import data.list open list setoid quot decidable nat namespace finset definition eqv {A : Type} (l₁ l₂ : list A) := ∀ a, a ∈ l₁ ↔ a ∈ l₂ infix `~` := eqv theorem eqv.refl {A : Type} (l : list A) : l ~ l := λ a, !iff.refl theorem eqv.symm {A : Type} {l₁ l₂ : list A} : l₁ ~ l₂ → l₂ ~ l₁ := λ H a, iff.symm (H a) theorem eqv.trans {A : Type} {l₁ l₂ l₃ : list A} : l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃ := λ H₁ H₂ a, iff.trans (H₁ a) (H₂ a) theorem eqv.is_equivalence (A : Type) : equivalence (@eqv A) := and.intro (@eqv.refl A) (and.intro (@eqv.symm A) (@eqv.trans A)) definition norep {A : Type} [H : decidable_eq A] : list A → list A | [] := [] | (x :: xs) := if x ∈ xs then norep xs else x :: norep xs definition eqv_norep {A : Type} [H : decidable_eq A] : ∀ l : list A, l ~ norep l | [] := (λ a, iff.rfl) | (x :: xs) := take y, assert ih : xs ~ norep xs, from eqv_norep xs, show y ∈ x :: xs ↔ y ∈ if x ∈ xs then norep xs else x :: norep xs, begin apply (@by_cases (x ∈ xs)), begin intro xin, rewrite (if_pos xin), apply iff.intro, {intro yinxxs, apply (or.elim (iff.mp !mem_cons_iff yinxxs)), intro yeqx, rewrite -yeqx at xin, exact (iff.mp (ih y) xin), intro yeqxs, exact (iff.mp (ih y) yeqxs)}, {intro yinnrep, show y ∈ x::xs, from or.inr (iff.mp' (ih y) yinnrep)} end, begin intro xnin, rewrite (if_neg xnin), apply iff.intro, {intro yinxxs, apply (or.elim (iff.mp !mem_cons_iff yinxxs)), intro yeqx, rewrite yeqx, apply mem_cons, intro yinxs, show y ∈ x:: norep xs, from or.inr (iff.mp (ih y) yinxs)}, {intro yinxnrep, apply (or.elim (iff.mp !mem_cons_iff yinxnrep)), intro yeqx, rewrite yeqx, apply mem_cons, intro yinrep, show y ∈ x::xs, from or.inr (iff.mp' (ih y) yinrep)} end end definition sub {A : Type} (l₁ l₂ : list A) := ∀ a, a ∈ l₁ → a ∈ l₂ infix ⊆ := sub theorem eqv_of_sub_of_sub {A : Type} {l₁ l₂ : list A} : l₁ ⊆ l₂ → l₂ ⊆ l₁ → l₁ ~ l₂ := assume h₁ h₂ a, iff.intro (h₁ a) (h₂ a) theorem sub_of_eqv_left {A : Type} {l₁ l₂ : list A} : l₁ ~ l₂ → l₁ ⊆ l₂ := assume h₁ a ainl₁, iff.mp (h₁ a) ainl₁ theorem sub_of_eqv_right {A : Type} {l₁ l₂ : list A} : l₁ ~ l₂ → l₂ ⊆ l₁ := assume h₁ a ainl₂, iff.mp' (h₁ a) ainl₂ definition sub_of_cons_sub {A : Type} {a : A} {l₁ l₂ : list A} : a :: l₁ ⊆ l₂ → l₁ ⊆ l₂ := assume s, take b, assume binl₁, s b (or.inr binl₁) definition decidable_sub [instance] {A : Type} [H : decidable_eq A] : ∀ l₁ l₂ : list A, decidable (l₁ ⊆ l₂) | [] ys := inl (λ a h, absurd h !not_mem_nil) | (x::xs) ys := if xinys : x ∈ ys then match decidable_sub xs ys with | (inl xs_sub_ys) := inl (λ y yinxxs, or.elim (iff.mp !mem_cons_iff yinxxs) (λ yeqx, by rewrite yeqx; exact xinys) (λ yinxs, xs_sub_ys y yinxs)) | (inr nxs_sub_ys) := inr (λ h, absurd (sub_of_cons_sub h) nxs_sub_ys) end else inr (λ h, absurd (h x !mem_cons) xinys) example : [(1 : nat), 2, 3] ⊆ [1,3,4,1,2] := dec_trivial definition decidable_eqv [instance] {A : Type} [H : decidable_eq A] : ∀ l₁ l₂ : list A, decidable (l₁ ~ l₂) := take l₁ l₂ : list A, match decidable_sub l₁ l₂ with | (inl s₁) := match decidable_sub l₂ l₁ with | (inl s₂) := inl (eqv_of_sub_of_sub s₁ s₂) | (inr n₂) := inr (λ h, absurd (sub_of_eqv_right h) n₂) end | (inr n₁) := inr (λ h, absurd (sub_of_eqv_left h) n₁) end example : [(1:nat), 2, 3, 2, 2, 2] ~ [1,3,3,1,2] := dec_trivial definition finset_setoid [instance] (A : Type) : setoid (list A) := setoid.mk (@eqv A) (eqv.is_equivalence A) definition finset (A : Type) : Type := quot (finset_setoid A) definition has_decidable_eq [instance] {A : Type} [H : decidable_eq A] : ∀ s₁ s₂ : finset A, decidable (s₁ = s₂) := take s₁ s₂, quot.rec_on_subsingleton₂ s₁ s₂ (take l₁ l₂, match decidable_eqv l₁ l₂ with | inl e := inl (quot.sound e) | inr d := inr (λ e, absurd (quot.exact e) d) end) definition to_finset {A : Type} (l : list A) : finset A := ⟦l⟧ definition mem {A : Type} (a : A) (s : finset A) : Prop := quot.lift_on s (λ l : list A, a ∈ l) (λ l₁ l₂ r, propext (r a)) infix ∈ := mem theorem mem_list {A : Type} {a : A} {l : list A} : a ∈ l → a ∈ ⟦l⟧ := λ H, H definition empty {A : Type} : finset A := ⟦nil⟧ notation `∅` := empty definition union {A : Type} (s₁ s₂ : finset A) : finset A := quot.lift_on₂ s₁ s₂ (λ l₁ l₂ : list A, ⟦l₁ ++ l₂⟧) (λ l₁ l₂ l₃ l₄ r₁ r₂, begin apply quot.sound, intro a, apply iff.intro, begin intro inl₁l₂, apply (or.elim (mem_or_mem_of_mem_append inl₁l₂)), intro inl₁, exact (mem_append_of_mem_or_mem (or.inl (iff.mp (r₁ a) inl₁))), intro inl₂, exact (mem_append_of_mem_or_mem (or.inr (iff.mp (r₂ a) inl₂))) end, begin intro inl₃l₄, apply (or.elim (mem_or_mem_of_mem_append inl₃l₄)), intro inl₃, exact (mem_append_of_mem_or_mem (or.inl (iff.mp' (r₁ a) inl₃))), intro inl₄, exact (mem_append_of_mem_or_mem (or.inr (iff.mp' (r₂ a) inl₄))) end, end) infix `∪` := union theorem mem_union_left {A : Type} (s₁ s₂ : finset A) (a : A) : a ∈ s₁ → a ∈ s₁ ∪ s₂ := quot.ind₂ (λ l₁ l₂ ainl₁, mem_append_left l₂ ainl₁) s₁ s₂ theorem mem_union_right {A : Type} (s₁ s₂ : finset A) (a : A) : a ∈ s₂ → a ∈ s₁ ∪ s₂ := quot.ind₂ (λ l₁ l₂ ainl₂, mem_append_right l₁ ainl₂) s₁ s₂ theorem union_empty {A : Type} (s : finset A) : s ∪ ∅ = s := quot.induction_on s (λ l, quot.sound (λ a, by rewrite append_nil_right)) theorem empty_union {A : Type} (s : finset A) : ∅ ∪ s = s := quot.induction_on s (λ l, quot.sound (λ a, by rewrite append_nil_left)) example : to_finset (1::2::nil) ∪ to_finset (2::3::nil) = ⟦1 :: 2 :: 2 :: 3 :: nil⟧ := rfl example : to_finset [(1:nat), 1, 2, 3] = to_finset [2, 3, 1, 2, 2, 3] := dec_trivial example : to_finset [(1:nat), 1, 4, 2, 3] ≠ to_finset [2, 3, 1, 2, 2, 3] := dec_trivial definition clean {A : Type} [H : decidable_eq A] (s : finset A) : finset A := quot.lift_on s (λ l, ⟦norep l⟧) (λ l₁ l₂ e, calc ⟦norep l₁⟧ = ⟦l₁⟧ : quot.sound (eqv_norep l₁) ... = ⟦l₂⟧ : quot.sound e ... = ⟦norep l₂⟧ : quot.sound (eqv_norep l₂)) theorem eq_clean {A : Type} [H : decidable_eq A] : ∀ s : finset A, clean s = s := take s, quot.induction_on s (λ l, eq.symm (quot.sound (eqv_norep l))) theorem eq_of_clean_eq_clean {A : Type} [H : decidable_eq A] : ∀ s₁ s₂ : finset A, clean s₁ = clean s₂ → s₁ = s₂ := take s₁ s₂, by rewrite *eq_clean; intro H; apply H example : to_finset [(1:nat), 1, 2, 3] = to_finset [1, 2, 2, 2, 3, 3] := !eq_of_clean_eq_clean rfl end finset