ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.prod_add_prod_le
lemma prod_add_prod_le {i : ι} {f g h : ι → R} (hi : i ∈ s) (h2i : g i + h i ≤ f i)
    (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j) (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) (hg : ∀ i ∈ s, 0 ≤ g i)
    (hh : ∀ i ∈ s, 0 ≤ h i) : ((∏ i ∈ s, g i) + ∏ i ∈ s, h i) ≤ ∏ i ∈ s, f i 
Arguments: {ι : Type u_1} {R : Type u_2} [OrderedCommSemiring R] {s : Finset ι} {i : ι} {f g h : ι → R} (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j) (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) (hg : ∀ i ∈ s, 0 ≤ g i) (hh : ∀ i ∈ s, 0 ≤ h i)
Goal: ∏ i ∈ s, g i + ∏ i ∈ s, h i ≤ ∏ i ∈ s, f i
-- Finset.sum_mul_sq_le_sq_mul_sq
lemma sum_mul_sq_le_sq_mul_sq (s : Finset ι) (f g : ι → R) :
    (∑ i ∈ s, f i * g i) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) * ∑ i ∈ s, g i ^ 2 
Arguments: {ι : Type u_1} {R : Type u_2} [LinearOrderedCommSemiring R] [ExistsAddOfLE R] (s : Finset ι) (f g : ι → R)
Goal: (∑ i ∈ s, f i * g i) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) * ∑ i ∈ s, g i ^ 2
-- Finset.abs_prod
lemma abs_prod [LinearOrderedCommRing R] (s : Finset ι) (f : ι → R) :
    |∏ x ∈ s, f x| = ∏ x ∈ s, |f x| 
Arguments: {ι : Type u_1} {R : Type u_2} [LinearOrderedCommRing R] (s : Finset ι) (f : ι → R)
Goal: |∏ x ∈ s, f x| = ∏ x ∈ s, |f x|
-- CanonicallyOrderedCommSemiring.prod_pos
@[simp] lemma _root_.CanonicallyOrderedCommSemiring.prod_pos [Nontrivial R] :
    0 < ∏ i ∈ s, f i ↔ (∀ i ∈ s, (0 : R) < f i) 
Arguments: {ι : Type u_1} {R : Type u_2} [CanonicallyOrderedCommSemiring R] {f : ι → R} {s : Finset ι} [Nontrivial R]
Goal: 0 < ∏ i ∈ s, f i ↔ ∀ i ∈ s, 0 < f i
-- Finset.prod_add_prod_le'
lemma prod_add_prod_le' (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j)
    (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j) : ((∏ i ∈ s, g i) + ∏ i ∈ s, h i) ≤ ∏ i ∈ s, f i 
Arguments: {ι : Type u_1} {R : Type u_2} [CanonicallyOrderedCommSemiring R] {f g h : ι → R} {s : Finset ι} {i : ι} (hi : i ∈ s) (h2i : g i + h i ≤ f i) (hgf : ∀ j ∈ s, j ≠ i → g j ≤ f j) (hhf : ∀ j ∈ s, j ≠ i → h j ≤ f j)
Goal: ∏ i ∈ s, g i + ∏ i ∈ s, h i ≤ ∏ i ∈ s, f i
-- AbsoluteValue.sum_le
lemma AbsoluteValue.sum_le [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S)
    (s : Finset ι) (f : ι → R) : abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i) 
Arguments: {ι : Type u_1} {R : Type u_2} {S : Type u_3} [Semiring R] [OrderedSemiring S] (abv : AbsoluteValue R S) (s : Finset ι) (f : ι → R)
Goal: abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i)
-- IsAbsoluteValue.abv_sum
lemma IsAbsoluteValue.abv_sum [Semiring R] [OrderedSemiring S] (abv : R → S) [IsAbsoluteValue abv]
    (f : ι → R) (s : Finset ι) : abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i) 
Arguments: {ι : Type u_1} {R : Type u_2} {S : Type u_3} [Semiring R] [OrderedSemiring S] (abv : R → S) [IsAbsoluteValue abv] (f : ι → R) (s : Finset ι)
Goal: abv (∑ i ∈ s, f i) ≤ ∑ i ∈ s, abv (f i)
-- AbsoluteValue.map_prod
