Ordinal arithmetic

Ordinals have an addition (corresponding to disjoint union) that turns them into an additive monoid, and a multiplication (corresponding to the lexicographic order on the product) that turns them into a monoid. One can also define correspondingly a subtraction, a division, a successor function, a power function and a logarithm function.

We also define limit ordinals and prove the basic induction principle on ordinals separating successor ordinals and limit ordinals, in limitRecOn.

Main definitions and results

• o₁ + o₂ is the order on the disjoint union of o₁ and o₂ obtained by declaring that every element of o₁ is smaller than every element of o₂.
• o₁ - o₂ is the unique ordinal o such that o₂ + o = o₁, when o₂ ≤ o₁.
• o₁ * o₂ is the lexicographic order on o₂ × o₁.
• o₁ / o₂ is the ordinal o such that o₁ = o₂ * o + o' with o' < o₂. We also define the divisibility predicate, and a modulo operation.
• Order.succ o = o + 1 is the successor of o.
• pred o if the predecessor of o. If o is not a successor, we set pred o = o.

We discuss the properties of casts of natural numbers of and of ω with respect to these operations.

Some properties of the operations are also used to discuss general tools on ordinals:

• IsLimit o: an ordinal is a limit ordinal if it is neither 0 nor a successor.
• limitRecOn is the main induction principle of ordinals: if one can prove a property by induction at successor ordinals and at limit ordinals, then it holds for all ordinals.
• IsNormal: a function f : Ordinal → Ordinal satisfies IsNormal if it is strictly increasing and order-continuous, i.e., the image f o of a limit ordinal o is the sup of f a for a < o.
• enumOrd: enumerates an unbounded set of ordinals by the ordinals themselves.
• sup, lsub: the supremum / least strict upper bound of an indexed family of ordinals in Type u, as an ordinal in Type u.
• bsup, blsub: the supremum / least strict upper bound of a set of ordinals indexed by ordinals less than a given ordinal o.

Various other basic arithmetic results are given in Principal.lean instead.

Further properties of addition on ordinals

@[simp]

theorem Ordinal.lift_add (a : Ordinal.{v}) (b : Ordinal.{v}) : Ordinal.lift.{u, v} (a + b) = Ordinal.lift.{u, v} a + Ordinal.lift.{u, v} b

@[simp]

theorem Ordinal.lift_succ (a : Ordinal.{v}) : Ordinal.lift.{u, v} (Order.succ a) = Order.succ (Ordinal.lift.{u, v} a)

instance Ordinal.add_contravariantClass_le : ContravariantClass Ordinal.{u} Ordinal.{u} (fun (x x_1 : Ordinal.{u}) => x + x_1) fun (x x_1 : Ordinal.{u}) => x ≤ x_1

Equations

• Ordinal.add_contravariantClass_le = ⋯

theorem Ordinal.add_left_cancel (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} : a + b = a + c ↔ b = c

instance Ordinal.add_covariantClass_lt : CovariantClass Ordinal.{u} Ordinal.{u} (fun (x x_1 : Ordinal.{u}) => x + x_1) fun (x x_1 : Ordinal.{u}) => x < x_1

Equations

• Ordinal.add_covariantClass_lt = ⋯

instance Ordinal.add_contravariantClass_lt : ContravariantClass Ordinal.{u} Ordinal.{u} (fun (x x_1 : Ordinal.{u}) => x + x_1) fun (x x_1 : Ordinal.{u}) => x < x_1

Equations

• Ordinal.add_contravariantClass_lt = ⋯

instance Ordinal.add_swap_contravariantClass_lt : ContravariantClass Ordinal.{u} Ordinal.{u} (Function.swap fun (x x_1 : Ordinal.{u}) => x + x_1) fun (x x_1 : Ordinal.{u}) => x < x_1

Equations

• Ordinal.add_swap_contravariantClass_lt = ⋯

theorem Ordinal.add_le_add_iff_right {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (n : ℕ) : a + ↑n ≤ b + ↑n ↔ a ≤ b

theorem Ordinal.add_right_cancel {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (n : ℕ) : a + ↑n = b + ↑n ↔ a = b

theorem Ordinal.add_eq_zero_iff {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : a + b = 0 ↔ a = 0 ∧ b = 0

theorem Ordinal.left_eq_zero_of_add_eq_zero {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : a + b = 0) : a = 0

theorem Ordinal.right_eq_zero_of_add_eq_zero {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : a + b = 0) : b = 0

The predecessor of an ordinal

def Ordinal.pred (o : Ordinal.{u_4}) : Ordinal.{u_4}

The ordinal predecessor of o is o' if o = succ o', and o otherwise.

Equations

• Ordinal.pred o = if h : ∃ (a : Ordinal.{u_4}), o = Order.succ a then Classical.choose h else o

@[simp]

theorem Ordinal.pred_succ (o : Ordinal.{u_4}) : Ordinal.pred (Order.succ o) = o

theorem Ordinal.pred_le_self (o : Ordinal.{u_4}) : Ordinal.pred o ≤ o

theorem Ordinal.pred_eq_iff_not_succ {o : Ordinal.{u_4}} : Ordinal.pred o = o ↔ ¬∃ (a : Ordinal.{u_4}), o = Order.succ a

theorem Ordinal.pred_eq_iff_not_succ' {o : Ordinal.{u_4}} : Ordinal.pred o = o ↔ ∀ (a : Ordinal.{u_4}), o ≠ Order.succ a

theorem Ordinal.pred_lt_iff_is_succ {o : Ordinal.{u_4}} : Ordinal.pred o < o ↔ ∃ (a : Ordinal.{u_4}), o = Order.succ a

@[simp]

theorem Ordinal.pred_zero : Ordinal.pred 0 = 0

theorem Ordinal.succ_pred_iff_is_succ {o : Ordinal.{u_4}} : Order.succ (Ordinal.pred o) = o ↔ ∃ (a : Ordinal.{u_4}), o = Order.succ a

theorem Ordinal.succ_lt_of_not_succ {o : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : ¬∃ (a : Ordinal.{u_4}), o = Order.succ a) : Order.succ b < o ↔ b < o

theorem Ordinal.lt_pred {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : a < Ordinal.pred b ↔ Order.succ a < b

theorem Ordinal.pred_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : Ordinal.pred a ≤ b ↔ a ≤ Order.succ b

@[simp]

theorem Ordinal.lift_is_succ {o : Ordinal.{v}} : (∃ (a : Ordinal.{max v u} ), Ordinal.lift.{u, v} o = Order.succ a) ↔ ∃ (a : Ordinal.{v}), o = Order.succ a

@[simp]

theorem Ordinal.lift_pred (o : Ordinal.{v}) : Ordinal.lift.{u, v} (Ordinal.pred o) = Ordinal.pred (Ordinal.lift.{u, v} o)

Limit ordinals

def Ordinal.IsLimit (o : Ordinal.{u_4}) : Prop

A limit ordinal is an ordinal which is not zero and not a successor.

Equations

• Ordinal.IsLimit o = (o ≠ 0 ∧ ∀ a < o, Order.succ a < o)

theorem Ordinal.IsLimit.succ_lt {o : Ordinal.{u_4}} {a : Ordinal.{u_4}} (h : Ordinal.IsLimit o) : a < o → Order.succ a < o

theorem Ordinal.not_zero_isLimit : ¬Ordinal.IsLimit 0

theorem Ordinal.not_succ_isLimit (o : Ordinal.{u_4}) : ¬Ordinal.IsLimit (Order.succ o)

theorem Ordinal.not_succ_of_isLimit {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) : ¬∃ (a : Ordinal.{u_4}), o = Order.succ a

theorem Ordinal.succ_lt_of_isLimit {o : Ordinal.{u_4}} {a : Ordinal.{u_4}} (h : Ordinal.IsLimit o) : Order.succ a < o ↔ a < o

theorem Ordinal.le_succ_of_isLimit {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) {a : Ordinal.{u_4}} : o ≤ Order.succ a ↔ o ≤ a

theorem Ordinal.limit_le {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) {a : Ordinal.{u_4}} : o ≤ a ↔ ∀ x < o, x ≤ a

theorem Ordinal.lt_limit {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) {a : Ordinal.{u_4}} : a < o ↔ ∃ x < o, a < x

@[simp]

theorem Ordinal.lift_isLimit (o : Ordinal.{u_4}) : Ordinal.IsLimit (Ordinal.lift.{u_5, u_4} o) ↔ Ordinal.IsLimit o

theorem Ordinal.IsLimit.pos {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) : 0 < o

theorem Ordinal.IsLimit.one_lt {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) : 1 < o

theorem Ordinal.IsLimit.nat_lt {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) (n : ℕ) : ↑n < o

theorem Ordinal.zero_or_succ_or_limit (o : Ordinal.{u_4}) : o = 0 ∨ (∃ (a : Ordinal.{u_4}), o = Order.succ a) ∨ Ordinal.IsLimit o

def Ordinal.limitRecOn {C : Ordinal.{u_5} → Sort u_4} (o : Ordinal.{u_5}) (H₁ : C 0) (H₂ : (o : Ordinal.{u_5}) → C o → C (Order.succ o)) (H₃ : (o : Ordinal.{u_5}) → Ordinal.IsLimit o → ((o' : Ordinal.{u_5}) → o' < o → C o') → C o) : C o

Main induction principle of ordinals: if one can prove a property by induction at successor ordinals and at limit ordinals, then it holds for all ordinals.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Ordinal.limitRecOn_zero {C : Ordinal.{u_4} → Sort u_5} (H₁ : C 0) (H₂ : (o : Ordinal.{u_4}) → C o → C (Order.succ o)) (H₃ : (o : Ordinal.{u_4}) → Ordinal.IsLimit o → ((o' : Ordinal.{u_4}) → o' < o → C o') → C o) : Ordinal.limitRecOn 0 H₁ H₂ H₃ = H₁

@[simp]

theorem Ordinal.limitRecOn_succ {C : Ordinal.{u_4} → Sort u_5} (o : Ordinal.{u_4}) (H₁ : C 0) (H₂ : (o : Ordinal.{u_4}) → C o → C (Order.succ o)) (H₃ : (o : Ordinal.{u_4}) → Ordinal.IsLimit o → ((o' : Ordinal.{u_4}) → o' < o → C o') → C o) : Ordinal.limitRecOn (Order.succ o) H₁ H₂ H₃ = H₂ o (Ordinal.limitRecOn o H₁ H₂ H₃)

