Natural numbers with infinity

The natural numbers and an extra top element ⊤. This implementation uses Part ℕ as an implementation. Use ℕ∞ instead unless you care about computability.

Main definitions

The following instances are defined:

• OrderedAddCommMonoid PartENat
• CanonicallyOrderedAddCommMonoid PartENat
• CompleteLinearOrder PartENat

There is no additive analogue of MonoidWithZero; if there were then PartENat could be an AddMonoidWithTop.

• toWithTop : the map from PartENat to ℕ∞, with theorems that it plays well with + and ≤.

• withTopAddEquiv : PartENat ≃+ ℕ∞

• withTopOrderIso : PartENat ≃o ℕ∞

Implementation details

PartENat is defined to be Part ℕ.

+ and ≤ are defined on PartENat, but there is an issue with * because it's not clear what 0 * ⊤ should be. mul is hence left undefined. Similarly ⊤ - ⊤ is ambiguous so there is no - defined on PartENat.

Before the open scoped Classical line, various proofs are made with decidability assumptions. This can cause issues -- see for example the non-simp lemma toWithTopZero proved by rfl, followed by @[simp] lemma toWithTopZero' whose proof uses convert.

Tags

PartENat, ℕ∞

def PartENat : Type

Type of natural numbers with infinity (⊤)

Equations

• PartENat = Part ℕ

def PartENat.some : ℕ → PartENat

The computable embedding ℕ → PartENat.

This coincides with the coercion coe : ℕ → PartENat, see PartENat.some_eq_natCast.

Equations

• PartENat.some = Part.some

instance PartENat.instZero : Zero PartENat

Equations

• PartENat.instZero = { zero := ↑0 }

instance PartENat.instInhabited : Inhabited PartENat

Equations

• PartENat.instInhabited = { default := 0 }

instance PartENat.instOne : One PartENat

Equations

• PartENat.instOne = { one := ↑1 }

instance PartENat.instAdd : Add PartENat

Equations

• PartENat.instAdd = { add := fun (x y : PartENat) => { Dom := x.Dom ∧ y.Dom, get := fun (h : x.Dom ∧ y.Dom) => x.get ⋯ + y.get ⋯ } }

instance PartENat.instDecidableDomNatSome (n : ℕ) : Decidable (↑n).Dom

Equations

• PartENat.instDecidableDomNatSome n = isTrue trivial

@[simp]

theorem PartENat.dom_some (x : ℕ) : (↑x).Dom

instance PartENat.addCommMonoid : AddCommMonoid PartENat

Equations

• PartENat.addCommMonoid = AddCommMonoid.mk PartENat.addCommMonoid.proof_6

instance PartENat.instAddCommMonoidWithOne : AddCommMonoidWithOne PartENat

Equations

• PartENat.instAddCommMonoidWithOne = AddCommMonoidWithOne.mk ⋯

theorem PartENat.some_eq_natCast (n : ℕ) : ↑n = ↑n

instance PartENat.instCharZero : CharZero PartENat

Equations

• PartENat.instCharZero = ⋯

theorem PartENat.natCast_inj {x : ℕ} {y : ℕ} : ↑x = ↑y ↔ x = y

Alias of Nat.cast_inj specialized to PartENat -

@[simp]

theorem PartENat.dom_natCast (x : ℕ) : (↑x).Dom

@[simp]

theorem PartENat.dom_ofNat (x : ℕ) [x.AtLeastTwo] : (OfNat.ofNat x).Dom

@[simp]

theorem PartENat.dom_zero : Part.Dom 0

@[simp]

theorem PartENat.dom_one : Part.Dom 1

instance PartENat.instCanLiftNatCastDom : CanLift PartENat ℕ Nat.cast Part.Dom

Equations

• PartENat.instCanLiftNatCastDom = ⋯

instance PartENat.instLE : LE PartENat

Equations

• PartENat.instLE = { le := fun (x y : PartENat) => ∃ (h : y.Dom → x.Dom), ∀ (hy : y.Dom), x.get ⋯ ≤ y.get hy }

