ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- norm_add_lt_of_not_sameRay
theorem norm_add_lt_of_not_sameRay (h : ¬SameRay ℝ x y) : ‖x + y‖ < ‖x‖ + ‖y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E} (h : ¬SameRay ℝ x y)
Goal: ‖x + y‖ < ‖x‖ + ‖y‖
-- lt_norm_sub_of_not_sameRay
theorem lt_norm_sub_of_not_sameRay (h : ¬SameRay ℝ x y) : ‖x‖ - ‖y‖ < ‖x - y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E} (h : ¬SameRay ℝ x y)
Goal: ‖x‖ - ‖y‖ < ‖x - y‖
-- abs_lt_norm_sub_of_not_sameRay
theorem abs_lt_norm_sub_of_not_sameRay (h : ¬SameRay ℝ x y) : |‖x‖ - ‖y‖| < ‖x - y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E} (h : ¬SameRay ℝ x y)
Goal: |‖x‖ - ‖y‖| < ‖x - y‖
-- sameRay_iff_norm_add
theorem sameRay_iff_norm_add : SameRay ℝ x y ↔ ‖x + y‖ = ‖x‖ + ‖y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E}
Goal: SameRay ℝ x y ↔ ‖x + y‖ = ‖x‖ + ‖y‖
-- eq_of_norm_eq_of_norm_add_eq
theorem eq_of_norm_eq_of_norm_add_eq (h₁ : ‖x‖ = ‖y‖) (h₂ : ‖x + y‖ = ‖x‖ + ‖y‖) : x = y 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E} (h₁ : ‖x‖ = ‖y‖) (h₂ : ‖x + y‖ = ‖x‖ + ‖y‖)
Goal: x = y
-- not_sameRay_iff_norm_add_lt
theorem not_sameRay_iff_norm_add_lt : ¬SameRay ℝ x y ↔ ‖x + y‖ < ‖x‖ + ‖y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E}
Goal: ¬SameRay ℝ x y ↔ ‖x + y‖ < ‖x‖ + ‖y‖
-- sameRay_iff_norm_sub
theorem sameRay_iff_norm_sub : SameRay ℝ x y ↔ ‖x - y‖ = |‖x‖ - ‖y‖| 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E}
Goal: SameRay ℝ x y ↔ ‖x - y‖ = |‖x‖ - ‖y‖|
-- not_sameRay_iff_abs_lt_norm_sub
theorem not_sameRay_iff_abs_lt_norm_sub : ¬SameRay ℝ x y ↔ |‖x‖ - ‖y‖| < ‖x - y‖ 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E}
Goal: ¬SameRay ℝ x y ↔ |‖x‖ - ‖y‖| < ‖x - y‖
-- norm_midpoint_lt_iff
theorem norm_midpoint_lt_iff (h : ‖x‖ = ‖y‖) : ‖(1 / 2 : ℝ) • (x + y)‖ < ‖x‖ ↔ x ≠ y 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℝ E] [StrictConvexSpace ℝ E] {x y : E} (h : ‖x‖ = ‖y‖)
Goal: ‖(1 / 2) • (x + y)‖ < ‖x‖ ↔ x ≠ y
-- SchwartzMap.smooth'
structure SchwartzMap where
  toFun : E → F
  smooth' : ContDiff ℝ ⊤ toFun
  decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n toFun x‖ ≤ C
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (self : SchwartzMap E F)
Goal: ContDiff ℝ ⊤ self.toFun
-- SchwartzMap.decay'
structure SchwartzMap where
  toFun : E → F
  smooth' : ContDiff ℝ ⊤ toFun
  decay' : ∀ k n : ℕ, ∃ C : ℝ, ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n toFun x‖ ≤ C
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (self : SchwartzMap E F) (k n : ℕ)
Goal: ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n self.toFun x‖ ≤ C
-- SchwartzMap.decay
theorem decay (f : 𝓢(E, F)) (k n : ℕ) :
    ∃ C : ℝ, 0 < C ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) (k n : ℕ)
Goal: ∃ C, 0 < C ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C
-- SchwartzMap.smooth
theorem smooth (f : 𝓢(E, F)) (n : ℕ∞) : ContDiff ℝ n f 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) (n : ℕ∞)
Goal: ContDiff ℝ n ⇑f
-- SchwartzMap.continuous
@[continuity]
protected theorem continuous (f : 𝓢(E, F)) : Continuous f 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F)
