Algebraic operations on upper/lower sets

Upper/lower sets are preserved under pointwise algebraic operations in ordered groups.

theorem IsUpperSet.vadd_subset {α : Type u_1} [OrderedAddCommMonoid α] {s : Set α} {x : α} (hs : IsUpperSet s) (hx : 0 ≤ x) : x +ᵥ s ⊆ s

theorem IsUpperSet.smul_subset {α : Type u_1} [OrderedCommMonoid α] {s : Set α} {x : α} (hs : IsUpperSet s) (hx : 1 ≤ x) : x • s ⊆ s

theorem IsLowerSet.vadd_subset {α : Type u_1} [OrderedAddCommMonoid α] {s : Set α} {x : α} (hs : IsLowerSet s) (hx : x ≤ 0) : x +ᵥ s ⊆ s

theorem IsLowerSet.smul_subset {α : Type u_1} [OrderedCommMonoid α] {s : Set α} {x : α} (hs : IsLowerSet s) (hx : x ≤ 1) : x • s ⊆ s

theorem IsUpperSet.vadd {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {a : α} (hs : IsUpperSet s) : IsUpperSet (a +ᵥ s)

theorem IsUpperSet.smul {α : Type u_1} [OrderedCommGroup α] {s : Set α} {a : α} (hs : IsUpperSet s) : IsUpperSet (a • s)

theorem IsLowerSet.vadd {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {a : α} (hs : IsLowerSet s) : IsLowerSet (a +ᵥ s)

theorem IsLowerSet.smul {α : Type u_1} [OrderedCommGroup α] {s : Set α} {a : α} (hs : IsLowerSet s) : IsLowerSet (a • s)

theorem Set.OrdConnected.vadd {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {a : α} (hs : s.OrdConnected) : (a +ᵥ s).OrdConnected

theorem Set.OrdConnected.smul {α : Type u_1} [OrderedCommGroup α] {s : Set α} {a : α} (hs : s.OrdConnected) : (a • s).OrdConnected

theorem IsUpperSet.add_left {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (ht : IsUpperSet t) : IsUpperSet (s + t)

theorem IsUpperSet.mul_left {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (ht : IsUpperSet t) : IsUpperSet (s * t)

theorem IsUpperSet.add_right {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (hs : IsUpperSet s) : IsUpperSet (s + t)

theorem IsUpperSet.mul_right {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (hs : IsUpperSet s) : IsUpperSet (s * t)

theorem IsLowerSet.add_left {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (ht : IsLowerSet t) : IsLowerSet (s + t)

theorem IsLowerSet.mul_left {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (ht : IsLowerSet t) : IsLowerSet (s * t)

theorem IsLowerSet.add_right {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (hs : IsLowerSet s) : IsLowerSet (s + t)

theorem IsLowerSet.mul_right {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (hs : IsLowerSet s) : IsLowerSet (s * t)

theorem IsUpperSet.neg {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} (hs : IsUpperSet s) : IsLowerSet (-s)

theorem IsUpperSet.inv {α : Type u_1} [OrderedCommGroup α] {s : Set α} (hs : IsUpperSet s) : IsLowerSet s⁻¹

theorem IsLowerSet.neg {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} (hs : IsLowerSet s) : IsUpperSet (-s)

theorem IsLowerSet.inv {α : Type u_1} [OrderedCommGroup α] {s : Set α} (hs : IsLowerSet s) : IsUpperSet s⁻¹

theorem IsUpperSet.sub_left {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (ht : IsUpperSet t) : IsLowerSet (s - t)

theorem IsUpperSet.div_left {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (ht : IsUpperSet t) : IsLowerSet (s / t)

theorem IsUpperSet.sub_right {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (hs : IsUpperSet s) : IsUpperSet (s - t)

theorem IsUpperSet.div_right {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (hs : IsUpperSet s) : IsUpperSet (s / t)

theorem IsLowerSet.sub_left {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (ht : IsLowerSet t) : IsUpperSet (s - t)

theorem IsLowerSet.div_left {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (ht : IsLowerSet t) : IsUpperSet (s / t)

theorem IsLowerSet.sub_right {α : Type u_1} [OrderedAddCommGroup α] {s : Set α} {t : Set α} (hs : IsLowerSet s) : IsLowerSet (s - t)

theorem IsLowerSet.div_right {α : Type u_1} [OrderedCommGroup α] {s : Set α} {t : Set α} (hs : IsLowerSet s) : IsLowerSet (s / t)

instance UpperSet.instZero {α : Type u_1} [OrderedAddCommGroup α] : Zero (UpperSet α)