instance PartENat.instTop : Top PartENat

Equations

• PartENat.instTop = { top := Part.none }

instance PartENat.instBot : Bot PartENat

Equations

• PartENat.instBot = { bot := 0 }

instance PartENat.instSup : Sup PartENat

Equations

• PartENat.instSup = { sup := fun (x y : PartENat) => { Dom := x.Dom ∧ y.Dom, get := fun (h : x.Dom ∧ y.Dom) => x.get ⋯ ⊔ y.get ⋯ } }

theorem PartENat.le_def (x : PartENat) (y : PartENat) : x ≤ y ↔ ∃ (h : y.Dom → x.Dom), ∀ (hy : y.Dom), x.get ⋯ ≤ y.get hy

theorem PartENat.casesOn' {P : PartENat → Prop} (a : PartENat) : P ⊤ → (∀ (n : ℕ), P ↑n) → P a

theorem PartENat.casesOn {P : PartENat → Prop} (a : PartENat) : P ⊤ → (∀ (n : ℕ), P ↑n) → P a

theorem PartENat.top_add (x : PartENat) : ⊤ + x = ⊤

theorem PartENat.add_top (x : PartENat) : x + ⊤ = ⊤

@[simp]

theorem PartENat.natCast_get {x : PartENat} (h : x.Dom) : ↑(x.get h) = x

@[simp]

theorem PartENat.get_natCast' (x : ℕ) (h : (↑x).Dom) : (↑x).get h = x

theorem PartENat.get_natCast {x : ℕ} : (↑x).get ⋯ = x

theorem PartENat.coe_add_get {x : ℕ} {y : PartENat} (h : (↑x + y).Dom) : (↑x + y).get h = x + y.get ⋯

@[simp]

theorem PartENat.get_add {x : PartENat} {y : PartENat} (h : (x + y).Dom) : (x + y).get h = x.get ⋯ + y.get ⋯

@[simp]

theorem PartENat.get_zero (h : Part.Dom 0) : Part.get 0 h = 0

@[simp]

theorem PartENat.get_one (h : Part.Dom 1) : Part.get 1 h = 1

@[simp]

theorem PartENat.get_ofNat' (x : ℕ) [x.AtLeastTwo] (h : (OfNat.ofNat x).Dom) : (OfNat.ofNat x).get h = OfNat.ofNat x

theorem PartENat.get_eq_iff_eq_some {a : PartENat} {ha : a.Dom} {b : ℕ} : a.get ha = b ↔ a = ↑b

theorem PartENat.get_eq_iff_eq_coe {a : PartENat} {ha : a.Dom} {b : ℕ} : a.get ha = b ↔ a = ↑b

theorem PartENat.dom_of_le_of_dom {x : PartENat} {y : PartENat} : x ≤ y → y.Dom → x.Dom

theorem PartENat.dom_of_le_some {x : PartENat} {y : ℕ} (h : x ≤ ↑y) : x.Dom

theorem PartENat.dom_of_le_natCast {x : PartENat} {y : ℕ} (h : x ≤ ↑y) : x.Dom

instance PartENat.decidableLe (x : PartENat) (y : PartENat) [Decidable x.Dom] [Decidable y.Dom] : Decidable (x ≤ y)

Equations

• x.decidableLe y = if hx : x.Dom then decidable_of_decidable_of_iff ⋯ else if hy : y.Dom then isFalse ⋯ else isTrue ⋯

instance PartENat.partialOrder : PartialOrder PartENat

Equations

• PartENat.partialOrder = PartialOrder.mk PartENat.partialOrder.proof_4

theorem PartENat.lt_def (x : PartENat) (y : PartENat) : x < y ↔ ∃ (hx : x.Dom), ∀ (hy : y.Dom), x.get hx < y.get hy

noncomputable instance PartENat.orderedAddCommMonoid : OrderedAddCommMonoid PartENat

