Theorem: FormalMultilinearSeries.comp_rightInv_aux1
File: Mathlib/Analysis/Analytic/Inverse.lean

Proof steps:
state_before: 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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
A : univ = {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}
⊢ (p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v
state_after: 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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
A : univ = {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}
B : Disjoint {c | 1 < c.length}.toFinset {Composition.single n hn}
⊢ (p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v
tactic: have B :
    Disjoint ({c | 1 < Composition.length c} : Set (Composition n)).toFinset
      {Composition.single n hn} := by
    simp [Set.mem_toFinset (s := {c | 1 < Composition.length c})]
state_before: 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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
A : univ = {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}
B : Disjoint {c | 1 < c.length}.toFinset {Composition.single n hn}
⊢ (p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v
state_after: 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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
A : univ = {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}
B : Disjoint {c | 1 < c.length}.toFinset {Composition.single n hn}
C : (p (Composition.single n hn).length) (q.applyComposition (Composition.single n hn) v) = (p 1) fun x => (q n) v
⊢ (p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v
tactic: have C :
    p (Composition.single n hn).length (q.applyComposition (Composition.single n hn) v) =
      p 1 fun _ : Fin 1 => q n v := by
    apply p.congr (Composition.single_length hn) fun j hj1 _ => ?_
    simp [applyComposition_single]
state_before: 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
n : ℕ
hn : 0 < n
p : FormalMultilinearSeries 𝕜 E F
q : FormalMultilinearSeries 𝕜 F E
v : Fin n → F
A : univ = {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}
B : Disjoint {c | 1 < c.length}.toFinset {Composition.single n hn}
C : (p (Composition.single n hn).length) (q.applyComposition (Composition.single n hn) v) = (p 1) fun x => (q n) v
⊢ (p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v
state_after: no goals
tactic: simp [FormalMultilinearSeries.comp, A, Finset.sum_union B, C, -Set.toFinset_setOf,
    -add_right_inj, -Composition.single_length]