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cardinality.v
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cardinality.v
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Require Export ssreflect ssrbool ssrfun naturals Setoid.
Set Warnings "-notation-overridden".
Open Scope set_scope.
Definition equinumerous A B := ∃ f, domain f = A ∧ range f = B ∧ bijective f.
Infix "~" := equinumerous (at level 70) : set_scope.
Theorem cardinality_refl : ∀ A, A ~ A.
Proof.
move=> A.
exists (@functionify A A id).
rewrite /bijective Injective_classification Surjective_classification
functionify_domain functionify_range; (repeat split; auto) =>
[x y H H0 | y H]; [ | exists y ];
rewrite (reify H) ? (reify H0) ? functionify_action //.
Qed.
Theorem cardinality_sym : ∀ A B, A ~ B → B ~ A.
Proof.
move=> A B [f [H [H0 H1]]].
exists (inverse f).
rewrite inverse_domain ? inverse_range; auto using inverse_bijective.
Qed.
Theorem cardinality_trans : ∀ A B C, A ~ B → B ~ C → A ~ C.
Proof.
move=> A B C [f [<- [<- [? ?]]]]
[g [/[dup] ? /Composition_classification [? [? ?]] [<- [*]]]].
exists (g ∘ f).
repeat split; eauto using composition_injective, composition_surjective.
Qed.
Theorem cardinality_is_equivalence :
∀ X, is_equivalence (P X) {z in P X × P X |
proj1 (P X) (P X) z ~ proj2 (P X) (P X) z}.
Proof.
(repeat split) =>
[? ? | ? ? ? ? /Specify_classification [H1] |
? ? ? ? ? ? /Specify_classification [?] /[swap] /Specify_classification [?]];
rewrite Specify_classification ? Product_classification ? proj1_eval
? proj2_eval;
intuition eauto using cardinality_refl, cardinality_sym, cardinality_trans.
Qed.
Add Parametric Relation : set equinumerous
reflexivity proved by (cardinality_refl)
symmetry proved by (cardinality_sym)
transitivity proved by (cardinality_trans) as set_equivalence.
Theorem cardinality_eq : ∀ A B, A = B → A ~ B.
Proof.
now move=> A B ->.
Qed.
Theorem injective_into_image : ∀ f, injective f → domain f ~ image f.
Proof.
move=> f /Injective_classification H.
exists (restriction_Y (Set_is_subset (image f))).
rewrite restriction_Y_domain restriction_Y_range /bijective
Injective_classification Surjective_classification
restriction_Y_domain restriction_Y_range ? restriction_Y_action.
(repeat split; auto) =>
[x y /[dup] H0 /H /[swap] /[dup] H1 /[swap] /[apply] |
y /Specify_classification [H0 [x [H1 H2]]]]; [ | exists x ];
rewrite ? restriction_Y_action //.
Qed.
Theorem injection_restriction :
∀ f S, injective f → S ⊂ domain f → S ~ push_forward f S.
Proof.
move=> f S /Injective_classification H H0.
have: image (restriction f S) ⊂ push_forward f S =>
[y /Specify_classification /and_comm [[x]] | H1].
{ rewrite restriction_domain restriction_range => /[dup] [[?]].
rewrite -restriction_action // => [? [? ?] ?].
apply Specify_classification, conj; eauto. }
exists (restriction_Y H1).
rewrite restriction_Y_domain restriction_Y_range restriction_domain
/bijective Injective_classification Surjective_classification.
(repeat split; auto; first by apply /Intersection_subset) => [x y | y].
- rewrite restriction_Y_domain restriction_domain =>
/[dup] H2 /[dup] /Intersection_right H3 /[swap]
/[dup] H4 /[dup] /Intersection_right H5.
rewrite ? restriction_Y_action ? restriction_domain
-? restriction_action; auto.
- rewrite restriction_Y_range restriction_Y_domain restriction_domain
Specify_classification => [[H2 [x [H3 H4]]]].
exists x.
rewrite restriction_Y_action ? restriction_domain -? restriction_action //.
Qed.
Theorem cardinality_of_subsets_of_n :
∀ m (n : N), m ⊊ n → ∃ n', n' < n ∧ m ~ n'.
Proof.
move => /[swap].
elim/Induction =>
[m /proper_subset_inhab [z [/Empty_set_classification H H0]]
| n IH m /[dup] [[H H0]] /proper_subset_inhab [z [H1 H2]]] //.
elim (classic (n ∈ m)).
- elim (function_construction m n (λ x, If x = n then z else x)) =>
[f [H3 [H4 H5]] | a].
+ have: injective f.