@[simp]

theorem Ordinal.limitRecOn_limit {C : Ordinal.{u_4} → Sort u_5} (o : Ordinal.{u_4}) (H₁ : C 0) (H₂ : (o : Ordinal.{u_4}) → C o → C (Order.succ o)) (H₃ : (o : Ordinal.{u_4}) → Ordinal.IsLimit o → ((o' : Ordinal.{u_4}) → o' < o → C o') → C o) (h : Ordinal.IsLimit o) : Ordinal.limitRecOn o H₁ H₂ H₃ = H₃ o h fun (x : Ordinal.{u_4}) (_h : x < o) => Ordinal.limitRecOn x H₁ H₂ H₃

instance Ordinal.orderTopOutSucc (o : Ordinal.{u_4}) : OrderTop (Quotient.out (Order.succ o)).α

Equations

• Ordinal.orderTopOutSucc o = OrderTop.mk ⋯

theorem Ordinal.enum_succ_eq_top {o : Ordinal.{u_4}} : Ordinal.enum (fun (x x_1 : (Quotient.out (Order.succ o)).α) => x < x_1) o ⋯ = ⊤

theorem Ordinal.has_succ_of_type_succ_lt {α : Type u_4} {r : α → α → Prop} [wo : IsWellOrder α r] (h : ∀ a < Ordinal.type r, Order.succ a < Ordinal.type r) (x : α) : ∃ (y : α), r x y

theorem Ordinal.out_no_max_of_succ_lt {o : Ordinal.{u_4}} (ho : ∀ a < o, Order.succ a < o) : NoMaxOrder (Quotient.out o).α

theorem Ordinal.bounded_singleton {α : Type u_1} {r : α → α → Prop} [IsWellOrder α r] (hr : Ordinal.IsLimit (Ordinal.type r)) (x : α) : Set.Bounded r {x}

theorem Ordinal.type_subrel_lt (o : Ordinal.{u}) : Ordinal.type (Subrel (fun (x x_1 : Ordinal.{u}) => x < x_1) {o' : Ordinal.{u} | o' < o}) = Ordinal.lift.{u + 1, u} o

theorem Ordinal.mk_initialSeg (o : Ordinal.{u}) : Cardinal.mk ↑{o' : Ordinal.{u} | o' < o} = Cardinal.lift.{u + 1, u} (Ordinal.card o)

Normal ordinal functions

def Ordinal.IsNormal (f : Ordinal.{u_4} → Ordinal.{u_5}) : Prop

A normal ordinal function is a strictly increasing function which is order-continuous, i.e., the image f o of a limit ordinal o is the sup of f a for a < o.

Equations

• Ordinal.IsNormal f = ((∀ (o : Ordinal.{u_4}), f o < f (Order.succ o)) ∧ ∀ (o : Ordinal.{u_4}), Ordinal.IsLimit o → ∀ (a : Ordinal.{u_5}), f o ≤ a ↔ ∀ b < o, f b ≤ a)

theorem Ordinal.IsNormal.limit_le {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) {o : Ordinal.{u_4}} : Ordinal.IsLimit o → ∀ {a : Ordinal.{u_5}}, f o ≤ a ↔ ∀ b < o, f b ≤ a

theorem Ordinal.IsNormal.limit_lt {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) {o : Ordinal.{u_4}} (h : Ordinal.IsLimit o) {a : Ordinal.{u_5}} : a < f o ↔ ∃ b < o, a < f b

theorem Ordinal.IsNormal.strictMono {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) : StrictMono f

theorem Ordinal.IsNormal.monotone {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) : Monotone f

theorem Ordinal.isNormal_iff_strictMono_limit (f : Ordinal.{u_4} → Ordinal.{u_5}) : Ordinal.IsNormal f ↔ StrictMono f ∧ ∀ (o : Ordinal.{u_4}), Ordinal.IsLimit o → ∀ (a : Ordinal.{u_5}), (∀ b < o, f b ≤ a) → f o ≤ a

theorem Ordinal.IsNormal.lt_iff {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : f a < f b ↔ a < b

theorem Ordinal.IsNormal.le_iff {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : f a ≤ f b ↔ a ≤ b

theorem Ordinal.IsNormal.inj {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : f a = f b ↔ a = b

theorem Ordinal.IsNormal.self_le {f : Ordinal.{u_4} → Ordinal.{u_4}} (H : Ordinal.IsNormal f) (a : Ordinal.{u_4}) : a ≤ f a

theorem Ordinal.IsNormal.le_set {f : Ordinal.{u_4} → Ordinal.{u_5}} {o : Ordinal.{u_5}} (H : Ordinal.IsNormal f) (p : Set Ordinal.{u_4}) (p0 : Set.Nonempty p) (b : Ordinal.{u_4}) (H₂ : ∀ (o : Ordinal.{u_4}), b ≤ o ↔ ∀ a ∈ p, a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f a ≤ o

theorem Ordinal.IsNormal.le_set' {α : Type u_1} {f : Ordinal.{u_4} → Ordinal.{u_5}} {o : Ordinal.{u_5}} (H : Ordinal.IsNormal f) (p : Set α) (p0 : Set.Nonempty p) (g : α → Ordinal.{u_4}) (b : Ordinal.{u_4}) (H₂ : ∀ (o : Ordinal.{u_4}), b ≤ o ↔ ∀ a ∈ p, g a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o

theorem Ordinal.IsNormal.refl : Ordinal.IsNormal id

theorem Ordinal.IsNormal.trans {f : Ordinal.{u_4} → Ordinal.{u_5}} {g : Ordinal.{u_6} → Ordinal.{u_4}} (H₁ : Ordinal.IsNormal f) (H₂ : Ordinal.IsNormal g) : Ordinal.IsNormal (f ∘ g)

theorem Ordinal.IsNormal.isLimit {f : Ordinal.{u_4} → Ordinal.{u_5}} (H : Ordinal.IsNormal f) {o : Ordinal.{u_4}} (l : Ordinal.IsLimit o) : Ordinal.IsLimit (f o)

theorem Ordinal.IsNormal.le_iff_eq {f : Ordinal.{u_4} → Ordinal.{u_4}} (H : Ordinal.IsNormal f) {a : Ordinal.{u_4}} : f a ≤ a ↔ f a = a

theorem Ordinal.add_le_of_limit {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : Ordinal.IsLimit b) : a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c

theorem Ordinal.add_isNormal (a : Ordinal.{u_4}) : Ordinal.IsNormal fun (x : Ordinal.{u_4}) => a + x

theorem Ordinal.add_isLimit (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} : Ordinal.IsLimit b → Ordinal.IsLimit (a + b)

theorem Ordinal.IsLimit.add (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} : Ordinal.IsLimit b → Ordinal.IsLimit (a + b)

Alias of Ordinal.add_isLimit.

Subtraction on ordinals

theorem Ordinal.sub_nonempty {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : Set.Nonempty {o : Ordinal.{u_4} | a ≤ b + o}

The set in the definition of subtraction is nonempty.

instance Ordinal.sub : Sub Ordinal.{u_4}

a - b is the unique ordinal satisfying b + (a - b) = a when b ≤ a.

Equations

• Ordinal.sub = { sub := fun (a b : Ordinal.{u_4}) => sInf {o : Ordinal.{u_4} | a ≤ b + o} }

theorem Ordinal.le_add_sub (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a ≤ b + (a - b)

theorem Ordinal.sub_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} : a - b ≤ c ↔ a ≤ b + c

theorem Ordinal.lt_sub {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} : a < b - c ↔ c + a < b

theorem Ordinal.add_sub_cancel (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a + b - a = b

theorem Ordinal.sub_eq_of_add_eq {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : a + b = c) : c - a = b

theorem Ordinal.sub_le_self (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a - b ≤ a

theorem Ordinal.add_sub_cancel_of_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : b ≤ a) : b + (a - b) = a

theorem Ordinal.le_sub_of_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : b ≤ a) : c ≤ a - b ↔ b + c ≤ a

theorem Ordinal.sub_lt_of_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : b ≤ a) : a - b < c ↔ a < b + c

instance Ordinal.existsAddOfLE : ExistsAddOfLE Ordinal.{u_4}

Equations

• Ordinal.existsAddOfLE = ⋯

@[simp]

theorem Ordinal.sub_zero (a : Ordinal.{u_4}) : a - 0 = a

@[simp]

theorem Ordinal.zero_sub (a : Ordinal.{u_4}) : 0 - a = 0

@[simp]

theorem Ordinal.sub_self (a : Ordinal.{u_4}) : a - a = 0

theorem Ordinal.sub_eq_zero_iff_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : a - b = 0 ↔ a ≤ b

theorem Ordinal.sub_sub (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) (c : Ordinal.{u_4}) : a - b - c = a - (b + c)

@[simp]

theorem Ordinal.add_sub_add_cancel (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) (c : Ordinal.{u_4}) : a + b - (a + c) = b - c

theorem Ordinal.sub_isLimit {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (l : Ordinal.IsLimit a) (h : b < a) : Ordinal.IsLimit (a - b)

theorem Ordinal.one_add_omega : 1 + Ordinal.omega = Ordinal.omega

@[simp]

theorem Ordinal.one_add_of_omega_le {o : Ordinal.{u_4}} (h : Ordinal.omega ≤ o) : 1 + o = o

Multiplication of ordinals

instance Ordinal.monoid : Monoid Ordinal.{u}

The multiplication of ordinals o₁ and o₂ is the (well founded) lexicographic order on o₂ × o₁.

