Theorem: constFormalMultilinearSeries_zero
File: Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean

tactic: ext n x
state_before: 𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
⊢ constFormalMultilinearSeries 𝕜 E 0 = 0
state_after: case h.H
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : Fin n → E
⊢ (constFormalMultilinearSeries 𝕜 E 0 n) x = (0 n) x

tactic: simp only [FormalMultilinearSeries.zero_apply, ContinuousMultilinearMap.zero_apply,
  constFormalMultilinearSeries]
state_before: case h.H
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : Fin n → E
⊢ (constFormalMultilinearSeries 𝕜 E 0 n) x = (0 n) x
state_after: case h.H
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : Fin n → E
⊢ (match n with
      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
      | x => 0)
      x =
    0

tactic: induction n
state_before: case h.H
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
n : ℕ
x : Fin n → E
⊢ (match n with
      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
      | x => 0)
      x =
    0
state_after: case h.H.zero
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
x : Fin 0 → E
⊢ (match 0 with
      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
      | x => 0)
      x =
    0

case h.H.succ
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
n✝ : ℕ
a✝ :
  ∀ (x : Fin n✝ → E),
    (match n✝ with
        | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
        | x => 0)
        x =
      0
x : Fin (n✝ + 1) → E
⊢ (match n✝ + 1 with
      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
      | x => 0)
      x =
    0

tactic: simp only [Nat.zero_eq, ContinuousMultilinearMap.curry0_apply]
state_before: case h.H.zero
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
x : Fin 0 → E
⊢ (match 0 with
      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
      | x => 0)
      x =
    0
state_after: no goals

tactic: simp only [constFormalMultilinearSeries.match_1.eq_2, ContinuousMultilinearMap.zero_apply]
state_before: case h.H.succ
𝕜 : Type u
𝕜' : Type u'
E : Type v
F : Type w
G : Type x
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 F
n✝ : ℕ
a✝ :
  ∀ (x : Fin n✝ → E),
    (match n✝ with
        | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
        | x => 0)
        x =
      0
x : Fin (n✝ + 1) → E
⊢ (match n✝ + 1 with
      | 0 => ContinuousMultilinearMap.curry0 𝕜 E 0
      | x => 0)
      x =
    0
state_after: no goals