178 lines
7.1 KiB
Text
178 lines
7.1 KiB
Text
/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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The notion of "finiteness" for sets. This approach is not computational: for example, just because
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an element s : set A satsifies finite s doesn't mean that we can compute the cardinality. For
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a computational representation, use the finset type.
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-/
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import data.set.function data.finset.to_set
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open nat classical
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variable {A : Type}
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namespace set
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definition finite [class] (s : set A) : Prop := ∃ (s' : finset A), s = finset.to_set s'
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theorem finite_finset [instance] (s : finset A) : finite (finset.to_set s) :=
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exists.intro s rfl
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/- to finset: casts every set to a finite set -/
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noncomputable definition to_finset (s : set A) : finset A :=
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if fins : finite s then some fins else finset.empty
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theorem to_finset_of_not_finite {s : set A} (nfins : ¬ finite s) : to_finset s = (#finset ∅) :=
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by rewrite [↑to_finset, dif_neg nfins]
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theorem to_set_to_finset (s : set A) [fins : finite s] : finset.to_set (to_finset s) = s :=
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by rewrite [↑to_finset, dif_pos fins]; exact eq.symm (some_spec fins)
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theorem mem_to_finset_eq (a : A) (s : set A) [fins : finite s] :
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(#finset a ∈ to_finset s) = (a ∈ s) :=
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by rewrite [-to_set_to_finset at {2}]
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theorem to_set_to_finset_of_not_finite {s : set A} (nfins : ¬ finite s) :
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finset.to_set (to_finset s) = ∅ :=
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by rewrite [to_finset_of_not_finite nfins]
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theorem to_finset_to_set (s : finset A) : to_finset (finset.to_set s) = s :=
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by rewrite [finset.eq_eq_to_set_eq, to_set_to_finset (finset.to_set s)]
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theorem to_finset_eq_of_to_set_eq {s : set A} {t : finset A} (H : finset.to_set t = s) :
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to_finset s = t :=
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finset.eq_of_to_set_eq_to_set (by subst [s]; rewrite to_finset_to_set)
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/- finiteness -/
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theorem finite_of_to_set_to_finset_eq {s : set A} (H : finset.to_set (to_finset s) = s) :
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finite s :=
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by rewrite -H; apply finite_finset
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theorem finite_empty [instance] : finite (∅ : set A) :=
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by rewrite [-finset.to_set_empty]; apply finite_finset
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theorem to_finset_empty : to_finset (∅ : set A) = (#finset ∅) :=
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to_finset_eq_of_to_set_eq !finset.to_set_empty
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theorem finite_insert [instance] (a : A) (s : set A) [fins : finite s] : finite (insert a s) :=
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exists.intro (finset.insert a (to_finset s))
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(by rewrite [finset.to_set_insert, to_set_to_finset])
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theorem to_finset_insert (a : A) (s : set A) [fins : finite s] :
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to_finset (insert a s) = finset.insert a (to_finset s) :=
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by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_insert, to_set_to_finset]
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example : finite '{1, 2, 3} := _
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theorem finite_union [instance] (s t : set A) [fins : finite s] [fint : finite t] :
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finite (s ∪ t) :=
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exists.intro (#finset to_finset s ∪ to_finset t)
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(by rewrite [finset.to_set_union, *to_set_to_finset])
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theorem to_finset_union (s t : set A) [fins : finite s] [fint : finite t] :
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to_finset (s ∪ t) = (#finset to_finset s ∪ to_finset t) :=
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by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_union, *to_set_to_finset]
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theorem finite_inter [instance] (s t : set A) [fins : finite s] [fint : finite t] :
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finite (s ∩ t) :=
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exists.intro (#finset to_finset s ∩ to_finset t)
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(by rewrite [finset.to_set_inter, *to_set_to_finset])
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theorem to_finset_inter (s t : set A) [fins : finite s] [fint : finite t] :
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to_finset (s ∩ t) = (#finset to_finset s ∩ to_finset t) :=
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by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_inter, *to_set_to_finset]
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theorem finite_sep [instance] (s : set A) (p : A → Prop) [h : decidable_pred p]
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[fins : finite s] :
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finite {x ∈ s | p x} :=
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exists.intro (finset.sep p (to_finset s))
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(by rewrite [finset.to_set_sep, *to_set_to_finset])
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theorem to_finset_sep (s : set A) (p : A → Prop) [h : decidable_pred p] [fins : finite s] :
