
λ ⟨ι, ⟨i⟩, f, hf, ha⟩, ⟨_, λ h, (type_eq_zero_iff_is_empty.1 h).elim i, bfamily_of_family f,
λ i hi, hf _, by rwa bsup_eq_sup⟩,
λ ⟨o, ho, f, hf, ha⟩, ⟨_, out_nonempty_iff_ne_zero.2 ho, family_of_bfamily o f,
λ i, hf _ _, by rwa sup_eq_bsup⟩⟩ | theorem | ordinal.mem_closure_iff_bsup | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"closure",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_iff_bsup (hs : is_closed s) :
a ∈ s ↔ ∃ {o : ordinal} (ho : o ≠ 0) (f : Π a < o, ordinal),
(∀ i hi, f i hi ∈ s) ∧ bsup.{u u} o f = a | by rw [←mem_closure_iff_bsup, hs.closure_eq] | theorem | ordinal.mem_closed_iff_bsup | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_closed",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_sup :
is_closed s ↔ ∀ {ι : Type u} (hι : nonempty ι) (f : ι → ordinal),
(∀ i, f i ∈ s) → sup.{u u} f ∈ s | begin
use λ hs ι hι f hf, (mem_closed_iff_sup hs).2 ⟨ι, hι, f, hf, rfl⟩,
rw ←closure_subset_iff_is_closed,
intros h x hx,
rcases mem_closure_iff_sup.1 hx with ⟨ι, hι, f, hf, rfl⟩,
exact h hι f hf
end | theorem | ordinal.is_closed_iff_sup | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_closed",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff_bsup :
is_closed s ↔ ∀ {o : ordinal} (ho : o ≠ 0) (f : Π a < o, ordinal),
(∀ i hi, f i hi ∈ s) → bsup.{u u} o f ∈ s | begin
rw is_closed_iff_sup,
refine ⟨λ H o ho f hf, H (out_nonempty_iff_ne_zero.2 ho) _ _, λ H ι hι f hf, _⟩,
{ exact λ i, hf _ _ },
{ rw ←bsup_eq_sup,
apply H (type_ne_zero_iff_nonempty.2 hι),
exact λ i hi, hf _ }
end | theorem | ordinal.is_closed_iff_bsup | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_closed",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_of_mem_frontier (ha : a ∈ frontier s) : is_limit a | begin
simp only [frontier_eq_closure_inter_closure, set.mem_inter_iff, mem_closure_iff] at ha,
by_contra h,
rw ←is_open_singleton_iff at h,
rcases ha.1 _ h rfl with ⟨b, hb, hb'⟩,
rcases ha.2 _ h rfl with ⟨c, hc, hc'⟩,
rw set.mem_singleton_iff at *,
subst hb, subst hc,
exact hc' hb'
end | theorem | ordinal.is_limit_of_mem_frontier | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"by_contra",
"frontier",
"frontier_eq_closure_inter_closure",
"mem_closure_iff",
"set.mem_inter_iff",
"set.mem_singleton_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal_iff_strict_mono_and_continuous (f : ordinal.{u} → ordinal.{u}) :
is_normal f ↔ strict_mono f ∧ continuous f | begin
refine ⟨λ h, ⟨h.strict_mono, _⟩, _⟩,
{ rw continuous_def,
intros s hs,
rw is_open_iff at *,
intros o ho ho',
rcases hs _ ho (h.is_limit ho') with ⟨a, ha, has⟩,
rw [←is_normal.bsup_eq.{u u} h ho', lt_bsup] at ha,
rcases ha with ⟨b, hb, hab⟩,
exact ⟨b, hb, λ c hc,
set.mem_preim... | theorem | ordinal.is_normal_iff_strict_mono_and_continuous | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"continuous",
"continuous_def",
"is_closed.preimage",
"is_closed_Iic",
"set.Iic",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
enum_ord_is_normal_iff_is_closed (hs : s.unbounded (<)) :
is_normal (enum_ord s) ↔ is_closed s | begin
have Hs := enum_ord_strict_mono hs,
refine ⟨λ h, is_closed_iff_sup.2 (λ ι hι f hf, _),
λ h, (is_normal_iff_strict_mono_limit _).2 ⟨Hs, λ a ha o H, _⟩⟩,
{ let g : ι → ordinal.{u} := λ i, (enum_ord_order_iso hs).symm ⟨_, hf i⟩,
suffices : enum_ord s (sup.{u u} g) = sup.{u u} f,
{ rw ←this, exact e... | theorem | ordinal.enum_ord_is_normal_iff_is_closed | set_theory.ordinal | src/set_theory/ordinal/topology.lean | [
"set_theory.ordinal.arithmetic",
"topology.order.basic"
] | [
"is_closed",
"order_iso.apply_symm_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric : pgame → Prop | | ⟨l, r, L, R⟩ :=
(∀ i j, L i < R j) ∧ (∀ i, numeric (L i)) ∧ (∀ j, numeric (R j)) | def | pgame.numeric | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | A pre-game is numeric if everything in the L set is less than everything in the R set,