Goal: Continuous ⇑f
-- SchwartzMap.differentiable
protected theorem differentiable (f : 𝓢(E, F)) : Differentiable ℝ f 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F)
Goal: Differentiable ℝ ⇑f
-- SchwartzMap.differentiableAt
protected theorem differentiableAt (f : 𝓢(E, F)) {x : E} : DifferentiableAt ℝ f x 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) {x : E}
Goal: DifferentiableAt ℝ (⇑f) x
-- SchwartzMap.ext
@[ext]
theorem ext {f g : 𝓢(E, F)} (h : ∀ x, (f : E → F) x = g x) : f = g 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f g : SchwartzMap E F} (h : ∀ (x : E), f x = g x)
Goal: f = g
-- SchwartzMap.isBigO_cocompact_zpow_neg_nat
theorem isBigO_cocompact_zpow_neg_nat (k : ℕ) :
    f =O[cocompact E] fun x => ‖x‖ ^ (-k : ℤ) 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) (k : ℕ)
Goal: ⇑f =O[Filter.cocompact E] fun x => ‖x‖ ^ (-↑k)
-- SchwartzMap.isBigO_cocompact_rpow
theorem isBigO_cocompact_rpow [ProperSpace E] (s : ℝ) :
    f =O[cocompact E] fun x => ‖x‖ ^ s 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) [ProperSpace E] (s : ℝ)
Goal: ⇑f =O[Filter.cocompact E] fun x => ‖x‖ ^ s
-- SchwartzMap.isBigO_cocompact_zpow
theorem isBigO_cocompact_zpow [ProperSpace E] (k : ℤ) :
    f =O[cocompact E] fun x => ‖x‖ ^ k 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) [ProperSpace E] (k : ℤ)
Goal: ⇑f =O[Filter.cocompact E] fun x => ‖x‖ ^ k
-- SchwartzMap.bounds_nonempty
theorem bounds_nonempty (k n : ℕ) (f : 𝓢(E, F)) :
    ∃ c : ℝ, c ∈ { c : ℝ | 0 ≤ c ∧ ∀ x : E, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ c } 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f : SchwartzMap E F)
Goal: ∃ c, c ∈ {c | 0 ≤ c ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ c}
-- SchwartzMap.bounds_bddBelow
theorem bounds_bddBelow (k n : ℕ) (f : 𝓢(E, F)) :
    BddBelow { c | 0 ≤ c ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ c } 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f : SchwartzMap E F)
Goal: BddBelow {c | 0 ≤ c ∧ ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ c}
-- SchwartzMap.decay_add_le_aux
theorem decay_add_le_aux (k n : ℕ) (f g : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n ((f : E → F) + (g : E → F)) x‖ ≤
      ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ + ‖x‖ ^ k * ‖iteratedFDeriv ℝ n g x‖ 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f g : SchwartzMap E F) (x : E)
Goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f + ⇑g) x‖ ≤
  ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ + ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑g) x‖
-- SchwartzMap.decay_neg_aux
theorem decay_neg_aux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-f : E → F) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f : SchwartzMap E F) (x : E)
Goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖
-- SchwartzMap.decay_smul_aux
theorem decay_smul_aux (k n : ℕ) (f : 𝓢(E, F)) (c : 𝕜) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • (f : E → F)) x‖ =
      ‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (k n : ℕ) (f : SchwartzMap E F) (c : 𝕜) (x : E)
Goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (c • ⇑f) x‖ = ‖c‖ * ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖
