ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- finrank_real_complex_fact'
theorem finrank_real_complex_fact' : Fact (finrank ℝ ℂ = 1 + 1) 
Goal: Fact (FiniteDimensional.finrank ℝ ℂ = 1 + 1)
-- contMDiff_expMapCircle
theorem contMDiff_expMapCircle : ContMDiff 𝓘(ℝ, ℝ) (𝓡 1) ∞ expMapCircle 
Goal: ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, EuclideanSpace ℝ (Fin 1)) ⊤ ⇑expMapCircle
-- InnerProductGeometry.mul_norm_eq_abs_sub_sq_norm
theorem mul_norm_eq_abs_sub_sq_norm {x y z : V} (h₁ : ∃ k : ℝ, k ≠ 1 ∧ x + y = k • (x - y))
    (h₂ : ‖z - y‖ = ‖z + y‖) : ‖x - y‖ * ‖x + y‖ = |‖z + y‖ ^ 2 - ‖z - x‖ ^ 2| 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {x y z : V} (h₁ : ∃ k, k ≠ 1 ∧ x + y = k • (x - y)) (h₂ : ‖z - y‖ = ‖z + y‖)
Goal: ‖x - y‖ * ‖x + y‖ = |‖z + y‖ ^ 2 - ‖z - x‖ ^ 2|
-- EuclideanGeometry.mul_dist_eq_abs_sub_sq_dist
theorem mul_dist_eq_abs_sub_sq_dist {a b p q : P} (hp : ∃ k : ℝ, k ≠ 1 ∧ b -ᵥ p = k • (a -ᵥ p))
    (hq : dist a q = dist b q) : dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2| 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {P : Type u_2} [MetricSpace P] [NormedAddTorsor V P] {a b p q : P} (hp : ∃ k, k ≠ 1 ∧ b -ᵥ p = k • (a -ᵥ p)) (hq : dist a q = dist b q)
Goal: dist a p * dist b p = |dist b q ^ 2 - dist p q ^ 2|
-- EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical
theorem mul_dist_eq_mul_dist_of_cospherical {a b c d p : P} (h : Cospherical ({a, b, c, d} : Set P))
    (hapb : ∃ k₁ : ℝ, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p))
    (hcpd : ∃ k₂ : ℝ, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p)) :
    dist a p * dist b p = dist c p * dist d p 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {P : Type u_2} [MetricSpace P] [NormedAddTorsor V P] {a b c d p : P} (h : EuclideanGeometry.Cospherical {a, b, c, d}) (hapb : ∃ k₁, k₁ ≠ 1 ∧ b -ᵥ p = k₁ • (a -ᵥ p)) (hcpd : ∃ k₂, k₂ ≠ 1 ∧ d -ᵥ p = k₂ • (c -ᵥ p))
Goal: dist a p * dist b p = dist c p * dist d p
-- EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi
theorem mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi {a b c d p : P}
    (h : Cospherical ({a, b, c, d} : Set P)) (hapb : ∠ a p b = π) (hcpd : ∠ c p d = π) :
    dist a p * dist b p = dist c p * dist d p 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {P : Type u_2} [MetricSpace P] [NormedAddTorsor V P] {a b c d p : P} (h : EuclideanGeometry.Cospherical {a, b, c, d}) (hapb : ∠ a p b = π) (hcpd : ∠ c p d = π)
Goal: dist a p * dist b p = dist c p * dist d p
-- EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero
theorem mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero {a b c d p : P}
    (h : Cospherical ({a, b, c, d} : Set P)) (hab : a ≠ b) (hcd : c ≠ d) (hapb : ∠ a p b = 0)
    (hcpd : ∠ c p d = 0) : dist a p * dist b p = dist c p * dist d p 
Arguments: {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] {P : Type u_2} [MetricSpace P] [NormedAddTorsor V P] {a b c d p : P} (h : EuclideanGeometry.Cospherical {a, b, c, d}) (hab : a ≠ b) (hcd : c ≠ d) (hapb : ∠ a p b = 0) (hcpd : ∠ c p d = 0)
Goal: dist a p * dist b p = dist c p * dist d p
-- ContextFreeRule.Rewrites.exists_parts
lemma Rewrites.exists_parts {r : ContextFreeRule T N} {u v : List (Symbol T N)}
    (hyp : r.Rewrites u v) :
    ∃ p q : List (Symbol T N),
      u = p ++ [Symbol.nonterminal r.input] ++ q ∧ v = p ++ r.output ++ q 
Arguments: {T : Type uT} {N : Type uN} {r : ContextFreeRule T N} {u v : List (Symbol T N)} (hyp : r.Rewrites u v)
Goal: ∃ p q, u = p ++ [Symbol.nonterminal r.input] ++ q ∧ v = p ++ r.output ++ q
-- ContextFreeRule.rewrites_of_exists_parts
