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
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|---|---|---|---|---|---|---|---|---|---|---|
sub_sub (a b c : ordinal) : a - b - c = a - (b + c) | eq_of_forall_ge_iff $ λ d, by rw [sub_le, sub_le, sub_le, add_assoc] | theorem | ordinal.sub_sub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_of_forall_ge_iff",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_sub_add_cancel (a b c : ordinal) : a + b - (a + c) = b - c | by rw [← sub_sub, add_sub_cancel] | theorem | ordinal.add_sub_add_cancel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_is_limit {a b} (l : is_limit a) (h : b < a) : is_limit (a - b) | ⟨ne_of_gt $ lt_sub.2 $ by rwa add_zero,
λ c h, by rw [lt_sub, add_succ]; exact l.2 _ (lt_sub.1 h)⟩ | theorem | ordinal.sub_is_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_omega : 1 + ω = ω | begin
refine le_antisymm _ (le_add_left _ _),
rw [omega, ← lift_one.{0}, ← lift_add, lift_le, ← type_unit, ← type_sum_lex],
refine ⟨rel_embedding.collapse (rel_embedding.of_monotone _ _)⟩,
{ apply sum.rec, exact λ _, 0, exact nat.succ },
{ intros a b, cases a; cases b; intro H; cases H with _ _ H _ _ H;
[... | theorem | ordinal.one_add_omega | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"rel_embedding.of_monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_of_omega_le {o} (h : ω ≤ o) : 1 + o = o | by rw [← ordinal.add_sub_cancel_of_le h, ← add_assoc, one_add_omega] | theorem | ordinal.one_add_of_omega_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
type_prod_lex {α β : Type u} (r : α → α → Prop) (s : β → β → Prop)
[is_well_order α r] [is_well_order β s] : type (prod.lex s r) = type r * type s | rfl | theorem | ordinal.type_prod_lex | 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 | |
mul_eq_zero' {a b : ordinal} : a * b = 0 ↔ a = 0 ∨ b = 0 | induction_on a $ λ α _ _, induction_on b $ λ β _ _, begin
simp_rw [←type_prod_lex, type_eq_zero_iff_is_empty],
rw or_comm,
exact is_empty_prod
end | theorem | ordinal.mul_eq_zero' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_empty_prod",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_mul (a b) : lift (a * b) = lift a * lift b | quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
quotient.sound ⟨(rel_iso.preimage equiv.ulift _).trans
(rel_iso.prod_lex_congr (rel_iso.preimage equiv.ulift _)
(rel_iso.preimage equiv.ulift _)).symm⟩ | theorem | ordinal.lift_mul | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"equiv.ulift",
"lift",
"rel_iso.preimage",
"rel_iso.prod_lex_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
card_mul (a b) : card (a * b) = card a * card b | quotient.induction_on₂ a b $ λ ⟨α, r, _⟩ ⟨β, s, _⟩,
mul_comm (mk β) (mk α) | theorem | ordinal.card_mul | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_succ (a b : ordinal) : a * succ b = a * b + a | mul_add_one a b | theorem | ordinal.mul_succ | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_add_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_covariant_class_le : covariant_class ordinal.{u} ordinal.{u} (*) (≤) | ⟨λ c a b, quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩, begin
resetI,
refine (rel_embedding.of_monotone (λ a : α × γ, (f a.1, a.2)) (λ a b h, _)).ordinal_type_le,
clear_,
cases h with a₁ b₁ a₂ b₂ h' a b₁ b₂ h',
{ exact prod.lex.left _ _ (f.to_rel_embedding.map_rel_iff.2 h') },
{ exact ... | instance | ordinal.mul_covariant_class_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"covariant_class",
"rel_embedding.of_monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_swap_covariant_class_le : covariant_class ordinal.{u} ordinal.{u} (swap (*)) (≤) | ⟨λ c a b, quotient.induction_on₃ a b c $ λ ⟨α, r, _⟩ ⟨β, s, _⟩ ⟨γ, t, _⟩ ⟨f⟩, begin
resetI,