and all the elements of L and R are also numeric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
numeric_def {x : pgame} : numeric x ↔ (∀ i j, x.move_left i < x.move_right j) ∧
(∀ i, numeric (x.move_left i)) ∧ (∀ j, numeric (x.move_right j)) | by { cases x, refl } | lemma | pgame.numeric_def | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk {x : pgame} (h₁ : ∀ i j, x.move_left i < x.move_right j)
(h₂ : ∀ i, numeric (x.move_left i)) (h₃ : ∀ j, numeric (x.move_right j)) : numeric x | numeric_def.2 ⟨h₁, h₂, h₃⟩ | lemma | pgame.numeric.mk | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_lt_right {x : pgame} (o : numeric x) (i : x.left_moves) (j : x.right_moves) :
x.move_left i < x.move_right j | by { cases x, exact o.1 i j } | lemma | pgame.numeric.left_lt_right | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left {x : pgame} (o : numeric x) (i : x.left_moves) :
numeric (x.move_left i) | by { cases x, exact o.2.1 i } | lemma | pgame.numeric.move_left | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_right {x : pgame} (o : numeric x) (j : x.right_moves) :
numeric (x.move_right j) | by { cases x, exact o.2.2 j } | lemma | pgame.numeric.move_right | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_rec {C : pgame → Prop}
(H : ∀ l r (L : l → pgame) (R : r → pgame),
(∀ i j, L i < R j) → (∀ i, numeric (L i)) → (∀ i, numeric (R i)) →
(∀ i, C (L i)) → (∀ i, C (R i)) → C ⟨l, r, L, R⟩) :
∀ x, numeric x → C x | | ⟨l, r, L, R⟩ ⟨h, hl, hr⟩ :=
H _ _ _ _ h hl hr (λ i, numeric_rec _ (hl i)) (λ i, numeric_rec _ (hr i)) | theorem | pgame.numeric_rec | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabelling.numeric_imp {x y : pgame} (r : x ≡r y) (ox : numeric x) : numeric y | begin
induction x using pgame.move_rec_on with x IHl IHr generalizing y,
apply numeric.mk (λ i j, _) (λ i, _) (λ j, _),
{ rw ←lt_congr (r.move_left_symm i).equiv (r.move_right_symm j).equiv,
apply ox.left_lt_right },
{ exact IHl _ (ox.move_left _) (r.move_left_symm i) },
{ exact IHr _ (ox.move_right _) (r... | theorem | pgame.relabelling.numeric_imp | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"equiv",
"pgame",
"pgame.move_rec_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
relabelling.numeric_congr {x y : pgame} (r : x ≡r y) : numeric x ↔ numeric y | ⟨r.numeric_imp, r.symm.numeric_imp⟩ | theorem | pgame.relabelling.numeric_congr | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | Relabellings preserve being numeric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lf_asymm {x y : pgame} (ox : numeric x) (oy : numeric y) : x ⧏ y → ¬ y ⧏ x | begin
refine numeric_rec (λ xl xr xL xR hx oxl oxr IHxl IHxr, _) x ox y oy,
refine numeric_rec (λ yl yr yL yR hy oyl oyr IHyl IHyr, _),
rw [mk_lf_mk, mk_lf_mk], rintro (⟨i, h₁⟩ | ⟨j, h₁⟩) (⟨i, h₂⟩ | ⟨j, h₂⟩),
{ exact IHxl _ _ (oyl _) (h₁.move_left_lf _) (h₂.move_left_lf _) },
{ exact (le_trans h₂ h₁).not_gf (... | theorem | pgame.lf_asymm | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_lf {x y : pgame} (h : x ⧏ y) (ox : numeric x) (oy : numeric y) : x ≤ y | not_lf.1 (lf_asymm ox oy h) | theorem | pgame.le_of_lf | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_of_lf {x y : pgame} (h : x ⧏ y) (ox : numeric x) (oy : numeric y) : x < y | (lt_or_fuzzy_of_lf h).resolve_right (not_fuzzy_of_le (h.le ox oy)) | theorem | pgame.lt_of_lf | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lf_iff_lt {x y : pgame} (ox : numeric x) (oy : numeric y) : x ⧏ y ↔ x < y | ⟨λ h, h.lt ox oy, lf_of_lt⟩ | theorem | pgame.lf_iff_lt | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_forall_lt {x y : pgame} (ox : x.numeric) (oy : y.numeric) :
x ≤ y ↔ (∀ i, x.move_left i < y) ∧ ∀ j, x < y.move_right j | begin
refine le_iff_forall_lf.trans (and_congr _ _);
refine forall_congr (λ i, lf_iff_lt _ _);