{ apply /Injective_classification => x y H6 H7.
rewrite ? H5; repeat case excluded_middle_informative; congruence. }
elim (classic (image f = range f)) => [ | H6].
* exists n.
rewrite -H3 -H4 -H6.
auto using succ_lt, injective_into_image.
* elim (IH (image f)) =>
[n' [H7 H8] | ]; rewrite -? H3 -? H4; repeat split;
eauto 6 using lt_trans, succ_lt, injective_into_image,
cardinality_trans, image_subset_range.
+ case excluded_middle_informative => [-> | ].
* move: H1 H2 => /in_S_succ /Pairwise_union_classification =>
[[H1 | /Singleton_classification ->]] //.
* move: H =>
/[swap] H3 /[apply] /in_S_succ /Pairwise_union_classification
[ | /Singleton_classification] //.
- elim (classic (m = n)) =>
[-> | H3 H4]; eauto using lt_trans, succ_lt, cardinality_refl.
suff: m ⊊ n => [/IH [n' [H5 H6]] | ]; eauto using lt_trans, succ_lt.
split; auto =>
x /[dup] /[swap] /H /in_S_succ /Pairwise_union_classification
[ | /Singleton_classification ->] //.
Qed.
Lemma equivalence_minus_element_singular : ∀ A x y,
x ∈ A → y ∈ A → A \ {x,x} ~ A \ {y,y}.
Proof.
move=> A x y H H0.
case (classic (x = y)) => [-> | H1]; auto using cardinality_refl.
elim (function_construction (A \ {x,x}) (A \ {y,y})
(λ z, If z = y then x else z))
=> [f' [H2 [H3 H4]] | a0 /Complement_classification H2].
- exists f'.
rewrite /bijective Injective_classification Surjective_classification H2 H3.
(repeat split; auto) =>
[x0 y0 /Complement_classification [H5 /Singleton_classification H6]
/Complement_classification [H7 /Singleton_classification H8] |
y0 /Complement_classification [H5 /Singleton_classification H6]].
+ rewrite ? H4 ? Complement_classification ? Singleton_classification //.
case (classic (x = x0)), (classic (x = y0));
repeat case excluded_middle_informative; congruence.
+ case (classic (x = y0)) =>
[<- | ]; [ exists y | exists y0 ];
rewrite H4 Complement_classification Singleton_classification;
repeat split; try case excluded_middle_informative; intuition.
- rewrite Complement_classification Singleton_classification.
case excluded_middle_informative => *; intuition.
Qed.
Theorem equivalence_minus_element : ∀ A B x y,
A ~ B → x ∈ A → y ∈ B → A \ {x,x} ~ B \ {y,y}.
Proof.
move=> A B x y [f [<- [<- [/Injective_classification ?
/Surjective_classification H]]]] ? ?.
enough ((domain f) \ {x,x} ~ (range f) \ {f x, f x});
eauto using cardinality_trans, equivalence_minus_element_singular,
function_maps_domain_to_range.
elim (function_construction ((domain f) \ {x,x})
((range f) \ {f x, f x}) (λ x, f x)) =>
[f' [H0 [H1 H2]] | a /Complement_classification
[H5 /Singleton_classification H6]].
- exists f'.
rewrite /bijective Injective_classification Surjective_classification H0 H1.
(repeat split; auto) =>
[x0 y0 /[dup] ? /Complement_classification [? /Singleton_classification ?]
/[dup] ? /Complement_classification [? /Singleton_classification ?] |
y0 /[dup] ? /Complement_classification
[/[dup] ? /H ? /Singleton_classification H8]]; try rewrite ? H2; auto.
elim (H y0) => [x0 [? ?] | ] //.
exists x0.
rewrite H2 Complement_classification Singleton_classification;
repeat split; auto; move: H8 => /[swap] <- /neq_sym //.
- rewrite Complement_classification Singleton_classification;
auto using function_maps_domain_to_range.
Qed.
Lemma two_sided_inverse_bijective : ∀ A B,
(∃ (f : elts A → elts B) (g : elts B → elts A),
((∀ a : elts A, g (f a) = a) ∧ (∀ b : elts B, f (g b) = b))) → A ~ B.
Proof.
move=> A B [f [g [H H0]]].
exists (functionify f).
rewrite /bijective Injective_classification Surjective_classification
functionify_range functionify_domain.
(repeat split; auto) => [x y H1 H2 | y H1].
- rewrite (reify H1) (reify H2) ? functionify_action
=> /set_proj_injective H3.
f_equal.
rewrite <-H, <-H3, H => //.
- exists (g (mkSet H1)).
rewrite functionify_action H0.
auto using elts_in_set.
Qed.