-- SchwartzMap.seminormAux_nonneg
theorem seminormAux_nonneg (k n : ℕ) (f : 𝓢(E, F)) : 0 ≤ f.seminormAux k n 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f : SchwartzMap E F)
Goal: 0 ≤ SchwartzMap.seminormAux k n f
-- SchwartzMap.le_seminormAux
theorem le_seminormAux (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ f.seminormAux k n 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f : SchwartzMap E F) (x : E)
Goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ SchwartzMap.seminormAux k n f
-- SchwartzMap.seminormAux_le_bound
theorem seminormAux_le_bound (k n : ℕ) (f : 𝓢(E, F)) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ M) : f.seminormAux k n ≤ M 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f : SchwartzMap E F) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ M)
Goal: SchwartzMap.seminormAux k n f ≤ M
-- SchwartzMap.smul_apply
@[simp]
theorem smul_apply {f : 𝓢(E, F)} {c : 𝕜} {x : E} : (c • f) x = c • f x 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {f : SchwartzMap E F} {c : 𝕜} {x : E}
Goal: (c • f) x = c • f x
-- SchwartzMap.seminormAux_smul_le
theorem seminormAux_smul_le (k n : ℕ) (c : 𝕜) (f : 𝓢(E, F)) :
    (c • f).seminormAux k n ≤ ‖c‖ * f.seminormAux k n 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (k n : ℕ) (c : 𝕜) (f : SchwartzMap E F)
Goal: SchwartzMap.seminormAux k n (c • f) ≤ ‖c‖ * SchwartzMap.seminormAux k n f
-- SchwartzMap.coe_zero
theorem coe_zero : DFunLike.coe (0 : 𝓢(E, F)) = (0 : E → F) 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: ⇑0 = 0
-- SchwartzMap.coeFn_zero
@[simp]
theorem coeFn_zero : ⇑(0 : 𝓢(E, F)) = (0 : E → F) 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: ⇑0 = 0
-- SchwartzMap.zero_apply
@[simp]
theorem zero_apply {x : E} : (0 : 𝓢(E, F)) x = 0 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {x : E}
Goal: 0 x = 0
-- SchwartzMap.seminormAux_zero
theorem seminormAux_zero (k n : ℕ) : (0 : 𝓢(E, F)).seminormAux k n = 0 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ)
Goal: SchwartzMap.seminormAux k n 0 = 0
-- SchwartzMap.add_apply
@[simp]
theorem add_apply {f g : 𝓢(E, F)} {x : E} : (f + g) x = f x + g x 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f g : SchwartzMap E F} {x : E}
Goal: (f + g) x = f x + g x
-- SchwartzMap.seminormAux_add_le
theorem seminormAux_add_le (k n : ℕ) (f g : 𝓢(E, F)) :
    (f + g).seminormAux k n ≤ f.seminormAux k n + g.seminormAux k n 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (k n : ℕ) (f g : SchwartzMap E F)
Goal: SchwartzMap.seminormAux k n (f + g) ≤ SchwartzMap.seminormAux k n f + SchwartzMap.seminormAux k n g
-- SchwartzMap.sub_apply
@[simp]
theorem sub_apply {f g : 𝓢(E, F)} {x : E} : (f - g) x = f x - g x 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f g : SchwartzMap E F} {x : E}
Goal: (f - g) x = f x - g x
-- SchwartzMap.coe_coeHom
theorem coe_coeHom : (coeHom E F : 𝓢(E, F) → E → F) = DFunLike.coe 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: ⇑(SchwartzMap.coeHom E F) = DFunLike.coe
-- SchwartzMap.coeHom_injective