apply_rules [numeric.move_left, numeric.move_right]
end | theorem | pgame.le_iff_forall_lt | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | Definition of `x ≤ y` on numeric pre-games, in terms of `<` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_iff_exists_le {x y : pgame} (ox : x.numeric) (oy : y.numeric) :
x < y ↔ (∃ i, x ≤ y.move_left i) ∨ ∃ j, x.move_right j ≤ y | by rw [←lf_iff_lt ox oy, lf_iff_exists_le] | theorem | pgame.lt_iff_exists_le | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | Definition of `x < y` on numeric pre-games, in terms of `≤` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_of_exists_le {x y : pgame} (ox : x.numeric) (oy : y.numeric) :
((∃ i, x ≤ y.move_left i) ∨ ∃ j, x.move_right j ≤ y) → x < y | (lt_iff_exists_le ox oy).2 | theorem | pgame.lt_of_exists_le | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_def {x y : pgame} (ox : x.numeric) (oy : y.numeric) : x < y ↔
(∃ i, (∀ i', x.move_left i' < y.move_left i) ∧ ∀ j, x < (y.move_left i).move_right j) ∨
∃ j, (∀ i, (x.move_right j).move_left i < y) ∧ ∀ j', x.move_right j < y.move_right j' | begin
rw [←lf_iff_lt ox oy, lf_def],
refine or_congr _ _;
refine exists_congr (λ x_1, _);
refine and_congr _ _;
refine (forall_congr $ λ i, lf_iff_lt _ _);
apply_rules [numeric.move_left, numeric.move_right]
end | theorem | pgame.lt_def | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | The definition of `x < y` on numeric pre-games, in terms of `<` two moves later. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_fuzzy {x y : pgame} (ox : numeric x) (oy : numeric y) : ¬ fuzzy x y | λ h, not_lf.2 ((lf_of_fuzzy h).le ox oy) h.2 | theorem | pgame.not_fuzzy | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_or_equiv_or_gt {x y : pgame} (ox : numeric x) (oy : numeric y) : x < y ∨ x ≈ y ∨ y < x | (lf_or_equiv_or_gf x y).imp (λ h, h.lt ox oy) $ or.imp_right $ λ h, h.lt oy ox | theorem | pgame.lt_or_equiv_or_gt | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_of_is_empty (x : pgame) [is_empty x.left_moves] [is_empty x.right_moves] :
numeric x | numeric.mk is_empty_elim is_empty_elim is_empty_elim | theorem | pgame.numeric_of_is_empty | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"is_empty",
"is_empty_elim",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_of_is_empty_left_moves (x : pgame) [is_empty x.left_moves] :
(∀ j, numeric (x.move_right j)) → numeric x | numeric.mk is_empty_elim is_empty_elim | theorem | pgame.numeric_of_is_empty_left_moves | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"is_empty",
"is_empty_elim",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_of_is_empty_right_moves (x : pgame) [is_empty x.right_moves]
(H : ∀ i, numeric (x.move_left i)) : numeric x | numeric.mk (λ _, is_empty_elim) H is_empty_elim | theorem | pgame.numeric_of_is_empty_right_moves | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"is_empty",
"is_empty_elim",
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_zero : numeric 0 | numeric_of_is_empty 0 | theorem | pgame.numeric_zero | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_one : numeric 1 | numeric_of_is_empty_right_moves 1 $ λ _, numeric_zero | theorem | pgame.numeric_one | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric.neg : Π {x : pgame} (o : numeric x), numeric (-x) | | ⟨l, r, L, R⟩ o := ⟨λ j i, neg_lt_neg_iff.2 (o.1 i j), λ j, (o.2.2 j).neg, λ i, (o.2.1 i).neg⟩ | theorem | pgame.numeric.neg | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_lt {x : pgame} (o : numeric x) (i) : x.move_left i < x | (move_left_lf i).lt (o.move_left i) o | theorem | pgame.numeric.move_left_lt | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
