ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LieModuleEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (e : M ≃ₗ⁅R,L⁆ N) : ∀ x, e.symm (e x) = x 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N) (x : M)
Goal: e.symm (e x) = x
-- LieModuleEquiv.apply_eq_iff_eq_symm_apply
theorem apply_eq_iff_eq_symm_apply {m : M} {n : N} (e : M ≃ₗ⁅R,L⁆ N) :
    e m = n ↔ m = e.symm n 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] {m : M} {n : N} (e : M ≃ₗ⁅R,L⁆ N)
Goal: e m = n ↔ m = e.symm n
-- LieModuleEquiv.symm_symm
@[simp]
theorem symm_symm (e : M ≃ₗ⁅R,L⁆ N) : e.symm.symm = e 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N)
Goal: e.symm.symm = e
-- LieModuleEquiv.symm_bijective
theorem symm_bijective :
    Function.Bijective (LieModuleEquiv.symm : (M ≃ₗ⁅R,L⁆ N) → N ≃ₗ⁅R,L⁆ M) 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N]
Goal: Function.Bijective LieModuleEquiv.symm
-- LieModuleEquiv.trans_apply
@[simp]
theorem trans_apply (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) (m : M) : (e₁.trans e₂) m = e₂ (e₁ m) 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) (m : M)
Goal: (e₁.trans e₂) m = e₂ (e₁ m)
-- LieModuleEquiv.symm_trans
@[simp]
theorem symm_trans (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P) :
    (e₁.trans e₂).symm = e₂.symm.trans e₁.symm 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} {P : Type w₂} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] [LieRingModule L M] [LieRingModule L N] [LieRingModule L P] (e₁ : M ≃ₗ⁅R,L⁆ N) (e₂ : N ≃ₗ⁅R,L⁆ P)
Goal: (e₁.trans e₂).symm = e₂.symm.trans e₁.symm
-- LieModuleEquiv.self_trans_symm
@[simp]
theorem self_trans_symm (e : M ≃ₗ⁅R,L⁆ N) : e.trans e.symm = refl 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N)
Goal: e.trans e.symm = LieModuleEquiv.refl
-- LieModuleEquiv.symm_trans_self
@[simp]
theorem symm_trans_self (e : M ≃ₗ⁅R,L⁆ N) : e.symm.trans e = refl 
Arguments: {R : Type u} {L : Type v} {M : Type w} {N : Type w₁} [CommRing R] [LieRing L] [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [LieRingModule L M] [LieRingModule L N] (e : M ≃ₗ⁅R,L⁆ N)
Goal: e.symm.trans e = LieModuleEquiv.refl
-- rat_add_continuous_lemma
theorem rat_add_continuous_lemma {ε : α} (ε0 : 0 < ε) :
    ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ →
      abv (a₁ + a₂ - (b₁ + b₂)) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] (abv : β → α) [IsAbsoluteValue abv] {ε : α} (ε0 : 0 < ε)
Goal: ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ + a₂ - (b₁ + b₂)) < ε
-- rat_mul_continuous_lemma
theorem rat_mul_continuous_lemma {ε K₁ K₂ : α} (ε0 : 0 < ε) :
    ∃ δ > 0, ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ →
      abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] (abv : β → α) [IsAbsoluteValue abv] {ε K₁ K₂ : α} (ε0 : 0 < ε)
Goal: ∃ δ > 0,
  ∀ {a₁ a₂ b₁ b₂ : β}, abv a₁ < K₁ → abv b₂ < K₂ → abv (a₁ - b₁) < δ → abv (a₂ - b₂) < δ → abv (a₁ * a₂ - b₁ * b₂) < ε
-- rat_inv_continuous_lemma
theorem rat_inv_continuous_lemma {β : Type*} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv]
    {ε K : α} (ε0 : 0 < ε) (K0 : 0 < K) :
    ∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε 
Arguments: {α : Type u_1} [LinearOrderedField α] {β : Type u_3} [DivisionRing β] (abv : β → α) [IsAbsoluteValue abv] {ε K : α} (ε0 : 0 < ε) (K0 : 0 < K)
Goal: ∃ δ > 0, ∀ {a b : β}, K ≤ abv a → K ≤ abv b → abv (a - b) < δ → abv (a⁻¹ - b⁻¹) < ε
-- IsCauSeq.cauchy₂
theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
    ∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε)
Goal: ∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε
-- IsCauSeq.cauchy₃