nonrec lemma AbsoluteValue.map_prod [CommSemiring R] [Nontrivial R] [LinearOrderedCommRing S]
    (abv : AbsoluteValue R S) (f : ι → R) (s : Finset ι) :
    abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i) 
Arguments: {ι : Type u_1} {R : Type u_2} {S : Type u_3} [CommSemiring R] [Nontrivial R] [LinearOrderedCommRing S] (abv : AbsoluteValue R S) (f : ι → R) (s : Finset ι)
Goal: abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i)
-- IsAbsoluteValue.map_prod
lemma IsAbsoluteValue.map_prod [CommSemiring R] [Nontrivial R] [LinearOrderedCommRing S]
    (abv : R → S) [IsAbsoluteValue abv] (f : ι → R) (s : Finset ι) :
    abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i) 
Arguments: {ι : Type u_1} {R : Type u_2} {S : Type u_3} [CommSemiring R] [Nontrivial R] [LinearOrderedCommRing S] (abv : R → S) [IsAbsoluteValue abv] (f : ι → R) (s : Finset ι)
Goal: abv (∏ i ∈ s, f i) = ∏ i ∈ s, abv (f i)
-- CauSeq.Completion.mk_eq_mk
@[simp]
theorem mk_eq_mk (f : CauSeq _ abv) : @Eq (Cauchy abv) ⟦f⟧ (mk f) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv)
Goal: ⟦f⟧ = CauSeq.Completion.mk f
-- CauSeq.Completion.mk_eq
theorem mk_eq {f g : CauSeq _ abv} : mk f = mk g ↔ f ≈ g 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv}
Goal: CauSeq.Completion.mk f = CauSeq.Completion.mk g ↔ f ≈ g
-- CauSeq.Completion.ofRat_zero
theorem ofRat_zero : (ofRat 0 : Cauchy abv) = 0 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: CauSeq.Completion.ofRat 0 = 0
-- CauSeq.Completion.ofRat_one
theorem ofRat_one : (ofRat 1 : Cauchy abv) = 1 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: CauSeq.Completion.ofRat 1 = 1
-- CauSeq.Completion.mk_eq_zero
@[simp]
theorem mk_eq_zero {f : CauSeq _ abv} : mk f = 0 ↔ LimZero f 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv}
Goal: CauSeq.Completion.mk f = 0 ↔ f.LimZero
-- CauSeq.Completion.mk_add
@[simp]
theorem mk_add (f g : CauSeq β abv) : mk f + mk g = mk (f + g) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv)
Goal: CauSeq.Completion.mk f + CauSeq.Completion.mk g = CauSeq.Completion.mk (f + g)
-- CauSeq.Completion.mk_neg
@[simp]
theorem mk_neg (f : CauSeq β abv) : -mk f = mk (-f) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv)
Goal: -CauSeq.Completion.mk f = CauSeq.Completion.mk (-f)
-- CauSeq.Completion.mk_mul
@[simp]
theorem mk_mul (f g : CauSeq β abv) : mk f * mk g = mk (f * g) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv)
Goal: CauSeq.Completion.mk f * CauSeq.Completion.mk g = CauSeq.Completion.mk (f * g)
-- CauSeq.Completion.mk_sub
@[simp]
theorem mk_sub (f g : CauSeq β abv) : mk f - mk g = mk (f - g) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv)
Goal: CauSeq.Completion.mk f - CauSeq.Completion.mk g = CauSeq.Completion.mk (f - g)
-- CauSeq.Completion.mk_smul
@[simp]
theorem mk_smul {γ : Type*} [SMul γ β] [IsScalarTower γ β β] (c : γ) (f : CauSeq β abv) :
    c • mk f = mk (c • f) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] {γ : Type u_3} [SMul γ β] [IsScalarTower γ β β] (c : γ) (f : CauSeq β abv)
Goal: c • CauSeq.Completion.mk f = CauSeq.Completion.mk (c • f)