lemma rewrites_of_exists_parts (r : ContextFreeRule T N) (p q : List (Symbol T N)) :
    r.Rewrites (p ++ [Symbol.nonterminal r.input] ++ q) (p ++ r.output ++ q) 
Arguments: {T : Type uT} {N : Type uN} (r : ContextFreeRule T N) (p q : List (Symbol T N))
Goal: r.Rewrites (p ++ [Symbol.nonterminal r.input] ++ q) (p ++ r.output ++ q)
-- ContextFreeRule.rewrites_iff
theorem rewrites_iff {r : ContextFreeRule T N} (u v : List (Symbol T N)) :
    r.Rewrites u v ↔ ∃ p q : List (Symbol T N),
      u = p ++ [Symbol.nonterminal r.input] ++ q ∧ v = p ++ r.output ++ q 
Arguments: {T : Type uT} {N : Type uN} {r : ContextFreeRule T N} (u v : List (Symbol T N))
Goal: r.Rewrites u v ↔ ∃ p q, u = p ++ [Symbol.nonterminal r.input] ++ q ∧ v = p ++ r.output ++ q
-- ContextFreeRule.Rewrites.append_left
lemma Rewrites.append_left {r : ContextFreeRule T N} {v w : List (Symbol T N)}
    (hvw : r.Rewrites v w) (p : List (Symbol T N)) : r.Rewrites (p ++ v) (p ++ w) 
Arguments: {T : Type uT} {N : Type uN} {r : ContextFreeRule T N} {v w : List (Symbol T N)} (hvw : r.Rewrites v w) (p : List (Symbol T N))
Goal: r.Rewrites (p ++ v) (p ++ w)
-- ContextFreeRule.Rewrites.append_right
lemma Rewrites.append_right {r : ContextFreeRule T N} {v w : List (Symbol T N)}
    (hvw : r.Rewrites v w) (p : List (Symbol T N)) : r.Rewrites (v ++ p) (w ++ p) 
Arguments: {T : Type uT} {N : Type uN} {r : ContextFreeRule T N} {v w : List (Symbol T N)} (hvw : r.Rewrites v w) (p : List (Symbol T N))
Goal: r.Rewrites (v ++ p) (w ++ p)
-- ContextFreeGrammar.mem_language_iff
@[simp]
lemma mem_language_iff (g : ContextFreeGrammar.{uN} T) (w : List T) :
    w ∈ g.language ↔ g.Derives [Symbol.nonterminal g.initial] (List.map Symbol.terminal w) 
Arguments: {T : Type uT} (g : ContextFreeGrammar T) (w : List T)
Goal: w ∈ g.language ↔ g.Derives [Symbol.nonterminal g.initial] (List.map Symbol.terminal w)
-- ContextFreeGrammar.Derives.refl
@[refl]
lemma Derives.refl (w : List (Symbol T g.NT)) : g.Derives w w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} (w : List (Symbol T g.NT))
Goal: g.Derives w w
-- ContextFreeGrammar.Produces.single
lemma Produces.single {v w : List (Symbol T g.NT)} (hvw : g.Produces v w) : g.Derives v w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {v w : List (Symbol T g.NT)} (hvw : g.Produces v w)
Goal: g.Derives v w
-- ContextFreeGrammar.Derives.trans
@[trans]
lemma Derives.trans {u v w : List (Symbol T g.NT)} (huv : g.Derives u v) (hvw : g.Derives v w) :
    g.Derives u w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {u v w : List (Symbol T g.NT)} (huv : g.Derives u v) (hvw : g.Derives v w)
Goal: g.Derives u w
-- ContextFreeGrammar.Derives.trans_produces
lemma Derives.trans_produces {u v w : List (Symbol T g.NT)}
    (huv : g.Derives u v) (hvw : g.Produces v w) :
    g.Derives u w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {u v w : List (Symbol T g.NT)} (huv : g.Derives u v) (hvw : g.Produces v w)
Goal: g.Derives u w
-- ContextFreeGrammar.Produces.trans_derives
lemma Produces.trans_derives {u v w : List (Symbol T g.NT)}
    (huv : g.Produces u v) (hvw : g.Derives v w) :
    g.Derives u w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {u v w : List (Symbol T g.NT)} (huv : g.Produces u v) (hvw : g.Derives v w)
Goal: g.Derives u w
-- ContextFreeGrammar.Derives.eq_or_head
lemma Derives.eq_or_head {u w : List (Symbol T g.NT)} (huw : g.Derives u w) :
    u = w ∨ ∃ v : List (Symbol T g.NT), g.Produces u v ∧ g.Derives v w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {u w : List (Symbol T g.NT)} (huw : g.Derives u w)
Goal: u = w ∨ ∃ v, g.Produces u v ∧ g.Derives v w
-- ContextFreeGrammar.Derives.eq_or_tail
