ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CauSeq.const_inv
theorem const_inv {x : β} (hx : x ≠ 0) :
    const abv x⁻¹ = inv (const abv x) (by rwa [const_limZero]) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {x : β} (hx : x ≠ 0)
Goal: CauSeq.const abv x⁻¹ = (CauSeq.const abv x).inv ⋯
-- CauSeq.not_limZero_of_pos
theorem not_limZero_of_pos {f : CauSeq α abs} : Pos f → ¬LimZero f
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs}
Goal: f.Pos → ¬f.LimZero
-- CauSeq.const_pos
theorem const_pos {x : α} : Pos (const x) ↔ 0 < x 
Arguments: {α : Type u_1} [LinearOrderedField α] {x : α}
Goal: (CauSeq.const abs x).Pos ↔ 0 < x
-- CauSeq.add_pos
theorem add_pos {f g : CauSeq α abs} : Pos f → Pos g → Pos (f + g)
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs}
Goal: f.Pos → g.Pos → (f + g).Pos
-- CauSeq.pos_add_limZero
theorem pos_add_limZero {f g : CauSeq α abs} : Pos f → LimZero g → Pos (f + g)
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs}
Goal: f.Pos → g.LimZero → (f + g).Pos
-- CauSeq.mul_pos
protected theorem mul_pos {f g : CauSeq α abs} : Pos f → Pos g → Pos (f * g)
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs}
Goal: f.Pos → g.Pos → (f * g).Pos
-- CauSeq.trichotomy
theorem trichotomy (f : CauSeq α abs) : Pos f ∨ LimZero f ∨ Pos (-f) 
Arguments: {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs)
Goal: f.Pos ∨ f.LimZero ∨ (-f).Pos
-- CauSeq.lt_of_lt_of_eq
theorem lt_of_lt_of_eq {f g h : CauSeq α abs} (fg : f < g) (gh : g ≈ h) : f < h 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g h : CauSeq α abs} (fg : f < g) (gh : g ≈ h)
Goal: f < h
-- CauSeq.lt_of_eq_of_lt
theorem lt_of_eq_of_lt {f g h : CauSeq α abs} (fg : f ≈ g) (gh : g < h) : f < h 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g h : CauSeq α abs} (fg : f ≈ g) (gh : g < h)
Goal: f < h
-- CauSeq.lt_trans
theorem lt_trans {f g h : CauSeq α abs} (fg : f < g) (gh : g < h) : f < h 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g h : CauSeq α abs} (fg : f < g) (gh : g < h)
Goal: f < h
-- CauSeq.lt_irrefl
theorem lt_irrefl {f : CauSeq α abs} : ¬f < f
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f : CauSeq α abs}
Goal: ¬f < f
-- CauSeq.le_of_eq_of_le
theorem le_of_eq_of_le {f g h : CauSeq α abs} (hfg : f ≈ g) (hgh : g ≤ h) : f ≤ h 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g h : CauSeq α abs} (hfg : f ≈ g) (hgh : g ≤ h)
Goal: f ≤ h
-- CauSeq.le_of_le_of_eq
theorem le_of_le_of_eq {f g h : CauSeq α abs} (hfg : f ≤ g) (hgh : g ≈ h) : f ≤ h 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g h : CauSeq α abs} (hfg : f ≤ g) (hgh : g ≈ h)
Goal: f ≤ h
-- CauSeq.le_antisymm
theorem le_antisymm {f g : CauSeq α abs} (fg : f ≤ g) (gf : g ≤ f) : f ≈ g 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs} (fg : f ≤ g) (gf : g ≤ f)
Goal: f ≈ g
-- CauSeq.lt_total
theorem lt_total (f g : CauSeq α abs) : f < g ∨ f ≈ g ∨ g < f 
Arguments: {α : Type u_1} [LinearOrderedField α] (f g : CauSeq α abs)
Goal: f < g ∨ f ≈ g ∨ g < f
-- CauSeq.le_total
theorem le_total (f g : CauSeq α abs) : f ≤ g ∨ g ≤ f 
