ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- exists_nat_pow_near_of_lt_one
theorem exists_nat_pow_near_of_lt_one (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1) :
    ∃ n : ℕ, y ^ (n + 1) < x ∧ x ≤ y ^ n 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] [Archimedean α] [ExistsAddOfLE α] {x y : α} (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1)
Goal: ∃ n, y ^ (n + 1) < x ∧ x ≤ y ^ n
-- exists_nat_one_div_lt
lemma exists_nat_one_div_lt (hε : 0 < ε) : ∃ n : ℕ, 1 / (n + 1 : α) < ε 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] [Archimedean α] {ε : α} (hε : 0 < ε)
Goal: ∃ n, 1 / (↑n + 1) < ε
-- exists_rat_gt
theorem exists_rat_gt (x : α) : ∃ q : ℚ, x < q 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] (x : α)
Goal: ∃ q, x < ↑q
-- exists_rat_lt
theorem exists_rat_lt (x : α) : ∃ q : ℚ, (q : α) < x 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] (x : α)
Goal: ∃ q, ↑q < x
-- exists_rat_btwn
theorem exists_rat_btwn {x y : α} (h : x < y) : ∃ q : ℚ, x < q ∧ (q : α) < y 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α} (h : x < y)
Goal: ∃ q, x < ↑q ∧ ↑q < y
-- le_of_forall_rat_lt_imp_le
theorem le_of_forall_rat_lt_imp_le (h : ∀ q : ℚ, (q : α) < x → (q : α) ≤ y) : x ≤ y 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α} (h : ∀ (q : ℚ), ↑q < x → ↑q ≤ y)
Goal: x ≤ y
-- le_of_forall_lt_rat_imp_le
theorem le_of_forall_lt_rat_imp_le (h : ∀ q : ℚ, y < q → x ≤ q) : x ≤ y 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α} (h : ∀ (q : ℚ), y < ↑q → x ≤ ↑q)
Goal: x ≤ y
-- le_iff_forall_rat_lt_imp_le
theorem le_iff_forall_rat_lt_imp_le : x ≤ y ↔ ∀ q : ℚ, (q : α) < x → (q : α) ≤ y 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α}
Goal: x ≤ y ↔ ∀ (q : ℚ), ↑q < x → ↑q ≤ y
-- le_iff_forall_lt_rat_imp_le
theorem le_iff_forall_lt_rat_imp_le : x ≤ y ↔ ∀ q : ℚ, y < q → x ≤ q 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α}
Goal: x ≤ y ↔ ∀ (q : ℚ), y < ↑q → x ≤ ↑q
-- eq_of_forall_rat_lt_iff_lt
theorem eq_of_forall_rat_lt_iff_lt (h : ∀ q : ℚ, (q : α) < x ↔ (q : α) < y) : x = y 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α} (h : ∀ (q : ℚ), ↑q < x ↔ ↑q < y)
Goal: x = y
-- eq_of_forall_lt_rat_iff_lt
theorem eq_of_forall_lt_rat_iff_lt (h : ∀ q : ℚ, x < q ↔ y < q) : x = y 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x y : α} (h : ∀ (q : ℚ), x < ↑q ↔ y < ↑q)
Goal: x = y
-- exists_pos_rat_lt
theorem exists_pos_rat_lt {x : α} (x0 : 0 < x) : ∃ q : ℚ, 0 < q ∧ (q : α) < x 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {x : α} (x0 : 0 < x)
Goal: ∃ q, 0 < q ∧ ↑q < x
-- exists_rat_near
theorem exists_rat_near (x : α) (ε0 : 0 < ε) : ∃ q : ℚ, |x - q| < ε 
Arguments: {α : Type u_1} [LinearOrderedField α] [Archimedean α] {ε : α} (x : α) (ε0 : 0 < ε)
Goal: ∃ q, |x - ↑q| < ε
-- archimedean_iff_nat_lt
theorem archimedean_iff_nat_lt : Archimedean α ↔ ∀ x : α, ∃ n : ℕ, x < n 
Arguments: {α : Type u_1} [LinearOrderedField α]
Goal: Archimedean α ↔ ∀ (x : α), ∃ n, x < ↑n
-- archimedean_iff_nat_le
theorem archimedean_iff_nat_le : Archimedean α ↔ ∀ x : α, ∃ n : ℕ, x ≤ n 
Arguments: {α : Type u_1} [LinearOrderedField α]
Goal: Archimedean α ↔ ∀ (x : α), ∃ n, x ≤ ↑n
-- archimedean_iff_int_lt
theorem archimedean_iff_int_lt : Archimedean α ↔ ∀ x : α, ∃ n : ℤ, x < n 
Arguments: {α : Type u_1} [LinearOrderedField α]
