ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ApproximatesLinearOn.surjective
protected theorem surjective [CompleteSpace E] (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) univ c)
    (hc : Subsingleton E ∨ c < N⁻¹) : Surjective f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {c : ℝ≥0} [CompleteSpace E] (hf : ApproximatesLinearOn f (↑f') Set.univ c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹)
Goal: Function.Surjective f
-- ApproximatesLinearOn.inverse_continuousOn
theorem inverse_continuousOn (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) : ContinuousOn (hf.toPartialEquiv hc).symm (f '' s) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹)
Goal: ContinuousOn (↑(hf.toPartialEquiv hc).symm) (f '' s)
-- ApproximatesLinearOn.to_inv
theorem to_inv (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c) (hc : Subsingleton E ∨ c < N⁻¹) :
    ApproximatesLinearOn (hf.toPartialEquiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
      (N * (N⁻¹ - c)⁻¹ * c) 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹)
Goal: ApproximatesLinearOn (↑(hf.toPartialEquiv hc).symm) (↑f'.symm) (f '' s) (‖↑f'.symm‖₊ * (‖↑f'.symm‖₊⁻¹ - c)⁻¹ * c)
-- ApproximatesLinearOn.toPartialHomeomorph_coe
@[simp]
theorem toPartialHomeomorph_coe (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
    (hf.toPartialHomeomorph f s hc hs : E → F) = f 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] (f : E → F) {f' : E ≃L[𝕜] F} (s : Set E) {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹) (hs : IsOpen s)
Goal: ↑(ApproximatesLinearOn.toPartialHomeomorph f s hf hc hs) = f
-- ApproximatesLinearOn.toPartialHomeomorph_source
@[simp]
theorem toPartialHomeomorph_source (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
    (hf.toPartialHomeomorph f s hc hs).source = s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] (f : E → F) {f' : E ≃L[𝕜] F} (s : Set E) {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹) (hs : IsOpen s)
Goal: (ApproximatesLinearOn.toPartialHomeomorph f s hf hc hs).source = s
-- ApproximatesLinearOn.toPartialHomeomorph_target
@[simp]
theorem toPartialHomeomorph_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) :
    (hf.toPartialHomeomorph f s hc hs).target = f '' s 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] [CompleteSpace E] (f : E → F) {f' : E ≃L[𝕜] F} (s : Set E) {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹) (hs : IsOpen s)
Goal: (ApproximatesLinearOn.toPartialHomeomorph f s hf hc hs).target = f '' s
-- ApproximatesLinearOn.closedBall_subset_target
theorem closedBall_subset_target (hf : ApproximatesLinearOn f (f' : E →L[𝕜] F) s c)
    (hc : Subsingleton E ∨ c < N⁻¹) (hs : IsOpen s) {b : E} (ε0 : 0 ≤ ε) (hε : closedBall b ε ⊆ s) :
    closedBall (f b) ((N⁻¹ - c) * ε) ⊆ (hf.toPartialHomeomorph f s hc hs).target 
Arguments: {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {ε : ℝ} [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {s : Set E} {c : ℝ≥0} (hf : ApproximatesLinearOn f (↑f') s c) (hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹) (hs : IsOpen s) {b : E} (ε0 : 0 ≤ ε) (hε : Metric.closedBall b ε ⊆ s)
Goal: Metric.closedBall (f b) ((↑‖↑f'.symm‖₊⁻¹ - ↑c) * ε) ⊆ (ApproximatesLinearOn.toPartialHomeomorph f s hf hc hs).target
-- Complex.HadamardThreeLines.sSupNormIm_nonneg
lemma sSupNormIm_nonneg (x : ℝ) : 0 ≤ sSupNormIm f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (x : ℝ)
Goal: 0 ≤ Complex.HadamardThreeLines.sSupNormIm f x
-- Complex.HadamardThreeLines.sSupNormIm_eps_pos
