37 lines
1.8 KiB
Text
37 lines
1.8 KiB
Text
constant bag_setoid : ∀ A, setoid (list A)
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attribute [instance] bag_setoid
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definition bag (A : Type) : Type :=
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quotient (bag_setoid A)
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constant subcount : ∀ {A}, list A → list A → bool
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constant list.count : ∀ {A}, A → list A → nat
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constant all_of_subcount_eq_tt : ∀ {A} {l₁ l₂ : list A}, subcount l₁ l₂ = tt → ∀ a, list.count a l₁ ≤ list.count a l₂
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constant ex_of_subcount_eq_ff : ∀ {A} {l₁ l₂ : list A}, subcount l₁ l₂ = ff → ∃ a, ¬ list.count a l₁ ≤ list.count a l₂
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noncomputable definition count {A} (a : A) (b : bag A) : nat :=
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quotient.lift_on b (λ l, list.count a l)
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(λ l₁ l₂ h, sorry)
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definition subbag {A} (b₁ b₂ : bag A) := ∀ a, count a b₁ ≤ count a b₂
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infix ⊆ := subbag
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noncomputable definition decidable_subbag_1 {A} (b₁ b₂ : bag A) : decidable (b₁ ⊆ b₂) :=
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quotient.rec_on_subsingleton₂ b₁ b₂ (λ l₁ l₂,
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match subcount l₁ l₂, rfl : ∀ (b : _), subcount l₁ l₂ = b → _ with
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| tt, H := is_true (all_of_subcount_eq_tt H)
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| ff, H := is_false (λ h, exists.elim (ex_of_subcount_eq_ff H) (λ w hw, _))
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end)
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noncomputable definition decidable_subbag_2 {A} (b₁ b₂ : bag A) : decidable (b₁ ⊆ b₂) :=
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quotient.rec_on_subsingleton₂ b₁ b₂ (λ l₁ l₂,
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match subcount l₁ l₂, rfl : ∀ (b : _), subcount l₁ l₂ = b → _ with
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| tt, H := is_true (all_of_subcount_eq_tt H)
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| ff, H := is_false (λ h, exists.elim (ex_of_subcount_eq_ff H) _)
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end)
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noncomputable definition decidable_subbag_3 {A} (b₁ b₂ : bag A) : decidable (b₁ ⊆ b₂) :=
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quotient.rec_on_subsingleton₂ b₁ b₂ (λ l₁ l₂,
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match subcount l₁ l₂, rfl : ∀ (b : _), subcount l₁ l₂ = b → _ with
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| tt, H := is_true (all_of_subcount_eq_tt H)
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| ff, H := is_false (λ h, _)
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end)
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