Equations

• UpperSet.instZero = { zero := UpperSet.Ici 0 }

instance UpperSet.instOne {α : Type u_1} [OrderedCommGroup α] : One (UpperSet α)

Equations

• UpperSet.instOne = { one := UpperSet.Ici 1 }

theorem UpperSet.instAdd.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : UpperSet α) (t : UpperSet α) : IsUpperSet (s.carrier + ↑t)

instance UpperSet.instAdd {α : Type u_1} [OrderedAddCommGroup α] : Add (UpperSet α)

Equations

• UpperSet.instAdd = { add := fun (s t : UpperSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x + x_1) ↑s ↑t, upper' := ⋯ } }

instance UpperSet.instMul {α : Type u_1} [OrderedCommGroup α] : Mul (UpperSet α)

Equations

• UpperSet.instMul = { mul := fun (s t : UpperSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x * x_1) ↑s ↑t, upper' := ⋯ } }

instance UpperSet.instSub {α : Type u_1} [OrderedAddCommGroup α] : Sub (UpperSet α)

Equations

• UpperSet.instSub = { sub := fun (s t : UpperSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x - x_1) ↑s ↑t, upper' := ⋯ } }

theorem UpperSet.instSub.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : UpperSet α) (t : UpperSet α) : IsUpperSet (s.carrier - ↑t)

instance UpperSet.instDiv {α : Type u_1} [OrderedCommGroup α] : Div (UpperSet α)

Equations

• UpperSet.instDiv = { div := fun (s t : UpperSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x / x_1) ↑s ↑t, upper' := ⋯ } }

instance UpperSet.instVAdd {α : Type u_1} [OrderedAddCommGroup α] : VAdd α (UpperSet α)

Equations

• UpperSet.instVAdd = { vadd := fun (a : α) (s : UpperSet α) => { carrier := (fun (x : α) => a +ᵥ x) '' ↑s, upper' := ⋯ } }

theorem UpperSet.instVAdd.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (a : α) (s : UpperSet α) : IsUpperSet (a +ᵥ s.carrier)

instance UpperSet.instSMul {α : Type u_1} [OrderedCommGroup α] : SMul α (UpperSet α)

Equations

• UpperSet.instSMul = { smul := fun (a : α) (s : UpperSet α) => { carrier := (fun (x : α) => a • x) '' ↑s, upper' := ⋯ } }

@[simp]

theorem UpperSet.coe_zero {α : Type u_1} [OrderedAddCommGroup α] : ↑0 = Set.Ici 0

@[simp]

theorem UpperSet.coe_one {α : Type u_1} [OrderedCommGroup α] : ↑1 = Set.Ici 1

@[simp]

theorem UpperSet.coe_add {α : Type u_1} [OrderedAddCommGroup α] (s : UpperSet α) (t : UpperSet α) : ↑(s + t) = ↑s + ↑t

@[simp]

theorem UpperSet.coe_mul {α : Type u_1} [OrderedCommGroup α] (s : UpperSet α) (t : UpperSet α) : ↑(s * t) = ↑s * ↑t

@[simp]

theorem UpperSet.coe_sub {α : Type u_1} [OrderedAddCommGroup α] (s : UpperSet α) (t : UpperSet α) : ↑(s - t) = ↑s - ↑t

@[simp]

theorem UpperSet.coe_div {α : Type u_1} [OrderedCommGroup α] (s : UpperSet α) (t : UpperSet α) : ↑(s / t) = ↑s / ↑t

@[simp]

theorem UpperSet.Ici_zero {α : Type u_1} [OrderedAddCommGroup α] : UpperSet.Ici 0 = 0

@[simp]

theorem UpperSet.Ici_one {α : Type u_1} [OrderedCommGroup α] : UpperSet.Ici 1 = 1

instance UpperSet.instAddAction {α : Type u_1} [OrderedAddCommGroup α] : AddAction α (UpperSet α)