-- CauSeq.Completion.mk_pow
@[simp]
theorem mk_pow (n : ℕ) (f : CauSeq β abv) : mk f ^ n = mk (f ^ n) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (n : ℕ) (f : CauSeq β abv)
Goal: CauSeq.Completion.mk f ^ n = CauSeq.Completion.mk (f ^ n)
-- CauSeq.Completion.ofRat_natCast
@[simp]
theorem ofRat_natCast (n : ℕ) : (ofRat n : Cauchy abv) = n 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (n : ℕ)
Goal: CauSeq.Completion.ofRat ↑n = ↑n
-- CauSeq.Completion.ofRat_intCast
@[simp]
theorem ofRat_intCast (z : ℤ) : (ofRat z : Cauchy abv) = z 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (z : ℤ)
Goal: CauSeq.Completion.ofRat ↑z = ↑z
-- CauSeq.Completion.ofRat_add
theorem ofRat_add (x y : β) :
    ofRat (x + y) = (ofRat x + ofRat y : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.Completion.ofRat (x + y) = CauSeq.Completion.ofRat x + CauSeq.Completion.ofRat y
-- CauSeq.Completion.ofRat_neg
theorem ofRat_neg (x : β) : ofRat (-x) = (-ofRat x : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β)
Goal: CauSeq.Completion.ofRat (-x) = -CauSeq.Completion.ofRat x
-- CauSeq.Completion.ofRat_mul
theorem ofRat_mul (x y : β) :
    ofRat (x * y) = (ofRat x * ofRat y : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.Completion.ofRat (x * y) = CauSeq.Completion.ofRat x * CauSeq.Completion.ofRat y
-- _private.Mathlib.Algebra.Order.CauSeq.Completion.0.CauSeq.Completion.zero_def
private theorem zero_def : 0 = mk (abv := abv) 0 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: 0 = CauSeq.Completion.mk 0
-- _private.Mathlib.Algebra.Order.CauSeq.Completion.0.CauSeq.Completion.one_def
private theorem one_def : 1 = mk (abv := abv) 1 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: 1 = CauSeq.Completion.mk 1
-- CauSeq.Completion.ofRatRingHom_apply
@[simps]
def ofRatRingHom : β →+* (Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β)
Goal: CauSeq.Completion.ofRatRingHom x = CauSeq.Completion.ofRat x
-- CauSeq.Completion.ofRat_sub
theorem ofRat_sub (x y : β) : ofRat (x - y) = (ofRat x - ofRat y : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.Completion.ofRat (x - y) = CauSeq.Completion.ofRat x - CauSeq.Completion.ofRat y
-- CauSeq.Completion.ofRat_nnratCast
@[simp, norm_cast] lemma ofRat_nnratCast (q : ℚ≥0) : ofRat (q : β) = (q : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (q : ℚ≥0)
Goal: CauSeq.Completion.ofRat ↑q = ↑q
-- CauSeq.Completion.ofRat_ratCast
@[simp, norm_cast] lemma ofRat_ratCast (q : ℚ) : ofRat (q : β) = (q : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (q : ℚ)
Goal: CauSeq.Completion.ofRat ↑q = ↑q
-- CauSeq.Completion.inv_zero
theorem inv_zero : (0 : (Cauchy abv))⁻¹ = 0 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv]
Goal: 0⁻¹ = 0
-- CauSeq.Completion.inv_mk
@[simp]
theorem inv_mk {f} (hf) : (mk (abv := abv) f)⁻¹ = mk (inv f hf) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero)
Goal: (CauSeq.Completion.mk f)⁻¹ = CauSeq.Completion.mk (f.inv hf)
-- CauSeq.Completion.cau_seq_zero_ne_one
