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 : Set.OrdConnected s) : Set.OrdConnected (a +ᵥ s)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

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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))

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

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

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

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