Equations

• Ordinal.monoid = Monoid.mk Ordinal.monoid.proof_4 Ordinal.monoid.proof_5 npowRec Ordinal.monoid.proof_7 Ordinal.monoid.proof_8

@[simp]

theorem Ordinal.type_prod_lex {α : Type u} {β : Type u} (r : α → α → Prop) (s : β → β → Prop) [IsWellOrder α r] [IsWellOrder β s] : Ordinal.type (Prod.Lex s r) = Ordinal.type r * Ordinal.type s

instance Ordinal.monoidWithZero : MonoidWithZero Ordinal.{u_4}

Equations

• Ordinal.monoidWithZero = let __src := Ordinal.monoid; MonoidWithZero.mk Ordinal.monoidWithZero.proof_1 Ordinal.monoidWithZero.proof_2

instance Ordinal.noZeroDivisors : NoZeroDivisors Ordinal.{u_4}

Equations

• Ordinal.noZeroDivisors = ⋯

@[simp]

theorem Ordinal.lift_mul (a : Ordinal.{v}) (b : Ordinal.{v}) : Ordinal.lift.{u, v} (a * b) = Ordinal.lift.{u, v} a * Ordinal.lift.{u, v} b

@[simp]

theorem Ordinal.card_mul (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : Ordinal.card (a * b) = Ordinal.card a * Ordinal.card b

instance Ordinal.leftDistribClass : LeftDistribClass Ordinal.{u}

Equations

• Ordinal.leftDistribClass = ⋯

theorem Ordinal.mul_succ (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a * Order.succ b = a * b + a

instance Ordinal.mul_covariantClass_le : CovariantClass Ordinal.{u} Ordinal.{u} (fun (x x_1 : Ordinal.{u}) => x * x_1) fun (x x_1 : Ordinal.{u}) => x ≤ x_1

Equations

• Ordinal.mul_covariantClass_le = ⋯

instance Ordinal.mul_swap_covariantClass_le : CovariantClass Ordinal.{u} Ordinal.{u} (Function.swap fun (x x_1 : Ordinal.{u}) => x * x_1) fun (x x_1 : Ordinal.{u}) => x ≤ x_1

Equations

• Ordinal.mul_swap_covariantClass_le = ⋯

theorem Ordinal.le_mul_left (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (hb : 0 < b) : a ≤ a * b

theorem Ordinal.le_mul_right (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (hb : 0 < b) : a ≤ b * a

theorem Ordinal.mul_le_of_limit {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : Ordinal.IsLimit b) : a * b ≤ c ↔ ∀ b' < b, a * b' ≤ c

theorem Ordinal.mul_isNormal {a : Ordinal.{u_4}} (h : 0 < a) : Ordinal.IsNormal fun (x : Ordinal.{u_4}) => a * x

theorem Ordinal.lt_mul_of_limit {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : Ordinal.IsLimit c) : a < b * c ↔ ∃ c' < c, a < b * c'

theorem Ordinal.mul_lt_mul_iff_left {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (a0 : 0 < a) : a * b < a * c ↔ b < c

theorem Ordinal.mul_le_mul_iff_left {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (a0 : 0 < a) : a * b ≤ a * c ↔ b ≤ c

theorem Ordinal.mul_lt_mul_of_pos_left {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : a < b) (c0 : 0 < c) : c * a < c * b

theorem Ordinal.mul_pos {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h₁ : 0 < a) (h₂ : 0 < b) : 0 < a * b

theorem Ordinal.mul_ne_zero {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : a ≠ 0 → b ≠ 0 → a * b ≠ 0

theorem Ordinal.le_of_mul_le_mul_left {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : c * a ≤ c * b) (h0 : 0 < c) : a ≤ b

theorem Ordinal.mul_right_inj {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (a0 : 0 < a) : a * b = a * c ↔ b = c

theorem Ordinal.mul_isLimit {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (a0 : 0 < a) : Ordinal.IsLimit b → Ordinal.IsLimit (a * b)

theorem Ordinal.mul_isLimit_left {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (l : Ordinal.IsLimit a) (b0 : 0 < b) : Ordinal.IsLimit (a * b)

theorem Ordinal.smul_eq_mul (n : ℕ) (a : Ordinal.{u_4}) : n • a = a * ↑n

Division on ordinals

theorem Ordinal.div_nonempty {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : b ≠ 0) : Set.Nonempty {o : Ordinal.{u_4} | a < b * Order.succ o}

The set in the definition of division is nonempty.

instance Ordinal.div : Div Ordinal.{u_4}

a / b is the unique ordinal o satisfying a = b * o + o' with o' < b.

Equations

• Ordinal.div = { div := fun (a b : Ordinal.{u_4}) => if _h : b = 0 then 0 else sInf {o : Ordinal.{u_4} | a < b * Order.succ o} }

@[simp]

theorem Ordinal.div_zero (a : Ordinal.{u_4}) : a / 0 = 0

theorem Ordinal.div_def (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (h : b ≠ 0) : a / b = sInf {o : Ordinal.{u_4} | a < b * Order.succ o}

theorem Ordinal.lt_mul_succ_div (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (h : b ≠ 0) : a < b * Order.succ (a / b)

theorem Ordinal.lt_mul_div_add (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (h : b ≠ 0) : a < b * (a / b) + b

theorem Ordinal.div_le {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (b0 : b ≠ 0) : a / b ≤ c ↔ a < b * Order.succ c

theorem Ordinal.lt_div {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : c ≠ 0) : a < b / c ↔ c * Order.succ a ≤ b

theorem Ordinal.div_pos {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : c ≠ 0) : 0 < b / c ↔ c ≤ b

theorem Ordinal.le_div {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (c0 : c ≠ 0) : a ≤ b / c ↔ c * a ≤ b

theorem Ordinal.div_lt {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (b0 : b ≠ 0) : a / b < c ↔ a < b * c

theorem Ordinal.div_le_of_le_mul {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : a ≤ b * c) : a / b ≤ c

theorem Ordinal.mul_lt_of_lt_div {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} : a < b / c → c * a < b

@[simp]

theorem Ordinal.zero_div (a : Ordinal.{u_4}) : 0 / a = 0

theorem Ordinal.mul_div_le (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : b * (a / b) ≤ a

theorem Ordinal.mul_add_div (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (b0 : b ≠ 0) (c : Ordinal.{u_4}) : (b * a + c) / b = a + c / b

theorem Ordinal.div_eq_zero_of_lt {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : a < b) : a / b = 0

@[simp]

theorem Ordinal.mul_div_cancel (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (b0 : b ≠ 0) : b * a / b = a

@[simp]

theorem Ordinal.div_one (a : Ordinal.{u_4}) : a / 1 = a

@[simp]

theorem Ordinal.div_self {a : Ordinal.{u_4}} (h : a ≠ 0) : a / a = 1

theorem Ordinal.mul_sub (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) (c : Ordinal.{u_4}) : a * (b - c) = a * b - a * c

theorem Ordinal.isLimit_add_iff {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : Ordinal.IsLimit (a + b) ↔ Ordinal.IsLimit b ∨ b = 0 ∧ Ordinal.IsLimit a

theorem Ordinal.dvd_add_iff {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} : a ∣ b → (a ∣ b + c ↔ a ∣ c)

theorem Ordinal.div_mul_cancel {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : a ≠ 0 → a ∣ b → a * (b / a) = b

theorem Ordinal.le_of_dvd {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : b ≠ 0 → a ∣ b → a ≤ b

theorem Ordinal.dvd_antisymm {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h₁ : a ∣ b) (h₂ : b ∣ a) : a = b

instance Ordinal.isAntisymm : IsAntisymm Ordinal.{u_4} fun (x x_1 : Ordinal.{u_4}) => x ∣ x_1

Equations

• Ordinal.isAntisymm = ⋯

instance Ordinal.mod : Mod Ordinal.{u_4}

a % b is the unique ordinal o' satisfying a = b * o + o' with o' < b.

Equations

• Ordinal.mod = { mod := fun (a b : Ordinal.{u_4}) => a - b * (a / b) }

theorem Ordinal.mod_def (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a % b = a - b * (a / b)

theorem Ordinal.mod_le (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a % b ≤ a

@[simp]

theorem Ordinal.mod_zero (a : Ordinal.{u_4}) : a % 0 = a

theorem Ordinal.mod_eq_of_lt {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (h : a < b) : a % b = a

@[simp]

theorem Ordinal.zero_mod (b : Ordinal.{u_4}) : 0 % b = 0

theorem Ordinal.div_add_mod (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : b * (a / b) + a % b = a

theorem Ordinal.mod_lt (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} (h : b ≠ 0) : a % b < b

@[simp]

theorem Ordinal.mod_self (a : Ordinal.{u_4}) : a % a = 0

@[simp]

theorem Ordinal.mod_one (a : Ordinal.{u_4}) : a % 1 = 0

theorem Ordinal.dvd_of_mod_eq_zero {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (H : a % b = 0) : b ∣ a

theorem Ordinal.mod_eq_zero_of_dvd {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} (H : b ∣ a) : a % b = 0

theorem Ordinal.dvd_iff_mod_eq_zero {a : Ordinal.{u_4}} {b : Ordinal.{u_4}} : b ∣ a ↔ a % b = 0

@[simp]

theorem Ordinal.mul_add_mod_self (x : Ordinal.{u_4}) (y : Ordinal.{u_4}) (z : Ordinal.{u_4}) : (x * y + z) % x = z % x

@[simp]

theorem Ordinal.mul_mod (x : Ordinal.{u_4}) (y : Ordinal.{u_4}) : x * y % x = 0

theorem Ordinal.mod_mod_of_dvd (a : Ordinal.{u_4}) {b : Ordinal.{u_4}} {c : Ordinal.{u_4}} (h : c ∣ b) : a % b % c = a % c

@[simp]

theorem Ordinal.mod_mod (a : Ordinal.{u_4}) (b : Ordinal.{u_4}) : a % b % b = a % b

Families of ordinals

There are two kinds of indexed families that naturally arise when dealing with ordinals: those indexed by some type in the appropriate universe, and those indexed by ordinals less than another. The following API allows one to convert from one kind of family to the other.

In many cases, this makes it easy to prove claims about one kind of family via the corresponding claim on the other.

def Ordinal.bfamilyOfFamily' {α : Type u_1} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) (a : Ordinal.{u}) : a < Ordinal.type r → α

Converts a family indexed by a Type u to one indexed by an Ordinal.{u} using a specified well-ordering.

Equations

• Ordinal.bfamilyOfFamily' r f a ha = f (Ordinal.enum r a ha)

def Ordinal.bfamilyOfFamily {α : Type u_1} {ι : Type u} : (ι → α) → (a : Ordinal.{u}) → a < Ordinal.type WellOrderingRel → α

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.

Equations

• Ordinal.bfamilyOfFamily = Ordinal.bfamilyOfFamily' WellOrderingRel

def Ordinal.familyOfBFamily' {α : Type u_1} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (f : (a : Ordinal.{u}) → a < o → α) : ι → α

Converts a family indexed by an Ordinal.{u} to one indexed by a Type u using a specified well-ordering.

Equations

• Ordinal.familyOfBFamily' r ho f i = f (Ordinal.typein r i) ⋯

def Ordinal.familyOfBFamily {α : Type u_1} (o : Ordinal.{u_4}) (f : (a : Ordinal.{u_4}) → a < o → α) : (Quotient.out o).α → α

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.