refine (rel_embedding.of_monotone (λ a : γ × α, (a.1, f a.2)) (λ a b h, _)).ordinal_type_le,
cases h with a₁ b₁ a₂ b₂ h' a b₁ b₂ h',
{ exact prod.lex.left _ _ h' },
{ exact prod.lex.right _ (f.to_rel_embedding.map_rel_... | instance | ordinal.mul_swap_covariant_class_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"covariant_class",
"rel_embedding.of_monotone"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_left (a : ordinal) {b : ordinal} (hb : 0 < b) : a ≤ a * b | by { convert mul_le_mul_left' (one_le_iff_pos.2 hb) a, rw mul_one a } | theorem | ordinal.le_mul_left | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_mul_left",
"mul_le_mul_left'",
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_right (a : ordinal) {b : ordinal} (hb : 0 < b) : a ≤ b * a | by { convert mul_le_mul_right' (one_le_iff_pos.2 hb) a, rw one_mul a } | theorem | ordinal.le_mul_right | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_mul_right",
"mul_le_mul_right'",
"one_mul",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_limit_aux {α β r s} [is_well_order α r] [is_well_order β s]
{c} (h : is_limit (type s)) (H : ∀ b' < type s, type r * b' ≤ c)
(l : c < type r * type s) : false | begin
suffices : ∀ a b, prod.lex s r (b, a) (enum _ _ l),
{ cases enum _ _ l with b a, exact irrefl _ (this _ _) },
intros a b,
rw [←typein_lt_typein (prod.lex s r), typein_enum],
have := H _ (h.2 _ (typein_lt_type s b)),
rw mul_succ at this,
have := ((add_lt_add_iff_left _).2 (typein_lt_type _ a)).trans_... | lemma | ordinal.mul_le_of_limit_aux | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"prod.lex_def",
"rel_embedding.of_monotone",
"subrel_val",
"subtype.mk_eq_mk",
"sum.lex_inl_inl",
"sum.lex_inr_inl",
"sum.lex_inr_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_of_limit {a b c : ordinal} (h : is_limit b) : a * b ≤ c ↔ ∀ b' < b, a * b' ≤ c | ⟨λ h b' l, (mul_le_mul_left' l.le _).trans h,
λ H, le_of_not_lt $ induction_on a (λ α r _, induction_on b $ λ β s _,
by exactI mul_le_of_limit_aux) h H⟩ | theorem | ordinal.mul_le_of_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_le_mul_left'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_is_normal {a : ordinal} (h : 0 < a) : is_normal ((*) a) | ⟨λ b, by rw mul_succ; simpa only [add_zero] using (add_lt_add_iff_left (a*b)).2 h,
λ b l c, mul_le_of_limit l⟩ | theorem | ordinal.mul_is_normal | 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_mul_of_limit {a b c : ordinal}
(h : is_limit c) : a < b * c ↔ ∃ c' < c, a < b * c' | by simpa only [not_ball, not_le] using not_congr (@mul_le_of_limit b c a h) | theorem | ordinal.lt_mul_of_limit | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_ball",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_iff_left {a b c : ordinal} (a0 : 0 < a) : a * b < a * c ↔ b < c | (mul_is_normal a0).lt_iff | theorem | ordinal.mul_lt_mul_iff_left | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_lt_mul_iff_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_mul_iff_left {a b c : ordinal} (a0 : 0 < a) : a * b ≤ a * c ↔ b ≤ c | (mul_is_normal a0).le_iff | theorem | ordinal.mul_le_mul_iff_left | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_le_mul_iff_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_mul_of_pos_left {a b c : ordinal}
(h : a < b) (c0 : 0 < c) : c * a < c * b | (mul_lt_mul_iff_left c0).2 h | theorem | ordinal.mul_lt_mul_of_pos_left | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_lt_mul_iff_left",
"mul_lt_mul_of_pos_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_pos {a b : ordinal} (h₁ : 0 < a) (h₂ : 0 < b) : 0 < a * b | by simpa only [mul_zero] using mul_lt_mul_of_pos_left h₂ h₁ | theorem | ordinal.mul_pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_lt_mul_of_pos_left",
