0ba60e62d7
Before this commit, an user could define their own prelude and change the types of quot, quot.mk, quot.lift or quot.ind. By doing that, they could prove false. This commit prevents this kind of abuse. It also modifies the definition of `quot` and avoids the `setoid` dependency. The previous `quot` type is now called `quotient`, and it is defined using the new `quot` type provided by the kernel. See discussion at #1330
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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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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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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