Arguments: {α : Type u_1} [LinearOrderedField α] (f g : CauSeq α abs)
Goal: f ≤ g ∨ g ≤ f
-- CauSeq.const_lt
theorem const_lt {x y : α} : const x < const y ↔ x < y 
Arguments: {α : Type u_1} [LinearOrderedField α] {x y : α}
Goal: CauSeq.const abs x < CauSeq.const abs y ↔ x < y
-- CauSeq.const_le
theorem const_le {x y : α} : const x ≤ const y ↔ x ≤ y 
Arguments: {α : Type u_1} [LinearOrderedField α] {x y : α}
Goal: CauSeq.const abs x ≤ CauSeq.const abs y ↔ x ≤ y
-- CauSeq.le_of_exists
theorem le_of_exists {f g : CauSeq α abs} (h : ∃ i, ∀ j ≥ i, f j ≤ g j) : f ≤ g 
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs} (h : ∃ i, ∀ j ≥ i, ↑f j ≤ ↑g j)
Goal: f ≤ g
-- CauSeq.exists_gt
theorem exists_gt (f : CauSeq α abs) : ∃ a : α, f < const a 
Arguments: {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs)
Goal: ∃ a, f < CauSeq.const abs a
-- CauSeq.exists_lt
theorem exists_lt (f : CauSeq α abs) : ∃ a : α, const a < f 
Arguments: {α : Type u_1} [LinearOrderedField α] (f : CauSeq α abs)
Goal: ∃ a, CauSeq.const abs a < f
-- CauSeq.rat_sup_continuous_lemma
theorem rat_sup_continuous_lemma {ε : α} {a₁ a₂ b₁ b₂ : α} :
    abs (a₁ - b₁) < ε → abs (a₂ - b₂) < ε → abs (a₁ ⊔ a₂ - b₁ ⊔ b₂) < ε 
Arguments: {α : Type u_1} [LinearOrderedField α] {ε a₁ a₂ b₁ b₂ : α}
Goal: |a₁ - b₁| < ε → |a₂ - b₂| < ε → |a₁ ⊔ a₂ - b₁ ⊔ b₂| < ε
-- CauSeq.rat_inf_continuous_lemma
theorem rat_inf_continuous_lemma {ε : α} {a₁ a₂ b₁ b₂ : α} :
    abs (a₁ - b₁) < ε → abs (a₂ - b₂) < ε → abs (a₁ ⊓ a₂ - b₁ ⊓ b₂) < ε 
Arguments: {α : Type u_1} [LinearOrderedField α] {ε a₁ a₂ b₁ b₂ : α}
Goal: |a₁ - b₁| < ε → |a₂ - b₂| < ε → |a₁ ⊓ a₂ - b₁ ⊓ b₂| < ε
-- CauSeq.coe_sup
@[simp, norm_cast]
theorem coe_sup (f g : CauSeq α abs) : ⇑(f ⊔ g) = (f : ℕ → α) ⊔ g 
Arguments: {α : Type u_1} [LinearOrderedField α] (f g : CauSeq α abs)
Goal: ↑(f ⊔ g) = ↑f ⊔ ↑g
-- CauSeq.coe_inf
@[simp, norm_cast]
theorem coe_inf (f g : CauSeq α abs) : ⇑(f ⊓ g) = (f : ℕ → α) ⊓ g 
Arguments: {α : Type u_1} [LinearOrderedField α] (f g : CauSeq α abs)
Goal: ↑(f ⊓ g) = ↑f ⊓ ↑g
-- CauSeq.sup_limZero
theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs} (hf : f.LimZero) (hg : g.LimZero)
Goal: (f ⊔ g).LimZero
-- CauSeq.inf_limZero
theorem inf_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊓ g)
  
Arguments: {α : Type u_1} [LinearOrderedField α] {f g : CauSeq α abs} (hf : f.LimZero) (hg : g.LimZero)
Goal: (f ⊓ g).LimZero
-- CauSeq.sup_equiv_sup
theorem sup_equiv_sup {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
    a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ 
Arguments: {α : Type u_1} [LinearOrderedField α] {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂)
Goal: a₁ ⊔ b₁ ≈ a₂ ⊔ b₂
-- CauSeq.inf_equiv_inf
theorem inf_equiv_inf {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂) :
    a₁ ⊓ b₁ ≈ a₂ ⊓ b₂ 
Arguments: {α : Type u_1} [LinearOrderedField α] {a₁ b₁ a₂ b₂ : CauSeq α abs} (ha : a₁ ≈ a₂) (hb : b₁ ≈ b₂)
Goal: a₁ ⊓ b₁ ≈ a₂ ⊓ b₂
-- CauSeq.sup_lt
protected theorem sup_lt {a b c : CauSeq α abs} (ha : a < c) (hb : b < c) : a ⊔ b < c 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b c : CauSeq α abs} (ha : a < c) (hb : b < c)