theorem coeHom_injective : Function.Injective (coeHom E F) 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: Function.Injective ⇑(SchwartzMap.coeHom E F)
-- SchwartzMap.seminorm_le_bound
theorem seminorm_le_bound (k n : ℕ) (f : 𝓢(E, F)) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ M) : SchwartzMap.seminorm 𝕜 k n f ≤ M 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (k n : ℕ) (f : SchwartzMap E F) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (x : E), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ M)
Goal: (SchwartzMap.seminorm 𝕜 k n) f ≤ M
-- SchwartzMap.seminorm_le_bound'
theorem seminorm_le_bound' (k n : ℕ) (f : 𝓢(ℝ, F)) {M : ℝ} (hMp : 0 ≤ M)
    (hM : ∀ x, |x| ^ k * ‖iteratedDeriv n f x‖ ≤ M) : SchwartzMap.seminorm 𝕜 k n f ≤ M 
Arguments: (𝕜 : Type u_1) {F : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (k n : ℕ) (f : SchwartzMap ℝ F) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (x : ℝ), |x| ^ k * ‖iteratedDeriv n (⇑f) x‖ ≤ M)
Goal: (SchwartzMap.seminorm 𝕜 k n) f ≤ M
-- SchwartzMap.le_seminorm
theorem le_seminorm (k n : ℕ) (f : 𝓢(E, F)) (x : E) :
    ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ SchwartzMap.seminorm 𝕜 k n f 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (k n : ℕ) (f : SchwartzMap E F) (x : E)
Goal: ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ (SchwartzMap.seminorm 𝕜 k n) f
-- SchwartzMap.le_seminorm'
theorem le_seminorm' (k n : ℕ) (f : 𝓢(ℝ, F)) (x : ℝ) :
    |x| ^ k * ‖iteratedDeriv n f x‖ ≤ SchwartzMap.seminorm 𝕜 k n f 
Arguments: (𝕜 : Type u_1) {F : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (k n : ℕ) (f : SchwartzMap ℝ F) (x : ℝ)
Goal: |x| ^ k * ‖iteratedDeriv n (⇑f) x‖ ≤ (SchwartzMap.seminorm 𝕜 k n) f
-- SchwartzMap.norm_iteratedFDeriv_le_seminorm
theorem norm_iteratedFDeriv_le_seminorm (f : 𝓢(E, F)) (n : ℕ) (x₀ : E) :
    ‖iteratedFDeriv ℝ n f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 n) f 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap E F) (n : ℕ) (x₀ : E)
Goal: ‖iteratedFDeriv ℝ n (⇑f) x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 n) f
-- SchwartzMap.norm_pow_mul_le_seminorm
theorem norm_pow_mul_le_seminorm (f : 𝓢(E, F)) (k : ℕ) (x₀ : E) :
    ‖x₀‖ ^ k * ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap E F) (k : ℕ) (x₀ : E)
Goal: ‖x₀‖ ^ k * ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f
-- SchwartzMap.norm_le_seminorm
theorem norm_le_seminorm (f : 𝓢(E, F)) (x₀ : E) : ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap E F) (x₀ : E)
Goal: ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f
-- SchwartzMap.schwartzSeminormFamily_apply
@[simp]
theorem schwartzSeminormFamily_apply (n k : ℕ) :
    schwartzSeminormFamily 𝕜 E F (n, k) = SchwartzMap.seminorm 𝕜 n k 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (n k : ℕ)
Goal: schwartzSeminormFamily 𝕜 E F (n, k) = SchwartzMap.seminorm 𝕜 n k
-- SchwartzMap.schwartzSeminormFamily_apply_zero
@[simp]
theorem schwartzSeminormFamily_apply_zero :
    schwartzSeminormFamily 𝕜 E F 0 = SchwartzMap.seminorm 𝕜 0 0 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F]
Goal: schwartzSeminormFamily 𝕜 E F 0 = SchwartzMap.seminorm 𝕜 0 0
-- SchwartzMap.one_add_le_sup_seminorm_apply