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ne_zero {a b : ordinal} : a ≠ 0 → b ≠ 0 → a * b ≠ 0 | by simpa only [ordinal.pos_iff_ne_zero] using mul_pos | theorem | ordinal.mul_ne_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_ne_zero",
"ordinal",
"ordinal.pos_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mul_le_mul_left {a b c : ordinal}
(h : c * a ≤ c * b) (h0 : 0 < c) : a ≤ b | le_imp_le_of_lt_imp_lt (λ h', mul_lt_mul_of_pos_left h' h0) h | theorem | ordinal.le_of_mul_le_mul_left | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_of_mul_le_mul_left",
"mul_lt_mul_of_pos_left",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_inj {a b c : ordinal} (a0 : 0 < a) : a * b = a * c ↔ b = c | (mul_is_normal a0).inj | theorem | ordinal.mul_right_inj | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_right_inj",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_is_limit {a b : ordinal}
(a0 : 0 < a) : is_limit b → is_limit (a * b) | (mul_is_normal a0).is_limit | theorem | ordinal.mul_is_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 | |
mul_is_limit_left {a b : ordinal}
(l : is_limit a) (b0 : 0 < b) : is_limit (a * b) | begin
rcases zero_or_succ_or_limit b with rfl|⟨b,rfl⟩|lb,
{ exact b0.false.elim },
{ rw mul_succ, exact add_is_limit _ l },
{ exact mul_is_limit l.pos lb }
end | theorem | ordinal.mul_is_limit_left | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_eq_mul : ∀ (n : ℕ) (a : ordinal), n • a = a * n | | 0 a := by rw [zero_smul, nat.cast_zero, mul_zero]
| (n + 1) a := by rw [succ_nsmul', nat.cast_add, mul_add, nat.cast_one, mul_one, smul_eq_mul] | theorem | ordinal.smul_eq_mul | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_one",
"mul_zero",
"nat.cast_add",
"nat.cast_one",
"nat.cast_zero",
"ordinal",
"smul_eq_mul",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_nonempty {a b : ordinal} (h : b ≠ 0) : {o | a < b * succ o}.nonempty | ⟨a, succ_le_iff.1 $
by simpa only [succ_zero, one_mul]
using mul_le_mul_right' (succ_le_of_lt (ordinal.pos_iff_ne_zero.2 h)) (succ a)⟩ | theorem | ordinal.div_nonempty | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_le_mul_right'",
"one_mul",
"ordinal"
] | The set in the definition of division is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_zero (a : ordinal) : a / 0 = 0 | dif_pos rfl | theorem | ordinal.div_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_def (a) {b : ordinal} (h : b ≠ 0) : a / b = Inf {o | a < b * succ o} | dif_neg h | lemma | ordinal.div_def | 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_mul_succ_div (a) {b : ordinal} (h : b ≠ 0) : a < b * succ (a / b) | by rw div_def a h; exact Inf_mem (div_nonempty h) | theorem | ordinal.lt_mul_succ_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"Inf_mem",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_mul_div_add (a) {b : ordinal} (h : b ≠ 0) : a < b * (a / b) + b | by simpa only [mul_succ] using lt_mul_succ_div a h | theorem | ordinal.lt_mul_div_add | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le {a b c : ordinal} (b0 : b ≠ 0) : a / b ≤ c ↔ a < b * succ c | ⟨λ h, (lt_mul_succ_div a b0).trans_le (mul_le_mul_left' (succ_le_succ_iff.2 h) _),
λ h, by rw div_def a b0; exact cInf_le' h⟩ | theorem | ordinal.div_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cInf_le'",
"mul_le_mul_left'",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_div {a b c : ordinal} (h : c ≠ 0) : a < b / c ↔ c * succ a ≤ b | by rw [← not_le, div_le h, not_lt] | theorem | ordinal.lt_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_pos {b c : ordinal} (h : c ≠ 0) : 0 < b / c ↔ c ≤ b | by simp [lt_div h] | theorem | ordinal.div_pos | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_pos",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_div {a b c : ordinal} (c0 : c ≠ 0) :
