constant bag_setoid : ∀ A, setoid (list A)
attribute [instance] bag_setoid


noncomputable definition bag (A : Type) : Type :=
quotient (bag_setoid A)

constant subcount : ∀ {A}, list A → list A → bool
constant list.count : ∀ {A}, A → list A → nat
constant all_of_subcount_eq_tt : ∀ {A} {l₁ l₂ : list A}, subcount l₁ l₂ = tt → ∀ a, list.count a l₁ ≤ list.count a l₂
constant ex_of_subcount_eq_ff : ∀ {A} {l₁ l₂ : list A}, subcount l₁ l₂ = ff → ∃ a, ¬ list.count a l₁ ≤ list.count a l₂
noncomputable definition count {A} (a : A) (b : bag A) : nat :=
quotient.lift_on b (λ l, list.count a l)
  (λ l₁ l₂ h, sorry)
noncomputable definition subbag {A} (b₁ b₂ : bag A) := ∀ a, count a b₁ ≤ count a b₂
infix ⊆ := subbag

noncomputable definition decidable_subbag_1 {A} (b₁ b₂ : bag A) : decidable (b₁ ⊆ b₂) :=
quotient.rec_on_subsingleton₂ b₁ b₂ (λ l₁ l₂,
  match subcount l₁ l₂, rfl : ∀ (b : _), subcount l₁ l₂ = b → _ with
  | tt, H := is_true (all_of_subcount_eq_tt H)
  | ff, H := is_false (λ h, exists.elim (ex_of_subcount_eq_ff H) (λ w hw, _))
  end)

noncomputable definition decidable_subbag_2 {A} (b₁ b₂ : bag A) : decidable (b₁ ⊆ b₂) :=
quotient.rec_on_subsingleton₂ b₁ b₂ (λ l₁ l₂,
  match subcount l₁ l₂, rfl : ∀ (b : _), subcount l₁ l₂ = b → _ with
  | tt, H := is_true (all_of_subcount_eq_tt H)
  | ff, H := is_false (λ h, exists.elim (ex_of_subcount_eq_ff H) _)
  end)

noncomputable definition decidable_subbag_3 {A} (b₁ b₂ : bag A) : decidable (b₁ ⊆ b₂) :=
quotient.rec_on_subsingleton₂ b₁ b₂ (λ l₁ l₂,
  match subcount l₁ l₂, rfl : ∀ (b : _), subcount l₁ l₂ = b → _ with
  | tt, H := is_true (all_of_subcount_eq_tt H)
  | ff, H := is_false (λ h, _)
  end)