Equations

• UpperSet.instAddAction = Function.Injective.addAction SetLike.coe ⋯ ⋯

theorem UpperSet.instAddAction.proof_1 {α : Type u_1} [OrderedAddCommGroup α] : Function.Injective SetLike.coe

theorem UpperSet.instAddAction.proof_2 {α : Type u_1} [OrderedAddCommGroup α] : ∀ (x : α) (x_1 : UpperSet α), ↑(x +ᵥ x_1) = ↑(x +ᵥ x_1)

instance UpperSet.instMulAction {α : Type u_1} [OrderedCommGroup α] : MulAction α (UpperSet α)

Equations

• UpperSet.instMulAction = Function.Injective.mulAction SetLike.coe ⋯ ⋯

theorem UpperSet.addCommSemigroup.proof_1 {α : Type u_1} [OrderedAddCommGroup α] : Function.Injective SetLike.coe

theorem UpperSet.addCommSemigroup.proof_2 {α : Type u_1} [OrderedAddCommGroup α] (a : UpperSet α) (b : UpperSet α) : a + b = b + a

instance UpperSet.addCommSemigroup {α : Type u_1} [OrderedAddCommGroup α] : AddCommSemigroup (UpperSet α)

Equations

• UpperSet.addCommSemigroup = let __src := Function.Injective.addCommSemigroup SetLike.coe ⋯ ⋯; AddCommSemigroup.mk ⋯

instance UpperSet.commSemigroup {α : Type u_1} [OrderedCommGroup α] : CommSemigroup (UpperSet α)

Equations

• UpperSet.commSemigroup = let __src := Function.Injective.commSemigroup SetLike.coe ⋯ ⋯; CommSemigroup.mk ⋯

theorem UpperSet.instAddCommMonoid.proof_2 {α : Type u_1} [OrderedAddCommGroup α] : ∀ (x : UpperSet α), nsmulRec 0 x = nsmulRec 0 x

instance UpperSet.instAddCommMonoid {α : Type u_1} [OrderedAddCommGroup α] : AddCommMonoid (UpperSet α)

Equations

• UpperSet.instAddCommMonoid = let __src := UpperSet.addCommSemigroup; AddCommMonoid.mk ⋯

theorem UpperSet.instAddCommMonoid.proof_4 {α : Type u_1} [OrderedAddCommGroup α] (a : UpperSet α) (b : UpperSet α) : a + b = b + a

theorem UpperSet.instAddCommMonoid.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : UpperSet α) : s + 0 = s

theorem UpperSet.instAddCommMonoid.proof_3 {α : Type u_1} [OrderedAddCommGroup α] : ∀ (n : ℕ) (x : UpperSet α), nsmulRec (n + 1) x = nsmulRec (n + 1) x

instance UpperSet.instCommMonoid {α : Type u_1} [OrderedCommGroup α] : CommMonoid (UpperSet α)

Equations

• UpperSet.instCommMonoid = let __src := UpperSet.commSemigroup; CommMonoid.mk ⋯

instance LowerSet.instZero {α : Type u_1} [OrderedAddCommGroup α] : Zero (LowerSet α)

Equations

• LowerSet.instZero = { zero := LowerSet.Iic 0 }

instance LowerSet.instOne {α : Type u_1} [OrderedCommGroup α] : One (LowerSet α)

Equations

• LowerSet.instOne = { one := LowerSet.Iic 1 }

instance LowerSet.instAdd {α : Type u_1} [OrderedAddCommGroup α] : Add (LowerSet α)

Equations

• LowerSet.instAdd = { add := fun (s t : LowerSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x + x_1) ↑s ↑t, lower' := ⋯ } }

theorem LowerSet.instAdd.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : LowerSet α) (t : LowerSet α) : IsLowerSet (s.carrier + ↑t)

instance LowerSet.instMul {α : Type u_1} [OrderedCommGroup α] : Mul (LowerSet α)

Equations

• LowerSet.instMul = { mul := fun (s t : LowerSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x * x_1) ↑s ↑t, lower' := ⋯ } }

instance LowerSet.instSub {α : Type u_1} [OrderedAddCommGroup α] : Sub (LowerSet α)

Equations

• LowerSet.instSub = { sub := fun (s t : LowerSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x - x_1) ↑s ↑t, lower' := ⋯ } }

theorem LowerSet.instSub.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : LowerSet α) (t : LowerSet α) : IsLowerSet (s.carrier - ↑t)

instance LowerSet.instDiv {α : Type u_1} [OrderedCommGroup α] : Div (LowerSet α)