Equations

• PartENat.orderedAddCommMonoid = OrderedAddCommMonoid.mk PartENat.orderedAddCommMonoid.proof_1

instance PartENat.semilatticeSup : SemilatticeSup PartENat

Equations

• PartENat.semilatticeSup = SemilatticeSup.mk PartENat.semilatticeSup.proof_1 PartENat.semilatticeSup.proof_2 PartENat.semilatticeSup.proof_3

instance PartENat.orderBot : OrderBot PartENat

Equations

• PartENat.orderBot = OrderBot.mk PartENat.orderBot.proof_1

instance PartENat.orderTop : OrderTop PartENat

Equations

• PartENat.orderTop = OrderTop.mk PartENat.orderTop.proof_1

instance PartENat.instZeroLEOneClass : ZeroLEOneClass PartENat

Equations

• PartENat.instZeroLEOneClass = ⋯

theorem PartENat.coe_le_coe {x : ℕ} {y : ℕ} : ↑x ≤ ↑y ↔ x ≤ y

Alias of Nat.cast_le specialized to PartENat -

theorem PartENat.coe_lt_coe {x : ℕ} {y : ℕ} : ↑x < ↑y ↔ x < y

Alias of Nat.cast_lt specialized to PartENat -

@[simp]

theorem PartENat.get_le_get {x : PartENat} {y : PartENat} {hx : x.Dom} {hy : y.Dom} : x.get hx ≤ y.get hy ↔ x ≤ y

theorem PartENat.le_coe_iff (x : PartENat) (n : ℕ) : x ≤ ↑n ↔ ∃ (h : x.Dom), x.get h ≤ n

theorem PartENat.lt_coe_iff (x : PartENat) (n : ℕ) : x < ↑n ↔ ∃ (h : x.Dom), x.get h < n

theorem PartENat.coe_le_iff (n : ℕ) (x : PartENat) : ↑n ≤ x ↔ ∀ (h : x.Dom), n ≤ x.get h

theorem PartENat.coe_lt_iff (n : ℕ) (x : PartENat) : ↑n < x ↔ ∀ (h : x.Dom), n < x.get h

theorem PartENat.eq_zero_iff {x : PartENat} : x = 0 ↔ x ≤ 0

theorem PartENat.ne_zero_iff {x : PartENat} : x ≠ 0 ↔ ⊥ < x

theorem PartENat.dom_of_lt {x : PartENat} {y : PartENat} : x < y → x.Dom

theorem PartENat.top_eq_none : ⊤ = Part.none

@[simp]

theorem PartENat.natCast_lt_top (x : ℕ) : ↑x < ⊤

@[simp]

theorem PartENat.zero_lt_top : 0 < ⊤

@[simp]

theorem PartENat.one_lt_top : 1 < ⊤

@[simp]

theorem PartENat.ofNat_lt_top (x : ℕ) [x.AtLeastTwo] : OfNat.ofNat x < ⊤

@[simp]

theorem PartENat.natCast_ne_top (x : ℕ) : ↑x ≠ ⊤

@[simp]

theorem PartENat.zero_ne_top : 0 ≠ ⊤

@[simp]

theorem PartENat.one_ne_top : 1 ≠ ⊤

@[simp]

theorem PartENat.ofNat_ne_top (x : ℕ) [x.AtLeastTwo] : OfNat.ofNat x ≠ ⊤

theorem PartENat.not_isMax_natCast (x : ℕ) : ¬IsMax ↑x

theorem PartENat.ne_top_iff {x : PartENat} : x ≠ ⊤ ↔ ∃ (n : ℕ), x = ↑n

theorem PartENat.ne_top_iff_dom {x : PartENat} : x ≠ ⊤ ↔ x.Dom

theorem PartENat.not_dom_iff_eq_top {x : PartENat} : ¬x.Dom ↔ x = ⊤

theorem PartENat.ne_top_of_lt {x : PartENat} {y : PartENat} (h : x < y) : x ≠ ⊤

theorem PartENat.eq_top_iff_forall_lt (x : PartENat) : x = ⊤ ↔ ∀ (n : ℕ), ↑n < x

theorem PartENat.eq_top_iff_forall_le (x : PartENat) : x = ⊤ ↔ ∀ (n : ℕ), ↑n ≤ x

theorem PartENat.pos_iff_one_le {x : PartENat} : 0 < x ↔ 1 ≤ x

instance PartENat.isTotal : IsTotal PartENat fun (x1 x2 : PartENat) => x1 ≤ x2

Equations

• PartENat.isTotal = ⋯

noncomputable instance PartENat.linearOrder : LinearOrder PartENat

Equations

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

instance PartENat.boundedOrder : BoundedOrder PartENat

Equations

• PartENat.boundedOrder = BoundedOrder.mk

noncomputable instance PartENat.lattice : Lattice PartENat

Equations

• PartENat.lattice = Lattice.mk ⋯ ⋯ ⋯

noncomputable instance PartENat.instCanonicallyOrderedAddCommMonoid : CanonicallyOrderedAddCommMonoid PartENat