theorem cauchy₃ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
    ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - f j) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε)
Goal: ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - f j) < ε
-- IsCauSeq.bounded
lemma bounded (hf : IsCauSeq abv f) : ∃ r, ∀ i, abv (f i) < r 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f)
Goal: ∃ r, ∀ (i : ℕ), abv (f i) < r
-- IsCauSeq.bounded'
lemma bounded' (hf : IsCauSeq abv f) (x : α) : ∃ r > x, ∀ i, abv (f i) < r 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β} (hf : IsCauSeq abv f) (x : α)
Goal: ∃ r > x, ∀ (i : ℕ), abv (f i) < r
-- IsCauSeq.const
lemma const (x : β) : IsCauSeq abv fun _ ↦ x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β)
Goal: IsCauSeq abv fun x_1 => x
-- IsCauSeq.add
theorem add (hf : IsCauSeq abv f) (hg : IsCauSeq abv g) : IsCauSeq abv (f + g) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : ℕ → β} (hf : IsCauSeq abv f) (hg : IsCauSeq abv g)
Goal: IsCauSeq abv (f + g)
-- IsCauSeq.mul
lemma mul (hf : IsCauSeq abv f) (hg : IsCauSeq abv g) : IsCauSeq abv (f * g) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : ℕ → β} (hf : IsCauSeq abv f) (hg : IsCauSeq abv g)
Goal: IsCauSeq abv (f * g)
-- isCauSeq_neg
@[simp] lemma _root_.isCauSeq_neg : IsCauSeq abv (-f) ↔ IsCauSeq abv f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : ℕ → β}
Goal: IsCauSeq abv (-f) ↔ IsCauSeq abv f
-- CauSeq.ext
@[ext]
theorem ext {f g : CauSeq β abv} (h : ∀ i, f i = g i) : f = g 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} {f g : CauSeq β abv} (h : ∀ (i : ℕ), ↑f i = ↑g i)
Goal: f = g
-- CauSeq.isCauSeq
theorem isCauSeq (f : CauSeq β abv) : IsCauSeq abv f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} (f : CauSeq β abv)
Goal: IsCauSeq abv ↑f
-- CauSeq.cauchy
theorem cauchy (f : CauSeq β abv) : ∀ {ε}, 0 < ε → ∃ i, ∀ j ≥ i, abv (f j - f i) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} (f : CauSeq β abv) {ε : α}
Goal: 0 < ε → ∃ i, ∀ j ≥ i, abv (↑f j - ↑f i) < ε
-- CauSeq.cauchy₂
theorem cauchy₂ (f : CauSeq β abv) {ε} :
    0 < ε → ∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) {ε : α}
Goal: 0 < ε → ∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (↑f j - ↑f k) < ε
-- CauSeq.cauchy₃
theorem cauchy₃ (f : CauSeq β abv) {ε} : 0 < ε → ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - f j) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) {ε : α}
Goal: 0 < ε → ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑f j) < ε
-- CauSeq.bounded
theorem bounded (f : CauSeq β abv) : ∃ r, ∀ i, abv (f i) < r 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv)
Goal: ∃ r, ∀ (i : ℕ), abv (↑f i) < r
-- CauSeq.bounded'
theorem bounded' (f : CauSeq β abv) (x : α) : ∃ r > x, ∀ i, abv (f i) < r 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (x : α)
Goal: ∃ r > x, ∀ (i : ℕ), abv (↑f i) < r
-- CauSeq.coe_add
@[simp, norm_cast]
theorem coe_add (f g : CauSeq β abv) : ⇑(f + g) = (f : ℕ → β) + g 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv)
Goal: ↑(f + g) = ↑f + ↑g
-- CauSeq.add_apply
@[simp, norm_cast]
theorem add_apply (f g : CauSeq β abv) (i : ℕ) : (f + g) i = f i + g i 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv) (i : ℕ)
Goal: ↑(f + g) i = ↑f i + ↑g i
-- CauSeq.coe_const
@[simp, norm_cast]
theorem coe_const (x : β) : (const x : ℕ → β) = Function.const ℕ x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β)
Goal: ↑(CauSeq.const abv x) = Function.const ℕ x
-- CauSeq.const_apply
@[simp, norm_cast]
theorem const_apply (x : β) (i : ℕ) : (const x : ℕ → β) i = x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (i : ℕ)
