internlm/Lean-Github huggingface dataset:
Theorem: W1.iteratedDeriv_sub
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n✝ n : ℕ
ih :
  ∀ {f g : ℝ → E},
    ContDiff ℝ (↑n) f → ContDiff ℝ (↑n) g → iteratedDeriv n (f - g) = iteratedDeriv n f - iteratedDeriv n g
f g : ℝ → E
hf : ContDiff ℝ (↑(n + 1)) f
hg : ContDiff ℝ (↑(n + 1)) g
hf' : ContDiff ℝ (↑n) (deriv f)
hg' : ContDiff ℝ (↑n) (deriv g)
x : ℝ
⊢ 1 ≤ ↑(n + 1)
Tactic:
simp
Proof state after:
no goals
Theorem: W1.iteratedDeriv_sub
Proof state before:
case h.hg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n✝ n : ℕ
ih :
  ∀ {f g : ℝ → E},
    ContDiff ℝ (↑n) f → ContDiff ℝ (↑n) g → iteratedDeriv n (f - g) = iteratedDeriv n f - iteratedDeriv n g
f g : ℝ → E
hf : ContDiff ℝ (↑(n + 1)) f
hg : ContDiff ℝ (↑(n + 1)) g
hf' : ContDiff ℝ (↑n) (deriv f)
hg' : ContDiff ℝ (↑n) (deriv g)
x : ℝ
⊢ DifferentiableAt ℝ (fun y => g y) x
Tactic:
exact (hg.differentiable (by simp)).differentiableAt
Proof state after:
no goals
Theorem: W1.integrable_iteratedDeriv_Schwarz
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : 𝓢(ℝ, ℂ)
⊢ Integrable (iteratedDeriv n ⇑f) volume
Tactic:
induction n generalizing f with
| zero => exact f.integrable
| succ n ih => simpa [iteratedDeriv_succ'] using ih (f := SchwartzMap.derivCLM ℝ f)
Proof state after:
no goals
Theorem: W1.integrable_iteratedDeriv_Schwarz
Proof state before:
case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : 𝓢(ℝ, ℂ)
⊢ Integrable (iteratedDeriv 0 ⇑f) volume
Tactic:
exact f.integrable
Proof state after:
no goals
Theorem: W1.integrable_iteratedDeriv_Schwarz
Proof state before:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n✝ n : ℕ
ih : ∀ {f : 𝓢(ℝ, ℂ)}, Integrable (iteratedDeriv n ⇑f) volume
f : 𝓢(ℝ, ℂ)
⊢ Integrable (iteratedDeriv (n + 1) ⇑f) volume
Tactic:
simpa [iteratedDeriv_succ'] using ih (f := SchwartzMap.derivCLM ℝ f)
Proof state after:
no goals
Theorem: W21.norm_nonneg
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f✝ : W21
f : ℝ → ℂ
t : ℝ
⊢ 0 t ≤ ‖f t‖
Tactic:
simp
Proof state after:
no goals
Theorem: W21.norm_nonneg
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f✝ : W21
f : ℝ → ℂ
⊢ 0 ≤ (4 * π ^ 2)⁻¹
Tactic:
positivity
Proof state after:
no goals
Theorem: W21.norm_nonneg
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f✝ : W21
f : ℝ → ℂ
t : ℝ
⊢ 0 t ≤ ‖deriv (deriv f) t‖
Tactic:
simp
Proof state after:
no goals
Theorem: W21.hf'
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f✝ f : W21
⊢ Integrable (deriv f.toFun) volume
Tactic:
simpa [iteratedDeriv_succ] using f.integrable one_le_two
Proof state after:
no goals
Theorem: W21.hf''
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f✝ f : W21
⊢ Integrable (deriv (deriv f.toFun)) volume
Tactic:
simpa [iteratedDeriv_succ] using f.integrable le_rfl
Proof state after:
no goals
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let f' := f.deriv
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let f'' := f'.deriv
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let g' := (g : CS 2 ℝ).deriv
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let g'' := g'.deriv
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let h R v := 1 - g.scale R v
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let h' R := - (g.scale R).deriv