Equations

• Ordinal.familyOfBFamily o f = Ordinal.familyOfBFamily' (fun (x x_1 : (Quotient.out o).α) => x < x_1) ⋯ f

@[simp]

theorem Ordinal.bfamilyOfFamily'_typein {α : Type u_1} {ι : Type u_4} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) (i : ι) : Ordinal.bfamilyOfFamily' r f (Ordinal.typein r i) ⋯ = f i

@[simp]

theorem Ordinal.bfamilyOfFamily_typein {α : Type u_1} {ι : Type u_4} (f : ι → α) (i : ι) : Ordinal.bfamilyOfFamily f (Ordinal.typein WellOrderingRel i) ⋯ = f i

@[simp]

theorem Ordinal.familyOfBFamily'_enum {α : Type u_1} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (f : (a : Ordinal.{u}) → a < o → α) (i : Ordinal.{u}) (hi : i < o) : Ordinal.familyOfBFamily' r ho f (Ordinal.enum r i ⋯) = f i hi

@[simp]

theorem Ordinal.familyOfBFamily_enum {α : Type u_1} (o : Ordinal.{u_4}) (f : (a : Ordinal.{u_4}) → a < o → α) (i : Ordinal.{u_4}) (hi : i < o) : Ordinal.familyOfBFamily o f (Ordinal.enum (fun (x x_1 : (Quotient.out o).α) => x < x_1) i ⋯) = f i hi

def Ordinal.brange {α : Type u_1} (o : Ordinal.{u_4}) (f : (a : Ordinal.{u_4}) → a < o → α) : Set α

The range of a family indexed by ordinals.

Equations

• Ordinal.brange o f = {a : α | ∃ (i : Ordinal.{u_4}) (hi : i < o), f i hi = a}

theorem Ordinal.mem_brange {α : Type u_1} {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < o → α} {a : α} : a ∈ Ordinal.brange o f ↔ ∃ (i : Ordinal.{u_4}) (hi : i < o), f i hi = a

theorem Ordinal.mem_brange_self {α : Type u_1} {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → α) (i : Ordinal.{u_4}) (hi : i < o) : f i hi ∈ Ordinal.brange o f

@[simp]

theorem Ordinal.range_familyOfBFamily' {α : Type u_1} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (f : (a : Ordinal.{u}) → a < o → α) : Set.range (Ordinal.familyOfBFamily' r ho f) = Ordinal.brange o f

@[simp]

theorem Ordinal.range_familyOfBFamily {α : Type u_1} {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → α) : Set.range (Ordinal.familyOfBFamily o f) = Ordinal.brange o f

@[simp]

theorem Ordinal.brange_bfamilyOfFamily' {α : Type u_1} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) : Ordinal.brange (Ordinal.type r) (Ordinal.bfamilyOfFamily' r f) = Set.range f

@[simp]

theorem Ordinal.brange_bfamilyOfFamily {α : Type u_1} {ι : Type u} (f : ι → α) : Ordinal.brange (Ordinal.type WellOrderingRel) (Ordinal.bfamilyOfFamily f) = Set.range f

@[simp]

theorem Ordinal.brange_const {α : Type u_1} {o : Ordinal.{u_4}} (ho : o ≠ 0) {c : α} : (Ordinal.brange o fun (x : Ordinal.{u_4}) (x : x < o) => c) = {c}

theorem Ordinal.comp_bfamilyOfFamily' {α : Type u_1} {β : Type u_2} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → α) (g : α → β) : (fun (i : Ordinal.{u}) (hi : i < Ordinal.type r) => g (Ordinal.bfamilyOfFamily' r f i hi)) = Ordinal.bfamilyOfFamily' r (g ∘ f)

theorem Ordinal.comp_bfamilyOfFamily {α : Type u_1} {β : Type u_2} {ι : Type u} (f : ι → α) (g : α → β) : (fun (i : Ordinal.{u}) (hi : i < Ordinal.type WellOrderingRel) => g (Ordinal.bfamilyOfFamily f i hi)) = Ordinal.bfamilyOfFamily (g ∘ f)

theorem Ordinal.comp_familyOfBFamily' {α : Type u_1} {β : Type u_2} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (f : (a : Ordinal.{u}) → a < o → α) (g : α → β) : g ∘ Ordinal.familyOfBFamily' r ho f = Ordinal.familyOfBFamily' r ho fun (i : Ordinal.{u}) (hi : i < o) => g (f i hi)

theorem Ordinal.comp_familyOfBFamily {α : Type u_1} {β : Type u_2} {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → α) (g : α → β) : g ∘ Ordinal.familyOfBFamily o f = Ordinal.familyOfBFamily o fun (i : Ordinal.{u_4}) (hi : i < o) => g (f i hi)

Supremum of a family of ordinals

def Ordinal.sup {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.{max u v}

The supremum of a family of ordinals

Equations

• Ordinal.sup f = iSup f

@[simp]

theorem Ordinal.sSup_eq_sup {ι : Type u} (f : ι → Ordinal.{max u v} ) : sSup (Set.range f) = Ordinal.sup f

theorem Ordinal.bddAbove_range {ι : Type u} (f : ι → Ordinal.{max u v} ) : BddAbove (Set.range f)

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.

theorem Ordinal.le_sup {ι : Type u} (f : ι → Ordinal.{max u v} ) (i : ι) : f i ≤ Ordinal.sup f

theorem Ordinal.sup_le_iff {ι : Type u} {f : ι → Ordinal.{max u v} } {a : Ordinal.{max u v} } : Ordinal.sup f ≤ a ↔ ∀ (i : ι), f i ≤ a

theorem Ordinal.sup_le {ι : Type u} {f : ι → Ordinal.{max u v} } {a : Ordinal.{max u v} } : (∀ (i : ι), f i ≤ a) → Ordinal.sup f ≤ a

theorem Ordinal.lt_sup {ι : Type u} {f : ι → Ordinal.{max u v} } {a : Ordinal.{max u v} } : a < Ordinal.sup f ↔ ∃ (i : ι), a < f i

theorem Ordinal.ne_sup_iff_lt_sup {ι : Type u} {f : ι → Ordinal.{max u v} } : (∀ (i : ι), f i ≠ Ordinal.sup f) ↔ ∀ (i : ι), f i < Ordinal.sup f

theorem Ordinal.sup_not_succ_of_ne_sup {ι : Type u} {f : ι → Ordinal.{max u v} } (hf : ∀ (i : ι), f i ≠ Ordinal.sup f) {a : Ordinal.{max u v} } (hao : a < Ordinal.sup f) : Order.succ a < Ordinal.sup f

@[simp]

theorem Ordinal.sup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v} } : Ordinal.sup f = 0 ↔ ∀ (i : ι), f i = 0

theorem Ordinal.IsNormal.sup {f : Ordinal.{max u v} → Ordinal.{max u w} } (H : Ordinal.IsNormal f) {ι : Type u} (g : ι → Ordinal.{max u v} ) [Nonempty ι] : f (Ordinal.sup g) = Ordinal.sup (f ∘ g)

@[simp]

theorem Ordinal.sup_empty {ι : Type u_4} [IsEmpty ι] (f : ι → Ordinal.{max u_5 u_4} ) : Ordinal.sup f = 0

@[simp]

theorem Ordinal.sup_const {ι : Type u_4} [_hι : Nonempty ι] (o : Ordinal.{max u_5 u_4} ) : (Ordinal.sup fun (x : ι) => o) = o

@[simp]

theorem Ordinal.sup_unique {ι : Type u_4} [Unique ι] (f : ι → Ordinal.{max u_5 u_4} ) : Ordinal.sup f = f default

theorem Ordinal.sup_le_of_range_subset {ι : Type u} {ι' : Type v} {f : ι → Ordinal.{max (max u v) w} } {g : ι' → Ordinal.{max (max u v) w} } (h : Set.range f ⊆ Set.range g) : Ordinal.sup f ≤ Ordinal.sup g

theorem Ordinal.sup_eq_of_range_eq {ι : Type u} {ι' : Type v} {f : ι → Ordinal.{max (max u v) w} } {g : ι' → Ordinal.{max (max u v) w} } (h : Set.range f = Set.range g) : Ordinal.sup f = Ordinal.sup g

@[simp]

theorem Ordinal.sup_sum {α : Type u} {β : Type v} (f : α ⊕ β → Ordinal.{max (max u v) w} ) : Ordinal.sup f = max (Ordinal.sup fun (a : α) => f (Sum.inl a)) (Ordinal.sup fun (b : β) => f (Sum.inr b))

theorem Ordinal.unbounded_range_of_sup_ge {α : Type u} {β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β → α) (h : Ordinal.type r ≤ Ordinal.sup (Ordinal.typein r ∘ f)) : Set.Unbounded r (Set.range f)

theorem Ordinal.le_sup_shrink_equiv {s : Set Ordinal.{u}} (hs : Small.{u, u + 1} ↑s) (a : Ordinal.{u}) (ha : a ∈ s) : a ≤ Ordinal.sup fun (x : Shrink.{u, u + 1} ↑s) => ↑((equivShrink ↑s).symm x)

instance Ordinal.small_Iio (o : Ordinal.{u}) : Small.{u, u + 1} ↑(Set.Iio o)

Equations

• ⋯ = ⋯

instance Ordinal.small_Iic (o : Ordinal.{u}) : Small.{u, u + 1} ↑(Set.Iic o)

Equations

• ⋯ = ⋯

theorem Ordinal.bddAbove_iff_small {s : Set Ordinal.{u}} : BddAbove s ↔ Small.{u, u + 1} ↑s

theorem Ordinal.bddAbove_of_small (s : Set Ordinal.{u}) [h : Small.{u, u + 1} ↑s] : BddAbove s

theorem Ordinal.sup_eq_sSup {s : Set Ordinal.{u}} (hs : Small.{u, u + 1} ↑s) : (Ordinal.sup fun (x : Shrink.{u, u + 1} ↑s) => ↑((equivShrink ↑s).symm x)) = sSup s

theorem Ordinal.sSup_ord {s : Set Cardinal.{u}} (hs : BddAbove s) : Cardinal.ord (sSup s) = sSup (Cardinal.ord '' s)

theorem Ordinal.iSup_ord {ι : Sort u_4} {f : ι → Cardinal.{u_5}} (hf : BddAbove (Set.range f)) : Cardinal.ord (iSup f) = ⨆ (i : ι), Cardinal.ord (f i)

theorem Ordinal.sup_eq_sup {ι : Type u} {ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop) [IsWellOrder ι r] [IsWellOrder ι' r'] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (ho' : Ordinal.type r' = o) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.sup (Ordinal.familyOfBFamily' r ho f) = Ordinal.sup (Ordinal.familyOfBFamily' r' ho' f)

def Ordinal.bsup (o : Ordinal.{u}) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.{max u v}

The supremum of a family of ordinals indexed by the set of ordinals less than some o : Ordinal.{u}. This is a special case of sup over the family provided by familyOfBFamily.