theorem cau_seq_zero_ne_one : ¬(0 : CauSeq _ abv) ≈ 1 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv]
Goal: ¬0 ≈ 1
-- CauSeq.Completion.zero_ne_one
theorem zero_ne_one : (0 : (Cauchy abv)) ≠ 1 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv]
Goal: 0 ≠ 1
-- CauSeq.Completion.inv_mul_cancel
protected theorem inv_mul_cancel {x : (Cauchy abv)} : x ≠ 0 → x⁻¹ * x = 1 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {x : CauSeq.Completion.Cauchy abv}
Goal: x ≠ 0 → x⁻¹ * x = 1
-- CauSeq.Completion.mul_inv_cancel
protected theorem mul_inv_cancel {x : (Cauchy abv)} : x ≠ 0 → x * x⁻¹ = 1 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {x : CauSeq.Completion.Cauchy abv}
Goal: x ≠ 0 → x * x⁻¹ = 1
-- CauSeq.Completion.ofRat_inv
theorem ofRat_inv (x : β) : ofRat x⁻¹ = ((ofRat x)⁻¹ : (Cauchy abv)) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (x : β)
Goal: CauSeq.Completion.ofRat x⁻¹ = (CauSeq.Completion.ofRat x)⁻¹
-- CauSeq.Completion.ofRat_div
lemma ofRat_div (x y : β) : ofRat (x / y) = (ofRat x / ofRat y : Cauchy abv) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.Completion.ofRat (x / y) = CauSeq.Completion.ofRat x / CauSeq.Completion.ofRat y
-- CauSeq.IsComplete.isComplete
class IsComplete : Prop where
    isComplete : ∀ s : CauSeq β abv, ∃ b : β, s ≈ const abv b
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [self : CauSeq.IsComplete β abv] (s : CauSeq β abv)
Goal: ∃ b, s ≈ CauSeq.const abv b
-- CauSeq.complete
theorem complete : ∀ s : CauSeq β abv, ∃ b : β, s ≈ const abv b 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (s : CauSeq β abv)
Goal: ∃ b, s ≈ CauSeq.const abv b
-- CauSeq.equiv_lim
theorem equiv_lim (s : CauSeq β abv) : s ≈ const abv (lim s) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (s : CauSeq β abv)
Goal: s ≈ CauSeq.const abv s.lim
-- CauSeq.eq_lim_of_const_equiv
theorem eq_lim_of_const_equiv {f : CauSeq β abv} {x : β} (h : CauSeq.const abv x ≈ f) : x = lim f 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] {f : CauSeq β abv} {x : β} (h : CauSeq.const abv x ≈ f)
Goal: x = f.lim
-- CauSeq.lim_eq_of_equiv_const
theorem lim_eq_of_equiv_const {f : CauSeq β abv} {x : β} (h : f ≈ CauSeq.const abv x) : lim f = x 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] {f : CauSeq β abv} {x : β} (h : f ≈ CauSeq.const abv x)
Goal: f.lim = x
-- CauSeq.lim_eq_lim_of_equiv
theorem lim_eq_lim_of_equiv {f g : CauSeq β abv} (h : f ≈ g) : lim f = lim g 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] {f g : CauSeq β abv} (h : f ≈ g)
Goal: f.lim = g.lim
-- CauSeq.lim_const
@[simp]
theorem lim_const (x : β) : lim (const abv x) = x 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (x : β)
Goal: (CauSeq.const abv x).lim = x
-- CauSeq.lim_add
theorem lim_add (f g : CauSeq β abv) : lim f + lim g = lim (f + g) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (f g : CauSeq β abv)
Goal: f.lim + g.lim = (f + g).lim
-- CauSeq.lim_mul_lim
theorem lim_mul_lim (f g : CauSeq β abv) : lim f * lim g = lim (f * g) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (f g : CauSeq β abv)