a ≤ b / c ↔ c * a ≤ b | begin
apply limit_rec_on a,
{ simp only [mul_zero, ordinal.zero_le] },
{ intros, rw [succ_le_iff, lt_div c0] },
{ simp only [mul_le_of_limit, limit_le, iff_self, forall_true_iff] {contextual := tt} }
end | theorem | ordinal.le_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"forall_true_iff",
"mul_zero",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_lt {a b c : ordinal} (b0 : b ≠ 0) :
a / b < c ↔ a < b * c | lt_iff_lt_of_le_iff_le $ le_div b0 | theorem | ordinal.div_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lt_iff_lt_of_le_iff_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_le_of_le_mul {a b c : ordinal} (h : a ≤ b * c) : a / b ≤ c | if b0 : b = 0 then by simp only [b0, div_zero, ordinal.zero_le] else
(div_le b0).2 $ h.trans_lt $ mul_lt_mul_of_pos_left (lt_succ c) (ordinal.pos_iff_ne_zero.2 b0) | theorem | ordinal.div_le_of_le_mul | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_zero",
"mul_lt_mul_of_pos_left",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_of_lt_div {a b c : ordinal} : a < b / c → c * a < b | lt_imp_lt_of_le_imp_le div_le_of_le_mul | theorem | ordinal.mul_lt_of_lt_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"lt_imp_lt_of_le_imp_le",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_div (a : ordinal) : 0 / a = 0 | ordinal.le_zero.1 $ div_le_of_le_mul $ ordinal.zero_le _ | theorem | ordinal.zero_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.zero_le",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_le (a b : ordinal) : b * (a / b) ≤ a | if b0 : b = 0 then by simp only [b0, zero_mul, ordinal.zero_le] else (le_div b0).1 le_rfl | theorem | ordinal.mul_div_le | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_rfl",
"ordinal",
"ordinal.zero_le",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_div (a) {b : ordinal} (b0 : b ≠ 0) (c) : (b * a + c) / b = a + c / b | begin
apply le_antisymm,
{ apply (div_le b0).2,
rw [mul_succ, mul_add, add_assoc, add_lt_add_iff_left],
apply lt_mul_div_add _ b0 },
{ rw [le_div b0, mul_add, add_le_add_iff_left],
apply mul_div_le }
end | theorem | ordinal.mul_add_div | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_zero_of_lt {a b : ordinal} (h : a < b) : a / b = 0 | begin
rw [←ordinal.le_zero, div_le $ ordinal.pos_iff_ne_zero.1 $ (ordinal.zero_le _).trans_lt h],
simpa only [succ_zero, mul_one] using h
end | theorem | ordinal.div_eq_zero_of_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_one",
"ordinal",
"ordinal.zero_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel (a) {b : ordinal} (b0 : b ≠ 0) : b * a / b = a | by simpa only [add_zero, zero_div] using mul_add_div a b0 0 | theorem | ordinal.mul_div_cancel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_div_cancel",
"ordinal",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_one (a : ordinal) : a / 1 = a | by simpa only [one_mul] using mul_div_cancel a ordinal.one_ne_zero | theorem | ordinal.div_one | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_one",
"mul_div_cancel",
"one_mul",
"ordinal",
"ordinal.one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_self {a : ordinal} (h : a ≠ 0) : a / a = 1 | by simpa only [mul_one] using mul_div_cancel 1 h | theorem | ordinal.div_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_self",
"mul_div_cancel",
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_sub (a b c : ordinal) : a * (b - c) = a * b - a * c | if a0 : a = 0 then by simp only [a0, zero_mul, sub_self] else
eq_of_forall_ge_iff $ λ d,
by rw [sub_le, ← le_div a0, sub_le, ← le_div a0, mul_add_div _ a0] | theorem | ordinal.mul_sub | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_of_forall_ge_iff",
"ordinal",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_limit_add_iff {a b} : is_limit (a + b) ↔ is_limit b ∨ (b = 0 ∧ is_limit a) | begin
