ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- FormalMultilinearSeries.comp_coeff_zero
theorem comp_coeff_zero (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
    (v : Fin 0 → E) (v' : Fin 0 → F) : (q.comp p) 0 v = q 0 v' 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (v : Fin 0 → E) (v' : Fin 0 → F)
Goal: (q.comp p 0) v = (q 0) v'
-- FormalMultilinearSeries.comp_coeff_zero'
@[simp]
theorem comp_coeff_zero' (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
    (v : Fin 0 → E) : (q.comp p) 0 v = q 0 fun _i => 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (v : Fin 0 → E)
Goal: (q.comp p 0) v = (q 0) fun _i => 0
-- FormalMultilinearSeries.comp_coeff_zero''
theorem comp_coeff_zero'' (q : FormalMultilinearSeries 𝕜 E F) (p : FormalMultilinearSeries 𝕜 E E) :
    (q.comp p) 0 = q 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (q : FormalMultilinearSeries 𝕜 E F) (p : FormalMultilinearSeries 𝕜 E E)
Goal: q.comp p 0 = q 0
-- FormalMultilinearSeries.comp_coeff_one
theorem comp_coeff_one (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
    (v : Fin 1 → E) : (q.comp p) 1 v = q 1 fun _i => p 1 v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (v : Fin 1 → E)
Goal: (q.comp p 1) v = (q 1) fun _i => (p 1) v
-- FormalMultilinearSeries.removeZero_comp_of_pos
theorem removeZero_comp_of_pos (q : FormalMultilinearSeries 𝕜 F G)
    (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n) :
    q.removeZero.comp p n = q.comp p n 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n)
Goal: q.removeZero.comp p n = q.comp p n
-- FormalMultilinearSeries.comp_removeZero
@[simp]
theorem comp_removeZero (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) :
    q.comp p.removeZero = q.comp p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
Goal: q.comp p.removeZero = q.comp p
-- FormalMultilinearSeries.compAlongComposition_bound
theorem compAlongComposition_bound {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n)
    (f : ContinuousMultilinearMap 𝕜 (fun _i : Fin c.length => F) G) (v : Fin n → E) :
    ‖f.compAlongComposition p c v‖ ≤ (‖f‖ * ∏ i, ‖p (c.blocksFun i)‖) * ∏ i : Fin n, ‖v i‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (f : ContinuousMultilinearMap 𝕜 (fun _i => F) G) (v : Fin n → E)
Goal: ‖(ContinuousMultilinearMap.compAlongComposition p c f) v‖ ≤
  (‖f‖ * ∏ i : Fin c.length, ‖p (c.blocksFun i)‖) * ∏ i : Fin n, ‖v i‖
-- FormalMultilinearSeries.compAlongComposition_norm
theorem compAlongComposition_norm {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
    (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) :
    ‖q.compAlongComposition p c‖ ≤ ‖q c.length‖ * ∏ i, ‖p (c.blocksFun i)‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n)
Goal: ‖q.compAlongComposition p c‖ ≤ ‖q c.length‖ * ∏ i : Fin c.length, ‖p (c.blocksFun i)‖
-- FormalMultilinearSeries.compAlongComposition_nnnorm
theorem compAlongComposition_nnnorm {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
    (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) :
    ‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n)
Goal: ‖q.compAlongComposition p c‖₊ ≤ ‖q c.length‖₊ * ∏ i : Fin c.length, ‖p (c.blocksFun i)‖₊
-- FormalMultilinearSeries.id_apply_one
@[simp]
theorem id_apply_one (v : Fin 1 → E) : (FormalMultilinearSeries.id 𝕜 E) 1 v = v 0 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] (v : Fin 1 → E)
Goal: (FormalMultilinearSeries.id 𝕜 E 1) v = v 0