Theorem two_sided_inverse_bijective_set:
∀ A B, (∃ f g, (∀ a, a ∈ A → (f a ∈ B ∧ g (f a) = a)) ∧
(∀ b, b ∈ B → (g b ∈ A ∧ f (g b) = b))) → A ~ B.
Proof.
move=> A B [f [g [/[dup] H /[swap] /[dup] H0]]].
elim (function_construction A B f) => [f' [<- [<- H1 H2 H3]] | a /H [H1]] //.
exists f'.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) =>
[x y /[dup] H4 /H3 [H5 H6] /[dup] H7 /H3 [H8 H9] | y /[dup] H4 /H2 [H5 H6]];
[ | exists (g y) ]; rewrite ? H1 // -{2} H6 -{2} H9 => -> //.
Qed.
Theorem proper_subsets_of_natural_numbers : ∀ m (n : N), m ⊊ n → ¬ n ~ m.
Proof.
move /[swap].
elim/Induction => [m /proper_subset_inhab [z [/Empty_set_classification]] |
n IH m /[dup] H [H0 H1] [f [H2 [H3 /[dup] ? [? ?]]]]] //.
have H4: S n \ {n,n} = n.
{ rewrite -S_is_succ /succ complement_disjoint_union ? disjoint_succ //. }
elim (classic (n ∈ m)) => [? | ?].
- (apply /(IH (m \ {n,n})); repeat split) =>
[x /Complement_classification
[/H0 /in_S_succ /Pairwise_union_classification [H5 | H5] ?] | H5 | ] //.
+ contradict H1.
apply Subset_equality_iff, conj; auto =>
x /in_S_succ /Pairwise_union_classification
[ | /Singleton_classification ->] //.
rewrite -H5 => /complement_subset //.
+ rewrite -{1}H4.
apply equivalence_minus_element; rewrite /equinumerous; eauto.
- apply /(IH (m \ {f n, f n})).
+ (apply /(subsetneq_subset_trans _ m); repeat split) =>
[x /Complement_classification [H5 ?] | H5 |
x /[dup] /H0 /in_S_succ /Pairwise_union_classification
[H5 | /Singleton_classification -> ] ] //.
move: (function_maps_domain_to_range f n) (in_succ n).
rewrite H2 in_S_succ H3 -H5 => /[apply].
rewrite Complement_classification Singleton_classification => [[?]] //.
+ rewrite -{1}H4.
apply /equivalence_minus_element; try apply /in_S_succ /in_succ.
* rewrite /equinumerous; eauto.
* move: (function_maps_domain_to_range f n) (in_succ n).
rewrite H2 in_S_succ H3 => /[apply] //.
Qed.
Theorem equiv_proper_subset_ω : ∃ S, S ⊊ ω ∧ S ~ ω.
Proof.
exists (ω \ {0,0}).
(repeat split) =>
[x /Complement_classification [?] | | ]
//; first by move: (elts_in_set 0) =>
/[swap] {2}<- /Complement_classification [_] /Singleton_classification //.
symmetry.
set (f := functionify (λ x, x + 1)).
elim (function_construction ω (ω \ {0,0}) (λ x, f x)) =>
[f' [H [H0 H1]] | a H].
- exists f'.
(repeat split; auto;
rewrite ? Injective_classification ? Surjective_classification) =>
[x y H2 H3 | y H2].
+ move: H H2 H3 -> => H2 H3.
(rewrite ? H1 ? (reify H2) ? (reify H3)
? functionify_action -? (add_comm 1); auto) =>
/set_proj_injective /cancellation_add -> //.
+ move: H0 H2 -> =>
/Complement_classification [H0] /Singleton_classification H2.
exists (mkSet H0 - 1).
(have: (mkSet H0) ≠ 0 by (contradict H2; now inversion H2)) =>
/succ_0 => [[m /[dup] H3]].
rewrite H H1 /f ? functionify_action 1? add_comm ? sub_abab H3;
repeat split; eauto using elts_in_set, one_le_succ.
rewrite -H3 //.
- rewrite /f (reify H) functionify_action Complement_classification
Singleton_classification add_1_r.
split; eauto using elts_in_set => /set_proj_injective /(@eq_sym N) /PA4 //.
Qed.
Definition finite S := ∃ n : N, S ~ n.
Definition infinite S := ¬ finite S.
Theorem naturals_are_finite : ∀ n : N, finite n.
Proof.
rewrite /finite.
eauto using cardinality_refl.
Qed.
Theorem infinite_ω : infinite ω.
Proof.
move: equiv_proper_subset_ω =>
[S [/[dup] /[dup] H [H0 H1] /proper_subset_inhab [z [H2 H3]] H4]]
[n /[dup] H5 [f [H6 [H7 [/Injective_classification H8
/Surjective_classification H9]]]]].