split; intro h,
{ by_cases h' : b = 0,
{ rw [h', add_zero] at h, right, exact ⟨h', h⟩ },
left, rw [←add_sub_cancel a b], apply sub_is_limit h,
suffices : a + 0 < a + b, simpa only [add_zero],
rwa [add_lt_add_iff_left, ordinal.pos_iff_ne_zero] },
rcases h with h|⟨rfl, h⟩, exact add_is_l... | theorem | ordinal.is_limit_add_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal.pos_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_add_iff : ∀ {a b c : ordinal}, a ∣ b → (a ∣ b + c ↔ a ∣ c) | | a _ c ⟨b, rfl⟩ :=
⟨λ ⟨d, e⟩, ⟨d - b, by rw [mul_sub, ← e, add_sub_cancel]⟩,
λ ⟨d, e⟩, by { rw [e, ← mul_add], apply dvd_mul_right }⟩ | theorem | ordinal.dvd_add_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"dvd_mul_right",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_cancel : ∀ {a b : ordinal}, a ≠ 0 → a ∣ b → a * (b / a) = b | | a _ a0 ⟨b, rfl⟩ := by rw [mul_div_cancel _ a0] | theorem | ordinal.div_mul_cancel | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_mul_cancel",
"mul_div_cancel",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_dvd : ∀ {a b : ordinal}, b ≠ 0 → a ∣ b → a ≤ b | | a _ b0 ⟨b, rfl⟩ := by simpa only [mul_one] using mul_le_mul_left'
(one_le_iff_ne_zero.2 (λ h : b = 0, by simpa only [h, mul_zero] using b0)) a | theorem | ordinal.le_of_dvd | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_le_mul_left'",
"mul_one",
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_antisymm {a b : ordinal} (h₁ : a ∣ b) (h₂ : b ∣ a) : a = b | if a0 : a = 0 then by subst a; exact (eq_zero_of_zero_dvd h₁).symm else
if b0 : b = 0 then by subst b; exact eq_zero_of_zero_dvd h₂ else
(le_of_dvd b0 h₁).antisymm (le_of_dvd a0 h₂) | theorem | ordinal.dvd_antisymm | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"dvd_antisymm",
"eq_zero_of_zero_dvd",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_def (a b : ordinal) : a % b = a - b * (a / b) | rfl | theorem | ordinal.mod_def | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_le (a b : ordinal) : a % b ≤ a | sub_le_self a _ | theorem | ordinal.mod_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 | |
mod_zero (a : ordinal) : a % 0 = a | by simp only [mod_def, div_zero, zero_mul, sub_zero] | theorem | ordinal.mod_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_zero",
"ordinal",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_eq_of_lt {a b : ordinal} (h : a < b) : a % b = a | by simp only [mod_def, div_eq_zero_of_lt h, mul_zero, sub_zero] | theorem | ordinal.mod_eq_of_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_zero",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_mod (b : ordinal) : 0 % b = 0 | by simp only [mod_def, zero_div, mul_zero, sub_self] | theorem | ordinal.zero_mod | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_zero",
"ordinal",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_add_mod (a b : ordinal) : b * (a / b) + a % b = a | ordinal.add_sub_cancel_of_le $ mul_div_le _ _ | theorem | ordinal.div_add_mod | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.add_sub_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_lt (a) {b : ordinal} (h : b ≠ 0) : a % b < b | (add_lt_add_iff_left (b * (a / b))).1 $
by rw div_add_mod; exact lt_mul_div_add a h | theorem | ordinal.mod_lt | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_self (a : ordinal) : a % a = 0 | if a0 : a = 0 then by simp only [a0, zero_mod] else
by simp only [mod_def, div_self a0, mul_one, sub_self] | theorem | ordinal.mod_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_self",
"mul_one",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_one (a : ordinal) : a % 1 = 0 | by simp only [mod_def, div_one, one_mul, sub_self] | theorem | ordinal.mod_one | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"div_one",