lemma Derives.eq_or_tail {u w : List (Symbol T g.NT)} (huw : g.Derives u w) :
    u = w ∨ ∃ v : List (Symbol T g.NT), g.Derives u v ∧ g.Produces v w 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {u w : List (Symbol T g.NT)} (huw : g.Derives u w)
Goal: u = w ∨ ∃ v, g.Derives u v ∧ g.Produces v w
-- ContextFreeGrammar.Produces.append_left
lemma Produces.append_left {v w : List (Symbol T g.NT)}
    (hvw : g.Produces v w) (p : List (Symbol T g.NT)) :
    g.Produces (p ++ v) (p ++ w) 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {v w : List (Symbol T g.NT)} (hvw : g.Produces v w) (p : List (Symbol T g.NT))
Goal: g.Produces (p ++ v) (p ++ w)
-- ContextFreeGrammar.Produces.append_right
lemma Produces.append_right {v w : List (Symbol T g.NT)}
    (hvw : g.Produces v w) (p : List (Symbol T g.NT)) :
    g.Produces (v ++ p) (w ++ p) 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {v w : List (Symbol T g.NT)} (hvw : g.Produces v w) (p : List (Symbol T g.NT))
Goal: g.Produces (v ++ p) (w ++ p)
-- ContextFreeGrammar.Derives.append_left
lemma Derives.append_left {v w : List (Symbol T g.NT)}
    (hvw : g.Derives v w) (p : List (Symbol T g.NT)) :
    g.Derives (p ++ v) (p ++ w) 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {v w : List (Symbol T g.NT)} (hvw : g.Derives v w) (p : List (Symbol T g.NT))
Goal: g.Derives (p ++ v) (p ++ w)
-- ContextFreeGrammar.Derives.append_right
lemma Derives.append_right {v w : List (Symbol T g.NT)}
    (hvw : g.Derives v w) (p : List (Symbol T g.NT)) :
    g.Derives (v ++ p) (w ++ p) 
Arguments: {T : Type uT} {g : ContextFreeGrammar T} {v w : List (Symbol T g.NT)} (hvw : g.Derives v w) (p : List (Symbol T g.NT))
Goal: g.Derives (v ++ p) (w ++ p)
-- ContextFreeRule.reverse_involutive
lemma ContextFreeRule.reverse_involutive {N : Type uN} :
    Function.Involutive (@ContextFreeRule.reverse T N) 
Arguments: {T : Type uT} {N : Type uN}
Goal: Function.Involutive ContextFreeRule.reverse
-- ContextFreeGrammar.reverse_involutive
lemma ContextFreeGrammar.reverse_involutive :
    Function.Involutive (@ContextFreeGrammar.reverse T) 
Arguments: {T : Type uT}
Goal: Function.Involutive ContextFreeGrammar.reverse
-- ContextFreeGrammar.reverse_derives
lemma ContextFreeGrammar.reverse_derives (g : ContextFreeGrammar T) {s : List (Symbol T g.NT)}
    (hgs : g.reverse.Derives [Symbol.nonterminal g.reverse.initial] s) :
    g.Derives [Symbol.nonterminal g.initial] s.reverse 
Arguments: {T : Type uT} (g : ContextFreeGrammar T) {s : List (Symbol T g.NT)} (hgs : g.reverse.Derives [Symbol.nonterminal g.reverse.initial] s)
Goal: g.Derives [Symbol.nonterminal g.initial] s.reverse
-- ContextFreeGrammar.reverse_mem_language_of_mem_reverse_language
lemma ContextFreeGrammar.reverse_mem_language_of_mem_reverse_language (g : ContextFreeGrammar T)
    {w : List T} (hgw : w ∈ g.reverse.language) :
    w.reverse ∈ g.language 
Arguments: {T : Type uT} (g : ContextFreeGrammar T) {w : List T} (hgw : w ∈ g.reverse.language)
Goal: w.reverse ∈ g.language
-- ContextFreeGrammar.mem_reverse_language_iff_reverse_mem_language
lemma ContextFreeGrammar.mem_reverse_language_iff_reverse_mem_language (g : ContextFreeGrammar T)
    (w : List T) :
    w ∈ g.reverse.language ↔ w.reverse ∈ g.language 
Arguments: {T : Type uT} (g : ContextFreeGrammar T) (w : List T)
Goal: w ∈ g.reverse.language ↔ w.reverse ∈ g.language
-- Language.IsContextFree.reverse
theorem Language.IsContextFree.reverse (L : Language T) :
    L.IsContextFree → L.reverse.IsContextFree 
Arguments: {T : Type uT} (L : Language T)
Goal: L.IsContextFree → L.reverse.IsContextFree