-- FormalMultilinearSeries.id_apply_one'
theorem id_apply_one' {n : ℕ} (h : n = 1) (v : Fin n → E) :
    (id 𝕜 E) n v = v ⟨0, h.symm ▸ zero_lt_one⟩ 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ} (h : n = 1) (v : Fin n → E)
Goal: (FormalMultilinearSeries.id 𝕜 E n) v = v ⟨0, ⋯⟩
-- FormalMultilinearSeries.id_apply_ne_one
@[simp]
theorem id_apply_ne_one {n : ℕ} (h : n ≠ 1) : (FormalMultilinearSeries.id 𝕜 E) n = 0 
Arguments: (𝕜 : Type u_1) (E : Type u_2) [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {n : ℕ} (h : n ≠ 1)
Goal: FormalMultilinearSeries.id 𝕜 E n = 0
-- FormalMultilinearSeries.comp_id
@[simp]
theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) : p.comp (id 𝕜 E) = p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F)
Goal: p.comp (FormalMultilinearSeries.id 𝕜 E) = p
-- FormalMultilinearSeries.id_comp
@[simp]
theorem id_comp (p : FormalMultilinearSeries 𝕜 E F) (h : p 0 = 0) : (id 𝕜 F).comp p = p 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (h : p 0 = 0)
Goal: (FormalMultilinearSeries.id 𝕜 F).comp p = p
-- FormalMultilinearSeries.comp_summable_nnreal
theorem comp_summable_nnreal (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
    (hq : 0 < q.radius) (hp : 0 < p.radius) :
    ∃ r > (0 : ℝ≥0),
      Summable fun i : Σ n, Composition n => ‖q.compAlongComposition p i.2‖₊ * r ^ i.1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (hq : 0 < q.radius) (hp : 0 < p.radius)
Goal: ∃ r > 0, Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst
-- FormalMultilinearSeries.le_comp_radius_of_summable
theorem le_comp_radius_of_summable (q : FormalMultilinearSeries 𝕜 F G)
    (p : FormalMultilinearSeries 𝕜 E F) (r : ℝ≥0)
    (hr : Summable fun i : Σ n, Composition n => ‖q.compAlongComposition p i.2‖₊ * r ^ i.1) :
    (r : ℝ≥0∞) ≤ (q.comp p).radius 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (r : ℝ≥0) (hr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst)
Goal: ↑r ≤ (q.comp p).radius
-- FormalMultilinearSeries.mem_compPartialSumSource_iff
@[simp]
theorem mem_compPartialSumSource_iff (m M N : ℕ) (i : Σ n, Fin n → ℕ) :
    i ∈ compPartialSumSource m M N ↔
      (m ≤ i.1 ∧ i.1 < M) ∧ ∀ a : Fin i.1, 1 ≤ i.2 a ∧ i.2 a < N 
Arguments: (m M N : ℕ) (i : (n : ℕ) × (Fin n → ℕ))
Goal: i ∈ FormalMultilinearSeries.compPartialSumSource m M N ↔
  (m ≤ i.fst ∧ i.fst < M) ∧ ∀ (a : Fin i.fst), 1 ≤ i.snd a ∧ i.snd a < N
-- FormalMultilinearSeries.compChangeOfVariables_length
@[simp]
theorem compChangeOfVariables_length (m M N : ℕ) {i : Σ n, Fin n → ℕ}
    (hi : i ∈ compPartialSumSource m M N) :
    Composition.length (compChangeOfVariables m M N i hi).2 = i.1 
Arguments: (m M N : ℕ) {i : (n : ℕ) × (Fin n → ℕ)} (hi : i ∈ FormalMultilinearSeries.compPartialSumSource m M N)
Goal: (FormalMultilinearSeries.compChangeOfVariables m M N i hi).snd.length = i.fst
-- FormalMultilinearSeries.compChangeOfVariables_blocksFun
theorem compChangeOfVariables_blocksFun (m M N : ℕ) {i : Σ n, Fin n → ℕ}
    (hi : i ∈ compPartialSumSource m M N) (j : Fin i.1) :
    (compChangeOfVariables m M N i hi).2.blocksFun
        ⟨j, (compChangeOfVariables_length m M N hi).symm ▸ j.2⟩ =
      i.2 j 
Arguments: (m M N : ℕ) {i : (n : ℕ) × (Fin n → ℕ)} (hi : i ∈ FormalMultilinearSeries.compPartialSumSource m M N) (j : Fin i.fst)
Goal: (FormalMultilinearSeries.compChangeOfVariables m M N i hi).snd.blocksFun ⟨↑j, ⋯⟩ = i.snd j
-- FormalMultilinearSeries.compPartialSumTargetSet_image_compPartialSumSource