"one_mul",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_of_mod_eq_zero {a b : ordinal} (H : a % b = 0) : b ∣ a | ⟨a / b, by simpa [H] using (div_add_mod a b).symm⟩ | theorem | ordinal.dvd_of_mod_eq_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_eq_zero_of_dvd {a b : ordinal} (H : b ∣ a) : a % b = 0 | begin
rcases H with ⟨c, rfl⟩,
rcases eq_or_ne b 0 with rfl | hb,
{ simp },
{ simp [mod_def, hb] }
end | theorem | ordinal.mod_eq_zero_of_dvd | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dvd_iff_mod_eq_zero {a b : ordinal} : b ∣ a ↔ a % b = 0 | ⟨mod_eq_zero_of_dvd, dvd_of_mod_eq_zero⟩ | theorem | ordinal.dvd_iff_mod_eq_zero | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_add_mod_self (x y z : ordinal) : (x * y + z) % x = z % x | begin
rcases eq_or_ne x 0 with rfl | hx,
{ simp },
{ rwa [mod_def, mul_add_div, mul_add, ←sub_sub, add_sub_cancel, mod_def] }
end | theorem | ordinal.mul_add_mod_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_or_ne",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mod (x y : ordinal) : x * y % x = 0 | by simpa using mul_add_mod_self x y 0 | theorem | ordinal.mul_mod | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_mod_of_dvd (a : ordinal) {b c : ordinal} (h : c ∣ b) : a % b % c = a % c | begin
nth_rewrite_rhs 0 ←div_add_mod a b,
rcases h with ⟨d, rfl⟩,
rw [mul_assoc, mul_add_mod_self]
end | theorem | ordinal.mod_mod_of_dvd | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"mul_assoc",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mod_mod (a b : ordinal) : a % b % b = a % b | mod_mod_of_dvd a dvd_rfl | theorem | ordinal.mod_mod | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"dvd_rfl",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bfamily_of_family' {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r] (f : ι → α) :
Π a < type r, α | λ a ha, f (enum r a ha) | def | ordinal.bfamily_of_family' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | Converts a family indexed by a `Type u` to one indexed by an `ordinal.{u}` using a specified
well-ordering. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bfamily_of_family {ι : Type u} : (ι → α) → Π a < type (@well_ordering_rel ι), α | bfamily_of_family' well_ordering_rel | def | ordinal.bfamily_of_family | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"well_ordering_rel"
] | Converts a family indexed by a `Type u` to one indexed by an `ordinal.{u}` using a well-ordering
given by the axiom of choice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
family_of_bfamily' {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r] {o} (ho : type r = o)
(f : Π a < o, α) : ι → α | λ i, f (typein r i) (by { rw ←ho, exact typein_lt_type r i }) | def | ordinal.family_of_bfamily' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order"
] | Converts a family indexed by an `ordinal.{u}` to one indexed by an `Type u` using a specified
well-ordering. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
family_of_bfamily (o : ordinal) (f : Π a < o, α) : o.out.α → α | family_of_bfamily' (<) (type_lt o) f | def | ordinal.family_of_bfamily | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | Converts a family indexed by an `ordinal.{u}` to one indexed by a `Type u` using a well-ordering
given by the axiom of choice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bfamily_of_family'_typein {ι} (r : ι → ι → Prop) [is_well_order ι r] (f : ι → α)
(i) : bfamily_of_family' r f (typein r i) (typein_lt_type r i) = f i | by simp only [bfamily_of_family', enum_typein] | theorem | ordinal.bfamily_of_family'_typein | 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 | |
bfamily_of_family_typein {ι} (f : ι → α) (i) :