Goal: ↑(CauSeq.const abv x) i = x
-- CauSeq.const_inj
theorem const_inj {x y : β} : (const x : CauSeq β abv) = const y ↔ x = y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {x y : β}
Goal: CauSeq.const abv x = CauSeq.const abv y ↔ x = y
-- CauSeq.coe_zero
@[simp, norm_cast]
theorem coe_zero : ⇑(0 : CauSeq β abv) = 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: ↑0 = 0
-- CauSeq.coe_one
@[simp, norm_cast]
theorem coe_one : ⇑(1 : CauSeq β abv) = 1 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: ↑1 = 1
-- CauSeq.zero_apply
@[simp, norm_cast]
theorem zero_apply (i) : (0 : CauSeq β abv) i = 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (i : ℕ)
Goal: ↑0 i = 0
-- CauSeq.one_apply
@[simp, norm_cast]
theorem one_apply (i) : (1 : CauSeq β abv) i = 1 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (i : ℕ)
Goal: ↑1 i = 1
-- CauSeq.const_zero
@[simp]
theorem const_zero : const 0 = 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: CauSeq.const abv 0 = 0
-- CauSeq.const_one
@[simp]
theorem const_one : const 1 = 1 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: CauSeq.const abv 1 = 1
-- CauSeq.const_add
theorem const_add (x y : β) : const (x + y) = const x + const y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.const abv (x + y) = CauSeq.const abv x + CauSeq.const abv y
-- CauSeq.coe_mul
@[simp, norm_cast]
theorem coe_mul (f g : CauSeq β abv) : ⇑(f * g) = (f : ℕ → β) * g 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv)
Goal: ↑(f * g) = ↑f * ↑g
-- CauSeq.mul_apply
@[simp, norm_cast]
theorem mul_apply (f g : CauSeq β abv) (i : ℕ) : (f * g) i = f i * g i 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv) (i : ℕ)
Goal: ↑(f * g) i = ↑f i * ↑g i
-- CauSeq.const_mul
theorem const_mul (x y : β) : const (x * y) = const x * const y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.const abv (x * y) = CauSeq.const abv x * CauSeq.const abv y
-- CauSeq.coe_neg
@[simp, norm_cast]
theorem coe_neg (f : CauSeq β abv) : ⇑(-f) = -f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv)
Goal: ↑(-f) = -↑f
-- CauSeq.neg_apply
@[simp, norm_cast]
theorem neg_apply (f : CauSeq β abv) (i) : (-f) i = -f i 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (i : ℕ)
Goal: ↑(-f) i = -↑f i
-- CauSeq.const_neg
theorem const_neg (x : β) : const (-x) = -const x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β)
Goal: CauSeq.const abv (-x) = -CauSeq.const abv x
-- CauSeq.coe_sub
@[simp, norm_cast]
theorem coe_sub (f g : CauSeq β abv) : ⇑(f - g) = (f : ℕ → β) - g 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv)
Goal: ↑(f - g) = ↑f - ↑g
-- CauSeq.sub_apply
@[simp, norm_cast]
theorem sub_apply (f g : CauSeq β abv) (i : ℕ) : (f - g) i = f i - g i 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f g : CauSeq β abv) (i : ℕ)
Goal: ↑(f - g) i = ↑f i - ↑g i
-- CauSeq.const_sub
theorem const_sub (x y : β) : const (x - y) = const x - const y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x y : β)
Goal: CauSeq.const abv (x - y) = CauSeq.const abv x - CauSeq.const abv y
-- CauSeq.coe_smul
@[simp, norm_cast]
theorem coe_smul (a : G) (f : CauSeq β abv) : ⇑(a • f) = a • (f : ℕ → β) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] (a : G) (f : CauSeq β abv)
Goal: ↑(a • f) = a • ↑f
-- CauSeq.smul_apply
@[simp, norm_cast]
theorem smul_apply (a : G) (f : CauSeq β abv) (i : ℕ) : (a • f) i = a • f i 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] (a : G) (f : CauSeq β abv) (i : ℕ)
Goal: ↑(a • f) i = a • ↑f i
-- CauSeq.const_smul
theorem const_smul (a : G) (x : β) : const (a • x) = a • const x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] (a : G) (x : β)