theorem compPartialSumTargetSet_image_compPartialSumSource (m M N : ℕ)
    (i : Σ n, Composition n) (hi : i ∈ compPartialSumTargetSet m M N) :
    ∃ (j : _) (hj : j ∈ compPartialSumSource m M N), compChangeOfVariables m M N j hj = i 
Arguments: (m M N : ℕ) (i : (n : ℕ) × Composition n) (hi : i ∈ FormalMultilinearSeries.compPartialSumTargetSet m M N)
Goal: ∃ j,
  ∃ (hj : j ∈ FormalMultilinearSeries.compPartialSumSource m M N),
    FormalMultilinearSeries.compChangeOfVariables m M N j hj = i
-- FormalMultilinearSeries.mem_compPartialSumTarget_iff
@[simp]
theorem mem_compPartialSumTarget_iff {m M N : ℕ} {a : Σ n, Composition n} :
    a ∈ compPartialSumTarget m M N ↔
      m ≤ a.2.length ∧ a.2.length < M ∧ ∀ j : Fin a.2.length, a.2.blocksFun j < N 
Arguments: {m M N : ℕ} {a : (n : ℕ) × Composition n}
Goal: a ∈ FormalMultilinearSeries.compPartialSumTarget m M N ↔
  m ≤ a.snd.length ∧ a.snd.length < M ∧ ∀ (j : Fin a.snd.length), a.snd.blocksFun j < N
-- FormalMultilinearSeries.compChangeOfVariables_sum
theorem compChangeOfVariables_sum {α : Type*} [AddCommMonoid α] (m M N : ℕ)
    (f : (Σ n : ℕ, Fin n → ℕ) → α) (g : (Σ n, Composition n) → α)
    (h : ∀ (e) (he : e ∈ compPartialSumSource m M N), f e = g (compChangeOfVariables m M N e he)) :
    ∑ e ∈ compPartialSumSource m M N, f e = ∑ e ∈ compPartialSumTarget m M N, g e 
Arguments: {α : Type u_6} [AddCommMonoid α] (m M N : ℕ) (f : (n : ℕ) × (Fin n → ℕ) → α) (g : (n : ℕ) × Composition n → α) (h :
  ∀ (e : (n : ℕ) × (Fin n → ℕ)) (he : e ∈ FormalMultilinearSeries.compPartialSumSource m M N),
    f e = g (FormalMultilinearSeries.compChangeOfVariables m M N e he))
Goal: ∑ e ∈ FormalMultilinearSeries.compPartialSumSource m M N, f e =
  ∑ e ∈ FormalMultilinearSeries.compPartialSumTarget m M N, g e
-- FormalMultilinearSeries.compPartialSumTarget_tendsto_atTop
theorem compPartialSumTarget_tendsto_atTop :
    Tendsto (fun N => compPartialSumTarget 0 N N) atTop atTop 
Goal: Filter.Tendsto (fun N => FormalMultilinearSeries.compPartialSumTarget 0 N N) Filter.atTop Filter.atTop
-- FormalMultilinearSeries.comp_partialSum
theorem comp_partialSum (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
    (N : ℕ) (z : E) :
    q.partialSum N (∑ i ∈ Finset.Ico 1 N, p i fun _j => z) =
      ∑ i ∈ compPartialSumTarget 0 N N, q.compAlongComposition p i.2 fun _j => z 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (N : ℕ) (z : E)
Goal: q.partialSum N (∑ i ∈ Finset.Ico 1 N, (p i) fun _j => z) =
  ∑ i ∈ FormalMultilinearSeries.compPartialSumTarget 0 N N, (q.compAlongComposition p i.snd) fun _j => z
-- HasFPowerSeriesAt.comp
theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
    {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hg : HasFPowerSeriesAt g q (f x))
    (hf : HasFPowerSeriesAt f p x) : HasFPowerSeriesAt (g ∘ f) (q.comp p) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G} {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hg : HasFPowerSeriesAt g q (f x)) (hf : HasFPowerSeriesAt f p x)
Goal: HasFPowerSeriesAt (g ∘ f) (q.comp p) x
-- AnalyticAt.comp
theorem AnalyticAt.comp {g : F → G} {f : E → F} {x : E} (hg : AnalyticAt 𝕜 g (f x))
    (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (g ∘ f) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : F → G} {f : E → F} {x : E} (hg : AnalyticAt 𝕜 g (f x)) (hf : AnalyticAt 𝕜 f x)
Goal: AnalyticAt 𝕜 (g ∘ f) x
-- AnalyticAt.comp_of_eq
