File: Mathlib/Algebra/Order/Pointwise.lean

full_name: sSup_zero
code: @[to_additive (attr := simp)]
theorem sSup_one : sSup (1 : Set α) = 1 
goal: sSup 0 = 0

full_name: sSup_one
code: @[to_additive (attr := simp)]
theorem sSup_one : sSup (1 : Set α) = 1 
goal: sSup 1 = 1

full_name: sInf_one
code: @[to_additive (attr := simp)]
theorem sInf_one : sInf (1 : Set α) = 1 
goal: sInf 1 = 1

full_name: sInf_zero
code: @[to_additive (attr := simp)]
theorem sInf_one : sInf (1 : Set α) = 1 
goal: sInf 0 = 0

full_name: sSup_inv
code: @[to_additive]
theorem sSup_inv (s : Set α) : sSup s⁻¹ = (sInf s)⁻¹ 
goal: sSup s⁻¹ = (sInf s)⁻¹

full_name: sSup_neg
code: @[to_additive]
theorem sSup_inv (s : Set α) : sSup s⁻¹ = (sInf s)⁻¹ 
goal: sSup (-s) = -sInf s

full_name: sInf_neg
code: @[to_additive]
theorem sInf_inv (s : Set α) : sInf s⁻¹ = (sSup s)⁻¹ 
goal: sInf (-s) = -sSup s

full_name: sInf_inv
code: @[to_additive]
theorem sInf_inv (s : Set α) : sInf s⁻¹ = (sSup s)⁻¹ 
goal: sInf s⁻¹ = (sSup s)⁻¹

full_name: sSup_mul
code: @[to_additive]
theorem sSup_mul : sSup (s * t) = sSup s * sSup t 
goal: sSup (s * t) = sSup s * sSup t

full_name: sSup_add
code: @[to_additive]
theorem sSup_mul : sSup (s * t) = sSup s * sSup t 
goal: sSup (s + t) = sSup s + sSup t

full_name: sInf_add
code: @[to_additive]
theorem sInf_mul : sInf (s * t) = sInf s * sInf t 
goal: sInf (s + t) = sInf s + sInf t

full_name: sInf_mul
code: @[to_additive]
theorem sInf_mul : sInf (s * t) = sInf s * sInf t 
goal: sInf (s * t) = sInf s * sInf t

full_name: sSup_div
code: @[to_additive]
theorem sSup_div : sSup (s / t) = sSup s / sInf t 
goal: sSup (s / t) = sSup s / sInf t

full_name: sSup_sub
code: @[to_additive]
theorem sSup_div : sSup (s / t) = sSup s / sInf t 
goal: sSup (s - t) = sSup s - sInf t

full_name: sInf_div
code: @[to_additive]
theorem sInf_div : sInf (s / t) = sInf s / sSup t 
goal: sInf (s / t) = sInf s / sSup t

full_name: sInf_sub
code: @[to_additive]
theorem sInf_div : sInf (s / t) = sInf s / sSup t 
goal: sInf (s - t) = sInf s - sSup t

full_name: csSup_zero
code: @[to_additive (attr := simp)]
theorem csSup_one : sSup (1 : Set α) = 1 
goal: sSup 0 = 0

full_name: csSup_one
code: @[to_additive (attr := simp)]
theorem csSup_one : sSup (1 : Set α) = 1 
goal: sSup 1 = 1

full_name: csInf_zero
code: @[to_additive (attr := simp)]
theorem csInf_one : sInf (1 : Set α) = 1 
goal: sInf 0 = 0

full_name: csInf_one
code: @[to_additive (attr := simp)]
theorem csInf_one : sInf (1 : Set α) = 1 
goal: sInf 1 = 1

full_name: csSup_neg
code: @[to_additive]
theorem csSup_inv (hs₀ : s.Nonempty) (hs₁ : BddBelow s) : sSup s⁻¹ = (sInf s)⁻¹ 
goal: sSup (-s) = -sInf s

full_name: csSup_inv
code: @[to_additive]
theorem csSup_inv (hs₀ : s.Nonempty) (hs₁ : BddBelow s) : sSup s⁻¹ = (sInf s)⁻¹ 
goal: sSup s⁻¹ = (sInf s)⁻¹

full_name: csInf_neg
code: @[to_additive]
theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ 
goal: sInf (-s) = -sSup s

full_name: csInf_inv
code: @[to_additive]
theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ 
goal: sInf s⁻¹ = (sSup s)⁻¹

full_name: csSup_mul
code: @[to_additive]
theorem csSup_mul (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sSup (s * t) = sSup s * sSup t 
goal: sSup (s * t) = sSup s * sSup t

full_name: csSup_add
code: @[to_additive]
theorem csSup_mul (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sSup (s * t) = sSup s * sSup t 
goal: sSup (s + t) = sSup s + sSup t

full_name: csInf_add
code: @[to_additive]
theorem csInf_mul (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sInf (s * t) = sInf s * sInf t 
goal: sInf (s + t) = sInf s + sInf t

full_name: csInf_mul
code: @[to_additive]
theorem csInf_mul (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sInf (s * t) = sInf s * sInf t 
goal: sInf (s * t) = sInf s * sInf t

full_name: csSup_div
code: @[to_additive]
theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sSup (s / t) = sSup s / sInf t 
goal: sSup (s / t) = sSup s / sInf t

full_name: csSup_sub
code: @[to_additive]
theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sSup (s / t) = sSup s / sInf t 
goal: sSup (s - t) = sSup s - sInf t

full_name: csInf_div
code: @[to_additive]
theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sInf (s / t) = sInf s / sSup t 
goal: sInf (s / t) = sInf s / sSup t

full_name: csInf_sub
code: @[to_additive]
theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sInf (s / t) = sInf s / sSup t 
goal: sInf (s - t) = sInf s - sSup t

full_name: LinearOrderedField.smul_Ioo
code: theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) 
goal: r • Set.Ioo a b = Set.Ioo (r • a) (r • b)

full_name: LinearOrderedField.smul_Icc
code: theorem smul_Icc : r • Icc a b = Icc (r • a) (r • b) 
goal: r • Set.Icc a b = Set.Icc (r • a) (r • b)

full_name: LinearOrderedField.smul_Ico
code: theorem smul_Ico : r • Ico a b = Ico (r • a) (r • b) 
goal: r • Set.Ico a b = Set.Ico (r • a) (r • b)

full_name: LinearOrderedField.smul_Ioc
code: theorem smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) 
goal: r • Set.Ioc a b = Set.Ioc (r • a) (r • b)

full_name: LinearOrderedField.smul_Ioi
code: theorem smul_Ioi : r • Ioi a = Ioi (r • a) 
goal: r • Set.Ioi a = Set.Ioi (r • a)

full_name: LinearOrderedField.smul_Iio
code: theorem smul_Iio : r • Iio a = Iio (r • a) 
goal: r • Set.Iio a = Set.Iio (r • a)

full_name: LinearOrderedField.smul_Ici
code: theorem smul_Ici : r • Ici a = Ici (r • a) 
goal: r • Set.Ici a = Set.Ici (r • a)

full_name: LinearOrderedField.smul_Iic
code: theorem smul_Iic : r • Iic a = Iic (r • a) 
goal: r • Set.Iic a = Set.Iic (r • a)