Equations

• LowerSet.instDiv = { div := fun (s t : LowerSet α) => { carrier := Set.image2 (fun (x x_1 : α) => x / x_1) ↑s ↑t, lower' := ⋯ } }

instance LowerSet.instVAdd {α : Type u_1} [OrderedAddCommGroup α] : VAdd α (LowerSet α)

Equations

• LowerSet.instVAdd = { vadd := fun (a : α) (s : LowerSet α) => { carrier := (fun (x : α) => a +ᵥ x) '' ↑s, lower' := ⋯ } }

theorem LowerSet.instVAdd.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (a : α) (s : LowerSet α) : IsLowerSet (a +ᵥ s.carrier)

instance LowerSet.instSMul {α : Type u_1} [OrderedCommGroup α] : SMul α (LowerSet α)

Equations

• LowerSet.instSMul = { smul := fun (a : α) (s : LowerSet α) => { carrier := (fun (x : α) => a • x) '' ↑s, lower' := ⋯ } }

@[simp]

theorem LowerSet.coe_add {α : Type u_1} [OrderedAddCommGroup α] (s : LowerSet α) (t : LowerSet α) : ↑(s + t) = ↑s + ↑t

@[simp]

theorem LowerSet.coe_mul {α : Type u_1} [OrderedCommGroup α] (s : LowerSet α) (t : LowerSet α) : ↑(s * t) = ↑s * ↑t

@[simp]

theorem LowerSet.coe_sub {α : Type u_1} [OrderedAddCommGroup α] (s : LowerSet α) (t : LowerSet α) : ↑(s - t) = ↑s - ↑t

@[simp]

theorem LowerSet.coe_div {α : Type u_1} [OrderedCommGroup α] (s : LowerSet α) (t : LowerSet α) : ↑(s / t) = ↑s / ↑t

@[simp]

theorem LowerSet.Iic_zero {α : Type u_1} [OrderedAddCommGroup α] : LowerSet.Iic 0 = 0

@[simp]

theorem LowerSet.Iic_one {α : Type u_1} [OrderedCommGroup α] : LowerSet.Iic 1 = 1

theorem LowerSet.instAddAction.proof_2 {α : Type u_1} [OrderedAddCommGroup α] : ∀ (x : α) (x_1 : LowerSet α), ↑(x +ᵥ x_1) = ↑(x +ᵥ x_1)

theorem LowerSet.instAddAction.proof_1 {α : Type u_1} [OrderedAddCommGroup α] : Function.Injective SetLike.coe

instance LowerSet.instAddAction {α : Type u_1} [OrderedAddCommGroup α] : AddAction α (LowerSet α)

Equations

• LowerSet.instAddAction = Function.Injective.addAction SetLike.coe ⋯ ⋯

instance LowerSet.instMulAction {α : Type u_1} [OrderedCommGroup α] : MulAction α (LowerSet α)

Equations

• LowerSet.instMulAction = Function.Injective.mulAction SetLike.coe ⋯ ⋯

instance LowerSet.addCommSemigroup {α : Type u_1} [OrderedAddCommGroup α] : AddCommSemigroup (LowerSet α)

Equations

• LowerSet.addCommSemigroup = let __src := Function.Injective.addCommSemigroup SetLike.coe ⋯ ⋯; AddCommSemigroup.mk ⋯

theorem LowerSet.addCommSemigroup.proof_1 {α : Type u_1} [OrderedAddCommGroup α] : Function.Injective SetLike.coe

theorem LowerSet.addCommSemigroup.proof_2 {α : Type u_1} [OrderedAddCommGroup α] (a : LowerSet α) (b : LowerSet α) : a + b = b + a

instance LowerSet.commSemigroup {α : Type u_1} [OrderedCommGroup α] : CommSemigroup (LowerSet α)

Equations

• LowerSet.commSemigroup = let __src := Function.Injective.commSemigroup SetLike.coe ⋯ ⋯; CommSemigroup.mk ⋯

instance LowerSet.instAddCommMonoid {α : Type u_1} [OrderedAddCommGroup α] : AddCommMonoid (LowerSet α)

