lean2/library/data/set/finite.lean

/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad

The notion of "finiteness" for sets. This approach is not computational: for example, just because
an element  s : set A  satsifies  finite s  doesn't mean that we can compute the cardinality. For
a computational representation, use the finset type.
-/
import data.finset.to_set .classical_inverse
open nat classical

variable {A : Type}

namespace set

definition finite [class] (s : set A) : Prop := ∃ (s' : finset A), s = finset.to_set s'

theorem finite_finset [instance] (s : finset A) : finite (finset.to_set s) :=
exists.intro s rfl

/- to finset: casts every set to a finite set -/

noncomputable definition to_finset (s : set A) : finset A :=
if fins : finite s then some fins else finset.empty

theorem to_finset_of_not_finite {s : set A} (nfins : ¬ finite s) : to_finset s = (#finset ∅) :=
by rewrite [↑to_finset, dif_neg nfins]

theorem to_set_to_finset (s : set A) [fins : finite s] : finset.to_set (to_finset s) = s :=
by rewrite [↑to_finset, dif_pos fins]; exact eq.symm (some_spec fins)

theorem mem_to_finset_eq (a : A) (s : set A) [finite s] :
  (#finset a ∈ to_finset s) = (a ∈ s) :=
by rewrite [-to_set_to_finset at {2}]

theorem to_set_to_finset_of_not_finite {s : set A} (nfins : ¬ finite s) :
  finset.to_set (to_finset s) = ∅ :=
by rewrite [to_finset_of_not_finite nfins]

theorem to_finset_to_set (s : finset A) : to_finset (finset.to_set s) = s :=
by rewrite [finset.eq_eq_to_set_eq, to_set_to_finset (finset.to_set s)]

theorem to_finset_eq_of_to_set_eq {s : set A} {t : finset A} (H : finset.to_set t = s) :
  to_finset s = t :=
finset.eq_of_to_set_eq_to_set (by subst [s]; rewrite to_finset_to_set)

/- finiteness -/

theorem finite_of_to_set_to_finset_eq {s : set A} (H : finset.to_set (to_finset s) = s) :
  finite s :=
by rewrite -H; apply finite_finset

theorem finite_empty [instance] : finite (∅ : set A) :=
by rewrite [-finset.to_set_empty]; apply finite_finset

theorem to_finset_empty : to_finset (∅ : set A) = (#finset ∅) :=
to_finset_eq_of_to_set_eq !finset.to_set_empty

theorem to_finset_eq_empty_of_eq_empty {s : set A} [fins : finite s] (H : s = ∅) :
  to_finset s = finset.empty := by rewrite [H, to_finset_empty]

theorem eq_empty_of_to_finset_eq_empty {s : set A} [fins : finite s]
    (H : to_finset s = finset.empty) :
  s = ∅ := by rewrite [-finset.to_set_empty, -H, to_set_to_finset]

theorem to_finset_eq_empty (s : set A) [fins : finite s] :
  (to_finset s = finset.empty) ↔ (s = ∅) :=
iff.intro eq_empty_of_to_finset_eq_empty to_finset_eq_empty_of_eq_empty

theorem finite_insert [instance] (a : A) (s : set A) [finite s] : finite (insert a s) :=
exists.intro (finset.insert a (to_finset s))
  (by rewrite [finset.to_set_insert, to_set_to_finset])

theorem to_finset_insert (a : A) (s : set A) [finite s] :
  to_finset (insert a s) = finset.insert a (to_finset s) :=
by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_insert, to_set_to_finset]

theorem finite_union [instance] (s t : set A) [finite s] [finite t] :
  finite (s ∪ t) :=
exists.intro (#finset to_finset s ∪ to_finset t)
  (by rewrite [finset.to_set_union, *to_set_to_finset])

theorem to_finset_union (s t : set A) [finite s] [finite t] :
  to_finset (s ∪ t) = (#finset to_finset s ∪ to_finset t) :=
by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_union, *to_set_to_finset]

theorem finite_inter [instance] (s t : set A) [finite s] [finite t] :
  finite (s ∩ t) :=
exists.intro (#finset to_finset s ∩ to_finset t)
  (by rewrite [finset.to_set_inter, *to_set_to_finset])

theorem to_finset_inter (s t : set A) [finite s] [finite t] :
  to_finset (s ∩ t) = (#finset to_finset s ∩ to_finset t) :=
by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_inter, *to_set_to_finset]

theorem finite_sep [instance] (s : set A) (p : A → Prop) [finite s] :
  finite {x ∈ s | p x}  :=
exists.intro (finset.sep p (to_finset s))
  (by rewrite [finset.to_set_sep, *to_set_to_finset])

theorem to_finset_sep (s : set A) (p : A → Prop) [finite s] :
  to_finset {x ∈ s | p x} = (#finset {x ∈ to_finset s | p x}) :=
by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_sep, to_set_to_finset]

theorem finite_image [instance] {B : Type} (f : A → B) (s : set A) [finite s] :
  finite (f ' s) :=
exists.intro (finset.image f (to_finset s))
  (by rewrite [finset.to_set_image, *to_set_to_finset])