lemma sSupNormIm_eps_pos {ε : ℝ} (hε : ε > 0) (x : ℝ) : 0 < ε + sSupNormIm f x 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) {ε : ℝ} (hε : ε > 0) (x : ℝ)
Goal: 0 < ε + Complex.HadamardThreeLines.sSupNormIm f x
-- Complex.HadamardThreeLines.abs_invInterpStrip
lemma abs_invInterpStrip {ε : ℝ} (hε : ε > 0) :
    abs (invInterpStrip f z ε) =
    (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) {ε : ℝ} (hε : ε > 0)
Goal: Complex.abs (Complex.HadamardThreeLines.invInterpStrip f z ε) =
  (ε + Complex.HadamardThreeLines.sSupNormIm f 0) ^ (z.re - 1) *
    (ε + Complex.HadamardThreeLines.sSupNormIm f 1) ^ (-z.re)
-- Complex.HadamardThreeLines.diffContOnCl_invInterpStrip
lemma diffContOnCl_invInterpStrip {ε : ℝ} (hε : ε > 0) :
    DiffContOnCl ℂ (fun z ↦ invInterpStrip f z ε) (verticalStrip 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) {ε : ℝ} (hε : ε > 0)
Goal: DiffContOnCl ℂ (fun z => Complex.HadamardThreeLines.invInterpStrip f z ε) (Complex.HadamardThreeLines.verticalStrip 0 1)
-- Complex.HadamardThreeLines.norm_le_sSupNormIm
lemma norm_le_sSupNormIm (f : ℂ → E) (z : ℂ) (hD : z ∈ verticalClosedStrip 0 1)
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) :
    ‖f z‖ ≤ sSupNormIm f (z.re) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) (hD : z ∈ Complex.HadamardThreeLines.verticalClosedStrip 0 1) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1))
Goal: ‖f z‖ ≤ Complex.HadamardThreeLines.sSupNormIm f z.re
-- Complex.HadamardThreeLines.norm_lt_sSupNormIm_eps
lemma norm_lt_sSupNormIm_eps (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (z : ℂ)
    (hD : z ∈ verticalClosedStrip 0 1) (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) :
    ‖f z‖ < ε + sSupNormIm f (z.re) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (z : ℂ) (hD : z ∈ Complex.HadamardThreeLines.verticalClosedStrip 0 1) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1))
Goal: ‖f z‖ < ε + Complex.HadamardThreeLines.sSupNormIm f z.re
-- Complex.HadamardThreeLines.F_BddAbove
lemma F_BddAbove (f : ℂ → E) (ε : ℝ) (hε : ε > 0)
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) :
    BddAbove ((norm ∘ (F f ε)) '' (verticalClosedStrip 0 1)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1))
Goal: BddAbove (norm ∘ Complex.HadamardThreeLines.F f ε '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)
-- Complex.HadamardThreeLines.F_edge_le_one
lemma F_edge_le_one (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (z : ℂ)
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) (hz : z ∈ re ⁻¹' {0, 1}) :
    ‖F f ε z‖ ≤ 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (z : ℂ) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)) (hz : z ∈ Complex.re ⁻¹' {0, 1})
Goal: ‖Complex.HadamardThreeLines.F f ε z‖ ≤ 1
-- Complex.HadamardThreeLines.norm_mul_invInterpStrip_le_one_of_mem_verticalClosedStrip
theorem norm_mul_invInterpStrip_le_one_of_mem_verticalClosedStrip (f : ℂ → E) (ε : ℝ) (hε : 0 < ε)
    (z : ℂ) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) (hz : z ∈ verticalClosedStrip 0 1) :
    ‖F f ε z‖ ≤ 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (ε : ℝ) (hε : 0 < ε) (z : ℂ) (hd : DiffContOnCl ℂ f (Complex.HadamardThreeLines.verticalStrip 0 1)) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)) (hz : z ∈ Complex.HadamardThreeLines.verticalClosedStrip 0 1)
Goal: ‖Complex.HadamardThreeLines.F f ε z‖ ≤ 1
-- Complex.HadamardThreeLines.interpStrip_eq_of_pos
lemma interpStrip_eq_of_pos (z : ℂ) (h0 : 0 < sSupNormIm f 0) (h1 : 0 < sSupNormIm f 1) :
