theorem perm.perm_erase_dup_of_perm [congr] : ∀ {A : Type} [H : decidable_eq A] {l₁ l₂ : list A}, l₁ ~ l₂ → erase_dup l₁ ~ erase_dup l₂ :=
λ (A : Type) (H : decidable_eq A) (l₁ l₂ : list A) (p : l₁ ~ l₂),
  perm_induction_on p nil
    (λ (x : A) (t₁ t₂ : list A) (p : t₁ ~ t₂) (r : erase_dup t₁ ~ erase_dup t₂),
       decidable.by_cases
         (λ (xint₁ : x ∈ t₁), assert xint₂ : x ∈ t₂, from mem_of_mem_erase_dup …, … …)
         (λ (nxint₁ : x ∉ t₁),
            assert nxint₂ : x ∉ t₂, from λ (xint₂ : x ∈ t₂), … nxint₁,
            eq.rec … (eq.symm …)))
    (λ (y x : A) (t₁ t₂ : list A) (p : t₁ ~ t₂) (r : erase_dup t₁ ~ erase_dup t₂),
       decidable.by_cases
         (λ (xinyt₁ : x ∈ y :: t₁),
            decidable.by_cases (λ (yint₁ : …), …)
              (λ (nyint₁ : y ∉ t₁), assert nyint₂ : …, from …, …))
         (λ (nxinyt₁ : x ∉ y :: t₁),
            have xney : x ≠ y, from ne_of_not_mem_cons nxinyt₁,
            have nxint₁ : x ∉ t₁, from not_mem_of_not_mem_cons nxinyt₁,
            assert nxint₂ : x ∉ t₂, from λ (xint₂ : …), …,
            … …))
    (λ (t₁ t₂ t₃ : list A) (p₁ : t₁ ~ t₂) (p₂ : t₂ ~ t₃), trans)