Equations

• Ordinal.bsup o f = Ordinal.sup (Ordinal.familyOfBFamily o f)

@[simp]

theorem Ordinal.sup_eq_bsup {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.sup (Ordinal.familyOfBFamily o f) = Ordinal.bsup o f

@[simp]

theorem Ordinal.sup_eq_bsup' {o : Ordinal.{u}} {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (ho : Ordinal.type r = o) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.sup (Ordinal.familyOfBFamily' r ho f) = Ordinal.bsup o f

@[simp]

theorem Ordinal.sSup_eq_bsup {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : sSup (Ordinal.brange o f) = Ordinal.bsup o f

@[simp]

theorem Ordinal.bsup_eq_sup' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → Ordinal.{max u v} ) : Ordinal.bsup (Ordinal.type r) (Ordinal.bfamilyOfFamily' r f) = Ordinal.sup f

theorem Ordinal.bsup_eq_bsup {ι : Type u} (r : ι → ι → Prop) (r' : ι → ι → Prop) [IsWellOrder ι r] [IsWellOrder ι r'] (f : ι → Ordinal.{max u v} ) : Ordinal.bsup (Ordinal.type r) (Ordinal.bfamilyOfFamily' r f) = Ordinal.bsup (Ordinal.type r') (Ordinal.bfamilyOfFamily' r' f)

@[simp]

theorem Ordinal.bsup_eq_sup {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.bsup (Ordinal.type WellOrderingRel) (Ordinal.bfamilyOfFamily f) = Ordinal.sup f

theorem Ordinal.bsup_congr {o₁ : Ordinal.{u}} {o₂ : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o₁ → Ordinal.{max u v} ) (ho : o₁ = o₂) : Ordinal.bsup o₁ f = Ordinal.bsup o₂ fun (a : Ordinal.{u}) (h : a < o₂) => f a ⋯

theorem Ordinal.bsup_le_iff {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} } {a : Ordinal.{max u v} } : Ordinal.bsup o f ≤ a ↔ ∀ (i : Ordinal.{u}) (h : i < o), f i h ≤ a

theorem Ordinal.bsup_le {o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} } {a : Ordinal.{max u v} } : (∀ (i : Ordinal.{u}) (h : i < o), f i h ≤ a) → Ordinal.bsup o f ≤ a

theorem Ordinal.le_bsup {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4} ) (i : Ordinal.{u_4}) (h : i < o) : f i h ≤ Ordinal.bsup o f

theorem Ordinal.lt_bsup {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) {a : Ordinal.{max u v} } : a < Ordinal.bsup o f ↔ ∃ (i : Ordinal.{u}) (hi : i < o), a < f i hi

theorem Ordinal.IsNormal.bsup {f : Ordinal.{max u v} → Ordinal.{max u w} } (H : Ordinal.IsNormal f) {o : Ordinal.{u}} (g : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : o ≠ 0 → f (Ordinal.bsup o g) = Ordinal.bsup o fun (a : Ordinal.{u}) (h : a < o) => f (g a h)

theorem Ordinal.lt_bsup_of_ne_bsup {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} } : (∀ (i : Ordinal.{u}) (h : i < o), f i h ≠ Ordinal.bsup o f) ↔ ∀ (i : Ordinal.{u}) (h : i < o), f i h < Ordinal.bsup o f

theorem Ordinal.bsup_not_succ_of_ne_bsup {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} } (hf : ∀ {i : Ordinal.{u}} (h : i < o), f i h ≠ Ordinal.bsup o f) (a : Ordinal.{max u v} ) : a < Ordinal.bsup o f → Order.succ a < Ordinal.bsup o f

@[simp]

theorem Ordinal.bsup_eq_zero_iff {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4} } : Ordinal.bsup o f = 0 ↔ ∀ (i : Ordinal.{u_4}) (hi : i < o), f i hi = 0

theorem Ordinal.lt_bsup_of_limit {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} } (hf : ∀ {a a' : Ordinal.{u_4}} (ha : a < o) (ha' : a' < o), a < a' → f a ha < f a' ha') (ho : ∀ a < o, Order.succ a < o) (i : Ordinal.{u_4}) (h : i < o) : f i h < Ordinal.bsup o f

theorem Ordinal.bsup_succ_of_mono {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < Order.succ o → Ordinal.{max u_4 u_5} } (hf : ∀ {i j : Ordinal.{u_4}} (hi : i < Order.succ o) (hj : j < Order.succ o), i ≤ j → f i hi ≤ f j hj) : Ordinal.bsup (Order.succ o) f = f o ⋯

@[simp]

theorem Ordinal.bsup_zero (f : (a : Ordinal.{u_4}) → a < 0 → Ordinal.{max u_4 u_5} ) : Ordinal.bsup 0 f = 0

theorem Ordinal.bsup_const {o : Ordinal.{u}} (ho : o ≠ 0) (a : Ordinal.{max u v} ) : (Ordinal.bsup o fun (x : Ordinal.{u}) (x : x < o) => a) = a

@[simp]

theorem Ordinal.bsup_one (f : (a : Ordinal.{u_4}) → a < 1 → Ordinal.{max u_4 u_5} ) : Ordinal.bsup 1 f = f 0 ⋯

theorem Ordinal.bsup_le_of_brange_subset {o : Ordinal.{u}} {o' : Ordinal.{v}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max (max u v) w} } {g : (a : Ordinal.{v}) → a < o' → Ordinal.{max (max u v) w} } (h : Ordinal.brange o f ⊆ Ordinal.brange o' g) : Ordinal.bsup o f ≤ Ordinal.bsup o' g

theorem Ordinal.bsup_eq_of_brange_eq {o : Ordinal.{u}} {o' : Ordinal.{v}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max (max u v) w} } {g : (a : Ordinal.{v}) → a < o' → Ordinal.{max (max u v) w} } (h : Ordinal.brange o f = Ordinal.brange o' g) : Ordinal.bsup o f = Ordinal.bsup o' g

def Ordinal.lsub {ι : Type u_4} (f : ι → Ordinal.{max u_5 u_4} ) : Ordinal.{max u_5 u_4}

The least strict upper bound of a family of ordinals.

Equations

• Ordinal.lsub f = Ordinal.sup (Order.succ ∘ f)

@[simp]

theorem Ordinal.sup_eq_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.sup (Order.succ ∘ f) = Ordinal.lsub f

theorem Ordinal.lsub_le_iff {ι : Type u} {f : ι → Ordinal.{max u v} } {a : Ordinal.{max v u} } : Ordinal.lsub f ≤ a ↔ ∀ (i : ι), f i < a

theorem Ordinal.lsub_le {ι : Type u_4} {f : ι → Ordinal.{max u_5 u_4} } {a : Ordinal.{max u_5 u_4} } : (∀ (i : ι), f i < a) → Ordinal.lsub f ≤ a

theorem Ordinal.lt_lsub {ι : Type u_4} (f : ι → Ordinal.{max u_5 u_4} ) (i : ι) : f i < Ordinal.lsub f

theorem Ordinal.lt_lsub_iff {ι : Type u} {f : ι → Ordinal.{max u v} } {a : Ordinal.{max v u} } : a < Ordinal.lsub f ↔ ∃ (i : ι), a ≤ f i

theorem Ordinal.sup_le_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.sup f ≤ Ordinal.lsub f

theorem Ordinal.lsub_le_sup_succ {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.lsub f ≤ Order.succ (Ordinal.sup f)

theorem Ordinal.sup_eq_lsub_or_sup_succ_eq_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.sup f = Ordinal.lsub f ∨ Order.succ (Ordinal.sup f) = Ordinal.lsub f

theorem Ordinal.sup_succ_le_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Order.succ (Ordinal.sup f) ≤ Ordinal.lsub f ↔ ∃ (i : ι), f i = Ordinal.sup f

theorem Ordinal.sup_succ_eq_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Order.succ (Ordinal.sup f) = Ordinal.lsub f ↔ ∃ (i : ι), f i = Ordinal.sup f

theorem Ordinal.sup_eq_lsub_iff_succ {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.sup f = Ordinal.lsub f ↔ ∀ a < Ordinal.lsub f, Order.succ a < Ordinal.lsub f

theorem Ordinal.sup_eq_lsub_iff_lt_sup {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.sup f = Ordinal.lsub f ↔ ∀ (i : ι), f i < Ordinal.sup f

@[simp]

theorem Ordinal.lsub_empty {ι : Type u_4} [h : IsEmpty ι] (f : ι → Ordinal.{max u_5 u_4} ) : Ordinal.lsub f = 0

theorem Ordinal.lsub_pos {ι : Type u} [h : Nonempty ι] (f : ι → Ordinal.{max u v} ) : 0 < Ordinal.lsub f

@[simp]

theorem Ordinal.lsub_eq_zero_iff {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.lsub f = 0 ↔ IsEmpty ι

@[simp]

theorem Ordinal.lsub_const {ι : Type u_4} [Nonempty ι] (o : Ordinal.{max u_5 u_4} ) : (Ordinal.lsub fun (x : ι) => o) = Order.succ o

@[simp]

theorem Ordinal.lsub_unique {ι : Type u_4} [Unique ι] (f : ι → Ordinal.{max u_5 u_4} ) : Ordinal.lsub f = Order.succ (f default)

theorem Ordinal.lsub_le_of_range_subset {ι : Type u} {ι' : Type v} {f : ι → Ordinal.{max (max u v) w} } {g : ι' → Ordinal.{max (max u v) w} } (h : Set.range f ⊆ Set.range g) : Ordinal.lsub f ≤ Ordinal.lsub g

theorem Ordinal.lsub_eq_of_range_eq {ι : Type u} {ι' : Type v} {f : ι → Ordinal.{max (max u v) w} } {g : ι' → Ordinal.{max (max u v) w} } (h : Set.range f = Set.range g) : Ordinal.lsub f = Ordinal.lsub g

@[simp]

theorem Ordinal.lsub_sum {α : Type u} {β : Type v} (f : α ⊕ β → Ordinal.{max (max u v) w} ) : Ordinal.lsub f = max (Ordinal.lsub fun (a : α) => f (Sum.inl a)) (Ordinal.lsub fun (b : β) => f (Sum.inr b))

theorem Ordinal.lsub_not_mem_range {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.lsub f ∉ Set.range f

theorem Ordinal.nonempty_compl_range {ι : Type u} (f : ι → Ordinal.{max u v} ) : Set.Nonempty (Set.range f)ᶜ

@[simp]

theorem Ordinal.lsub_typein (o : Ordinal.{u}) : Ordinal.lsub (Ordinal.typein fun (x x_1 : (Quotient.out o).α) => x < x_1) = o

theorem Ordinal.sup_typein_limit {o : Ordinal.{u}} (ho : ∀ a < o, Order.succ a < o) : Ordinal.sup (Ordinal.typein fun (x x_1 : (Quotient.out o).α) => x < x_1) = o

@[simp]

theorem Ordinal.sup_typein_succ {o : Ordinal.{u}} : Ordinal.sup (Ordinal.typein fun (x x_1 : (Quotient.out (Order.succ o)).α) => x < x_1) = o

def Ordinal.blsub (o : Ordinal.{u}) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.{max u v}

The least strict upper bound of a family of ordinals indexed by the set of ordinals less than some o : Ordinal.{u}.