    interpStrip f z = (sSupNormIm f 0) ^ (1 - z) * (sSupNormIm f 1) ^ z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) (h0 : 0 < Complex.HadamardThreeLines.sSupNormIm f 0) (h1 : 0 < Complex.HadamardThreeLines.sSupNormIm f 1)
Goal: Complex.HadamardThreeLines.interpStrip f z =
  ↑(Complex.HadamardThreeLines.sSupNormIm f 0) ^ (1 - z) * ↑(Complex.HadamardThreeLines.sSupNormIm f 1) ^ z
-- Complex.HadamardThreeLines.interpStrip_eq_of_zero
lemma interpStrip_eq_of_zero (z : ℂ) (h : sSupNormIm f 0 = 0 ∨ sSupNormIm f 1 = 0) :
    interpStrip f z = 0 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) (h : Complex.HadamardThreeLines.sSupNormIm f 0 = 0 ∨ Complex.HadamardThreeLines.sSupNormIm f 1 = 0)
Goal: Complex.HadamardThreeLines.interpStrip f z = 0
-- Complex.HadamardThreeLines.interpStrip_eq_of_mem_verticalStrip
lemma interpStrip_eq_of_mem_verticalStrip (z : ℂ) (hz : z ∈ verticalStrip 0 1) :
    interpStrip f z = (sSupNormIm f 0) ^ (1 - z) * (sSupNormIm f 1) ^ z 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E) (z : ℂ) (hz : z ∈ Complex.HadamardThreeLines.verticalStrip 0 1)
Goal: Complex.HadamardThreeLines.interpStrip f z =
  ↑(Complex.HadamardThreeLines.sSupNormIm f 0) ^ (1 - z) * ↑(Complex.HadamardThreeLines.sSupNormIm f 1) ^ z
-- Complex.HadamardThreeLines.diffContOnCl_interpStrip
lemma diffContOnCl_interpStrip :
    DiffContOnCl ℂ (interpStrip f) (verticalStrip 0 1) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] (f : ℂ → E)
Goal: DiffContOnCl ℂ (Complex.HadamardThreeLines.interpStrip f) (Complex.HadamardThreeLines.verticalStrip 0 1)
-- Complex.HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip_eps
lemma norm_le_interpStrip_of_mem_verticalClosedStrip_eps (ε : ℝ) (hε : ε > 0) (z : ℂ)
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1)))
    (hd : DiffContOnCl ℂ f (verticalStrip 0 1)) (hz : z ∈ verticalClosedStrip 0 1) :
    ‖f z‖ ≤  ‖((ε + sSupNormIm f 0) ^ (1-z) * (ε + sSupNormIm f 1) ^ z : ℂ)‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (z : ℂ) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)) (hd : DiffContOnCl ℂ f (Complex.HadamardThreeLines.verticalStrip 0 1)) (hz : z ∈ Complex.HadamardThreeLines.verticalClosedStrip 0 1)
Goal: ‖f z‖ ≤
  ‖(↑ε + ↑(Complex.HadamardThreeLines.sSupNormIm f 0)) ^ (1 - z) *
      (↑ε + ↑(Complex.HadamardThreeLines.sSupNormIm f 1)) ^ z‖
-- Complex.HadamardThreeLines.eventuallyle
lemma eventuallyle (z : ℂ) (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1)))
    (hd : DiffContOnCl ℂ f (verticalStrip 0 1)) (hz : z ∈ verticalStrip 0 1) :
    (fun _ : ℝ ↦ ‖f z‖) ≤ᶠ[𝓝[>] 0]
    (fun ε ↦ ‖((ε + sSupNormIm f 0) ^ (1 - z) * (ε + sSupNormIm f 1) ^ z : ℂ)‖) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)) (hd : DiffContOnCl ℂ f (Complex.HadamardThreeLines.verticalStrip 0 1)) (hz : z ∈ Complex.HadamardThreeLines.verticalStrip 0 1)
Goal: (fun x => ‖f z‖) ≤ᶠ[𝓝[>] 0] fun ε =>
  ‖(↑ε + ↑(Complex.HadamardThreeLines.sSupNormIm f 0)) ^ (1 - z) *
      (↑ε + ↑(Complex.HadamardThreeLines.sSupNormIm f 1)) ^ z‖
-- Complex.HadamardThreeLines.norm_le_interpStrip_of_mem_verticalStrip_zero
lemma norm_le_interpStrip_of_mem_verticalStrip_zero (z : ℂ)
    (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) (hz : z ∈ verticalStrip 0 1) :
    ‖f z‖ ≤ ‖interpStrip f z‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) (z : ℂ) (hd : DiffContOnCl ℂ f (Complex.HadamardThreeLines.verticalStrip 0 1)) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)) (hz : z ∈ Complex.HadamardThreeLines.verticalStrip 0 1)
Goal: ‖f z‖ ≤ ‖Complex.HadamardThreeLines.interpStrip f z‖