bfamily_of_family f (typein _ i) (typein_lt_type _ i) = f i | bfamily_of_family'_typein _ f i | theorem | ordinal.bfamily_of_family_typein | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
family_of_bfamily'_enum {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r] {o}
(ho : type r = o) (f : Π a < o, α) (i hi) :
family_of_bfamily' r ho f (enum r i (by rwa ho)) = f i hi | by simp only [family_of_bfamily', typein_enum] | theorem | ordinal.family_of_bfamily'_enum | 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 | |
family_of_bfamily_enum (o : ordinal) (f : Π a < o, α) (i hi) :
family_of_bfamily o f (enum (<) i (by { convert hi, exact type_lt _ })) = f i hi | family_of_bfamily'_enum _ (type_lt o) f _ _ | theorem | ordinal.family_of_bfamily_enum | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
brange (o : ordinal) (f : Π a < o, α) : set α | {a | ∃ i hi, f i hi = a} | def | ordinal.brange | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | The range of a family indexed by ordinals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_brange {o : ordinal} {f : Π a < o, α} {a} : a ∈ brange o f ↔ ∃ i hi, f i hi = a | iff.rfl | theorem | ordinal.mem_brange | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_brange_self {o} (f : Π a < o, α) (i hi) : f i hi ∈ brange o f | ⟨i, hi, rfl⟩ | theorem | ordinal.mem_brange_self | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_family_of_bfamily' {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r] {o}
(ho : type r = o) (f : Π a < o, α) : range (family_of_bfamily' r ho f) = brange o f | begin
refine set.ext (λ a, ⟨_, _⟩),
{ rintro ⟨b, rfl⟩,
apply mem_brange_self },
{ rintro ⟨i, hi, rfl⟩,
exact ⟨_, family_of_bfamily'_enum _ _ _ _ _⟩ }
end | theorem | ordinal.range_family_of_bfamily' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_family_of_bfamily {o} (f : Π a < o, α) :
range (family_of_bfamily o f) = brange o f | range_family_of_bfamily' _ _ f | theorem | ordinal.range_family_of_bfamily | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
brange_bfamily_of_family' {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r]
(f : ι → α) : brange _ (bfamily_of_family' r f) = range f | begin
refine set.ext (λ a, ⟨_, _⟩),
{ rintro ⟨i, hi, rfl⟩,
apply mem_range_self },
{ rintro ⟨b, rfl⟩,
exact ⟨_, _, bfamily_of_family'_typein _ _ _⟩ },
end | theorem | ordinal.brange_bfamily_of_family' | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"is_well_order",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
brange_bfamily_of_family {ι : Type u} (f : ι → α) :
brange _ (bfamily_of_family f) = range f | brange_bfamily_of_family' _ _ | theorem | ordinal.brange_bfamily_of_family | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
brange_const {o : ordinal} (ho : o ≠ 0) {c : α} : brange o (λ _ _, c) = {c} | begin
rw ←range_family_of_bfamily,
exact @set.range_const _ o.out.α (out_nonempty_iff_ne_zero.2 ho) c
end | theorem | ordinal.brange_const | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"set.range_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_bfamily_of_family' {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r] (f : ι → α)
(g : α → β) : (λ i hi, g (bfamily_of_family' r f i hi)) = bfamily_of_family' r (g ∘ f) | rfl | theorem | ordinal.comp_bfamily_of_family' | 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 | |
comp_bfamily_of_family {ι : Type u} (f : ι → α) (g : α → β) :
(λ i hi, g (bfamily_of_family f i hi)) = bfamily_of_family (g ∘ f) | rfl | theorem | ordinal.comp_bfamily_of_family | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_family_of_bfamily' {ι : Type u} (r : ι → ι → Prop) [is_well_order ι r] {o}
(ho : type r = o) (f : Π a < o, α) (g : α → β) :
g ∘ (family_of_bfamily' r ho f) = family_of_bfamily' r ho (λ i hi, g (f i hi)) | rfl | theorem | ordinal.comp_family_of_bfamily' | 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 | |