Goal: Archimedean α ↔ ∀ (x : α), ∃ n, x < ↑n
-- archimedean_iff_int_le
theorem archimedean_iff_int_le : Archimedean α ↔ ∀ x : α, ∃ n : ℤ, x ≤ n 
Arguments: {α : Type u_1} [LinearOrderedField α]
Goal: Archimedean α ↔ ∀ (x : α), ∃ n, x ≤ ↑n
-- archimedean_iff_rat_lt
theorem archimedean_iff_rat_lt : Archimedean α ↔ ∀ x : α, ∃ q : ℚ, x < q 
Arguments: {α : Type u_1} [LinearOrderedField α]
Goal: Archimedean α ↔ ∀ (x : α), ∃ q, x < ↑q
-- archimedean_iff_rat_le
theorem archimedean_iff_rat_le : Archimedean α ↔ ∀ x : α, ∃ q : ℚ, x ≤ q 
Arguments: {α : Type u_1} [LinearOrderedField α]
Goal: Archimedean α ↔ ∀ (x : α), ∃ q, x ≤ ↑q
-- sSup_zero
@[to_additive (attr := simp)]
theorem sSup_one : sSup (1 : Set α) = 1 
Arguments: {α : Type u_1} [CompleteLattice α] [Zero α]
Goal: sSup 0 = 0
-- sSup_one
@[to_additive (attr := simp)]
theorem sSup_one : sSup (1 : Set α) = 1 
Arguments: {α : Type u_1} [CompleteLattice α] [One α]
Goal: sSup 1 = 1
-- sInf_one
@[to_additive (attr := simp)]
theorem sInf_one : sInf (1 : Set α) = 1 
Arguments: {α : Type u_1} [CompleteLattice α] [One α]
Goal: sInf 1 = 1
-- sInf_zero
@[to_additive (attr := simp)]
theorem sInf_one : sInf (1 : Set α) = 1 
Arguments: {α : Type u_1} [CompleteLattice α] [Zero α]
Goal: sInf 0 = 0
-- sSup_inv
@[to_additive]
theorem sSup_inv (s : Set α) : sSup s⁻¹ = (sInf s)⁻¹ 
Arguments: {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s : Set α)
Goal: sSup s⁻¹ = (sInf s)⁻¹
-- sSup_neg
@[to_additive]
theorem sSup_inv (s : Set α) : sSup s⁻¹ = (sInf s)⁻¹ 
Arguments: {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s : Set α)
Goal: sSup (-s) = -sInf s
-- sInf_neg
@[to_additive]
theorem sInf_inv (s : Set α) : sInf s⁻¹ = (sSup s)⁻¹ 
Arguments: {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s : Set α)
Goal: sInf (-s) = -sSup s
-- sInf_inv
@[to_additive]
theorem sInf_inv (s : Set α) : sInf s⁻¹ = (sSup s)⁻¹ 
Arguments: {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s : Set α)
Goal: sInf s⁻¹ = (sSup s)⁻¹
-- sSup_mul
@[to_additive]
theorem sSup_mul : sSup (s * t) = sSup s * sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sSup (s * t) = sSup s * sSup t
-- sSup_add
@[to_additive]
theorem sSup_mul : sSup (s * t) = sSup s * sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sSup (s + t) = sSup s + sSup t
-- sInf_add
@[to_additive]
theorem sInf_mul : sInf (s * t) = sInf s * sInf t 
Arguments: {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sInf (s + t) = sInf s + sInf t
-- sInf_mul
@[to_additive]
theorem sInf_mul : sInf (s * t) = sInf s * sInf t 
Arguments: {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sInf (s * t) = sInf s * sInf t
-- sSup_div
@[to_additive]
theorem sSup_div : sSup (s / t) = sSup s / sInf t 
Arguments: {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sSup (s / t) = sSup s / sInf t
-- sSup_sub
@[to_additive]
theorem sSup_div : sSup (s / t) = sSup s / sInf t 
Arguments: {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sSup (s - t) = sSup s - sInf t
-- sInf_div
@[to_additive]
theorem sInf_div : sInf (s / t) = sInf s / sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sInf (s / t) = sInf s / sSup t
-- sInf_sub
@[to_additive]
theorem sInf_div : sInf (s / t) = sInf s / sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (s t : Set α)
Goal: sInf (s - t) = sInf s - sSup t