Equations

• PartENat.instCanonicallyOrderedAddCommMonoid = CanonicallyOrderedAddCommMonoid.mk (@PartENat.instCanonicallyOrderedAddCommMonoid.proof_1) PartENat.instCanonicallyOrderedAddCommMonoid.proof_2

theorem PartENat.eq_natCast_sub_of_add_eq_natCast {x : PartENat} {y : PartENat} {n : ℕ} (h : x + y = ↑n) : x = ↑(n - y.get ⋯)

theorem PartENat.add_lt_add_right {x : PartENat} {y : PartENat} {z : PartENat} (h : x < y) (hz : z ≠ ⊤) : x + z < y + z

theorem PartENat.add_lt_add_iff_right {x : PartENat} {y : PartENat} {z : PartENat} (hz : z ≠ ⊤) : x + z < y + z ↔ x < y

theorem PartENat.add_lt_add_iff_left {x : PartENat} {y : PartENat} {z : PartENat} (hz : z ≠ ⊤) : z + x < z + y ↔ x < y

theorem PartENat.lt_add_iff_pos_right {x : PartENat} {y : PartENat} (hx : x ≠ ⊤) : x < x + y ↔ 0 < y

theorem PartENat.lt_add_one {x : PartENat} (hx : x ≠ ⊤) : x < x + 1

theorem PartENat.le_of_lt_add_one {x : PartENat} {y : PartENat} (h : x < y + 1) : x ≤ y

theorem PartENat.add_one_le_of_lt {x : PartENat} {y : PartENat} (h : x < y) : x + 1 ≤ y

theorem PartENat.add_one_le_iff_lt {x : PartENat} {y : PartENat} (hx : x ≠ ⊤) : x + 1 ≤ y ↔ x < y

theorem PartENat.coe_succ_le_iff {n : ℕ} {e : PartENat} : ↑n.succ ≤ e ↔ ↑n < e

theorem PartENat.lt_add_one_iff_lt {x : PartENat} {y : PartENat} (hx : x ≠ ⊤) : x < y + 1 ↔ x ≤ y

theorem PartENat.lt_coe_succ_iff_le {x : PartENat} {n : ℕ} (hx : x ≠ ⊤) : x < ↑n.succ ↔ x ≤ ↑n

theorem PartENat.add_eq_top_iff {a : PartENat} {b : PartENat} : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤

theorem PartENat.add_right_cancel_iff {a : PartENat} {b : PartENat} {c : PartENat} (hc : c ≠ ⊤) : a + c = b + c ↔ a = b

theorem PartENat.add_left_cancel_iff {a : PartENat} {b : PartENat} {c : PartENat} (ha : a ≠ ⊤) : a + b = a + c ↔ b = c

def PartENat.toWithTop (x : PartENat) [Decidable x.Dom] : ℕ∞

Computably converts a PartENat to a ℕ∞.

Equations

• x.toWithTop = Part.toOption x

theorem PartENat.toWithTop_top : let_fun this := Part.noneDecidable; ⊤.toWithTop = ⊤

@[simp]

theorem PartENat.toWithTop_top' {h : Decidable ⊤.Dom} : ⊤.toWithTop = ⊤

theorem PartENat.toWithTop_zero : let_fun this := Part.someDecidable 0; PartENat.toWithTop 0 = 0

@[simp]

theorem PartENat.toWithTop_zero' {h : Decidable (Part.Dom 0)} : PartENat.toWithTop 0 = 0

theorem PartENat.toWithTop_one : let_fun this := Part.someDecidable 1; PartENat.toWithTop 1 = 1

@[simp]

theorem PartENat.toWithTop_one' {h : Decidable (Part.Dom 1)} : PartENat.toWithTop 1 = 1

theorem PartENat.toWithTop_some (n : ℕ) : (↑n).toWithTop = ↑n

theorem PartENat.toWithTop_natCast (n : ℕ) : ∀ {x : Decidable (↑n).Dom}, (↑n).toWithTop = ↑n