Goal: a ⊔ b < c
-- CauSeq.lt_inf
protected theorem lt_inf {a b c : CauSeq α abs} (hb : a < b) (hc : a < c) : a < b ⊓ c 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b c : CauSeq α abs} (hb : a < b) (hc : a < c)
Goal: a < b ⊓ c
-- CauSeq.sup_idem
@[simp]
protected theorem sup_idem (a : CauSeq α abs) : a ⊔ a = a 
Arguments: {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs)
Goal: a ⊔ a = a
-- CauSeq.inf_idem
@[simp]
protected theorem inf_idem (a : CauSeq α abs) : a ⊓ a = a 
Arguments: {α : Type u_1} [LinearOrderedField α] (a : CauSeq α abs)
Goal: a ⊓ a = a
-- CauSeq.sup_comm
protected theorem sup_comm (a b : CauSeq α abs) : a ⊔ b = b ⊔ a 
Arguments: {α : Type u_1} [LinearOrderedField α] (a b : CauSeq α abs)
Goal: a ⊔ b = b ⊔ a
-- CauSeq.inf_comm
protected theorem inf_comm (a b : CauSeq α abs) : a ⊓ b = b ⊓ a 
Arguments: {α : Type u_1} [LinearOrderedField α] (a b : CauSeq α abs)
Goal: a ⊓ b = b ⊓ a
-- CauSeq.sup_eq_right
protected theorem sup_eq_right {a b : CauSeq α abs} (h : a ≤ b) : a ⊔ b ≈ b 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs} (h : a ≤ b)
Goal: a ⊔ b ≈ b
-- CauSeq.inf_eq_right
protected theorem inf_eq_right {a b : CauSeq α abs} (h : b ≤ a) : a ⊓ b ≈ b 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs} (h : b ≤ a)
Goal: a ⊓ b ≈ b
-- CauSeq.sup_eq_left
protected theorem sup_eq_left {a b : CauSeq α abs} (h : b ≤ a) : a ⊔ b ≈ a 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs} (h : b ≤ a)
Goal: a ⊔ b ≈ a
-- CauSeq.inf_eq_left
protected theorem inf_eq_left {a b : CauSeq α abs} (h : a ≤ b) : a ⊓ b ≈ a 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs} (h : a ≤ b)
Goal: a ⊓ b ≈ a
-- CauSeq.le_sup_left
protected theorem le_sup_left {a b : CauSeq α abs} : a ≤ a ⊔ b 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs}
Goal: a ≤ a ⊔ b
-- CauSeq.inf_le_left
protected theorem inf_le_left {a b : CauSeq α abs} : a ⊓ b ≤ a 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs}
Goal: a ⊓ b ≤ a
-- CauSeq.le_sup_right
protected theorem le_sup_right {a b : CauSeq α abs} : b ≤ a ⊔ b 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs}
Goal: b ≤ a ⊔ b
-- CauSeq.inf_le_right
protected theorem inf_le_right {a b : CauSeq α abs} : a ⊓ b ≤ b 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b : CauSeq α abs}
Goal: a ⊓ b ≤ b
-- CauSeq.sup_le
protected theorem sup_le {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c) : a ⊔ b ≤ c 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b c : CauSeq α abs} (ha : a ≤ c) (hb : b ≤ c)
Goal: a ⊔ b ≤ c
-- CauSeq.le_inf
protected theorem le_inf {a b c : CauSeq α abs} (hb : a ≤ b) (hc : a ≤ c) : a ≤ b ⊓ c 
Arguments: {α : Type u_1} [LinearOrderedField α] {a b c : CauSeq α abs} (hb : a ≤ b) (hc : a ≤ c)
Goal: a ≤ b ⊓ c
-- CauSeq.sup_inf_distrib_left
protected theorem sup_inf_distrib_left (a b c : CauSeq α abs) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) 
Arguments: {α : Type u_1} [LinearOrderedField α] (a b c : CauSeq α abs)
Goal: a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