-- LeftInvariantDerivation.left_invariant''
structure LeftInvariantDerivation extends Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯ where
  left_invariant'' :
    ∀ g, 𝒅ₕ (smoothLeftMul_one I g) (Derivation.evalAt 1 toDerivation) =
      Derivation.evalAt g toDerivation
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (self : LeftInvariantDerivation I G) (g : G)
Goal: (𝒅ₕ ⋯) ((Derivation.evalAt 1) ↑self) = (Derivation.evalAt g) ↑self
-- LeftInvariantDerivation.toDerivation_injective
theorem toDerivation_injective :
    Function.Injective (toDerivation : LeftInvariantDerivation I G → _) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G]
Goal: Function.Injective LeftInvariantDerivation.toDerivation
-- LeftInvariantDerivation.toFun_eq_coe
theorem toFun_eq_coe : X.toFun = ⇑X 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] {X : LeftInvariantDerivation I G}
Goal: (↑↑X).toFun = ⇑X
-- LeftInvariantDerivation.coe_injective
theorem coe_injective :
    @Function.Injective (LeftInvariantDerivation I G) (_ → C^∞⟮I, G; 𝕜⟯) DFunLike.coe 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G]
Goal: Function.Injective DFunLike.coe
-- LeftInvariantDerivation.ext
@[ext]
theorem ext (h : ∀ f, X f = Y f) : X = Y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] {X Y : LeftInvariantDerivation I G} (h : ∀ (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯), X f = Y f)
Goal: X = Y
-- LeftInvariantDerivation.coe_derivation
theorem coe_derivation :
    ⇑(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = (X : C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G)
Goal: ⇑↑X = ⇑X
-- LeftInvariantDerivation.left_invariant'
theorem left_invariant' :
    𝒅ₕ (smoothLeftMul_one I g) (Derivation.evalAt (1 : G) ↑X) = Derivation.evalAt g ↑X 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (g : G) (X : LeftInvariantDerivation I G)
Goal: (𝒅ₕ ⋯) ((Derivation.evalAt 1) ↑X) = (Derivation.evalAt g) ↑X
-- LeftInvariantDerivation.map_add
protected theorem map_add : X (f + f') = X f + X f' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯) {f' : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯}
Goal: X (f + f') = X f + X f'
-- LeftInvariantDerivation.map_zero
protected theorem map_zero : X 0 = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G)
Goal: X 0 = 0
-- LeftInvariantDerivation.map_neg
protected theorem map_neg : X (-f) = -X f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: X (-f) = -X f
-- LeftInvariantDerivation.map_sub
protected theorem map_sub : X (f - f') = X f - X f' 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯) {f' : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯}
Goal: X (f - f') = X f - X f'
-- LeftInvariantDerivation.map_smul
protected theorem map_smul : X (r • f) = r • X f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] {r : 𝕜} (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: X (r • f) = r • X f
-- LeftInvariantDerivation.leibniz
@[simp]
theorem leibniz : X (f * f') = f • X f' + f' • X f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯) {f' : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯}
Goal: X (f * f') = f • X f' + f' • X f
-- LeftInvariantDerivation.coe_add
@[simp]
theorem coe_add : ⇑(X + Y) = X + Y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X Y : LeftInvariantDerivation I G)
Goal: ⇑(X + Y) = ⇑X + ⇑Y
-- LeftInvariantDerivation.coe_zero
@[simp]
theorem coe_zero : ⇑(0 : LeftInvariantDerivation I G) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G]
Goal: ⇑0 = 0
-- LeftInvariantDerivation.coe_neg
@[simp]
theorem coe_neg : ⇑(-X) = -X 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G)