@[simp]

theorem PartENat.toWithTop_natCast' (n : ℕ) : ∀ {x : Decidable (↑n).Dom}, (↑n).toWithTop = ↑n

@[simp]

theorem PartENat.toWithTop_ofNat (n : ℕ) [n.AtLeastTwo] : ∀ {x : Decidable (Part.Dom (OfNat.ofNat n))}, (OfNat.ofNat n).toWithTop = OfNat.ofNat n

@[simp]

theorem PartENat.toWithTop_le {x : PartENat} {y : PartENat} [hx : Decidable x.Dom] [hy : Decidable y.Dom] : x.toWithTop ≤ y.toWithTop ↔ x ≤ y

@[simp]

theorem PartENat.toWithTop_lt {x : PartENat} {y : PartENat} [Decidable x.Dom] [Decidable y.Dom] : x.toWithTop < y.toWithTop ↔ x < y

def PartENat.ofENat : ℕ∞ → PartENat

Coercion from ℕ∞ to PartENat.

Equations

• ↑none = Part.none
• ↑(some n) = ↑n

instance PartENat.instCoeENat : Coe ℕ∞ PartENat

Equations

• PartENat.instCoeENat = { coe := PartENat.ofENat }

@[simp]

theorem PartENat.ofENat_top : ↑⊤ = ⊤

@[simp]

theorem PartENat.ofENat_coe (n : ℕ) : ↑↑n = ↑n

@[simp]

theorem PartENat.ofENat_zero : ↑0 = 0

@[simp]

theorem PartENat.ofENat_one : ↑1 = 1

@[simp]

theorem PartENat.ofENat_ofNat (n : ℕ) [n.AtLeastTwo] : ↑(OfNat.ofNat n) = OfNat.ofNat n

@[simp]

theorem PartENat.toWithTop_ofENat (n : ℕ∞) : ∀ {x : Decidable (↑n).Dom}, (↑n).toWithTop = n

@[simp]

theorem PartENat.ofENat_toWithTop (x : PartENat) : ∀ {x_1 : Decidable x.Dom}, ↑x.toWithTop = x

@[simp]

theorem PartENat.ofENat_le {x : ℕ∞} {y : ℕ∞} : ↑x ≤ ↑y ↔ x ≤ y

@[simp]

theorem PartENat.ofENat_lt {x : ℕ∞} {y : ℕ∞} : ↑x < ↑y ↔ x < y

@[simp]

theorem PartENat.toWithTop_add {x : PartENat} {y : PartENat} : (x + y).toWithTop = x.toWithTop + y.toWithTop

noncomputable def PartENat.withTopEquiv : PartENat ≃ ℕ∞

Equiv between PartENat and ℕ∞ (for the order isomorphism see withTopOrderIso).

Equations

• PartENat.withTopEquiv = { toFun := fun (x : PartENat) => x.toWithTop, invFun := fun (x : ℕ∞) => ↑x, left_inv := PartENat.withTopEquiv.proof_1, right_inv := PartENat.withTopEquiv.proof_2 }

@[simp]

theorem PartENat.withTopEquiv_symm_apply (x : ℕ∞) : PartENat.withTopEquiv.symm x = ↑x

@[simp]

theorem PartENat.withTopEquiv_apply (x : PartENat) : PartENat.withTopEquiv x = x.toWithTop

theorem PartENat.withTopEquiv_top : PartENat.withTopEquiv ⊤ = ⊤

theorem PartENat.withTopEquiv_natCast (n : ℕ) : PartENat.withTopEquiv ↑n = ↑n

theorem PartENat.withTopEquiv_zero : PartENat.withTopEquiv 0 = 0

theorem PartENat.withTopEquiv_one : PartENat.withTopEquiv 1 = 1

theorem PartENat.withTopEquiv_ofNat (n : ℕ) [n.AtLeastTwo] : PartENat.withTopEquiv (OfNat.ofNat n) = OfNat.ofNat n

theorem PartENat.withTopEquiv_le {x : PartENat} {y : PartENat} : PartENat.withTopEquiv x ≤ PartENat.withTopEquiv y ↔ x ≤ y

theorem PartENat.withTopEquiv_lt {x : PartENat} {y : PartENat} : PartENat.withTopEquiv x < PartENat.withTopEquiv y ↔ x < y