Goal: f.lim * g.lim = (f * g).lim
-- CauSeq.lim_mul
theorem lim_mul (f : CauSeq β abv) (x : β) : lim f * x = lim (f * const abv x) 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (f : CauSeq β abv) (x : β)
Goal: f.lim * x = (f * CauSeq.const abv x).lim
-- CauSeq.lim_neg
theorem lim_neg (f : CauSeq β abv) : lim (-f) = -lim f 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (f : CauSeq β abv)
Goal: (-f).lim = -f.lim
-- CauSeq.lim_eq_zero_iff
theorem lim_eq_zero_iff (f : CauSeq β abv) : lim f = 0 ↔ LimZero f 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Ring β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] (f : CauSeq β abv)
Goal: f.lim = 0 ↔ f.LimZero
-- CauSeq.lim_inv
theorem lim_inv {f : CauSeq β abv} (hf : ¬LimZero f) : lim (inv f hf) = (lim f)⁻¹ 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_2} [Field β] {abv : β → α} [IsAbsoluteValue abv] [CauSeq.IsComplete β abv] {f : CauSeq β abv} (hf : ¬f.LimZero)
Goal: (f.inv hf).lim = f.lim⁻¹
-- CauSeq.lim_le
theorem lim_le {f : CauSeq α abs} {x : α} (h : f ≤ CauSeq.const abs x) : lim f ≤ x 
Arguments: {α : Type u_1} [LinearOrderedField α] [CauSeq.IsComplete α abs] {f : CauSeq α abs} {x : α} (h : f ≤ CauSeq.const abs x)
Goal: f.lim ≤ x
-- CauSeq.le_lim
theorem le_lim {f : CauSeq α abs} {x : α} (h : CauSeq.const abs x ≤ f) : x ≤ lim f 
Arguments: {α : Type u_1} [LinearOrderedField α] [CauSeq.IsComplete α abs] {f : CauSeq α abs} {x : α} (h : CauSeq.const abs x ≤ f)
Goal: x ≤ f.lim
-- CauSeq.lt_lim
theorem lt_lim {f : CauSeq α abs} {x : α} (h : CauSeq.const abs x < f) : x < lim f 
Arguments: {α : Type u_1} [LinearOrderedField α] [CauSeq.IsComplete α abs] {f : CauSeq α abs} {x : α} (h : CauSeq.const abs x < f)
Goal: x < f.lim
-- CauSeq.lim_lt
theorem lim_lt {f : CauSeq α abs} {x : α} (h : f < CauSeq.const abs x) : lim f < x 
Arguments: {α : Type u_1} [LinearOrderedField α] [CauSeq.IsComplete α abs] {f : CauSeq α abs} {x : α} (h : f < CauSeq.const abs x)
Goal: f.lim < x
-- _private.Mathlib.Algebra.Lie.CartanMatrix.0.CartanMatrix.Relations.adE_of_eq_eq_zero
private theorem adE_of_eq_eq_zero (i : B) (h : A i i = 2) : adE R A ⟨i, i⟩ = 0 
Arguments: (R : Type u) {B : Type v} [CommRing R] (A : Matrix B B ℤ) (i : B) (h : A i i = 2)
Goal: CartanMatrix.Relations.adE R A (i, i) = 0
-- _private.Mathlib.Algebra.Lie.CartanMatrix.0.CartanMatrix.Relations.adF_of_eq_eq_zero
private theorem adF_of_eq_eq_zero (i : B) (h : A i i = 2) : adF R A ⟨i, i⟩ = 0 
Arguments: (R : Type u) {B : Type v} [CommRing R] (A : Matrix B B ℤ) (i : B) (h : A i i = 2)
Goal: CartanMatrix.Relations.adF R A (i, i) = 0
-- lieEquivMatrix'_apply
@[simp]
theorem lieEquivMatrix'_apply (f : Module.End R (n → R)) :
    lieEquivMatrix' f = LinearMap.toMatrix' f 
Arguments: {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (f : Module.End R (n → R))
Goal: lieEquivMatrix' f = LinearMap.toMatrix' f
-- lieEquivMatrix'_symm_apply
@[simp]
theorem lieEquivMatrix'_symm_apply (A : Matrix n n R) :
    (@lieEquivMatrix' R _ n _ _).symm A = Matrix.toLin' A 