Goal: ⇑(-X) = -⇑X
-- LeftInvariantDerivation.coe_sub
@[simp]
theorem coe_sub : ⇑(X - Y) = X - Y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X Y : LeftInvariantDerivation I G)
Goal: ⇑(X - Y) = ⇑X - ⇑Y
-- LeftInvariantDerivation.lift_add
@[simp, norm_cast]
theorem lift_add : (↑(X + Y) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = X + Y 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X Y : LeftInvariantDerivation I G)
Goal: ↑(X + Y) = ↑X + ↑Y
-- LeftInvariantDerivation.lift_zero
@[simp, norm_cast]
theorem lift_zero :
    (↑(0 : LeftInvariantDerivation I G) : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) = 0 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G]
Goal: ↑0 = 0
-- LeftInvariantDerivation.coe_smul
@[simp]
theorem coe_smul : ⇑(r • X) = r • ⇑X 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (r : 𝕜) (X : LeftInvariantDerivation I G)
Goal: ⇑(r • X) = r • ⇑X
-- LeftInvariantDerivation.lift_smul
@[simp]
theorem lift_smul (k : 𝕜) : (k • X).1 = k • X.1 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X : LeftInvariantDerivation I G) (k : 𝕜)
Goal: ↑(k • X) = k • ↑X
-- LeftInvariantDerivation.coeFnAddMonoidHom_apply
@[simps]
def coeFnAddMonoidHom : LeftInvariantDerivation I G →+ C^∞⟮I, G; 𝕜⟯ → C^∞⟮I, G; 𝕜⟯ 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) (G : Type u_4) [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G]
Goal: ∀ (a : LeftInvariantDerivation I G) (a_1 : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯),
  (LeftInvariantDerivation.coeFnAddMonoidHom I G) a a_1 = a a_1
-- LeftInvariantDerivation.evalAt_apply
theorem evalAt_apply : evalAt g X f = (X f) g 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (g : G) (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: ((LeftInvariantDerivation.evalAt g) X) f = (X f) g
-- LeftInvariantDerivation.evalAt_coe
@[simp]
theorem evalAt_coe : Derivation.evalAt g ↑X = evalAt g X 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (g : G) (X : LeftInvariantDerivation I G)
Goal: (Derivation.evalAt g) ↑X = (LeftInvariantDerivation.evalAt g) X
-- LeftInvariantDerivation.left_invariant
theorem left_invariant : 𝒅ₕ (smoothLeftMul_one I g) (evalAt (1 : G) X) = evalAt g X 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (g : G) (X : LeftInvariantDerivation I G)
Goal: (𝒅ₕ ⋯) ((LeftInvariantDerivation.evalAt 1) X) = (LeftInvariantDerivation.evalAt g) X
-- LeftInvariantDerivation.evalAt_mul
theorem evalAt_mul : evalAt (g * h) X = 𝒅ₕ (L_apply I g h) (evalAt h X) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (g h : G) (X : LeftInvariantDerivation I G)
Goal: (LeftInvariantDerivation.evalAt (g * h)) X = (𝒅ₕ ⋯) ((LeftInvariantDerivation.evalAt h) X)
-- LeftInvariantDerivation.comp_L
theorem comp_L : (X f).comp (𝑳 I g) = X (f.comp (𝑳 I g)) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (g : G) (X : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: (X f).comp (𝑳 I g) = X (f.comp (𝑳 I g))
-- LeftInvariantDerivation.commutator_coe_derivation
@[simp]
theorem commutator_coe_derivation :
    ⇑⁅X, Y⁆ =
      (⁅(X : Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯), Y⁆ :
        Derivation 𝕜 C^∞⟮I, G; 𝕜⟯ C^∞⟮I, G; 𝕜⟯) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X Y : LeftInvariantDerivation I G)
Goal: ⇑⁅X, Y⁆ = ⇑⁅↑X, ↑Y⁆
-- LeftInvariantDerivation.commutator_apply
theorem commutator_apply : ⁅X, Y⁆ f = X (Y f) - Y (X f) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {G : Type u_4} [TopologicalSpace G] [ChartedSpace H G] [Monoid G] [SmoothMul I G] (X Y : LeftInvariantDerivation I G) (f : C^⊤⟮I, G; 𝓘(𝕜, 𝕜), 𝕜⟯)
Goal: ⁅X, Y⁆ f = X (Y f) - Y (X f)