This is to lsub as bsup is to sup.

Equations

• Ordinal.blsub o f = Ordinal.bsup o fun (a : Ordinal.{u}) (ha : a < o) => Order.succ (f a ha)

@[simp]

theorem Ordinal.bsup_eq_blsub (o : Ordinal.{u}) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : (Ordinal.bsup o fun (a : Ordinal.{u}) (ha : a < o) => Order.succ (f a ha)) = Ordinal.blsub o f

theorem Ordinal.lsub_eq_blsub' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.lsub (Ordinal.familyOfBFamily' r ho f) = Ordinal.blsub o f

theorem Ordinal.lsub_eq_lsub {ι : Type u} {ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop) [IsWellOrder ι r] [IsWellOrder ι' r'] {o : Ordinal.{u}} (ho : Ordinal.type r = o) (ho' : Ordinal.type r' = o) (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.lsub (Ordinal.familyOfBFamily' r ho f) = Ordinal.lsub (Ordinal.familyOfBFamily' r' ho' f)

@[simp]

theorem Ordinal.lsub_eq_blsub {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.lsub (Ordinal.familyOfBFamily o f) = Ordinal.blsub o f

@[simp]

theorem Ordinal.blsub_eq_lsub' {ι : Type u} (r : ι → ι → Prop) [IsWellOrder ι r] (f : ι → Ordinal.{max u v} ) : Ordinal.blsub (Ordinal.type r) (Ordinal.bfamilyOfFamily' r f) = Ordinal.lsub f

theorem Ordinal.blsub_eq_blsub {ι : Type u} (r : ι → ι → Prop) (r' : ι → ι → Prop) [IsWellOrder ι r] [IsWellOrder ι r'] (f : ι → Ordinal.{max u v} ) : Ordinal.blsub (Ordinal.type r) (Ordinal.bfamilyOfFamily' r f) = Ordinal.blsub (Ordinal.type r') (Ordinal.bfamilyOfFamily' r' f)

@[simp]

theorem Ordinal.blsub_eq_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.blsub (Ordinal.type WellOrderingRel) (Ordinal.bfamilyOfFamily f) = Ordinal.lsub f

theorem Ordinal.blsub_congr {o₁ : Ordinal.{u}} {o₂ : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o₁ → Ordinal.{max u v} ) (ho : o₁ = o₂) : Ordinal.blsub o₁ f = Ordinal.blsub o₂ fun (a : Ordinal.{u}) (h : a < o₂) => f a ⋯

theorem Ordinal.blsub_le_iff {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} } {a : Ordinal.{max u v} } : Ordinal.blsub o f ≤ a ↔ ∀ (i : Ordinal.{u}) (h : i < o), f i h < a

theorem Ordinal.blsub_le {o : Ordinal.{u_4}} {f : (b : Ordinal.{u_4}) → b < o → Ordinal.{max u_4 u_5} } {a : Ordinal.{max u_4 u_5} } : (∀ (i : Ordinal.{u_4}) (h : i < o), f i h < a) → Ordinal.blsub o f ≤ a

theorem Ordinal.lt_blsub {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4} ) (i : Ordinal.{u_4}) (h : i < o) : f i h < Ordinal.blsub o f

theorem Ordinal.lt_blsub_iff {o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} } {a : Ordinal.{max u v} } : a < Ordinal.blsub o f ↔ ∃ (i : Ordinal.{u}) (hi : i < o), a ≤ f i hi

theorem Ordinal.bsup_le_blsub {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.bsup o f ≤ Ordinal.blsub o f

theorem Ordinal.blsub_le_bsup_succ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.blsub o f ≤ Order.succ (Ordinal.bsup o f)

theorem Ordinal.bsup_eq_blsub_or_succ_bsup_eq_blsub {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.bsup o f = Ordinal.blsub o f ∨ Order.succ (Ordinal.bsup o f) = Ordinal.blsub o f

theorem Ordinal.bsup_succ_le_blsub {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Order.succ (Ordinal.bsup o f) ≤ Ordinal.blsub o f ↔ ∃ (i : Ordinal.{u}) (hi : i < o), f i hi = Ordinal.bsup o f

theorem Ordinal.bsup_succ_eq_blsub {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Order.succ (Ordinal.bsup o f) = Ordinal.blsub o f ↔ ∃ (i : Ordinal.{u}) (hi : i < o), f i hi = Ordinal.bsup o f

theorem Ordinal.bsup_eq_blsub_iff_succ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.bsup o f = Ordinal.blsub o f ↔ ∀ a < Ordinal.blsub o f, Order.succ a < Ordinal.blsub o f

theorem Ordinal.bsup_eq_blsub_iff_lt_bsup {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.bsup o f = Ordinal.blsub o f ↔ ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < Ordinal.bsup o f

theorem Ordinal.bsup_eq_blsub_of_lt_succ_limit {o : Ordinal.{u}} (ho : Ordinal.IsLimit o) {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} } (hf : ∀ (a : Ordinal.{u}) (ha : a < o), f a ha < f (Order.succ a) ⋯) : Ordinal.bsup o f = Ordinal.blsub o f

theorem Ordinal.blsub_succ_of_mono {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < Order.succ o → Ordinal.{max u v} } (hf : ∀ {i j : Ordinal.{u}} (hi : i < Order.succ o) (hj : j < Order.succ o), i ≤ j → f i hi ≤ f j hj) : Ordinal.blsub (Order.succ o) f = Order.succ (f o ⋯)

@[simp]

theorem Ordinal.blsub_eq_zero_iff {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4} } : Ordinal.blsub o f = 0 ↔ o = 0

@[simp]

theorem Ordinal.blsub_zero (f : (a : Ordinal.{u_4}) → a < 0 → Ordinal.{max u_4 u_5} ) : Ordinal.blsub 0 f = 0

theorem Ordinal.blsub_pos {o : Ordinal.{u_4}} (ho : 0 < o) (f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} ) : 0 < Ordinal.blsub o f

theorem Ordinal.blsub_type {α : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : (a : Ordinal.{u}) → a < Ordinal.type r → Ordinal.{max u v} ) : Ordinal.blsub (Ordinal.type r) f = Ordinal.lsub fun (a : α) => f (Ordinal.typein r a) ⋯

theorem Ordinal.blsub_const {o : Ordinal.{u}} (ho : o ≠ 0) (a : Ordinal.{max u v} ) : (Ordinal.blsub o fun (x : Ordinal.{u}) (x : x < o) => a) = Order.succ a

@[simp]

theorem Ordinal.blsub_one (f : (a : Ordinal.{u_4}) → a < 1 → Ordinal.{max u_4 u_5} ) : Ordinal.blsub 1 f = Order.succ (f 0 ⋯)

@[simp]

theorem Ordinal.blsub_id (o : Ordinal.{u}) : (Ordinal.blsub o fun (x : Ordinal.{u}) (x_1 : x < o) => x) = o

theorem Ordinal.bsup_id_limit {o : Ordinal.{u}} : (∀ a < o, Order.succ a < o) → (Ordinal.bsup o fun (x : Ordinal.{u}) (x_1 : x < o) => x) = o

@[simp]

theorem Ordinal.bsup_id_succ (o : Ordinal.{u}) : (Ordinal.bsup (Order.succ o) fun (x : Ordinal.{u}) (x_1 : x < Order.succ o) => x) = o

theorem Ordinal.blsub_le_of_brange_subset {o : Ordinal.{u}} {o' : Ordinal.{v}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max (max u v) w} } {g : (a : Ordinal.{v}) → a < o' → Ordinal.{max (max u v) w} } (h : Ordinal.brange o f ⊆ Ordinal.brange o' g) : Ordinal.blsub o f ≤ Ordinal.blsub o' g

theorem Ordinal.blsub_eq_of_brange_eq {o : Ordinal.{u}} {o' : Ordinal.{v}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max (max u v) w} } {g : (a : Ordinal.{v}) → a < o' → Ordinal.{max (max u v) w} } (h : {o_1 : Ordinal.{max (max u v) w} | ∃ (i : Ordinal.{u}) (hi : i < o), f i hi = o_1} = {o : Ordinal.{max (max u v) w} | ∃ (i : Ordinal.{v}) (hi : i < o'), g i hi = o}) : Ordinal.blsub o f = Ordinal.blsub o' g

theorem Ordinal.bsup_comp {o : Ordinal.{max u v} } {o' : Ordinal.{max u v} } {f : (a : Ordinal.{max u v} ) → a < o → Ordinal.{max u v w} } (hf : ∀ {i j : Ordinal.{max u v} } (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj) {g : (a : Ordinal.{max u v} ) → a < o' → Ordinal.{max u v} } (hg : Ordinal.blsub o' g = o) : (Ordinal.bsup o' fun (a : Ordinal.{max u v} ) (ha : a < o') => f (g a ha) ⋯) = Ordinal.bsup o f

theorem Ordinal.blsub_comp {o : Ordinal.{max u v} } {o' : Ordinal.{max u v} } {f : (a : Ordinal.{max u v} ) → a < o → Ordinal.{max u v w} } (hf : ∀ {i j : Ordinal.{max u v} } (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj) {g : (a : Ordinal.{max u v} ) → a < o' → Ordinal.{max u v} } (hg : Ordinal.blsub o' g = o) : (Ordinal.blsub o' fun (a : Ordinal.{max u v} ) (ha : a < o') => f (g a ha) ⋯) = Ordinal.blsub o f