theorem AnalyticAt.comp_of_eq {g : F → G} {f : E → F} {y : F} {x : E} (hg : AnalyticAt 𝕜 g y)
    (hf : AnalyticAt 𝕜 f x) (hy : f x = y) : AnalyticAt 𝕜 (g ∘ f) x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {g : F → G} {f : E → F} {y : F} {x : E} (hg : AnalyticAt 𝕜 g y) (hf : AnalyticAt 𝕜 f x) (hy : f x = y)
Goal: AnalyticAt 𝕜 (g ∘ f) x
-- AnalyticOn.comp'
theorem AnalyticOn.comp' {s : Set E} {g : F → G} {f : E → F} (hg : AnalyticOn 𝕜 g (s.image f))
    (hf : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (g ∘ f) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {g : F → G} {f : E → F} (hg : AnalyticOn 𝕜 g (f '' s)) (hf : AnalyticOn 𝕜 f s)
Goal: AnalyticOn 𝕜 (g ∘ f) s
-- AnalyticOn.comp
theorem AnalyticOn.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : AnalyticOn 𝕜 g t)
    (hf : AnalyticOn 𝕜 f s) (st : Set.MapsTo f s t) : AnalyticOn 𝕜 (g ∘ f) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : AnalyticOn 𝕜 g t) (hf : AnalyticOn 𝕜 f s) (st : Set.MapsTo f s t)
Goal: AnalyticOn 𝕜 (g ∘ f) s
-- Composition.sigma_composition_eq_iff
theorem sigma_composition_eq_iff (i j : Σ a : Composition n, Composition a.length) :
    i = j ↔ i.1.blocks = j.1.blocks ∧ i.2.blocks = j.2.blocks 
Arguments: {n : ℕ} (i j : (a : Composition n) × Composition a.length)
Goal: i = j ↔ i.fst.blocks = j.fst.blocks ∧ i.snd.blocks = j.snd.blocks
-- Composition.sigma_pi_composition_eq_iff
theorem sigma_pi_composition_eq_iff
    (u v : Σ c : Composition n, ∀ i : Fin c.length, Composition (c.blocksFun i)) :
    u = v ↔ (ofFn fun i => (u.2 i).blocks) = ofFn fun i => (v.2 i).blocks 
Arguments: {n : ℕ} (u v : (c : Composition n) × ((i : Fin c.length) → Composition (c.blocksFun i)))
Goal: u = v ↔ (List.ofFn fun i => (u.snd i).blocks) = List.ofFn fun i => (v.snd i).blocks
-- Composition.length_gather
theorem length_gather (a : Composition n) (b : Composition a.length) :
    length (a.gather b) = b.length 
Arguments: {n : ℕ} (a : Composition n) (b : Composition a.length)
Goal: (a.gather b).length = b.length
-- Composition.length_sigmaCompositionAux
theorem length_sigmaCompositionAux (a : Composition n) (b : Composition a.length)
    (i : Fin b.length) :
    Composition.length (Composition.sigmaCompositionAux a b ⟨i, (length_gather a b).symm ▸ i.2⟩) =
      Composition.blocksFun b i 
Arguments: {n : ℕ} (a : Composition n) (b : Composition a.length) (i : Fin b.length)
Goal: (a.sigmaCompositionAux b ⟨↑i, ⋯⟩).length = b.blocksFun i
-- Composition.blocksFun_sigmaCompositionAux
theorem blocksFun_sigmaCompositionAux (a : Composition n) (b : Composition a.length)
    (i : Fin b.length) (j : Fin (blocksFun b i)) :
    blocksFun (sigmaCompositionAux a b ⟨i, (length_gather a b).symm ▸ i.2⟩)
        ⟨j, (length_sigmaCompositionAux a b i).symm ▸ j.2⟩ =
      blocksFun a (embedding b i j) 
Arguments: {n : ℕ} (a : Composition n) (b : Composition a.length) (i : Fin b.length) (j : Fin (b.blocksFun i))
Goal: (a.sigmaCompositionAux b ⟨↑i, ⋯⟩).blocksFun ⟨↑j, ⋯⟩ = a.blocksFun ((b.embedding i) j)
-- Composition.sizeUpTo_sizeUpTo_add
theorem sizeUpTo_sizeUpTo_add (a : Composition n) (b : Composition a.length) {i j : ℕ}
    (hi : i < b.length) (hj : j < blocksFun b ⟨i, hi⟩) :
    sizeUpTo a (sizeUpTo b i + j) =
      sizeUpTo (a.gather b) i +
        sizeUpTo (sigmaCompositionAux a b ⟨i, (length_gather a b).symm ▸ hi⟩) j 
Arguments: {n : ℕ} (a : Composition n) (b : Composition a.length) {i j : ℕ} (hi : i < b.length) (hj : j < b.blocksFun ⟨i, hi⟩)