comp_family_of_bfamily {o} (f : Π a < o, α) (g : α → β) :
g ∘ (family_of_bfamily o f) = family_of_bfamily o (λ i hi, g (f i hi)) | rfl | theorem | ordinal.comp_family_of_bfamily | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup {ι : Type u} (f : ι → ordinal.{max u v}) : ordinal.{max u v} | supr f | def | ordinal.sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"supr"
] | The supremum of a family of ordinals | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Sup_eq_sup {ι : Type u} (f : ι → ordinal.{max u v}) : Sup (set.range f) = sup f | rfl | theorem | ordinal.Sup_eq_sup | 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 | |
bdd_above_range {ι : Type u} (f : ι → ordinal.{max u v}) : bdd_above (set.range f) | ⟨(supr (succ ∘ card ∘ f)).ord, begin
rintros a ⟨i, rfl⟩,
exact le_of_lt (cardinal.lt_ord.2 ((lt_succ _).trans_le (le_csupr (bdd_above_range _) _)))
end⟩ | theorem | ordinal.bdd_above_range | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"bdd_above",
"le_csupr",
"set.range",
"supr"
] | The range of an indexed ordinal function, whose outputs live in a higher universe than the
inputs, is always bounded above. See `ordinal.lsub` for an explicit bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_sup {ι} (f : ι → ordinal) : ∀ i, f i ≤ sup f | λ i, le_cSup (bdd_above_range f) (mem_range_self i) | theorem | ordinal.le_sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"le_cSup",
"ordinal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le_iff {ι} {f : ι → ordinal} {a} : sup f ≤ a ↔ ∀ i, f i ≤ a | (cSup_le_iff' (bdd_above_range f)).trans (by simp) | theorem | ordinal.sup_le_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"cSup_le_iff'",
"ordinal",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_le {ι} {f : ι → ordinal} {a} : (∀ i, f i ≤ a) → sup f ≤ a | sup_le_iff.2 | theorem | ordinal.sup_le | 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 | |
lt_sup {ι} {f : ι → ordinal} {a} : a < sup f ↔ ∃ i, a < f i | by simpa only [not_forall, not_le] using not_congr (@sup_le_iff _ f a) | theorem | ordinal.lt_sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"not_forall",
"ordinal",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_sup_iff_lt_sup {ι} {f : ι → ordinal} : (∀ i, f i ≠ sup f) ↔ ∀ i, f i < sup f | ⟨λ hf _, lt_of_le_of_ne (le_sup _ _) (hf _), λ hf _, ne_of_lt (hf _)⟩ | theorem | ordinal.ne_sup_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 | |
sup_not_succ_of_ne_sup {ι} {f : ι → ordinal} (hf : ∀ i, f i ≠ sup f) {a}
(hao : a < sup f) : succ a < sup f | begin
by_contra' hoa,
exact hao.not_le (sup_le $ λ i, le_of_lt_succ $
(lt_of_le_of_ne (le_sup _ _) (hf i)).trans_le hoa)
end | theorem | ordinal.sup_not_succ_of_ne_sup | 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 | |
sup_eq_zero_iff {ι} {f : ι → ordinal} : sup f = 0 ↔ ∀ i, f i = 0 | begin
refine ⟨λ h i, _, λ h, le_antisymm
(sup_le (λ i, ordinal.le_zero.2 (h i))) (ordinal.zero_le _)⟩,
rw [←ordinal.le_zero, ←h],
exact le_sup f i
end | theorem | ordinal.sup_eq_zero_iff | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"ordinal",
"ordinal.zero_le",
"sup_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_normal.sup {f} (H : is_normal f) {ι} (g : ι → ordinal) [nonempty ι] :
f (sup g) = sup (f ∘ g) | eq_of_forall_ge_iff $ λ a,
by rw [sup_le_iff, comp, H.le_set' set.univ set.univ_nonempty g]; simp [sup_le_iff] | theorem | ordinal.is_normal.sup | set_theory.ordinal | src/set_theory/ordinal/arithmetic.lean | [
"set_theory.ordinal.basic",
"tactic.by_contra"
] | [
"eq_of_forall_ge_iff",
"ordinal",
"set.univ_nonempty",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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