-- CauSeq.sup_inf_distrib_right
protected theorem sup_inf_distrib_right (a b c : CauSeq α abs) : a ⊓ b ⊔ c = (a ⊔ c) ⊓ (b ⊔ c) 
Arguments: {α : Type u_1} [LinearOrderedField α] (a b c : CauSeq α abs)
Goal: a ⊓ b ⊔ c = (a ⊔ c) ⊓ (b ⊔ c)
-- LieSubalgebra.lie_mem'
structure LieSubalgebra extends Submodule R L where
    lie_mem' : ∀ {x y}, x ∈ carrier → y ∈ carrier → ⁅x, y⁆ ∈ carrier
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (self : LieSubalgebra R L) {x y : L}
Goal: x ∈ self.carrier → y ∈ self.carrier → ⁅x, y⁆ ∈ self.carrier
-- LieSubalgebra.zero_mem
@[simp]
protected theorem zero_mem : (0 : L) ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L)
Goal: 0 ∈ L'
-- LieSubalgebra.add_mem
protected theorem add_mem {x y : L} : x ∈ L' → y ∈ L' → (x + y : L) ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x y : L}
Goal: x ∈ L' → y ∈ L' → x + y ∈ L'
-- LieSubalgebra.sub_mem
protected theorem sub_mem {x y : L} : x ∈ L' → y ∈ L' → (x - y : L) ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x y : L}
Goal: x ∈ L' → y ∈ L' → x - y ∈ L'
-- LieSubalgebra.smul_mem
theorem smul_mem (t : R) {x : L} (h : x ∈ L') : t • x ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (t : R) {x : L} (h : x ∈ L')
Goal: t • x ∈ L'
-- LieSubalgebra.lie_mem
theorem lie_mem {x y : L} (hx : x ∈ L') (hy : y ∈ L') : (⁅x, y⁆ : L) ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x y : L} (hx : x ∈ L') (hy : y ∈ L')
Goal: ⁅x, y⁆ ∈ L'
-- LieSubalgebra.mem_carrier
theorem mem_carrier {x : L} : x ∈ L'.carrier ↔ x ∈ (L' : Set L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L}
Goal: x ∈ L'.carrier ↔ x ∈ ↑L'
-- LieSubalgebra.mem_mk_iff
@[simp]
theorem mem_mk_iff (S : Set L) (h₁ h₂ h₃ h₄) {x : L} :
    x ∈ (⟨⟨⟨⟨S, h₁⟩, h₂⟩, h₃⟩, h₄⟩ : LieSubalgebra R L) ↔ x ∈ S 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ :
  ∀ (c : R) {x : L},
    x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier →
      c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ :
  ∀ {x y : L},
    x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier →
      y ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier →
        ⁅x, y⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier) {x : L}
Goal: x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } ↔ x ∈ S
-- LieSubalgebra.mem_coe_submodule
@[simp]
theorem mem_coe_submodule {x : L} : x ∈ (L' : Submodule R L) ↔ x ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L}
Goal: x ∈ L'.toSubmodule ↔ x ∈ L'
-- LieSubalgebra.mem_coe
theorem mem_coe {x : L} : x ∈ (L' : Set L) ↔ x ∈ L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {x : L}
Goal: x ∈ ↑L' ↔ x ∈ L'
-- LieSubalgebra.coe_bracket
@[simp, norm_cast]
theorem coe_bracket (x y : L') : (↑⁅x, y⁆ : L) = ⁅(↑x : L), ↑y⁆ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (x y : ↥L')
Goal: ↑⁅x, y⁆ = ⁅↑x, ↑y⁆
-- LieSubalgebra.ext_iff
theorem ext_iff (x y : L') : x = y ↔ (x : L) = y 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (x y : ↥L')
Goal: x = y ↔ ↑x = ↑y
-- LieSubalgebra.coe_zero_iff_zero