theorem PartENat.withTopEquiv_symm_top : PartENat.withTopEquiv.symm ⊤ = ⊤

theorem PartENat.withTopEquiv_symm_coe (n : ℕ) : PartENat.withTopEquiv.symm ↑n = ↑n

theorem PartENat.withTopEquiv_symm_zero : PartENat.withTopEquiv.symm 0 = 0

theorem PartENat.withTopEquiv_symm_one : PartENat.withTopEquiv.symm 1 = 1

theorem PartENat.withTopEquiv_symm_ofNat (n : ℕ) [n.AtLeastTwo] : PartENat.withTopEquiv.symm (OfNat.ofNat n) = OfNat.ofNat n

theorem PartENat.withTopEquiv_symm_le {x : ℕ∞} {y : ℕ∞} : PartENat.withTopEquiv.symm x ≤ PartENat.withTopEquiv.symm y ↔ x ≤ y

theorem PartENat.withTopEquiv_symm_lt {x : ℕ∞} {y : ℕ∞} : PartENat.withTopEquiv.symm x < PartENat.withTopEquiv.symm y ↔ x < y

noncomputable def PartENat.withTopOrderIso : PartENat ≃o ℕ∞

toWithTop induces an order isomorphism between PartENat and ℕ∞.

Equations

• PartENat.withTopOrderIso = { toEquiv := PartENat.withTopEquiv, map_rel_iff' := ⋯ }

noncomputable def PartENat.withTopAddEquiv : PartENat ≃+ ℕ∞

toWithTop induces an additive monoid isomorphism between PartENat and ℕ∞.

Equations

• PartENat.withTopAddEquiv = { toEquiv := PartENat.withTopEquiv, map_add' := PartENat.withTopAddEquiv.proof_1 }

theorem PartENat.lt_wf : WellFounded fun (x1 x2 : PartENat) => x1 < x2

instance PartENat.instWellFoundedLT : WellFoundedLT PartENat

Equations

• PartENat.instWellFoundedLT = ⋯

instance PartENat.isWellOrder : IsWellOrder PartENat fun (x1 x2 : PartENat) => x1 < x2

Equations

• PartENat.isWellOrder = ⋯

instance PartENat.wellFoundedRelation : WellFoundedRelation PartENat

Equations

• PartENat.wellFoundedRelation = { rel := fun (x1 x2 : PartENat) => x1 < x2, wf := PartENat.lt_wf }

def PartENat.find (P : ℕ → Prop) [DecidablePred P] : PartENat

The smallest PartENat satisfying a (decidable) predicate P : ℕ → Prop

Equations

• PartENat.find P = { Dom := ∃ (n : ℕ), P n, get := Nat.find }

@[simp]

theorem PartENat.find_get (P : ℕ → Prop) [DecidablePred P] (h : (PartENat.find P).Dom) : (PartENat.find P).get h = Nat.find h

theorem PartENat.find_dom (P : ℕ → Prop) [DecidablePred P] (h : ∃ (n : ℕ), P n) : (PartENat.find P).Dom

theorem PartENat.lt_find (P : ℕ → Prop) [DecidablePred P] (n : ℕ) (h : ∀ m ≤ n, ¬P m) : ↑n < PartENat.find P

theorem PartENat.lt_find_iff (P : ℕ → Prop) [DecidablePred P] (n : ℕ) : ↑n < PartENat.find P ↔ ∀ m ≤ n, ¬P m

theorem PartENat.find_le (P : ℕ → Prop) [DecidablePred P] (n : ℕ) (h : P n) : PartENat.find P ≤ ↑n

theorem PartENat.find_eq_top_iff (P : ℕ → Prop) [DecidablePred P] : PartENat.find P = ⊤ ↔ ∀ (n : ℕ), ¬P n

noncomputable instance PartENat.instLinearOrderedAddCommMonoidWithTop : LinearOrderedAddCommMonoidWithTop PartENat

Equations

• PartENat.instLinearOrderedAddCommMonoidWithTop = LinearOrderedAddCommMonoidWithTop.mk PartENat.top_add

noncomputable instance PartENat.instCompleteLinearOrder : CompleteLinearOrder PartENat

Equations

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