-- Mathlib/Analysis/Analytic/Inverse.lean
theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
    (leftInv p i x).comp p = id 𝕜 E x

Proof state:
case convert_2.e_f.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j => (p 1) fun x => v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j => (p 1) fun x => v j
c : Composition (n + 2)
⊢ (p.leftInv i x c.length) (p.applyComposition c (⇑↑i.symm ∘ fun j => (p 1) fun x => v j)) =
    (p.leftInv i x c.length) (p.applyComposition c v)

Tactic:
congr

Result:
case convert_2.e_f.h.h.e_6.h
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j => (p 1) fun x => v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j => (p 1) fun x => v j
c : Composition (n + 2)
⊢ p.applyComposition c (⇑↑i.symm ∘ fun j => (p 1) fun x => v j) = p.applyComposition c v

Proof state:
case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h✝ : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
c : Composition (n + 2)
x✝ : c ∈ univ
h : ¬c.length < n + 2
⊢ c ∈ {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}

Tactic:
simp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)]

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
⊢ ((p.leftInv i x).comp p (n + 2)) v = (id 𝕜 E x (n + 2)) v

Tactic:
have B :
  Disjoint ({c | Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset
    {Composition.ones (n + 2)} := by
  simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})]

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
⊢ ((p.leftInv i x).comp p (n + 2)) v = (id 𝕜 E x (n + 2)) v

Proof state:
case convert_2
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j => (p 1) fun x => v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j => (p 1) fun x => v j
⊢ (∑ a ∈ {x | x.length < n + 2}.toFinset,
      (ContinuousMultilinearMap.compAlongComposition (p.compContinuousLinearMap ↑i.symm) a (p.leftInv i x a.length))
        fun j => (p 1) fun x => v j) =
    ∑ c ∈ {c | c.length < n + 2}.toFinset, (p.leftInv i x c.length) (p.applyComposition c v)

Tactic:
simp only [compContinuousLinearMap_applyComposition,
  ContinuousMultilinearMap.compAlongComposition_apply]

Result:
case convert_2
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
A : univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}
B : Disjoint {c | c.length < n + 2}.toFinset {Composition.ones (n + 2)}
C :
  ((p.leftInv i x (Composition.ones (n + 2)).length) fun j => (p 1) fun x => v (Fin.castLE ⋯ j)) =
    (p.leftInv i x (n + 2)) fun j => (p 1) fun x => v j
⊢ ∑ x_1 ∈ {x | x.length < n + 2}.toFinset,
      (p.leftInv i x x_1.length) (p.applyComposition x_1 (⇑↑i.symm ∘ fun j => (p 1) fun x => v j)) =
    ∑ c ∈ {x | x.length < n + 2}.toFinset, (p.leftInv i x c.length) (p.applyComposition c v)

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n : ℕ
v : Fin 0 → E
⊢ ((p.leftInv i x).comp p 0) v = (id 𝕜 E x 0) v

Tactic:
simp only [comp_coeff_zero', leftInv_coeff_zero, ContinuousMultilinearMap.uncurry0_apply,
  id_apply_zero]

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
⊢ univ = {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}

Tactic:
refine Subset.antisymm (fun c _ => ?_) (subset_univ _)

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n✝ n : ℕ
v : Fin (n + 2) → E
c : Composition (n + 2)
x✝ : c ∈ univ
⊢ c ∈ {c | c.length < n + 2}.toFinset ∪ {Composition.ones (n + 2)}

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
x : E
h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i
n : ℕ
v : Fin 1 → E
⊢ ((p.leftInv i x).comp p 1) v = (id 𝕜 E x 1) v

Tactic:
simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
  ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply]

Result:
no goals