theorem coe_zero_iff_zero (x : L') : (x : L) = 0 ↔ x = 0 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) (x : ↥L')
Goal: ↑x = 0 ↔ x = 0
-- LieSubalgebra.ext
@[ext]
theorem ext (L₁' L₂' : LieSubalgebra R L) (h : ∀ x, x ∈ L₁' ↔ x ∈ L₂') : L₁' = L₂' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' L₂' : LieSubalgebra R L) (h : ∀ (x : L), x ∈ L₁' ↔ x ∈ L₂')
Goal: L₁' = L₂'
-- LieSubalgebra.ext_iff'
theorem ext_iff' (L₁' L₂' : LieSubalgebra R L) : L₁' = L₂' ↔ ∀ x, x ∈ L₁' ↔ x ∈ L₂' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' L₂' : LieSubalgebra R L)
Goal: L₁' = L₂' ↔ ∀ (x : L), x ∈ L₁' ↔ x ∈ L₂'
-- LieSubalgebra.mk_coe
@[simp]
theorem mk_coe (S : Set L) (h₁ h₂ h₃ h₄) :
    ((⟨⟨⟨⟨S, h₁⟩, h₂⟩, h₃⟩, h₄⟩ : LieSubalgebra R L) : Set L) = S 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set L) (h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S) (h₂ : S 0) (h₃ :
  ∀ (c : R) {x : L},
    x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier →
      c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier) (h₄ :
  ∀ {x y : L},
    x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier →
      y ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier →
        ⁅x, y⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier)
Goal: ↑{ carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } = S
-- LieSubalgebra.coe_to_submodule_mk
theorem coe_to_submodule_mk (p : Submodule R L) (h) :
    (({ p with lie_mem' := h } : LieSubalgebra R L) : Submodule R L) = p 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : Submodule R L) (h : ∀ {x y : L}, x ∈ p.carrier → y ∈ p.carrier → ⁅x, y⁆ ∈ p.carrier)
Goal: { toSubmodule := p, lie_mem' := h }.toSubmodule = p
-- LieSubalgebra.coe_injective
theorem coe_injective : Function.Injective ((↑) : LieSubalgebra R L → Set L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: Function.Injective SetLike.coe
-- LieSubalgebra.coe_set_eq
@[norm_cast]
theorem coe_set_eq (L₁' L₂' : LieSubalgebra R L) : (L₁' : Set L) = L₂' ↔ L₁' = L₂' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' L₂' : LieSubalgebra R L)
Goal: ↑L₁' = ↑L₂' ↔ L₁' = L₂'
-- LieSubalgebra.to_submodule_injective
theorem to_submodule_injective : Function.Injective ((↑) : LieSubalgebra R L → Submodule R L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: Function.Injective LieSubalgebra.toSubmodule
-- LieSubalgebra.coe_to_submodule_eq_iff
@[simp]
theorem coe_to_submodule_eq_iff (L₁' L₂' : LieSubalgebra R L) :
    (L₁' : Submodule R L) = (L₂' : Submodule R L) ↔ L₁' = L₂' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L₁' L₂' : LieSubalgebra R L)
Goal: L₁'.toSubmodule = L₂'.toSubmodule ↔ L₁' = L₂'
-- LieSubalgebra.coe_to_submodule
theorem coe_to_submodule : ((L' : Submodule R L) : Set L) = L' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L)
Goal: ↑L'.toSubmodule = ↑L'
-- LieSubalgebra.coe_bracket_of_module
@[simp]
theorem coe_bracket_of_module (x : L') (m : M) : ⁅x, m⁆ = ⁅(x : L), m⁆ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {M : Type w} [AddCommGroup M] [LieRingModule L M] (x : ↥L') (m : M)
Goal: ⁅x, m⁆ = ⁅↑x, m⁆