theorem one_add_le_sup_seminorm_apply {m : ℕ × ℕ} {k n : ℕ} (hk : k ≤ m.1) (hn : n ≤ m.2)
    (f : 𝓢(E, F)) (x : E) :
    (1 + ‖x‖) ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤
      2 ^ m.1 * (Finset.Iic m).sup (fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f 
Arguments: {𝕜 : Type u_1} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {m : ℕ × ℕ} {k n : ℕ} (hk : k ≤ m.1) (hn : n ≤ m.2) (f : SchwartzMap E F) (x : E)
Goal: (1 + ‖x‖) ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ 2 ^ m.1 * ((Finset.Iic m).sup fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f
-- schwartz_withSeminorms
theorem _root_.schwartz_withSeminorms : WithSeminorms (schwartzSeminormFamily 𝕜 E F) 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F]
Goal: WithSeminorms (schwartzSeminormFamily 𝕜 E F)
-- Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux
theorem _root_.Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux {f : E → F}
    (hf_temperate : f.HasTemperateGrowth) (n : ℕ) :
    ∃ (k : ℕ) (C : ℝ), 0 ≤ C ∧ ∀ N ≤ n, ∀ x : E, ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} (hf_temperate : Function.HasTemperateGrowth f) (n : ℕ)
Goal: ∃ k C, 0 ≤ C ∧ ∀ N ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k
-- Function.HasTemperateGrowth.of_fderiv
lemma _root_.Function.HasTemperateGrowth.of_fderiv {f : E → F}
    (h'f : Function.HasTemperateGrowth (fderiv ℝ f)) (hf : Differentiable ℝ f) {k : ℕ} {C : ℝ}
    (h : ∀ x, ‖f x‖ ≤ C * (1 + ‖x‖) ^ k) :
    Function.HasTemperateGrowth f 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F} (h'f : Function.HasTemperateGrowth (fderiv ℝ f)) (hf : Differentiable ℝ f) {k : ℕ} {C : ℝ} (h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k)
Goal: Function.HasTemperateGrowth f
-- Function.HasTemperateGrowth.zero
lemma _root_.Function.HasTemperateGrowth.zero :
    Function.HasTemperateGrowth (fun _ : E ↦ (0 : F)) 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F]
Goal: Function.HasTemperateGrowth fun x => 0
-- Function.HasTemperateGrowth.const
lemma _root_.Function.HasTemperateGrowth.const (c : F) :
    Function.HasTemperateGrowth (fun _ : E ↦ c) 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (c : F)
Goal: Function.HasTemperateGrowth fun x => c
-- ContinuousLinearMap.hasTemperateGrowth
lemma _root_.ContinuousLinearMap.hasTemperateGrowth (f : E →L[ℝ] F) :
    Function.HasTemperateGrowth f 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : E →L[ℝ] F)
Goal: Function.HasTemperateGrowth ⇑f
-- MeasureTheory.Measure.HasTemperateGrowth.exists_integrable
class _root_.MeasureTheory.Measure.HasTemperateGrowth (μ : Measure D) : Prop :=
  exists_integrable : ∃ (n : ℕ), Integrable (fun x ↦ (1 + ‖x‖) ^ (- (n : ℝ))) μ
Arguments: {D : Type u_3} [NormedAddCommGroup D] [MeasurableSpace D] {μ : MeasureTheory.Measure D} [self : μ.HasTemperateGrowth]
Goal: ∃ n, MeasureTheory.Integrable (fun x => (1 + ‖x‖) ^ (-↑n)) μ
-- SchwartzMap.integrable_pow_neg_integrablePower
lemma integrable_pow_neg_integrablePower
    (μ : Measure D) [h : μ.HasTemperateGrowth] :
    Integrable (fun x ↦ (1 + ‖x‖) ^ (- (μ.integrablePower : ℝ))) μ 