-- csSup_zero
@[to_additive (attr := simp)]
theorem csSup_one : sSup (1 : Set α) = 1 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Zero α]
Goal: sSup 0 = 0
-- csSup_one
@[to_additive (attr := simp)]
theorem csSup_one : sSup (1 : Set α) = 1 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [One α]
Goal: sSup 1 = 1
-- csInf_zero
@[to_additive (attr := simp)]
theorem csInf_one : sInf (1 : Set α) = 1 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Zero α]
Goal: sInf 0 = 0
-- csInf_one
@[to_additive (attr := simp)]
theorem csInf_one : sInf (1 : Set α) = 1 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [One α]
Goal: sInf 1 = 1
-- csSup_neg
@[to_additive]
theorem csSup_inv (hs₀ : s.Nonempty) (hs₁ : BddBelow s) : sSup s⁻¹ = (sInf s)⁻¹ 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {s : Set α} (hs₀ : s.Nonempty) (hs₁ : BddBelow s)
Goal: sSup (-s) = -sInf s
-- csSup_inv
@[to_additive]
theorem csSup_inv (hs₀ : s.Nonempty) (hs₁ : BddBelow s) : sSup s⁻¹ = (sInf s)⁻¹ 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {s : Set α} (hs₀ : s.Nonempty) (hs₁ : BddBelow s)
Goal: sSup s⁻¹ = (sInf s)⁻¹
-- csInf_neg
@[to_additive]
theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {s : Set α} (hs₀ : s.Nonempty) (hs₁ : BddAbove s)
Goal: sInf (-s) = -sSup s
-- csInf_inv
@[to_additive]
theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {s : Set α} (hs₀ : s.Nonempty) (hs₁ : BddAbove s)
Goal: sInf s⁻¹ = (sSup s)⁻¹
-- csSup_mul
@[to_additive]
theorem csSup_mul (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sSup (s * t) = sSup s * sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t)
Goal: sSup (s * t) = sSup s * sSup t
-- csSup_add
@[to_additive]
theorem csSup_mul (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sSup (s * t) = sSup s * sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t)
Goal: sSup (s + t) = sSup s + sSup t
-- csInf_add
@[to_additive]
theorem csInf_mul (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sInf (s * t) = sInf s * sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t)
Goal: sInf (s + t) = sInf s + sInf t
-- csInf_mul
@[to_additive]
theorem csInf_mul (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sInf (s * t) = sInf s * sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t)
Goal: sInf (s * t) = sInf s * sInf t
-- csSup_div
@[to_additive]
theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sSup (s / t) = sSup s / sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t)
Goal: sSup (s / t) = sSup s / sInf t
-- csSup_sub
@[to_additive]
theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
    sSup (s / t) = sSup s / sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t)
Goal: sSup (s - t) = sSup s - sInf t
-- csInf_div
@[to_additive]
theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sInf (s / t) = sInf s / sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t)
Goal: sInf (s / t) = sInf s / sSup t
-- csInf_sub
@[to_additive]
theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
    sInf (s / t) = sInf s / sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {s t : Set α} (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t)
Goal: sInf (s - t) = sInf s - sSup t
-- LinearOrderedField.smul_Ioo
theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a b r : K} (hr : 0 < r)