move_left_le {x : pgame} (o : numeric x) (i) : x.move_left i ≤ x | (o.move_left_lt i).le | theorem | pgame.numeric.move_left_le | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_move_right {x : pgame} (o : numeric x) (j) : x < x.move_right j | (lf_move_right j).lt o (o.move_right j) | theorem | pgame.numeric.lt_move_right | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_move_right {x : pgame} (o : numeric x) (j) : x ≤ x.move_right j | (o.lt_move_right j).le | theorem | pgame.numeric.le_move_right | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add : Π {x y : pgame} (ox : numeric x) (oy : numeric y), numeric (x + y) | | ⟨xl, xr, xL, xR⟩ ⟨yl, yr, yL, yR⟩ ox oy :=
⟨begin
rintros (ix|iy) (jx|jy),
{ exact add_lt_add_right (ox.1 ix jx) _ },
{ exact (add_lf_add_of_lf_of_le (lf_mk _ _ ix) (oy.le_move_right jy)).lt
((ox.move_left ix).add oy) (ox.add (oy.move_right jy)) },
{ exact (add_lf_add_of_lf_of_le (mk_lf _ _ jx) (oy.m... | theorem | pgame.numeric.add | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub {x y : pgame} (ox : numeric x) (oy : numeric y) : numeric (x - y) | ox.add oy.neg | lemma | pgame.numeric.sub | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_nat : Π (n : ℕ), numeric n | | 0 := numeric_zero
| (n + 1) := (numeric_nat n).add numeric_one | theorem | pgame.numeric_nat | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | Pre-games defined by natural numbers are numeric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
numeric_to_pgame (o : ordinal) : o.to_pgame.numeric | begin
induction o using ordinal.induction with o IH,
apply numeric_of_is_empty_right_moves,
simpa using λ i, IH _ (ordinal.to_left_moves_to_pgame_symm_lt i)
end | theorem | pgame.numeric_to_pgame | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"ordinal",
"ordinal.induction",
"ordinal.to_left_moves_to_pgame_symm_lt"
] | Ordinal games are numeric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surreal | quotient (subtype.setoid numeric) | def | surreal | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | The type of surreal numbers. These are the numeric pre-games quotiented
by the equivalence relation `x ≈ y ↔ x ≤ y ∧ y ≤ x`. In the quotient,
the order becomes a total order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk (x : pgame) (h : x.numeric) : surreal | ⟦⟨x, h⟩⟧ | def | surreal.mk | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"pgame",
"surreal"
] | Construct a surreal number from a numeric pre-game. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift {α} (f : ∀ x, numeric x → α)
(H : ∀ {x y} (hx : numeric x) (hy : numeric y), x.equiv y → f x hx = f y hy) : surreal → α | quotient.lift (λ x : {x // numeric x}, f x.1 x.2) (λ x y, H x.2 y.2) | def | surreal.lift | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"lift",
"surreal"
] | Lift an equivalence-respecting function on pre-games to surreals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift₂ {α} (f : ∀ x y, numeric x → numeric y → α)
(H : ∀ {x₁ y₁ x₂ y₂} (ox₁ : numeric x₁) (oy₁ : numeric y₁) (ox₂ : numeric x₂) (oy₂ : numeric y₂),
x₁.equiv x₂ → y₁.equiv y₂ → f x₁ y₁ ox₁ oy₁ = f x₂ y₂ ox₂ oy₂) : surreal → surreal → α | lift (λ x ox, lift (λ y oy, f x y ox oy) (λ y₁ y₂ oy₁ oy₂, H _ _ _ _ equiv_rfl))
(λ x₁ x₂ ox₁ ox₂ h, funext $ quotient.ind $ by exact λ ⟨y, oy⟩, H _ _ _ _ h equiv_rfl) | def | surreal.lift₂ | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"lift",
"surreal"
] | Lift a binary equivalence-respecting function on pre-games to surreals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_game : surreal →+o game | { to_fun := lift (λ x _, ⟦x⟧) (λ x y ox oy, quot.sound),
map_zero' := rfl,
map_add' := by { rintros ⟨_, _⟩ ⟨_, _⟩, refl },
monotone' := by { rintros ⟨_, _⟩ ⟨_, _⟩, exact id } } | def | surreal.to_game | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"game",
"lift",
"surreal"