Arguments: {D : Type u_3} [NormedAddCommGroup D] [MeasurableSpace D] (μ : MeasureTheory.Measure D) [h : μ.HasTemperateGrowth]
Goal: MeasureTheory.Integrable (fun x => (1 + ‖x‖) ^ (-↑μ.integrablePower)) μ
-- SchwartzMap.pow_mul_le_of_le_of_pow_mul_le
lemma pow_mul_le_of_le_of_pow_mul_le {C₁ C₂ : ℝ} {k l : ℕ} {x f : ℝ} (hx : 0 ≤ x) (hf : 0 ≤ f)
    (h₁ : f ≤ C₁) (h₂ : x ^ (k + l) * f ≤ C₂) :
    x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (- (l : ℝ)) 
Arguments: {C₁ C₂ : ℝ} {k l : ℕ} {x f : ℝ} (hx : 0 ≤ x) (hf : 0 ≤ f) (h₁ : f ≤ C₁) (h₂ : x ^ (k + l) * f ≤ C₂)
Goal: x ^ k * f ≤ 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l)
-- SchwartzMap.integrable_of_le_of_pow_mul_le
lemma integrable_of_le_of_pow_mul_le
    {μ : Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ}
    (hf : ∀ x, ‖f x‖ ≤ C₁) (h'f : ∀ x, ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂)
    (h''f : AEStronglyMeasurable f μ) :
    Integrable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ 
Arguments: {D : Type u_3} {E : Type u_4} [NormedAddCommGroup E] [NormedAddCommGroup D] [MeasurableSpace D] [BorelSpace D] [SecondCountableTopology D] {μ : MeasureTheory.Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ} (hf : ∀ (x : D), ‖f x‖ ≤ C₁) (h'f : ∀ (x : D), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂) (h''f : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.Integrable (fun x => ‖x‖ ^ k * ‖f x‖) μ
-- SchwartzMap.integral_pow_mul_le_of_le_of_pow_mul_le
lemma integral_pow_mul_le_of_le_of_pow_mul_le
    {μ : Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ}
    (hf : ∀ x, ‖f x‖ ≤ C₁) (h'f : ∀ x, ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂) :
    ∫ x, ‖x‖ ^ k * ‖f x‖ ∂μ ≤ 2 ^ μ.integrablePower *
      (∫ x, (1 + ‖x‖) ^ (- (μ.integrablePower : ℝ)) ∂μ) * (C₁ + C₂) 
Arguments: {D : Type u_3} {E : Type u_4} [NormedAddCommGroup E] [NormedAddCommGroup D] [MeasurableSpace D] {μ : MeasureTheory.Measure D} [μ.HasTemperateGrowth] {f : D → E} {C₁ C₂ : ℝ} {k : ℕ} (hf : ∀ (x : D), ‖f x‖ ≤ C₁) (h'f : ∀ (x : D), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂)
Goal: ∫ (x : D), ‖x‖ ^ k * ‖f x‖ ∂μ ≤ (2 ^ μ.integrablePower * ∫ (x : D), (1 + ‖x‖) ^ (-↑μ.integrablePower) ∂μ) * (C₁ + C₂)
-- SchwartzMap.compCLM_apply
@[simp] lemma compCLM_apply {g : D → E} (hg : g.HasTemperateGrowth)
    (hg_upper : ∃ (k : ℕ) (C : ℝ), ∀ x, ‖x‖ ≤ C * (1 + ‖g x‖) ^ k) (f : 𝓢(E, F)) :
    compCLM 𝕜 hg hg_upper f = f ∘ g 
Arguments: (𝕜 : Type u_1) {D : Type u_3} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {g : D → E} (hg : Function.HasTemperateGrowth g) (hg_upper : ∃ k C, ∀ (x : D), ‖x‖ ≤ C * (1 + ‖g x‖) ^ k) (f : SchwartzMap E F)
Goal: ⇑((SchwartzMap.compCLM 𝕜 hg hg_upper) f) = ⇑f ∘ g
-- SchwartzMap.compCLMOfAntilipschitz_apply
@[simp] lemma compCLMOfAntilipschitz_apply {K : ℝ≥0} {g : D → E} (hg : g.HasTemperateGrowth)
    (h'g : AntilipschitzWith K g) (f : 𝓢(E, F)) :
    compCLMOfAntilipschitz 𝕜 hg h'g f = f ∘ g 
Arguments: (𝕜 : Type u_1) {D : Type u_3} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {K : ℝ≥0} {g : D → E} (hg : Function.HasTemperateGrowth g) (h'g : AntilipschitzWith K g) (f : SchwartzMap E F)
Goal: ⇑((SchwartzMap.compCLMOfAntilipschitz 𝕜 hg h'g) f) = ⇑f ∘ g