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to_finset {x ∈ s | p x} = (#finset {x ∈ to_finset s | p x}) :=
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by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_sep, to_set_to_finset]
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theorem finite_image [instance] {B : Type} [h : decidable_eq B] (f : A → B) (s : set A)
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[fins : finite s] :
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finite (f '[s]) :=
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exists.intro (finset.image f (to_finset s))
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(by rewrite [finset.to_set_image, *to_set_to_finset])
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theorem to_finset_image {B : Type} [h : decidable_eq B] (f : A → B) (s : set A)
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[fins : finite s] :
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to_finset (f '[s]) = (#finset f '[to_finset s]) :=
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by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_image, to_set_to_finset]
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theorem finite_diff [instance] (s t : set A) [fins : finite s] : finite (s \ t) :=
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!finite_sep
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theorem to_finset_diff (s t : set A) [fins : finite s] [fint : finite t] :
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to_finset (s \ t) = (#finset to_finset s \ to_finset t) :=
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by apply to_finset_eq_of_to_set_eq; rewrite [finset.to_set_diff, *to_set_to_finset]
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theorem finite_subset {s t : set A} [fint : finite t] (ssubt : s ⊆ t) : finite s :=
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by rewrite (eq_sep_of_subset ssubt); apply finite_sep
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theorem to_finset_subset_to_finset_eq (s t : set A) [fins : finite s] [fint : finite t] :
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(#finset to_finset s ⊆ to_finset t) = (s ⊆ t) :=
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by rewrite [finset.subset_eq_to_set_subset, *to_set_to_finset]
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theorem finite_of_finite_insert {s : set A} {a : A} (finias : finite (insert a s)) : finite s :=
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finite_subset (subset_insert a s)
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theorem finite_upto [instance] (n : ℕ) : finite {i | i < n} :=
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by rewrite [-finset.to_set_upto n]; apply finite_finset
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theorem to_finset_upto (n : ℕ) : to_finset {i | i < n} = finset.upto n :=
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by apply (to_finset_eq_of_to_set_eq !finset.to_set_upto)
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set_option pp.notation false
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theorem finite_powerset (s : set A) [fins : finite s] : finite 𝒫 s :=
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assert H : 𝒫 s = finset.to_set '[finset.to_set (#finset 𝒫 (to_finset s))],
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from ext (take t, iff.intro
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(suppose t ∈ 𝒫 s,
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assert t ⊆ s, from this,
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assert finite t, from finite_subset this,
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assert (#finset to_finset t ∈ 𝒫 (to_finset s)),
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by rewrite [finset.mem_powerset_iff_subset, to_finset_subset_to_finset_eq]; apply `t ⊆ s`,
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assert to_finset t ∈ (finset.to_set (finset.powerset (to_finset s))), from this,
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mem_image this (by rewrite to_set_to_finset))
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(assume H',
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obtain t' [(tmem : (#finset t' ∈ 𝒫 (to_finset s))) (teq : finset.to_set t' = t)],
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from H',
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show t ⊆ s,
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begin
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rewrite [-teq, finset.mem_powerset_iff_subset at tmem, -to_set_to_finset s],
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rewrite -finset.subset_eq_to_set_subset, assumption
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end)),
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by rewrite H; apply finite_image
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/- induction for finite sets -/
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theorem induction_finite [recursor 6] {P : set A → Prop}
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(H1 : P ∅)
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(H2 : ∀ ⦃a : A⦄, ∀ {s : set A} [fins : finite s], a ∉ s → P s → P (insert a s)) :
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∀ (s : set A) [fins : finite s], P s :=
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begin
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intro s fins,
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rewrite [-to_set_to_finset s],
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generalize to_finset s,
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intro s',
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induction s' using finset.induction with a s' nains ih,
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{rewrite finset.to_set_empty, apply H1},
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rewrite [finset.to_set_insert],
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apply H2,
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{rewrite -finset.mem_eq_mem_to_set, assumption},
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exact ih
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end
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theorem induction_on_finite {P : set A → Prop} (s : set A) [fins : finite s]
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(H1 : P ∅)
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(H2 : ∀ ⦃a : A⦄, ∀ {s : set A} [fins : finite s], a ∉ s → P s → P (insert a s)) :
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P s :=
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induction_finite H1 H2 s
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end set
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