-- Complex.HadamardThreeLines.norm_le_interpStrip_of_mem_verticalClosedStrip
lemma norm_le_interpStrip_of_mem_verticalClosedStrip (f : ℂ → E) {z : ℂ}
    (hz : z ∈ verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1))) :
    ‖f z‖ ≤ ‖interpStrip f z‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) {z : ℂ} (hz : z ∈ Complex.HadamardThreeLines.verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (Complex.HadamardThreeLines.verticalStrip 0 1)) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1))
Goal: ‖f z‖ ≤ ‖Complex.HadamardThreeLines.interpStrip f z‖
-- Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip'
lemma norm_le_interp_of_mem_verticalClosedStrip' (f : ℂ → E) {z : ℂ} {a b : ℝ}
    (hz : z ∈ verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (verticalStrip 0 1))
    (hB : BddAbove ((norm ∘ f) '' (verticalClosedStrip 0 1)))
    (ha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a) (hb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b) :
    ‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℂ → E) {z : ℂ} {a b : ℝ} (hz : z ∈ Complex.HadamardThreeLines.verticalClosedStrip 0 1) (hd : DiffContOnCl ℂ f (Complex.HadamardThreeLines.verticalStrip 0 1)) (hB : BddAbove (norm ∘ f '' Complex.HadamardThreeLines.verticalClosedStrip 0 1)) (ha : ∀ z ∈ Complex.re ⁻¹' {0}, ‖f z‖ ≤ a) (hb : ∀ z ∈ Complex.re ⁻¹' {1}, ‖f z‖ ≤ b)
Goal: ‖f z‖ ≤ a ^ (1 - z.re) * b ^ z.re
-- FormalMultilinearSeries.zero_apply
@[simp, nolint simpNF]
theorem zero_apply (n : ℕ) : (0 : FormalMultilinearSeries 𝕜 E F) n = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (n : ℕ)
Goal: 0 n = 0
-- FormalMultilinearSeries.neg_apply
@[simp]
theorem neg_apply (f : FormalMultilinearSeries 𝕜 E F) (n : ℕ) : (-f) n = - f n 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (f : FormalMultilinearSeries 𝕜 E F) (n : ℕ)
Goal: (-f) n = -f n
-- FormalMultilinearSeries.ext
@[ext]
protected theorem ext {p q : FormalMultilinearSeries 𝕜 E F} (h : ∀ n, p n = q n) : p = q 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p q : FormalMultilinearSeries 𝕜 E F} (h : ∀ (n : ℕ), p n = q n)
Goal: p = q
-- FormalMultilinearSeries.ext_iff
protected theorem ext_iff {p q : FormalMultilinearSeries 𝕜 E F} : p = q ↔ ∀ n, p n = q n 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p q : FormalMultilinearSeries 𝕜 E F}
Goal: p = q ↔ ∀ (n : ℕ), p n = q n
-- FormalMultilinearSeries.ne_iff
protected theorem ne_iff {p q : FormalMultilinearSeries 𝕜 E F} : p ≠ q ↔ ∃ n, p n ≠ q n 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p q : FormalMultilinearSeries 𝕜 E F}
Goal: p ≠ q ↔ ∃ n, p n ≠ q n
-- FormalMultilinearSeries.removeZero_coeff_zero
@[simp]
theorem removeZero_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) : p.removeZero 0 = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F)
Goal: p.removeZero 0 = 0
-- FormalMultilinearSeries.removeZero_coeff_succ
@[simp]
theorem removeZero_coeff_succ (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    p.removeZero (n + 1) = p (n + 1) 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ)
Goal: p.removeZero (n + 1) = p (n + 1)
-- FormalMultilinearSeries.removeZero_of_pos
theorem removeZero_of_pos (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (h : 0 < n) :
    p.removeZero n = p n 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (h : 0 < n)
Goal: p.removeZero n = p n
-- FormalMultilinearSeries.congr
theorem congr (p : FormalMultilinearSeries 𝕜 E F) {m n : ℕ} {v : Fin m → E} {w : Fin n → E}
    (h1 : m = n) (h2 : ∀ (i : ℕ) (him : i < m) (hin : i < n), v ⟨i, him⟩ = w ⟨i, hin⟩) :