theorem Ordinal.IsNormal.bsup_eq {f : Ordinal.{u} → Ordinal.{max u v} } (H : Ordinal.IsNormal f) {o : Ordinal.{u}} (h : Ordinal.IsLimit o) : (Ordinal.bsup o fun (x : Ordinal.{u}) (x_1 : x < o) => f x) = f o

theorem Ordinal.IsNormal.blsub_eq {f : Ordinal.{u} → Ordinal.{max u v} } (H : Ordinal.IsNormal f) {o : Ordinal.{u}} (h : Ordinal.IsLimit o) : (Ordinal.blsub o fun (x : Ordinal.{u}) (x_1 : x < o) => f x) = f o

theorem Ordinal.isNormal_iff_lt_succ_and_bsup_eq {f : Ordinal.{u} → Ordinal.{max u v} } : Ordinal.IsNormal f ↔ (∀ (a : Ordinal.{u}), f a < f (Order.succ a)) ∧ ∀ (o : Ordinal.{u}), Ordinal.IsLimit o → (Ordinal.bsup o fun (x : Ordinal.{u}) (x_1 : x < o) => f x) = f o

theorem Ordinal.isNormal_iff_lt_succ_and_blsub_eq {f : Ordinal.{u} → Ordinal.{max u v} } : Ordinal.IsNormal f ↔ (∀ (a : Ordinal.{u}), f a < f (Order.succ a)) ∧ ∀ (o : Ordinal.{u}), Ordinal.IsLimit o → (Ordinal.blsub o fun (x : Ordinal.{u}) (x_1 : x < o) => f x) = f o

theorem Ordinal.IsNormal.eq_iff_zero_and_succ {f : Ordinal.{u} → Ordinal.{u}} {g : Ordinal.{u} → Ordinal.{u}} (hf : Ordinal.IsNormal f) (hg : Ordinal.IsNormal g) : f = g ↔ f 0 = g 0 ∧ ∀ (a : Ordinal.{u}), f a = g a → f (Order.succ a) = g (Order.succ a)

def Ordinal.blsub₂ (o₁ : Ordinal.{u_4}) (o₂ : Ordinal.{u_5}) (op : {a : Ordinal.{u_4}} → a < o₁ → {b : Ordinal.{u_5}} → b < o₂ → Ordinal.{max (max u_6 u_4) u_5} ) : Ordinal.{max u_6 u_4 u_5}

A two-argument version of Ordinal.blsub. We don't develop a full API for this, since it's only used in a handful of existence results.

Equations

• Ordinal.blsub₂ o₁ o₂ op = Ordinal.lsub fun (x : (Quotient.out o₁).α × (Quotient.out o₂).α) => op ⋯ ⋯

theorem Ordinal.lt_blsub₂ {o₁ : Ordinal.{u_4}} {o₂ : Ordinal.{u_5}} (op : {a : Ordinal.{u_4}} → a < o₁ → {b : Ordinal.{u_5}} → b < o₂ → Ordinal.{max (max u_4 u_5) u_6} ) {a : Ordinal.{u_4}} {b : Ordinal.{u_5}} (ha : a < o₁) (hb : b < o₂) : op ha hb < Ordinal.blsub₂ o₁ o₂ fun {a : Ordinal.{u_4}} => op

Minimum excluded ordinals

def Ordinal.mex {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.{max u v}

The minimum excluded ordinal in a family of ordinals.

Equations

• Ordinal.mex f = sInf (Set.range f)ᶜ

theorem Ordinal.mex_not_mem_range {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.mex f ∉ Set.range f

theorem Ordinal.le_mex_of_forall {ι : Type u} {f : ι → Ordinal.{max u v} } {a : Ordinal.{max u v} } (H : ∀ b < a, ∃ (i : ι), f i = b) : a ≤ Ordinal.mex f

theorem Ordinal.ne_mex {ι : Type u} (f : ι → Ordinal.{max u v} ) (i : ι) : f i ≠ Ordinal.mex f

theorem Ordinal.mex_le_of_ne {ι : Type u_4} {f : ι → Ordinal.{max u_5 u_4} } {a : Ordinal.{max u_5 u_4} } (ha : ∀ (i : ι), f i ≠ a) : Ordinal.mex f ≤ a

theorem Ordinal.exists_of_lt_mex {ι : Type u_4} {f : ι → Ordinal.{max u_5 u_4} } {a : Ordinal.{max u_4 u_5} } (ha : a < Ordinal.mex f) : ∃ (i : ι), f i = a

theorem Ordinal.mex_le_lsub {ι : Type u} (f : ι → Ordinal.{max u v} ) : Ordinal.mex f ≤ Ordinal.lsub f

theorem Ordinal.mex_monotone {α : Type u} {β : Type u} {f : α → Ordinal.{max u v} } {g : β → Ordinal.{max u v} } (h : Set.range f ⊆ Set.range g) : Ordinal.mex f ≤ Ordinal.mex g

theorem Ordinal.mex_lt_ord_succ_mk {ι : Type u} (f : ι → Ordinal.{u}) : Ordinal.mex f < Cardinal.ord (Order.succ (Cardinal.mk ι))

def Ordinal.bmex (o : Ordinal.{u_4}) (f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} ) : Ordinal.{max u_5 u_4}

The minimum excluded ordinal of a family of ordinals indexed by the set of ordinals less than some o : Ordinal.{u}. This is a special case of mex over the family provided by familyOfBFamily.

This is to mex as bsup is to sup.

Equations

• Ordinal.bmex o f = Ordinal.mex (Ordinal.familyOfBFamily o f)

theorem Ordinal.bmex_not_mem_brange {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} ) : Ordinal.bmex o f ∉ Ordinal.brange o f

theorem Ordinal.le_bmex_of_forall {o : Ordinal.{u_4}} (f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} ) {a : Ordinal.{max u_4 u_5} } (H : ∀ b < a, ∃ (i : Ordinal.{u_4}) (hi : i < o), f i hi = b) : a ≤ Ordinal.bmex o f

theorem Ordinal.ne_bmex {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) {i : Ordinal.{u}} (hi : i < o) : f i hi ≠ Ordinal.bmex o f

theorem Ordinal.bmex_le_of_ne {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} } {a : Ordinal.{max u_4 u_5} } (ha : ∀ (i : Ordinal.{u_4}) (hi : i < o), f i hi ≠ a) : Ordinal.bmex o f ≤ a

theorem Ordinal.exists_of_lt_bmex {o : Ordinal.{u_4}} {f : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_4 u_5} } {a : Ordinal.{max u_5 u_4} } (ha : a < Ordinal.bmex o f) : ∃ (i : Ordinal.{u_4}) (hi : i < o), f i hi = a

theorem Ordinal.bmex_le_blsub {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} ) : Ordinal.bmex o f ≤ Ordinal.blsub o f

theorem Ordinal.bmex_monotone {o : Ordinal.{u}} {o' : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} } {g : (a : Ordinal.{u}) → a < o' → Ordinal.{max u v} } (h : Ordinal.brange o f ⊆ Ordinal.brange o' g) : Ordinal.bmex o f ≤ Ordinal.bmex o' g

theorem Ordinal.bmex_lt_ord_succ_card {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}) : Ordinal.bmex o f < Cardinal.ord (Order.succ (Ordinal.card o))

Results about injectivity and surjectivity

theorem not_surjective_of_ordinal {α : Type u} (f : α → Ordinal.{u}) : ¬Function.Surjective f

theorem not_injective_of_ordinal {α : Type u} (f : Ordinal.{u} → α) : ¬Function.Injective f

theorem not_surjective_of_ordinal_of_small {α : Type v} [Small.{u, v} α] (f : α → Ordinal.{u}) : ¬Function.Surjective f

theorem not_injective_of_ordinal_of_small {α : Type v} [Small.{u, v} α] (f : Ordinal.{u} → α) : ¬Function.Injective f

theorem not_small_ordinal : ¬Small.{u, max (u + 1) (v + 1)} Ordinal.{max u v}

The type of ordinals in universe u is not Small.{u}. This is the type-theoretic analog of the Burali-Forti paradox.

Enumerating unbounded sets of ordinals with ordinals

def Ordinal.enumOrd (S : Set Ordinal.{u}) : Ordinal.{u} → Ordinal.{u}

Enumerator function for an unbounded set of ordinals.

Equations

• Ordinal.enumOrd S = WellFounded.fix Ordinal.lt_wf fun (o : Ordinal.{u}) (f : (y : Ordinal.{u}) → y < o → Ordinal.{u}) => sInf (S ∩ Set.Ici (Ordinal.blsub o f))

theorem Ordinal.enumOrd_def' {S : Set Ordinal.{u}} (o : Ordinal.{u}) : Ordinal.enumOrd S o = sInf (S ∩ Set.Ici (Ordinal.blsub o fun (a : Ordinal.{u}) (x : a < o) => Ordinal.enumOrd S a))

The equation that characterizes enumOrd definitionally. This isn't the nicest expression to work with, so consider using enumOrd_def instead.

theorem Ordinal.enumOrd_def'_nonempty {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) (a : Ordinal.{u}) : Set.Nonempty (S ∩ Set.Ici a)

The set in enumOrd_def' is nonempty.

theorem Ordinal.enumOrd_mem {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) (o : Ordinal.{u}) : Ordinal.enumOrd S o ∈ S

theorem Ordinal.blsub_le_enumOrd {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) (o : Ordinal.{u}) : (Ordinal.blsub o fun (c : Ordinal.{u}) (x : c < o) => Ordinal.enumOrd S c) ≤ Ordinal.enumOrd S o

theorem Ordinal.enumOrd_strictMono {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) : StrictMono (Ordinal.enumOrd S)

theorem Ordinal.enumOrd_def {S : Set Ordinal.{u}} (o : Ordinal.{u}) : Ordinal.enumOrd S o = sInf (S ∩ {b : Ordinal.{u} | ∀ c < o, Ordinal.enumOrd S c < b})

A more workable definition for enumOrd.

theorem Ordinal.enumOrd_def_nonempty {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) {o : Ordinal.{u}} : Set.Nonempty {x : Ordinal.{u} | x ∈ S ∧ ∀ c < o, Ordinal.enumOrd S c < x}

The set in enumOrd_def is nonempty.