Goal: CauSeq.const abv (a • x) = a • CauSeq.const abv x
-- CauSeq.coe_pow
@[simp, norm_cast]
theorem coe_pow (f : CauSeq β abv) (n : ℕ) : ⇑(f ^ n) = (f : ℕ → β) ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (n : ℕ)
Goal: ↑(f ^ n) = ↑f ^ n
-- CauSeq.pow_apply
@[simp, norm_cast]
theorem pow_apply (f : CauSeq β abv) (n i : ℕ) : (f ^ n) i = f i ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) (n i : ℕ)
Goal: ↑(f ^ n) i = ↑f i ^ n
-- CauSeq.const_pow
theorem const_pow (x : β) (n : ℕ) : const (x ^ n) = const x ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (x : β) (n : ℕ)
Goal: CauSeq.const abv (x ^ n) = CauSeq.const abv x ^ n
-- CauSeq.add_limZero
theorem add_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f + g)
  
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (hf : f.LimZero) (hg : g.LimZero)
Goal: (f + g).LimZero
-- CauSeq.mul_limZero_right
theorem mul_limZero_right (f : CauSeq β abv) {g} (hg : LimZero g) : LimZero (f * g)
  
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : CauSeq β abv) {g : CauSeq β abv} (hg : g.LimZero)
Goal: (f * g).LimZero
-- CauSeq.mul_limZero_left
theorem mul_limZero_left {f} (g : CauSeq β abv) (hg : LimZero f) : LimZero (f * g)
  
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (g : CauSeq β abv) (hg : f.LimZero)
Goal: (f * g).LimZero
-- CauSeq.neg_limZero
theorem neg_limZero {f : CauSeq β abv} (hf : LimZero f) : LimZero (-f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : f.LimZero)
Goal: (-f).LimZero
-- CauSeq.sub_limZero
theorem sub_limZero {f g : CauSeq β abv} (hf : LimZero f) (hg : LimZero g) : LimZero (f - g) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (hf : f.LimZero) (hg : g.LimZero)
Goal: (f - g).LimZero
-- CauSeq.limZero_sub_rev
theorem limZero_sub_rev {f g : CauSeq β abv} (hfg : LimZero (f - g)) : LimZero (g - f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (hfg : (f - g).LimZero)
Goal: (g - f).LimZero
-- CauSeq.zero_limZero
theorem zero_limZero : LimZero (0 : CauSeq β abv)
  
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv]
Goal: CauSeq.LimZero 0
-- CauSeq.const_limZero
theorem const_limZero {x : β} : LimZero (const x) ↔ x = 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {x : β}
Goal: (CauSeq.const abv x).LimZero ↔ x = 0
-- CauSeq.add_equiv_add
theorem add_equiv_add {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
    f1 + g1 ≈ f2 + g2 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2)
Goal: f1 + g1 ≈ f2 + g2
-- CauSeq.neg_equiv_neg
theorem neg_equiv_neg {f g : CauSeq β abv} (hf : f ≈ g) : -f ≈ -g 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (hf : f ≈ g)
Goal: -f ≈ -g
-- CauSeq.sub_equiv_sub
theorem sub_equiv_sub {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
    f1 - g1 ≈ f2 - g2 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2)
Goal: f1 - g1 ≈ f2 - g2
-- CauSeq.equiv_def₃
theorem equiv_def₃ {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε) :
    ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (f k - g j) < ε 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (h : f ≈ g) {ε : α} (ε0 : 0 < ε)
Goal: ∃ i, ∀ j ≥ i, ∀ k ≥ j, abv (↑f k - ↑g j) < ε
-- CauSeq.limZero_congr
theorem limZero_congr {f g : CauSeq β abv} (h : f ≈ g) : LimZero f ↔ LimZero g 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (h : f ≈ g)
Goal: f.LimZero ↔ g.LimZero
-- CauSeq.abv_pos_of_not_limZero
theorem abv_pos_of_not_limZero {f : CauSeq β abv} (hf : ¬LimZero f) :
    ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (f j) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero)
Goal: ∃ K > 0, ∃ i, ∀ j ≥ i, K ≤ abv (↑f j)