    p m v = p n w 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {m n : ℕ} {v : Fin m → E} {w : Fin n → E} (h1 : m = n) (h2 : ∀ (i : ℕ) (him : i < m) (hin : i < n), v ⟨i, him⟩ = w ⟨i, hin⟩)
Goal: (p m) v = (p n) w
-- FormalMultilinearSeries.compContinuousLinearMap_apply
@[simp]
theorem compContinuousLinearMap_apply (p : FormalMultilinearSeries 𝕜 F G) (u : E →L[𝕜] F) (n : ℕ)
    (v : Fin n → E) : (p.compContinuousLinearMap u) n v = p n (u ∘ v) 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [AddCommGroup G] [Module 𝕜 G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (p : FormalMultilinearSeries 𝕜 F G) (u : E →L[𝕜] F) (n : ℕ) (v : Fin n → E)
Goal: (p.compContinuousLinearMap u n) v = (p n) (⇑u ∘ v)
-- ContinuousLinearMap.compFormalMultilinearSeries_apply
@[simp]
theorem compFormalMultilinearSeries_apply (f : F →L[𝕜] G) (p : FormalMultilinearSeries 𝕜 E F)
    (n : ℕ) : (f.compFormalMultilinearSeries p) n = f.compContinuousMultilinearMap (p n) 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [AddCommGroup G] [Module 𝕜 G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (f : F →L[𝕜] G) (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ)
Goal: f.compFormalMultilinearSeries p n = f.compContinuousMultilinearMap (p n)
-- ContinuousLinearMap.compFormalMultilinearSeries_apply'
theorem compFormalMultilinearSeries_apply' (f : F →L[𝕜] G) (p : FormalMultilinearSeries 𝕜 E F)
    (n : ℕ) (v : Fin n → E) : (f.compFormalMultilinearSeries p) n v = f (p n v) 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} {G : Type x} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [AddCommGroup G] [Module 𝕜 G] [TopologicalSpace G] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] (f : F →L[𝕜] G) (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) (v : Fin n → E)
Goal: (f.compFormalMultilinearSeries p n) v = f ((p n) v)
-- FormalMultilinearSeries.order_zero
@[simp]
theorem order_zero : (0 : FormalMultilinearSeries 𝕜 E F).order = 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F]
Goal: FormalMultilinearSeries.order 0 = 0
-- FormalMultilinearSeries.ne_zero_of_order_ne_zero
theorem ne_zero_of_order_ne_zero (hp : p.order ≠ 0) : p ≠ 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} (hp : p.order ≠ 0)
Goal: p ≠ 0
-- FormalMultilinearSeries.order_eq_find
theorem order_eq_find [DecidablePred fun n => p n ≠ 0] (hp : ∃ n, p n ≠ 0) :
    p.order = Nat.find hp 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} [DecidablePred fun n => p n ≠ 0] (hp : ∃ n, p n ≠ 0)
Goal: p.order = Nat.find hp
-- FormalMultilinearSeries.order_eq_find'
theorem order_eq_find' [DecidablePred fun n => p n ≠ 0] (hp : p ≠ 0) :
    p.order = Nat.find (FormalMultilinearSeries.ne_iff.mp hp) 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} [DecidablePred fun n => p n ≠ 0] (hp : p ≠ 0)
Goal: p.order = Nat.find ⋯
-- FormalMultilinearSeries.order_eq_zero_iff'
theorem order_eq_zero_iff' : p.order = 0 ↔ p = 0 ∨ p 0 ≠ 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F}
Goal: p.order = 0 ↔ p = 0 ∨ p 0 ≠ 0
-- FormalMultilinearSeries.order_eq_zero_iff
theorem order_eq_zero_iff (hp : p ≠ 0) : p.order = 0 ↔ p 0 ≠ 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} (hp : p ≠ 0)
Goal: p.order = 0 ↔ p 0 ≠ 0
-- FormalMultilinearSeries.apply_order_ne_zero
theorem apply_order_ne_zero (hp : p ≠ 0) : p p.order ≠ 0 
Arguments: {𝕜 : Type u} {E : Type v} {F : Type w} [Ring 𝕜] [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {p : FormalMultilinearSeries 𝕜 E F} (hp : p ≠ 0)
Goal: p p.order ≠ 0