@[simp]

theorem Ordinal.enumOrd_range {f : Ordinal.{u_1} → Ordinal.{u_1}} (hf : StrictMono f) : Ordinal.enumOrd (Set.range f) = f

@[simp]

theorem Ordinal.enumOrd_univ : Ordinal.enumOrd Set.univ = id

@[simp]

theorem Ordinal.enumOrd_zero {S : Set Ordinal.{u}} : Ordinal.enumOrd S 0 = sInf S

theorem Ordinal.enumOrd_succ_le {S : Set Ordinal.{u}} {a : Ordinal.{u}} {b : Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) (ha : a ∈ S) (hb : Ordinal.enumOrd S b < a) : Ordinal.enumOrd S (Order.succ b) ≤ a

theorem Ordinal.enumOrd_le_of_subset {S : Set Ordinal.{u_1}} {T : Set Ordinal.{u_1}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u_1}) => x < x_1) S) (hST : S ⊆ T) (a : Ordinal.{u_1}) : Ordinal.enumOrd T a ≤ Ordinal.enumOrd S a

theorem Ordinal.enumOrd_surjective {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) (s : Ordinal.{u}) : s ∈ S → ∃ (a : Ordinal.{u}), Ordinal.enumOrd S a = s

def Ordinal.enumOrdOrderIso {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) : Ordinal.{u} ≃o ↑S

An order isomorphism between an unbounded set of ordinals and the ordinals.

Equations

• Ordinal.enumOrdOrderIso hS = StrictMono.orderIsoOfSurjective (fun (o : Ordinal.{u}) => { val := Ordinal.enumOrd S o, property := ⋯ }) ⋯ ⋯

theorem Ordinal.range_enumOrd {S : Set Ordinal.{u}} (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) : Set.range (Ordinal.enumOrd S) = S

theorem Ordinal.eq_enumOrd {S : Set Ordinal.{u}} (f : Ordinal.{u} → Ordinal.{u}) (hS : Set.Unbounded (fun (x x_1 : Ordinal.{u}) => x < x_1) S) : StrictMono f ∧ Set.range f = S ↔ f = Ordinal.enumOrd S

A characterization of enumOrd: it is the unique strict monotonic function with range S.

Casting naturals into ordinals, compatibility with operations

@[simp]

theorem Ordinal.one_add_natCast (m : ℕ) : 1 + ↑m = ↑(Order.succ m)

@[simp]

theorem Ordinal.one_add_ofNat (m : ℕ) [Nat.AtLeastTwo m] : 1 + OfNat.ofNat m = Order.succ (OfNat.ofNat m)

@[simp]

theorem Ordinal.natCast_mul (m : ℕ) (n : ℕ) : ↑(m * n) = ↑m * ↑n

theorem Ordinal.natCast_le {m : ℕ} {n : ℕ} : ↑m ≤ ↑n ↔ m ≤ n

Alias of Nat.cast_le, specialized to Ordinal -

theorem Ordinal.natCast_inj {m : ℕ} {n : ℕ} : ↑m = ↑n ↔ m = n

Alias of Nat.cast_inj, specialized to Ordinal -

instance Ordinal.charZero : CharZero Ordinal.{u_1}

Equations

• Ordinal.charZero = ⋯

theorem Ordinal.natCast_lt {m : ℕ} {n : ℕ} : ↑m < ↑n ↔ m < n

Alias of Nat.cast_lt, specialized to Ordinal -

theorem Ordinal.natCast_eq_zero {n : ℕ} : ↑n = 0 ↔ n = 0

Alias of Nat.cast_eq_zero, specialized to Ordinal -

theorem Ordinal.natCast_ne_zero {n : ℕ} : ↑n ≠ 0 ↔ n ≠ 0

Alias of Nat.cast_eq_zero, specialized to Ordinal -

theorem Ordinal.natCast_pos {n : ℕ} : 0 < ↑n ↔ 0 < n

Alias of Nat.cast_pos', specialized to Ordinal -

@[simp]

theorem Ordinal.natCast_sub (m : ℕ) (n : ℕ) : ↑(m - n) = ↑m - ↑n

@[simp]

theorem Ordinal.natCast_div (m : ℕ) (n : ℕ) : ↑(m / n) = ↑m / ↑n

@[simp]

theorem Ordinal.natCast_mod (m : ℕ) (n : ℕ) : ↑(m % n) = ↑m % ↑n

@[simp]

theorem Ordinal.lift_natCast (n : ℕ) : Ordinal.lift.{u, v} ↑n = ↑n

@[simp]

theorem Ordinal.lift_ofNat (n : ℕ) [Nat.AtLeastTwo n] : Ordinal.lift.{u, v} (OfNat.ofNat n) = OfNat.ofNat n

Properties of omega

@[simp]

theorem Cardinal.ord_aleph0 : Cardinal.ord Cardinal.aleph0 = Ordinal.omega

@[simp]

theorem Cardinal.add_one_of_aleph0_le {c : Cardinal.{u_1}} (h : Cardinal.aleph0 ≤ c) : c + 1 = c

theorem Ordinal.lt_add_of_limit {a : Ordinal.{u}} {b : Ordinal.{u}} {c : Ordinal.{u}} (h : Ordinal.IsLimit c) : a < b + c ↔ ∃ c' < c, a < b + c'

theorem Ordinal.lt_omega {o : Ordinal.{u_1}} : o < Ordinal.omega ↔ ∃ (n : ℕ), o = ↑n

theorem Ordinal.nat_lt_omega (n : ℕ) : ↑n < Ordinal.omega

theorem Ordinal.omega_pos : 0 < Ordinal.omega

theorem Ordinal.omega_ne_zero : Ordinal.omega ≠ 0

theorem Ordinal.one_lt_omega : 1 < Ordinal.omega

theorem Ordinal.omega_isLimit : Ordinal.IsLimit Ordinal.omega

theorem Ordinal.omega_le {o : Ordinal.{u_1}} : Ordinal.omega ≤ o ↔ ∀ (n : ℕ), ↑n ≤ o

@[simp]

theorem Ordinal.sup_natCast : Ordinal.sup Nat.cast = Ordinal.omega

theorem Ordinal.nat_lt_limit {o : Ordinal.{u_1}} (h : Ordinal.IsLimit o) (n : ℕ) : ↑n < o

theorem Ordinal.omega_le_of_isLimit {o : Ordinal.{u_1}} (h : Ordinal.IsLimit o) : Ordinal.omega ≤ o

theorem Ordinal.isLimit_iff_omega_dvd {a : Ordinal.{u_1}} : Ordinal.IsLimit a ↔ a ≠ 0 ∧ Ordinal.omega ∣ a

theorem Ordinal.add_mul_limit_aux {a : Ordinal.{u_1}} {b : Ordinal.{u_1}} {c : Ordinal.{u_1}} (ba : b + a = a) (l : Ordinal.IsLimit c) (IH : ∀ c' < c, (a + b) * Order.succ c' = a * Order.succ c' + b) : (a + b) * c = a * c

theorem Ordinal.add_mul_succ {a : Ordinal.{u_1}} {b : Ordinal.{u_1}} (c : Ordinal.{u_1}) (ba : b + a = a) : (a + b) * Order.succ c = a * Order.succ c + b

theorem Ordinal.add_mul_limit {a : Ordinal.{u_1}} {b : Ordinal.{u_1}} {c : Ordinal.{u_1}} (ba : b + a = a) (l : Ordinal.IsLimit c) : (a + b) * c = a * c

theorem Ordinal.add_le_of_forall_add_lt {a : Ordinal.{u_1}} {b : Ordinal.{u_1}} {c : Ordinal.{u_1}} (hb : 0 < b) (h : ∀ d < b, a + d < c) : a + b ≤ c

theorem Ordinal.IsNormal.apply_omega {f : Ordinal.{u} → Ordinal.{u}} (hf : Ordinal.IsNormal f) : Ordinal.sup (f ∘ Nat.cast) = f Ordinal.omega

@[simp]

theorem Ordinal.sup_add_nat (o : Ordinal.{u_1}) : (Ordinal.sup fun (n : ℕ) => o + ↑n) = o + Ordinal.omega

@[simp]

theorem Ordinal.sup_mul_nat (o : Ordinal.{u_1}) : (Ordinal.sup fun (n : ℕ) => o * ↑n) = o * Ordinal.omega

noncomputable def Acc.rank {α : Type u} {r : α → α → Prop} {a : α} (h : Acc r a) : Ordinal.{u}

The rank of an element a accessible under a relation r is defined inductively as the smallest ordinal greater than the ranks of all elements below it (i.e. elements b such that r b a).

Equations

• Acc.rank h = Acc.recOn h fun (a : α) (_h : ∀ (y : α), r y a → Acc r y) (ih : (y : α) → r y a → Ordinal.{u}) => Ordinal.sup fun (b : { b : α // r b a }) => Order.succ (ih ↑b ⋯)

theorem Acc.rank_eq {α : Type u} {r : α → α → Prop} {a : α} (h : Acc r a) : Acc.rank h = Ordinal.sup fun (b : { b : α // r b a }) => Order.succ (Acc.rank ⋯)

theorem Acc.rank_lt_of_rel {α : Type u} {r : α → α → Prop} {a : α} {b : α} (hb : Acc r b) (h : r a b) : Acc.rank ⋯ < Acc.rank hb

if r a b then the rank of a is less than the rank of b.

noncomputable def WellFounded.rank {α : Type u} {r : α → α → Prop} (hwf : WellFounded r) (a : α) : Ordinal.{u}

The rank of an element a under a well-founded relation r is defined inductively as the smallest ordinal greater than the ranks of all elements below it (i.e. elements b such that r b a).

Equations

• WellFounded.rank hwf a = Acc.rank ⋯

theorem WellFounded.rank_eq {α : Type u} {r : α → α → Prop} {a : α} (hwf : WellFounded r) : WellFounded.rank hwf a = Ordinal.sup fun (b : { b : α // r b a }) => Order.succ (WellFounded.rank hwf ↑b)

theorem WellFounded.rank_lt_of_rel {α : Type u} {r : α → α → Prop} {a : α} {b : α} (hwf : WellFounded r) (h : r a b) : WellFounded.rank hwf a < WellFounded.rank hwf b

theorem WellFounded.rank_strictMono {α : Type u} [Preorder α] [WellFoundedLT α] : StrictMono (WellFounded.rank ⋯)

theorem WellFounded.rank_strictAnti {α : Type u} [Preorder α] [WellFoundedGT α] : StrictAnti (WellFounded.rank ⋯)