feat(library/data/countable): choice function for countable types
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@ -7,7 +7,7 @@ Author: Leonardo de Moura
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Type class for countable types
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-/
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import data.fintype data.list data.sum data.nat
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import data.fintype data.list data.sum data.nat data.subtype
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open option list nat
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structure countable [class] (A : Type) :=
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@ -214,3 +214,90 @@ countable.mk
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esimp [option.cases_on],
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rewrite [linv]
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end)
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/-
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Choice function for countable types and decidable predicates.
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We provide the following API
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choose {A : Type} {p : A → Prop} [c : countable A] [d : decidable_pred p] : (∃ x, p x) → A :=
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choose_spec {A : Type} {p : A → Prop} [c : countable A] [d : decidable_pred p] (ex : ∃ x, p x) : p (choose ex) :=
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-/
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section find_a
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parameters {A : Type} {p : A → Prop} [c : countable A] [d : decidable_pred p]
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include c
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include d
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private definition pn (n : nat) : Prop :=
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match unpickle A n with
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| some a := p a
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| none := false
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end
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private definition decidable_pn : decidable_pred pn :=
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λ n,
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match unpickle A n with
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| some a := λ e : unpickle A n = some a,
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match d a with
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| decidable.inl t :=
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begin
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unfold pn, rewrite e, esimp [option.cases_on],
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exact (decidable.inl t)
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end
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| decidable.inr f :=
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begin
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unfold pn, rewrite e, esimp [option.cases_on],
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exact (decidable.inr f)
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end
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end
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| none := λ e : unpickle A n = none,
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begin
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unfold pn, rewrite e, esimp [option.cases_on],
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exact decidable_false
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end
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end (eq.refl (unpickle A n))
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private definition ex_pn_of_ex : (∃ x, p x) → (∃ x, pn x) :=
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assume ex,
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obtain (w : A) (pw : p w), from ex,
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exists.intro (pickle w)
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begin
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unfold pn, rewrite [picklek], esimp, exact pw
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end
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private lemma unpickle_ne_none_of_pn {n : nat} : pn n → unpickle A n ≠ none :=
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assume pnn e,
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begin
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rewrite [▸ (match unpickle A n with | some a := p a | none := false end) at pnn],
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rewrite [e at pnn], esimp [option.cases_on] at pnn,
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exact (false.elim pnn)
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end
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open subtype
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private lemma of_nat (n : nat) : pn n → { a : A | p a } :=
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match unpickle A n with
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| some a := λ (e : unpickle A n = some a),
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begin
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unfold pn, rewrite e, esimp [option.cases_on], intro pa,
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exact (tag a pa)
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end
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| none := λ (e : unpickle A n = none) h, absurd e (unpickle_ne_none_of_pn h)
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end (eq.refl (unpickle A n))
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private definition find_a : (∃ x, p x) → {a : A | p a} :=
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assume ex : ∃ x, p x,
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have exn : ∃ x, pn x, from ex_pn_of_ex ex,
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let r : nat := @nat.choose pn decidable_pn exn in
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have pnr : pn r, from @nat.choose_spec pn decidable_pn exn,
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of_nat r pnr
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end find_a
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namespace countable
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open subtype
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definition choose {A : Type} {p : A → Prop} [c : countable A] [d : decidable_pred p] : (∃ x, p x) → A :=
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assume ex, elt_of (find_a ex)
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theorem choose_spec {A : Type} {p : A → Prop} [c : countable A] [d : decidable_pred p] (ex : ∃ x, p x) : p (choose ex) :=
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has_property (find_a ex)
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end countable
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