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
let h'' R := - (g.scale R).deriv.deriv
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have ch {R} : Continuous (fun v => (h R v : ℂ)) := continuous_ofReal.comp <| continuous_const.sub (CS.continuous _)
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have ch' {R} : Continuous (fun v => (h' R v : ℂ)) := continuous_ofReal.comp (CS.continuous _)
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have ch'' {R} : Continuous (fun v => (h'' R v : ℂ)) := continuous_ofReal.comp (CS.continuous _)
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have dh R v : HasDerivAt (h R) (h' R v) v := by
  convert CS.hasDerivAt_scale (g : CS 2 ℝ) R v |>.const_sub 1 using 1
  simp [h', CS.deriv_scale'] ; left ; rfl
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have dh' R v : HasDerivAt (h' R) (h'' R v) v := ((g.scale R).deriv.hasDerivAt v).neg
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have hh1 R v : |h R v| ≤ 1 := by
  by_cases hR : R = 0 <;> simp [h, hR, CS.scale, funscale] ; rw [abs_le] ; constructor <;>
  linarith [g.le_one (R⁻¹ * v), g.nonneg (R⁻¹ * v)]
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
have vR v : Tendsto (fun R : ℝ => v * R⁻¹) atTop (𝓝 0) := by simpa using tendsto_inv_atTop_zero.const_mul v
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) atTop (𝓝 0)
Tactic:
convert_to Tendsto (fun R => W21.norm (fun v => h R v * f v)) atTop (𝓝 0)
Proof state after:
case h.e'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ (fun R => ‖f - W21.ofCS2 (g.scale R * f)‖) = fun R => W21.norm fun v => ↑(h R v) * f.toFun v

E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => W21.norm fun v => ↑(h R v) * f.toFun v) atTop (𝓝 0)
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => W21.norm fun v => ↑(h R v) * f.toFun v) atTop (𝓝 0)
Tactic:
rw [show (0 : ℝ) = 0 + ((4 * π ^ 2)⁻¹ : ℝ) * 0 by simp]
Proof state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => W21.norm fun v => ↑(h R v) * f.toFun v) atTop (𝓝 (0 + (4 * π ^ 2)⁻¹ * 0))
Theorem: W21_approximation
Proof state before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => W21.norm fun v => ↑(h R v) * f.toFun v) atTop (𝓝 (0 + (4 * π ^ 2)⁻¹ * 0))
Tactic:
refine Tendsto.add ?_ (Tendsto.const_mul _ ?_)
Proof state after:
case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun R => ∫ (v : ℝ), ‖(fun v => ↑(h R v) * f.toFun v) v‖) atTop (𝓝 0)

case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
n : ℕ
f : W21
g : trunc
f' : W1 1 ℂ := W1.deriv f
f'' : W1 0 ℂ := f'.deriv
g' : CS 1 ℝ := g.deriv
g'' : CS 0 ℝ := g'.deriv
h : ℝ → ℝ → ℝ := fun R v => 1 - (g.scale R).toFun v
h' : ℝ → CS 1 ℝ := fun R => -(g.scale R).deriv
h'' : ℝ → CS 0 ℝ := fun R => -(g.scale R).deriv.deriv
ch : ∀ {R : ℝ}, Continuous fun v => ↑(h R v)
ch' : ∀ {R : ℝ}, Continuous fun v => ↑((h' R).toFun v)
ch'' : ∀ {R : ℝ}, Continuous fun v => ↑((h'' R).toFun v)
dh : ∀ (R v : ℝ), HasDerivAt (h R) ((h' R).toFun v) v
dh' : ∀ (R v : ℝ), HasDerivAt (h' R).toFun ((h'' R).toFun v) v
hh1 : ∀ (R v : ℝ), |h R v| ≤ 1
vR : ∀ (v : ℝ), Tendsto (fun R => v * R⁻¹) atTop (𝓝 0)
⊢ Tendsto (fun k => ∫ (v : ℝ), ‖deriv (deriv fun v => ↑(h k v) * f.toFun v) v‖) atTop (𝓝 0)