theorem to_finset_image {B : Type}  (f : A → B) (s : set A)
    [fins : finite s] :
  to_finset (f ' s) = (#finset f ' (to_finset s)) :=
by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_image, to_set_to_finset]

theorem finite_diff [instance] (s t : set A) [finite s] : finite (s \ t) :=
!finite_sep

theorem to_finset_diff (s t : set A) [finite s] [finite t] :
        to_finset (s \ t) = (#finset to_finset s \ to_finset t) :=
by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_diff, *to_set_to_finset]

theorem finite_subset {s t : set A} [finite t] (ssubt : s ⊆ t) : finite s :=
by rewrite (eq_sep_of_subset ssubt); apply finite_sep

theorem to_finset_subset_to_finset_eq (s t : set A) [finite s] [finite t] :
  (#finset to_finset s ⊆ to_finset t) = (s ⊆ t) :=
by rewrite [finset.subset_eq_to_set_subset, *to_set_to_finset]

theorem finite_of_finite_insert {s : set A} {a : A} (finias : finite (insert a s)) : finite s :=
finite_subset (subset_insert a s)

theorem finite_upto [instance] (n : ℕ) : finite {i | i < n} :=
by rewrite [-finset.to_set_upto n]; apply finite_finset

theorem to_finset_upto (n : ℕ) : to_finset {i | i < n} = finset.upto n :=
by apply (to_finset_eq_of_to_set_eq !finset.to_set_upto)

theorem finite_of_surj_on {B : Type} {f : A → B} {s : set A} [finite s] {t : set B}
                          (H : surj_on f s t) :
        finite t :=
finite_subset H

theorem finite_of_inj_on {B : Type} {f : A → B} {s : set A} {t : set B} [finite t]
                         (mapsto : maps_to f s t) (injf : inj_on f s) :
        finite s :=
if H : s = ∅ then
  by rewrite H; apply _
else
  obtain (dflt : A) (xs : dflt ∈ s), from exists_mem_of_ne_empty H,
  let finv := inv_fun f s dflt in
  have surj_on finv t s, from surj_on_inv_fun_of_inj_on dflt mapsto injf,
  finite_of_surj_on this

theorem finite_of_bij_on {B : Type} {f : A → B} {s : set A} {t : set B} [finite s]
                         (bijf : bij_on f s t) :
        finite t :=
finite_of_surj_on (surj_on_of_bij_on bijf)

theorem finite_of_bij_on' {B : Type} {f : A → B} {s : set A} {t : set B} [finite t]
                         (bijf : bij_on f s t) :
        finite s :=
finite_of_inj_on (maps_to_of_bij_on bijf) (inj_on_of_bij_on bijf)

theorem finite_iff_finite_of_bij_on {B : Type} {f : A → B} {s : set A} {t : set B}
                                    (bijf : bij_on f s t) :
        finite s ↔ finite t :=
iff.intro (assume fs, finite_of_bij_on bijf) (assume ft, finite_of_bij_on' bijf)

theorem finite_powerset (s : set A) [finite s] : finite 𝒫 s :=
assert H : 𝒫 s = finset.to_set ' (finset.to_set (#finset 𝒫 (to_finset s))),
  from ext (take t, iff.intro
    (suppose t ∈ 𝒫 s,
      assert t ⊆ s, from this,
      assert finite t, from finite_subset this,
      assert (#finset to_finset t ∈ 𝒫 (to_finset s)),
        by rewrite [finset.mem_powerset_iff_subset, to_finset_subset_to_finset_eq]; apply `t ⊆ s`,
      assert to_finset t ∈ (finset.to_set (finset.powerset (to_finset s))), from this,
      mem_image this (by rewrite to_set_to_finset))
    (assume H',
      obtain t' [(tmem : (#finset t' ∈ 𝒫 (to_finset s))) (teq : finset.to_set t' = t)],
        from H',
      show t ⊆ s,
      begin
        rewrite [-teq, finset.mem_powerset_iff_subset at tmem, -to_set_to_finset s],
        rewrite -finset.subset_eq_to_set_subset, assumption
     end)),
by rewrite H; apply finite_image

/- induction for finite sets -/

theorem induction_finite [recursor 6] {P : set A → Prop}
    (H1 : P ∅) (H2 : ∀ ⦃a : A⦄, ∀ {s : set A} [finite s], a ∉ s → P s → P (insert a s)) :
  ∀ (s : set A) [finite s], P s :=
begin
  intro s fins,
  rewrite [-to_set_to_finset s],
  generalize to_finset s,
  intro s',
  induction s' using finset.induction with a s' nains ih,
    {rewrite finset.to_set_empty, apply H1},
  rewrite [finset.to_set_insert],
  apply H2,
    {rewrite -finset.mem_eq_mem_to_set, assumption},
  exact ih
end

theorem induction_on_finite {P : set A → Prop} (s : set A) [finite s]
    (H1 : P ∅) (H2 : ∀ ⦃a : A⦄, ∀ {s : set A} [finite s], a ∉ s → P s → P (insert a s)) :
  P s :=
induction_finite H1 H2 s

end set