Goal: r • Set.Ioo a b = Set.Ioo (r • a) (r • b)
-- LinearOrderedField.smul_Icc
theorem smul_Icc : r • Icc a b = Icc (r • a) (r • b) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a b r : K} (hr : 0 < r)
Goal: r • Set.Icc a b = Set.Icc (r • a) (r • b)
-- LinearOrderedField.smul_Ico
theorem smul_Ico : r • Ico a b = Ico (r • a) (r • b) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a b r : K} (hr : 0 < r)
Goal: r • Set.Ico a b = Set.Ico (r • a) (r • b)
-- LinearOrderedField.smul_Ioc
theorem smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a b r : K} (hr : 0 < r)
Goal: r • Set.Ioc a b = Set.Ioc (r • a) (r • b)
-- LinearOrderedField.smul_Ioi
theorem smul_Ioi : r • Ioi a = Ioi (r • a) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a r : K} (hr : 0 < r)
Goal: r • Set.Ioi a = Set.Ioi (r • a)
-- LinearOrderedField.smul_Iio
theorem smul_Iio : r • Iio a = Iio (r • a) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a r : K} (hr : 0 < r)
Goal: r • Set.Iio a = Set.Iio (r • a)
-- LinearOrderedField.smul_Ici
theorem smul_Ici : r • Ici a = Ici (r • a) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a r : K} (hr : 0 < r)
Goal: r • Set.Ici a = Set.Ici (r • a)
-- LinearOrderedField.smul_Iic
theorem smul_Iic : r • Iic a = Iic (r • a) 
Arguments: {K : Type u_2} [LinearOrderedField K] {a r : K} (hr : 0 < r)
Goal: r • Set.Iic a = Set.Iic (r • a)
-- LieAlgebra.lieCharacter_apply_lie
theorem lieCharacter_apply_lie (χ : LieCharacter R L) (x y : L) : χ ⁅x, y⁆ = 0 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieAlgebra.LieCharacter R L) (x y : L)
Goal: χ ⁅x, y⁆ = 0
-- LieAlgebra.lieCharacter_apply_lie'
@[simp]
theorem lieCharacter_apply_lie' (χ : LieCharacter R L) (x y : L) : ⁅χ x, χ y⁆ = 0 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieAlgebra.LieCharacter R L) (x y : L)
Goal: ⁅χ x, χ y⁆ = 0
-- LieAlgebra.lieCharacter_apply_of_mem_derived
theorem lieCharacter_apply_of_mem_derived (χ : LieCharacter R L) {x : L}
    (h : x ∈ derivedSeries R L 1) : χ x = 0 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (χ : LieAlgebra.LieCharacter R L) {x : L} (h : x ∈ LieAlgebra.derivedSeries R L 1)
Goal: χ x = 0
-- LieAlgebra.lieCharacterEquivLinearDual_apply
@[simps! apply symm_apply]
def lieCharacterEquivLinearDual [IsLieAbelian L] : LieCharacter R L ≃ Module.Dual R L 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] (χ : LieAlgebra.LieCharacter R L)
Goal: LieAlgebra.lieCharacterEquivLinearDual χ = ↑χ
-- LieAlgebra.lieCharacterEquivLinearDual_symm_apply
@[simps! apply symm_apply]
def lieCharacterEquivLinearDual [IsLieAbelian L] : LieCharacter R L ≃ Module.Dual R L 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] [IsLieAbelian L] (ψ : Module.Dual R L)
Goal: LieAlgebra.lieCharacterEquivLinearDual.symm ψ = { toLinearMap := ψ, map_lie' := ⋯ }
-- LinearMap.BilinForm.lieInvariant_iff
lemma _root_.LinearMap.BilinForm.lieInvariant_iff [LieModule R L M] :
    Φ.lieInvariant L ↔ Φ ∈ LieModule.maxTrivSubmodule R L (LinearMap.BilinForm R M) 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M] [LieRingModule L M] (Φ : LinearMap.BilinForm R M) [LieModule R L M]
Goal: LinearMap.BilinForm.lieInvariant L Φ ↔ Φ ∈ LieModule.maxTrivSubmodule R L (LinearMap.BilinForm R M)
-- LieAlgebra.InvariantForm.orthogonal_carrier
@[simps!]