] | Casts a `surreal` number into a `game`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_to_game : to_game 0 = 0 | rfl | theorem | surreal.zero_to_game | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_to_game : to_game 1 = 1 | rfl | theorem | surreal.one_to_game | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_to_game : ∀ n : ℕ, to_game n = n | map_nat_cast' _ one_to_game | theorem | surreal.nat_to_game | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"map_nat_cast'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
upper_bound_numeric {ι : Type u} {f : ι → pgame.{u}} (H : ∀ i, (f i).numeric) :
(upper_bound f).numeric | numeric_of_is_empty_right_moves _ $ λ i, (H _).move_left _ | theorem | surreal.upper_bound_numeric | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lower_bound_numeric {ι : Type u} {f : ι → pgame.{u}} (H : ∀ i, (f i).numeric) :
(lower_bound f).numeric | numeric_of_is_empty_left_moves _ $ λ i, (H _).move_right _ | theorem | surreal.lower_bound_numeric | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_of_small (s : set surreal.{u}) [small.{u} s] : bdd_above s | begin
let g := subtype.val ∘ quotient.out ∘ subtype.val ∘ (equiv_shrink s).symm,
refine ⟨mk (upper_bound g) (upper_bound_numeric $ λ i, subtype.prop _), λ i hi, _⟩,
rw ←quotient.out_eq i,
show i.out.1 ≤ _,
simpa [g] using le_upper_bound g (equiv_shrink s ⟨i, hi⟩)
end | lemma | surreal.bdd_above_of_small | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"bdd_above",
"equiv_shrink",
"quotient.out",
"subtype.prop"
] | A small set `s` of surreals is bounded above. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bdd_below_of_small (s : set surreal.{u}) [small.{u} s] : bdd_below s | begin
let g := subtype.val ∘ quotient.out ∘ subtype.val ∘ (equiv_shrink s).symm,
refine ⟨mk (lower_bound g) (lower_bound_numeric $ λ i, subtype.prop _), λ i hi, _⟩,
rw ←quotient.out_eq i,
show _ ≤ i.out.1,
simpa [g] using lower_bound_le g (equiv_shrink s ⟨i, hi⟩)
end | lemma | surreal.bdd_below_of_small | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"bdd_below",
"equiv_shrink",
"quotient.out",
"subtype.prop"
] | A small set `s` of surreals is bounded below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_surreal : ordinal ↪o surreal | { to_fun := λ o, mk _ (numeric_to_pgame o),
inj' := λ a b h, to_pgame_equiv_iff.1 (quotient.exact h),
map_rel_iff' := @to_pgame_le_iff } | def | ordinal.to_surreal | set_theory.surreal | src/set_theory/surreal/basic.lean | [
"algebra.order.hom.monoid",
"set_theory.game.ordinal"
] | [
"ordinal",
"surreal"
] | Converts an ordinal into the corresponding surreal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_half : ℕ → pgame | | 0 := 1
| (n + 1) := ⟨punit, punit, 0, λ _, pow_half n⟩ | def | pgame.pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"pgame"
] | For a natural number `n`, the pre-game `pow_half (n + 1)` is recursively defined as
`{0 | pow_half n}`. These are the explicit expressions of powers of `1 / 2`. By definition, we have
`pow_half 0 = 1` and `pow_half 1 ≈ 1 / 2` and we prove later on that
`pow_half (n + 1) + pow_half (n + 1) ≈ pow_half n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_half_zero : pow_half 0 = 1 | rfl | lemma | pgame.pow_half_zero | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_left_moves (n) : (pow_half n).left_moves = punit | by cases n; refl | lemma | pgame.pow_half_left_moves | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_zero_right_moves : (pow_half 0).right_moves = pempty | rfl | lemma | pgame.pow_half_zero_right_moves | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"pempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_succ_right_moves (n) : (pow_half (n + 1)).right_moves = punit | rfl | lemma | pgame.pow_half_succ_right_moves | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_move_left (n i) : (pow_half n).move_left i = 0 | by cases n; cases i; refl | lemma | pgame.pow_half_move_left | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_succ_move_right (n i) : (pow_half (n + 1)).move_right i = pow_half n | rfl | lemma | pgame.pow_half_succ_move_right | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_pow_half_left_moves (n) : unique (pow_half n).left_moves | by cases n; exact punit.unique | instance | pgame.unique_pow_half_left_moves | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"punit.unique",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_empty_pow_half_zero_right_moves : is_empty (pow_half 0).right_moves | pempty.is_empty | instance | pgame.is_empty_pow_half_zero_right_moves | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_pow_half_succ_right_moves (n) : unique (pow_half (n + 1)).right_moves | punit.unique | instance | pgame.unique_pow_half_succ_right_moves | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"punit.unique",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
birthday_half : birthday (pow_half 1) = 2 | by { rw birthday_def, dsimp, simpa using order.le_succ (1 : ordinal) } | theorem | pgame.birthday_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"order.le_succ",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
numeric_pow_half (n) : (pow_half n).numeric | begin
induction n with n hn,
{ exact numeric_one },
{ split,
{ simpa using hn.move_left_lt default },
{ exact ⟨λ _, numeric_zero, λ _, hn⟩ } }
end | theorem | pgame.numeric_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | For all natural numbers `n`, the pre-games `pow_half n` are numeric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_half_succ_lt_pow_half (n : ℕ) : pow_half (n + 1) < pow_half n | (numeric_pow_half (n + 1)).lt_move_right default | theorem | pgame.pow_half_succ_lt_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_succ_le_pow_half (n : ℕ) : pow_half (n + 1) ≤ pow_half n | (pow_half_succ_lt_pow_half n).le | theorem | pgame.pow_half_succ_le_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_le_one (n : ℕ) : pow_half n ≤ 1 | begin
induction n with n hn,
{ exact le_rfl },
{ exact (pow_half_succ_le_pow_half n).trans hn }
end | theorem | pgame.pow_half_le_one | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_succ_lt_one (n : ℕ) : pow_half (n + 1) < 1 | (pow_half_succ_lt_pow_half n).trans_le $ pow_half_le_one n | theorem | pgame.pow_half_succ_lt_one | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half_pos (n : ℕ) : 0 < pow_half n | by { rw [←lf_iff_lt numeric_zero (numeric_pow_half n), zero_lf_le], simp } | theorem | pgame.pow_half_pos | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_le_pow_half (n : ℕ) : 0 ≤ pow_half n | (pow_half_pos n).le | theorem | pgame.zero_le_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_pow_half_succ_self_eq_pow_half (n) : pow_half (n + 1) + pow_half (n + 1) ≈ pow_half n | begin
induction n using nat.strong_induction_on with n hn,
{ split; rw le_iff_forall_lf; split,
{ rintro (⟨⟨ ⟩⟩ | ⟨⟨ ⟩⟩); apply lf_of_lt,
{ calc 0 + pow_half n.succ ≈ pow_half n.succ : zero_add_equiv _
... < pow_half n : pow_half_succ_lt_pow_half n },
{ calc pow_hal... | theorem | pgame.add_pow_half_succ_self_eq_pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"forall_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
half_add_half_equiv_one : pow_half 1 + pow_half 1 ≈ 1 | add_pow_half_succ_self_eq_pow_half 0 | theorem | pgame.half_add_half_equiv_one | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_half (n : ℕ) : surreal | ⟦⟨pgame.pow_half n, pgame.numeric_pow_half n⟩⟧ | def | surreal.pow_half | set_theory.surreal | src/set_theory/surreal/dyadic.lean | [
"algebra.algebra.basic",
"set_theory.game.birthday",
"set_theory.surreal.basic",
"ring_theory.localization.basic"
] | [
"pgame.numeric_pow_half",
"surreal"
] | Powers of the surreal number `half`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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