Goal: a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j
-- FormalMultilinearSeries.comp_assoc
theorem comp_assoc (r : FormalMultilinearSeries 𝕜 G H) (q : FormalMultilinearSeries 𝕜 F G)
    (p : FormalMultilinearSeries 𝕜 E F) : (r.comp q).comp p = r.comp (q.comp p) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {H : Type u_5} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [NormedAddCommGroup H] [NormedSpace 𝕜 H] (r : FormalMultilinearSeries 𝕜 G H) (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
Goal: (r.comp q).comp p = r.comp (q.comp p)
-- ContinuousLinearMap.adjointAux_apply
@[simp]
theorem adjointAux_apply (A : E →L[𝕜] F) (x : F) :
    adjointAux A x = ((toDual 𝕜 E).symm : NormedSpace.Dual 𝕜 E → E) ((toSesqForm A) x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] (A : E →L[𝕜] F) (x : F)
Goal: (ContinuousLinearMap.adjointAux A) x = (InnerProductSpace.toDual 𝕜 E).symm ((ContinuousLinearMap.toSesqForm A) x)
-- ContinuousLinearMap.adjointAux_inner_left
theorem adjointAux_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪adjointAux A y, x⟫ = ⟪y, A x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] (A : E →L[𝕜] F) (x : E) (y : F)
Goal: ⟪(ContinuousLinearMap.adjointAux A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜
-- ContinuousLinearMap.adjointAux_inner_right
theorem adjointAux_inner_right (A : E →L[𝕜] F) (x : E) (y : F) :
    ⟪x, adjointAux A y⟫ = ⟪A x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] (A : E →L[𝕜] F) (x : E) (y : F)
Goal: ⟪x, (ContinuousLinearMap.adjointAux A) y⟫_𝕜 = ⟪A x, y⟫_𝕜
-- ContinuousLinearMap.adjointAux_adjointAux
theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F)
Goal: ContinuousLinearMap.adjointAux (ContinuousLinearMap.adjointAux A) = A
-- ContinuousLinearMap.adjointAux_norm
@[simp]
theorem adjointAux_norm (A : E →L[𝕜] F) : ‖adjointAux A‖ = ‖A‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F)
Goal: ‖ContinuousLinearMap.adjointAux A‖ = ‖A‖
-- ContinuousLinearMap.adjoint_inner_left
theorem adjoint_inner_left (A : E →L[𝕜] F) (x : E) (y : F) : ⟪(A†) y, x⟫ = ⟪y, A x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (x : E) (y : F)
Goal: ⟪(ContinuousLinearMap.adjoint A) y, x⟫_𝕜 = ⟪y, A x⟫_𝕜
-- ContinuousLinearMap.adjoint_inner_right
theorem adjoint_inner_right (A : E →L[𝕜] F) (x : E) (y : F) : ⟪x, (A†) y⟫ = ⟪A x, y⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (x : E) (y : F)
Goal: ⟪x, (ContinuousLinearMap.adjoint A) y⟫_𝕜 = ⟪A x, y⟫_𝕜
-- ContinuousLinearMap.adjoint_adjoint
@[simp]
theorem adjoint_adjoint (A : E →L[𝕜] F) : A†† = A 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F)
Goal: ContinuousLinearMap.adjoint (ContinuousLinearMap.adjoint A) = A
-- ContinuousLinearMap.adjoint_comp
@[simp]
theorem adjoint_comp (A : F →L[𝕜] G) (B : E →L[𝕜] F) : (A ∘L B)† = B† ∘L A† 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [InnerProductSpace 𝕜 G] [CompleteSpace E] [CompleteSpace G] [CompleteSpace F] (A : F →L[𝕜] G) (B : E →L[𝕜] F)
Goal: ContinuousLinearMap.adjoint (A.comp B) = (ContinuousLinearMap.adjoint B).comp (ContinuousLinearMap.adjoint A)
-- ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left
theorem apply_norm_sq_eq_inner_adjoint_left (A : E →L[𝕜] F) (x : E) :
    ‖A x‖ ^ 2 = re ⟪(A† ∘L A) x, x⟫ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 F] [CompleteSpace E] [CompleteSpace F] (A : E →L[𝕜] F) (x : E)
Goal: ‖A x‖ ^ 2 = RCLike.re ⟪((ContinuousLinearMap.adjoint A).comp A) x, x⟫_𝕜