-- LieModuleHom.coe_restrictLie
@[simp]
theorem _root_.LieModuleHom.coe_restrictLie (f : M →ₗ⁅R,L⁆ N) : ⇑(f.restrictLie L') = f 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L) {M : Type w} [AddCommGroup M] [LieRingModule L M] {N : Type w₁} [AddCommGroup N] [LieRingModule L N] [Module R N] [Module R M] (f : M →ₗ⁅R,L⁆ N)
Goal: ⇑(f.restrictLie L') = ⇑f
-- LieSubalgebra.coe_incl
@[simp]
theorem coe_incl : ⇑L'.incl = ((↑) : L' → L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L)
Goal: ⇑L'.incl = Subtype.val
-- LieSubalgebra.coe_incl'
@[simp]
theorem coe_incl' : ⇑L'.incl' = ((↑) : L' → L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (L' : LieSubalgebra R L)
Goal: ⇑L'.incl' = Subtype.val
-- LieHom.range_coe
@[simp]
theorem range_coe : (f.range : Set L₂) = Set.range f 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂)
Goal: ↑f.range = Set.range ⇑f
-- LieHom.mem_range
@[simp]
theorem mem_range (x : L₂) : x ∈ f.range ↔ ∃ y : L, f y = x 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L₂)
Goal: x ∈ f.range ↔ ∃ y, f y = x
-- LieHom.mem_range_self
theorem mem_range_self (x : L) : f x ∈ f.range 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L)
Goal: f x ∈ f.range
-- LieHom.rangeRestrict_apply
@[simp]
theorem rangeRestrict_apply (x : L) : f.rangeRestrict x = ⟨f x, f.mem_range_self x⟩ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (x : L)
Goal: f.rangeRestrict x = ⟨f x, ⋯⟩
-- LieHom.surjective_rangeRestrict
theorem surjective_rangeRestrict : Function.Surjective f.rangeRestrict 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂)
Goal: Function.Surjective ⇑f.rangeRestrict
-- LieHom.equivRangeOfInjective_apply
@[simp]
theorem equivRangeOfInjective_apply (h : Function.Injective f) (x : L) :
    f.equivRangeOfInjective h x = ⟨f x, mem_range_self f x⟩ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (h : Function.Injective ⇑f) (x : L)
Goal: (f.equivRangeOfInjective h) x = ⟨f x, ⋯⟩
-- Submodule.exists_lieSubalgebra_coe_eq_iff
theorem Submodule.exists_lieSubalgebra_coe_eq_iff (p : Submodule R L) :
    (∃ K : LieSubalgebra R L, ↑K = p) ↔ ∀ x y : L, x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (p : Submodule R L)
Goal: (∃ K, K.toSubmodule = p) ↔ ∀ (x y : L), x ∈ p → y ∈ p → ⁅x, y⁆ ∈ p
-- LieSubalgebra.incl_range
@[simp]
theorem incl_range : K.incl.range = K 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K : LieSubalgebra R L)
Goal: K.incl.range = K
-- LieSubalgebra.mem_map
@[simp]
theorem mem_map (x : L₂) : x ∈ K.map f ↔ ∃ y : L, y ∈ K ∧ f y = x 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂) (K : LieSubalgebra R L) (x : L₂)
Goal: x ∈ LieSubalgebra.map f K ↔ ∃ y ∈ K, f y = x
-- LieSubalgebra.mem_map_submodule
theorem mem_map_submodule (e : L ≃ₗ⁅R⁆ L₂) (x : L₂) :
    x ∈ K.map (e : L →ₗ⁅R⁆ L₂) ↔ x ∈ (K : Submodule R L).map (e : L →ₗ[R] L₂) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (K : LieSubalgebra R L) (e : L ≃ₗ⁅R⁆ L₂) (x : L₂)
Goal: x ∈ LieSubalgebra.map e.toLieHom K ↔ x ∈ Submodule.map (↑e.toLinearEquiv) K.toSubmodule
-- LieSubalgebra.le_def