Equations

• LowerSet.instAddCommMonoid = let __src := LowerSet.addCommSemigroup; AddCommMonoid.mk ⋯

theorem LowerSet.instAddCommMonoid.proof_3 {α : Type u_1} [OrderedAddCommGroup α] : ∀ (n : ℕ) (x : LowerSet α), nsmulRec (n + 1) x = nsmulRec (n + 1) x

theorem LowerSet.instAddCommMonoid.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : LowerSet α) : s + 0 = s

theorem LowerSet.instAddCommMonoid.proof_4 {α : Type u_1} [OrderedAddCommGroup α] (a : LowerSet α) (b : LowerSet α) : a + b = b + a

theorem LowerSet.instAddCommMonoid.proof_2 {α : Type u_1} [OrderedAddCommGroup α] : ∀ (x : LowerSet α), nsmulRec 0 x = nsmulRec 0 x

instance LowerSet.instCommMonoid {α : Type u_1} [OrderedCommGroup α] : CommMonoid (LowerSet α)

Equations

• LowerSet.instCommMonoid = let __src := LowerSet.commSemigroup; CommMonoid.mk ⋯

@[simp]

theorem upperClosure_zero {α : Type u_1} [OrderedAddCommGroup α] : upperClosure 0 = 0

@[simp]

theorem upperClosure_one {α : Type u_1} [OrderedCommGroup α] : upperClosure 1 = 1

@[simp]

theorem lowerClosure_zero {α : Type u_1} [OrderedAddCommGroup α] : lowerClosure 0 = 0

@[simp]

theorem lowerClosure_one {α : Type u_1} [OrderedCommGroup α] : lowerClosure 1 = 1

@[simp]

theorem upperClosure_vadd {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (a : α) : upperClosure (a +ᵥ s) = a +ᵥ upperClosure s

@[simp]

theorem upperClosure_smul {α : Type u_1} [OrderedCommGroup α] (s : Set α) (a : α) : upperClosure (a • s) = a • upperClosure s

@[simp]

theorem lowerClosure_vadd {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (a : α) : lowerClosure (a +ᵥ s) = a +ᵥ lowerClosure s

@[simp]

theorem lowerClosure_smul {α : Type u_1} [OrderedCommGroup α] (s : Set α) (a : α) : lowerClosure (a • s) = a • lowerClosure s

theorem add_upperClosure {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (t : Set α) : s + ↑(upperClosure t) = ↑(upperClosure (s + t))

theorem mul_upperClosure {α : Type u_1} [OrderedCommGroup α] (s : Set α) (t : Set α) : s * ↑(upperClosure t) = ↑(upperClosure (s * t))

theorem add_lowerClosure {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (t : Set α) : s + ↑(lowerClosure t) = ↑(lowerClosure (s + t))

theorem mul_lowerClosure {α : Type u_1} [OrderedCommGroup α] (s : Set α) (t : Set α) : s * ↑(lowerClosure t) = ↑(lowerClosure (s * t))

theorem upperClosure_add {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (t : Set α) : ↑(upperClosure s) + t = ↑(upperClosure (s + t))

theorem upperClosure_mul {α : Type u_1} [OrderedCommGroup α] (s : Set α) (t : Set α) : ↑(upperClosure s) * t = ↑(upperClosure (s * t))

theorem lowerClosure_add {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (t : Set α) : ↑(lowerClosure s) + t = ↑(lowerClosure (s + t))

theorem lowerClosure_mul {α : Type u_1} [OrderedCommGroup α] (s : Set α) (t : Set α) : ↑(lowerClosure s) * t = ↑(lowerClosure (s * t))

@[simp]

theorem upperClosure_add_distrib {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (t : Set α) : upperClosure (s + t) = upperClosure s + upperClosure t

@[simp]

theorem upperClosure_mul_distrib {α : Type u_1} [OrderedCommGroup α] (s : Set α) (t : Set α) : upperClosure (s * t) = upperClosure s * upperClosure t

@[simp]

theorem lowerClosure_add_distrib {α : Type u_1} [OrderedAddCommGroup α] (s : Set α) (t : Set α) : lowerClosure (s + t) = lowerClosure s + lowerClosure t

@[simp]

theorem lowerClosure_mul_distrib {α : Type u_1} [OrderedCommGroup α] (s : Set α) (t : Set α) : lowerClosure (s * t) = lowerClosure s * lowerClosure t