Arguments: {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (A : Matrix n n R)
Goal: lieEquivMatrix'.symm A = Matrix.toLin' A
-- Matrix.lieConj_apply
@[simp]
theorem Matrix.lieConj_apply (P A : Matrix n n R) (h : Invertible P) :
    P.lieConj h A = P * A * P⁻¹ 
Arguments: {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P A : Matrix n n R) (h : Invertible P)
Goal: (P.lieConj h) A = P * A * P⁻¹
-- Matrix.lieConj_symm_apply
@[simp]
theorem Matrix.lieConj_symm_apply (P A : Matrix n n R) (h : Invertible P) :
    (P.lieConj h).symm A = P⁻¹ * A * P 
Arguments: {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] (P A : Matrix n n R) (h : Invertible P)
Goal: (P.lieConj h).symm A = P⁻¹ * A * P
-- Matrix.reindexLieEquiv_apply
@[simp]
theorem Matrix.reindexLieEquiv_apply (M : Matrix n n R) :
    Matrix.reindexLieEquiv e M = Matrix.reindex e e M 
Arguments: {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] {m : Type w₁} [DecidableEq m] [Fintype m] (e : n ≃ m) (M : Matrix n n R)
Goal: (Matrix.reindexLieEquiv e) M = (Matrix.reindex e e) M
-- Matrix.reindexLieEquiv_symm
@[simp]
theorem Matrix.reindexLieEquiv_symm :
    (Matrix.reindexLieEquiv e : _ ≃ₗ⁅R⁆ _).symm = Matrix.reindexLieEquiv e.symm 
Arguments: {R : Type u} [CommRing R] {n : Type w} [DecidableEq n] [Fintype n] {m : Type w₁} [DecidableEq m] [Fintype m] (e : n ≃ m)
Goal: (Matrix.reindexLieEquiv e).symm = Matrix.reindexLieEquiv e.symm
-- Archimedean.arch
class Archimedean (α) [OrderedAddCommMonoid α] : Prop where
    arch : ∀ (x : α) {y : α}, 0 < y → ∃ n : ℕ, x ≤ n • y
Arguments: {α : Type u_2} [OrderedAddCommMonoid α] [self : Archimedean α] (x : α) {y : α}
Goal: 0 < y → ∃ n, x ≤ n • y
-- exists_lt_nsmul
theorem exists_lt_nsmul [OrderedAddCommMonoid M] [Archimedean M]
    [CovariantClass M M (· + ·) (· < ·)] {a : M} (ha : 0 < a) (b : M) :
    ∃ n : ℕ, b < n • a 
Arguments: {M : Type u_2} [OrderedAddCommMonoid M] [Archimedean M] [CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] {a : M} (ha : 0 < a) (b : M)
Goal: ∃ n, b < n • a
-- existsUnique_zsmul_near_of_pos
theorem existsUnique_zsmul_near_of_pos {a : α} (ha : 0 < a) (g : α) :
    ∃! k : ℤ, k • a ≤ g ∧ g < (k + 1) • a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {a : α} (ha : 0 < a) (g : α)
Goal: ∃! k, k • a ≤ g ∧ g < (k + 1) • a
-- existsUnique_zsmul_near_of_pos'
theorem existsUnique_zsmul_near_of_pos' {a : α} (ha : 0 < a) (g : α) :
    ∃! k : ℤ, 0 ≤ g - k • a ∧ g - k • a < a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {a : α} (ha : 0 < a) (g : α)
Goal: ∃! k, 0 ≤ g - k • a ∧ g - k • a < a
-- existsUnique_sub_zsmul_mem_Ico
theorem existsUnique_sub_zsmul_mem_Ico {a : α} (ha : 0 < a) (b c : α) :
    ∃! m : ℤ, b - m • a ∈ Set.Ico c (c + a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {a : α} (ha : 0 < a) (b c : α)
Goal: ∃! m, b - m • a ∈ Set.Ico c (c + a)
-- existsUnique_add_zsmul_mem_Ico
theorem existsUnique_add_zsmul_mem_Ico {a : α} (ha : 0 < a) (b c : α) :
    ∃! m : ℤ, b + m • a ∈ Set.Ico c (c + a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {a : α} (ha : 0 < a) (b c : α)