-- CauSeq.of_near
theorem of_near (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - g j) < ε) :
    IsCauSeq abv f
  
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (f : ℕ → β) (g : CauSeq β abv) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε)
Goal: IsCauSeq abv f
-- CauSeq.not_limZero_of_not_congr_zero
theorem not_limZero_of_not_congr_zero {f : CauSeq _ abv} (hf : ¬f ≈ 0) : ¬LimZero f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f ≈ 0)
Goal: ¬f.LimZero
-- CauSeq.mul_equiv_zero
theorem mul_equiv_zero (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : g * f ≈ 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (g : CauSeq β abv) {f : CauSeq β abv} (hf : f ≈ 0)
Goal: g * f ≈ 0
-- CauSeq.mul_equiv_zero'
theorem mul_equiv_zero' (g : CauSeq _ abv) {f : CauSeq _ abv} (hf : f ≈ 0) : f * g ≈ 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] (g : CauSeq β abv) {f : CauSeq β abv} (hf : f ≈ 0)
Goal: f * g ≈ 0
-- CauSeq.mul_not_equiv_zero
theorem mul_not_equiv_zero {f g : CauSeq _ abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) : ¬f * g ≈ 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f g : CauSeq β abv} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0)
Goal: ¬f * g ≈ 0
-- CauSeq.const_equiv
theorem const_equiv {x y : β} : const x ≈ const y ↔ x = y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {x y : β}
Goal: CauSeq.const abv x ≈ CauSeq.const abv y ↔ x = y
-- CauSeq.mul_equiv_mul
theorem mul_equiv_mul {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2) :
    f1 * g1 ≈ f2 * g2 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 f2 g1 g2 : CauSeq β abv} (hf : f1 ≈ f2) (hg : g1 ≈ g2)
Goal: f1 * g1 ≈ f2 * g2
-- CauSeq.smul_equiv_smul
theorem smul_equiv_smul {G : Type*} [SMul G β] [IsScalarTower G β β] {f1 f2 : CauSeq β abv} (c : G)
    (hf : f1 ≈ f2) : c • f1 ≈ c • f2 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {G : Type u_3} [SMul G β] [IsScalarTower G β β] {f1 f2 : CauSeq β abv} (c : G) (hf : f1 ≈ f2)
Goal: c • f1 ≈ c • f2
-- CauSeq.pow_equiv_pow
theorem pow_equiv_pow {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ) : f1 ^ n ≈ f2 ^ n 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] {abv : β → α} [IsAbsoluteValue abv] {f1 f2 : CauSeq β abv} (hf : f1 ≈ f2) (n : ℕ)
Goal: f1 ^ n ≈ f2 ^ n
-- CauSeq.one_not_equiv_zero
theorem one_not_equiv_zero : ¬const abv 1 ≈ const abv 0 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [Ring β] [IsDomain β] (abv : β → α) [IsAbsoluteValue abv]
Goal: ¬CauSeq.const abv 1 ≈ CauSeq.const abv 0
-- CauSeq.inv_aux
theorem inv_aux {f : CauSeq β abv} (hf : ¬LimZero f) :
    ∀ ε > 0, ∃ i, ∀ j ≥ i, abv ((f j)⁻¹ - (f i)⁻¹) < ε
  
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) (ε : α)
Goal: ε > 0 → ∃ i, ∀ j ≥ i, abv ((↑f j)⁻¹ - (↑f i)⁻¹) < ε
-- CauSeq.coe_inv
@[simp, norm_cast]
theorem coe_inv {f : CauSeq β abv} (hf) : ⇑(inv f hf) = (f : ℕ → β)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero)
Goal: ↑(f.inv hf) = (↑f)⁻¹
-- CauSeq.inv_apply
@[simp, norm_cast]
theorem inv_apply {f : CauSeq β abv} (hf i) : inv f hf i = (f i)⁻¹ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero) (i : ℕ)
Goal: ↑(f.inv hf) i = (↑f i)⁻¹
-- CauSeq.inv_mul_cancel
theorem inv_mul_cancel {f : CauSeq β abv} (hf) : inv f hf * f ≈ 1 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero)
Goal: f.inv hf * f ≈ 1
-- CauSeq.mul_inv_cancel
theorem mul_inv_cancel {f : CauSeq β abv} (hf) : f * inv f hf ≈ 1 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrderedField α] [DivisionRing β] {abv : β → α} [IsAbsoluteValue abv] {f : CauSeq β abv} (hf : ¬f.LimZero)
Goal: f * f.inv hf ≈ 1