73 lines
2.7 KiB
Text
73 lines
2.7 KiB
Text
definition foo (a b : bool) : bool :=
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match a, b with
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| tt, ff := tt
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| tt, tt := tt
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| ff, tt := tt
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| ff, ff := ff
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end
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example : foo tt tt = tt := rfl
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example : foo tt ff = tt := rfl
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example : foo ff tt = tt := rfl
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example : foo ff ff = ff := rfl
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inductive vec (A : Type) : nat → Type
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| nil {} : vec nat.zero
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| cons : ∀ {n}, A → vec n → vec (nat.succ n)
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open vec
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definition boo (n : nat) (v : vec bool n) : vec bool n :=
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match n, v : ∀ (n : _), vec bool n → _ with
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| 0, nil := nil
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| n+1, cons a v := cons (bnot a) v
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end
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constant bag_setoid : ∀ A, setoid (list A)
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attribute [instance] bag_setoid
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noncomputable 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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noncomputable definition subbag {A} (b₁ b₂ : bag A) := ∀ a, count a b₁ ≤ count a b₂
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infix ⊆ := subbag
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attribute [instance]
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noncomputable definition decidable_subbag {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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exists.elim (ex_of_subcount_eq_ff H)
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(λ w hw, absurd (h w) hw))
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end)
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attribute [instance]
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noncomputable definition decidable_subbag2 {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 psigma.mk (subcount l₁ l₂) rfl : (Σ' (b : _), subcount l₁ l₂ = b) → _ with
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| psigma.mk tt H := is_true (all_of_subcount_eq_tt H)
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| psigma.mk ff H := is_false (λ h,
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exists.elim (ex_of_subcount_eq_ff H)
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(λ w hw, absurd (h w) hw))
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end)
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local notation ⟦ a , b ⟧ := psigma.mk a b
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attribute [instance]
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noncomputable definition decidable_subbag3 {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⟧ : (Σ' (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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exists.elim (ex_of_subcount_eq_ff H)
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(λ w hw, absurd (h w) hw))
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end)
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