theorem le_def : K ≤ K' ↔ (K : Set L) ⊆ K' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K K' : LieSubalgebra R L)
Goal: K ≤ K' ↔ ↑K ⊆ ↑K'
-- LieSubalgebra.coe_submodule_le_coe_submodule
@[simp]
theorem coe_submodule_le_coe_submodule : (K : Submodule R L) ≤ K' ↔ K ≤ K' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K K' : LieSubalgebra R L)
Goal: K.toSubmodule ≤ K'.toSubmodule ↔ K ≤ K'
-- LieSubalgebra.bot_coe
@[simp]
theorem bot_coe : ((⊥ : LieSubalgebra R L) : Set L) = {0} 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: ↑⊥ = {0}
-- LieSubalgebra.bot_coe_submodule
@[simp]
theorem bot_coe_submodule : ((⊥ : LieSubalgebra R L) : Submodule R L) = ⊥ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: ⊥.toSubmodule = ⊥
-- LieSubalgebra.mem_bot
@[simp]
theorem mem_bot (x : L) : x ∈ (⊥ : LieSubalgebra R L) ↔ x = 0 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L)
Goal: x ∈ ⊥ ↔ x = 0
-- LieSubalgebra.top_coe
@[simp]
theorem top_coe : ((⊤ : LieSubalgebra R L) : Set L) = univ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: ↑⊤ = Set.univ
-- LieSubalgebra.top_coe_submodule
@[simp]
theorem top_coe_submodule : ((⊤ : LieSubalgebra R L) : Submodule R L) = ⊤ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L]
Goal: ⊤.toSubmodule = ⊤
-- LieSubalgebra.mem_top
@[simp]
theorem mem_top (x : L) : x ∈ (⊤ : LieSubalgebra R L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (x : L)
Goal: x ∈ ⊤
-- LieHom.range_eq_map
theorem _root_.LieHom.range_eq_map : f.range = map f ⊤ 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂] (f : L →ₗ⁅R⁆ L₂)
Goal: f.range = LieSubalgebra.map f ⊤
-- LieSubalgebra.inf_coe
@[simp]
theorem inf_coe : (↑(K ⊓ K') : Set L) = (K : Set L) ∩ (K' : Set L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K K' : LieSubalgebra R L)
Goal: ↑(K ⊓ K') = ↑K ∩ ↑K'
-- LieSubalgebra.sInf_coe_to_submodule
@[simp]
theorem sInf_coe_to_submodule (S : Set (LieSubalgebra R L)) :
    (↑(sInf S) : Submodule R L) = sInf {(s : Submodule R L) | s ∈ S} 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set (LieSubalgebra R L))
Goal: (sInf S).toSubmodule = sInf {x | ∃ s ∈ S, s.toSubmodule = x}
-- LieSubalgebra.sInf_coe
@[simp]
theorem sInf_coe (S : Set (LieSubalgebra R L)) : (↑(sInf S) : Set L) = ⋂ s ∈ S, (s : Set L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set (LieSubalgebra R L))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- LieSubalgebra.sInf_glb
theorem sInf_glb (S : Set (LieSubalgebra R L)) : IsGLB S (sInf S) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (S : Set (LieSubalgebra R L))
Goal: IsGLB S (sInf S)
-- LieSubalgebra.add_eq_sup
@[simp]
theorem add_eq_sup : K + K' = K ⊔ K' 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K K' : LieSubalgebra R L)
Goal: K + K' = K ⊔ K'
-- LieSubalgebra.inf_coe_to_submodule
@[simp]
theorem inf_coe_to_submodule :
    (↑(K ⊓ K') : Submodule R L) = (K : Submodule R L) ⊓ (K' : Submodule R L) 
Arguments: {R : Type u} {L : Type v} [CommRing R] [LieRing L] [LieAlgebra R L] (K K' : LieSubalgebra R L)
Goal: (K ⊓ K').toSubmodule = K.toSubmodule ⊓ K'.toSubmodule