Goal: ∃! m, b + m • a ∈ Set.Ico c (c + a)
-- existsUnique_add_zsmul_mem_Ioc
theorem existsUnique_add_zsmul_mem_Ioc {a : α} (ha : 0 < a) (b c : α) :
    ∃! m : ℤ, b + m • a ∈ Set.Ioc c (c + a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {a : α} (ha : 0 < a) (b c : α)
Goal: ∃! m, b + m • a ∈ Set.Ioc c (c + a)
-- existsUnique_sub_zsmul_mem_Ioc
theorem existsUnique_sub_zsmul_mem_Ioc {a : α} (ha : 0 < a) (b c : α) :
    ∃! m : ℤ, b - m • a ∈ Set.Ioc c (c + a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {a : α} (ha : 0 < a) (b c : α)
Goal: ∃! m, b - m • a ∈ Set.Ioc c (c + a)
-- exists_nat_ge
theorem exists_nat_ge [OrderedSemiring α] [Archimedean α] (x : α) : ∃ n : ℕ, x ≤ n 
Arguments: {α : Type u_1} [OrderedSemiring α] [Archimedean α] (x : α)
Goal: ∃ n, x ≤ ↑n
-- exists_nat_gt
lemma exists_nat_gt (x : α) : ∃ n : ℕ, x < n 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] [Archimedean α] (x : α)
Goal: ∃ n, x < ↑n
-- add_one_pow_unbounded_of_pos
theorem add_one_pow_unbounded_of_pos (x : α) (hy : 0 < y) : ∃ n : ℕ, x < (y + 1) ^ n 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] [Archimedean α] {y : α} (x : α) (hy : 0 < y)
Goal: ∃ n, x < (y + 1) ^ n
-- pow_unbounded_of_one_lt
lemma pow_unbounded_of_one_lt [ExistsAddOfLE α] (x : α) (hy1 : 1 < y) : ∃ n : ℕ, x < y ^ n 
Arguments: {α : Type u_1} [StrictOrderedSemiring α] [Archimedean α] {y : α} [ExistsAddOfLE α] (x : α) (hy1 : 1 < y)
Goal: ∃ n, x < y ^ n
-- exists_int_gt
theorem exists_int_gt (x : α) : ∃ n : ℤ, x < n 
Arguments: {α : Type u_1} [StrictOrderedRing α] [Archimedean α] (x : α)
Goal: ∃ n, x < ↑n
-- exists_int_lt
theorem exists_int_lt (x : α) : ∃ n : ℤ, (n : α) < x 
Arguments: {α : Type u_1} [StrictOrderedRing α] [Archimedean α] (x : α)
Goal: ∃ n, ↑n < x
-- exists_floor
theorem exists_floor (x : α) : ∃ fl : ℤ, ∀ z : ℤ, z ≤ fl ↔ (z : α) ≤ x 
Arguments: {α : Type u_1} [StrictOrderedRing α] [Archimedean α] (x : α)
Goal: ∃ fl, ∀ (z : ℤ), z ≤ fl ↔ ↑z ≤ x
-- exists_nat_pow_near
theorem exists_nat_pow_near (hx : 1 ≤ x) (hy : 1 < y) : ∃ n : ℕ, y ^ n ≤ x ∧ x < y ^ (n + 1) 
Arguments: {α : Type u_1} [LinearOrderedSemiring α] [Archimedean α] [ExistsAddOfLE α] {x y : α} (hx : 1 ≤ x) (hy : 1 < y)
Goal: ∃ n, y ^ n ≤ x ∧ x < y ^ (n + 1)
-- exists_mem_Ico_zpow
theorem exists_mem_Ico_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ico (y ^ n) (y ^ (n + 1)) 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] [Archimedean α] [ExistsAddOfLE α] {x y : α} (hx : 0 < x) (hy : 1 < y)
Goal: ∃ n, x ∈ Set.Ico (y ^ n) (y ^ (n + 1))
-- exists_mem_Ioc_zpow
theorem exists_mem_Ioc_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ioc (y ^ n) (y ^ (n + 1)) 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] [Archimedean α] [ExistsAddOfLE α] {x y : α} (hx : 0 < x) (hy : 1 < y)
Goal: ∃ n, x ∈ Set.Ioc (y ^ n) (y ^ (n + 1))
-- exists_pow_lt_of_lt_one
theorem exists_pow_lt_of_lt_one (hx : 0 < x) (hy : y < 1) : ∃ n : ℕ, y ^ n < x 
Arguments: {α : Type u_1} [LinearOrderedSemifield α] [Archimedean α] [ExistsAddOfLE α] {x y : α} (hx : 0 < x) (hy : y < 1)
Goal: ∃ n, y ^ n < x