def orthogonal (N : LieSubmodule R L M) : LieSubmodule R L M 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (Φ : LinearMap.BilinForm R M) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (N : LieSubmodule R L M)
Goal: ↑(LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv N) = {m | ∀ n ∈ N, Φ.IsOrtho n m}
-- LieAlgebra.InvariantForm.orthogonal_toSubmodule
@[simp]
lemma orthogonal_toSubmodule (N : LieSubmodule R L M) :
    (orthogonal Φ hΦ_inv N).toSubmodule = Φ.orthogonal N.toSubmodule 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (Φ : LinearMap.BilinForm R M) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (N : LieSubmodule R L M)
Goal: ↑(LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv N) = Φ.orthogonal ↑N
-- LieAlgebra.InvariantForm.mem_orthogonal
lemma mem_orthogonal (N : LieSubmodule R L M) (y : M) :
    y ∈ orthogonal Φ hΦ_inv N ↔ ∀ x ∈ N, Φ x y = 0 
Arguments: {R : Type u_1} {L : Type u_2} {M : Type u_3} [CommRing R] [LieRing L] [AddCommGroup M] [Module R M] [LieRingModule L M] (Φ : LinearMap.BilinForm R M) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (N : LieSubmodule R L M) (y : M)
Goal: y ∈ LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv N ↔ ∀ x ∈ N, (Φ x) y = 0
-- LieAlgebra.InvariantForm.orthogonal_disjoint
lemma orthogonal_disjoint
    (Φ : LinearMap.BilinForm R L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : Φ.lieInvariant L)
    -- TODO: replace the following assumption by a typeclass assumption `[HasNonAbelianAtoms]`
    (hL : ∀ I : LieIdeal R L, IsAtom I → ¬IsLieAbelian I)
    (I : LieIdeal R L) (hI : IsAtom I) :
    Disjoint I (orthogonal Φ hΦ_inv I) 
Arguments: {R : Type u_1} {L : Type u_2} [CommRing R] [LieRing L] [LieAlgebra R L] (Φ : LinearMap.BilinForm R L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hL : ∀ (I : LieIdeal R L), IsAtom I → ¬IsLieAbelian ↥↑I) (I : LieIdeal R L) (hI : IsAtom I)
Goal: Disjoint I (LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv I)
-- LieAlgebra.InvariantForm.orthogonal_isCompl_coe_submodule
lemma orthogonal_isCompl_coe_submodule (I : LieIdeal K L) (hI : IsAtom I) :
    IsCompl I.toSubmodule (orthogonal Φ hΦ_inv I).toSubmodule 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (Φ : LinearMap.BilinForm K L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hΦ_refl : Φ.IsRefl) (hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I) (I : LieIdeal K L) (hI : IsAtom I)
Goal: IsCompl ↑I ↑(LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv I)
-- LieAlgebra.InvariantForm.orthogonal_isCompl
lemma orthogonal_isCompl (I : LieIdeal K L) (hI : IsAtom I) :
    IsCompl I (orthogonal Φ hΦ_inv I) 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (Φ : LinearMap.BilinForm K L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hΦ_refl : Φ.IsRefl) (hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I) (I : LieIdeal K L) (hI : IsAtom I)
Goal: IsCompl I (LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv I)
-- LieAlgebra.InvariantForm.restrict_nondegenerate
lemma restrict_nondegenerate (I : LieIdeal K L) (hI : IsAtom I) :
    (Φ.restrict I).Nondegenerate 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (Φ : LinearMap.BilinForm K L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hΦ_refl : Φ.IsRefl) (hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I) (I : LieIdeal K L) (hI : IsAtom I)
Goal: (Φ.restrict (lieIdealSubalgebra K L I).toSubmodule).Nondegenerate
-- LieAlgebra.InvariantForm.restrict_orthogonal_nondegenerate
lemma restrict_orthogonal_nondegenerate (I : LieIdeal K L) (hI : IsAtom I) :
    (Φ.restrict (orthogonal Φ hΦ_inv I)).Nondegenerate 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (Φ : LinearMap.BilinForm K L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hΦ_refl : Φ.IsRefl) (hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I) (I : LieIdeal K L) (hI : IsAtom I)
Goal: (Φ.restrict (lieIdealSubalgebra K L (LieAlgebra.InvariantForm.orthogonal Φ hΦ_inv I)).toSubmodule).Nondegenerate
-- LieAlgebra.InvariantForm.atomistic
lemma atomistic : ∀ I : LieIdeal K L, sSup {J : LieIdeal K L | IsAtom J ∧ J ≤ I} = I 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (Φ : LinearMap.BilinForm K L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hΦ_refl : Φ.IsRefl) (hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I) (I : LieIdeal K L)
Goal: sSup {J | IsAtom J ∧ J ≤ I} = I
-- LieAlgebra.InvariantForm.isSemisimple_of_nondegenerate
theorem isSemisimple_of_nondegenerate : IsSemisimple K L 
Arguments: {K : Type u_1} {L : Type u_2} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L] (Φ : LinearMap.BilinForm K L) (hΦ_nondeg : Φ.Nondegenerate) (hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ) (hΦ_refl : Φ.IsRefl) (hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I)
Goal: LieAlgebra.IsSemisimple K L
-- LieAlgebra.IsKilling.killingCompl_top_eq_bot
class IsKilling : Prop :=
    killingCompl_top_eq_bot : LieIdeal.killingCompl R L ⊤ = ⊥
Arguments: {R : Type u_1} {L : Type u_3} [CommRing R] [LieRing L] [LieAlgebra R L] [self : LieAlgebra.IsKilling R L]
Goal: LieIdeal.killingCompl R L ⊤ = ⊥