(contradiction
(proper_subsets_of_natural_numbers {x in n | ∃ s, s ∈ S ∧ x = f s} n);
repeat split) => [x /Specify_classification [H10] | H10 | ] //.
- absurd (f z ∈ n).
+ rewrite -H10 Specify_classification => [[H11 [s [H12 /H8]]]] H13.
move: H6 H13 H2 (H12) (H12) -> =>
/[apply] /[swap] /H0 /[swap] /[apply] <- //.
+ move: H6 H7 H2 (function_maps_domain_to_range f z) -> =>
-> /[swap] /[apply] //.
- rewrite -{1}H5 -{1}H4.
elim (function_construction
S {x in n | ∃ s : set, s ∈ S ∧ x = f s} (λ x, f x))
=> [f' [H10 [H11 H12]] | a /[dup] H10].
+ exists f'.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) => [x y | y].
* move: H10 (H12) -> => /[swap] H13 /[dup] -> // /[swap] H14 -> // /H8.
move: H6 H13 H14 -> => /H0 /[swap] /H0; tauto.
* rewrite H11 => /Specify_classification => [[H13 [s] /[dup] [[H14]] _]].
rewrite -H10 => [[H15 ->]]; eauto.
+ move: H6 H7 (function_maps_domain_to_range f a) -> =>
-> /[swap] /H0 /[swap] /[apply] H11.
apply /Specify_classification /conj; eauto.
Qed.
Theorem natural_cardinality_uniqueness : ∀ m n : N, m ~ n → m = n.
Proof.
move=> m n.
wlog: m n / n < m =>
[x H | /lt_is_subsetneq /proper_subsets_of_natural_numbers] //.
elim (lt_trichotomy m n) => [ | []]; auto using eq_sym, cardinality_sym.
Qed.
Theorem finite_cardinality_uniqueness : ∀ S, finite S → exists ! n : N, S ~ n.
Proof.
move=> S [n H].
exists n.
split; eauto using natural_cardinality_uniqueness,
cardinality_trans, cardinality_sym.
Qed.
Definition finite_cardinality (S : set) : N.
Proof.
elim (excluded_middle_informative (finite S)) =>
[/finite_cardinality_uniqueness
/constructive_definite_description [n H] | H].
- exact n.
- exact 0.
Defined.
Notation " # E " := (finite_cardinality E) (at level 45) : set_scope.
Theorem card_equiv : ∀ x, finite x → x ~ # x.
Proof.
rewrite /finite_cardinality => x H.
(case excluded_middle_informative => /=; try tauto) => H0.
elim constructive_definite_description => //.
Qed.
Theorem card_equiv_l : ∀ S, finite S → # S ~ S.
Proof.
move=> S /card_equiv /cardinality_sym //.
Qed.
Add Morphism finite with signature equinumerous ==> iff as finiteness_equiv.
Proof.
split => [[n]|[n]]; exists n; eauto using cardinality_sym, cardinality_trans.
Qed.
Theorem card_of_natural : ∀ n : N, # n = n.
Proof.
eauto using eq_sym, natural_cardinality_uniqueness, card_equiv,
naturals_are_finite.
Qed.
Add Morphism finite_cardinality with signature
equinumerous ==> eq as finite_cardinality_equiv.
Proof.
rewrite /finite_cardinality => x y H.
(repeat case excluded_middle_informative => /=; auto) =>
[* | ? /[dup] | ? /[dup]]; try rewrite {1}H //.
repeat elim constructive_definite_description.
eauto using natural_cardinality_uniqueness, cardinality_trans,
cardinality_sym.
Qed.
Theorem equivalence_to_card : ∀ S (n : N), S ~ n → # S = n.
Proof.
move=> S n ->.
auto using card_of_natural.
Qed.
Theorem equivalence_to_bijection : ∀ S (n : N),
finite S → # S = n → ∃ f, domain f = S ∧ range f = n ∧ bijective f.
Proof.
move=> S n /card_equiv /[swap] -> [f H].
eauto.
Qed.
Lemma subset_equinumerosity :
∀ A B C D, A ~ B → B ⊂ C → C ~ D → ∃ E, E ⊂ D ∧ A ~ E.
Proof.
move=> A B C D [f [<- [<- [/Injective_classification H
/Surjective_classification H0]]]]
/[swap] [[g [<- [<- [/Injective_classification H1
/Surjective_classification H2]]]]] H3.
exists ({d in range g | ∃ a, a ∈ domain f ∧ g (f a) = d}).
split => [d /Specify_classification [H4] | ] //.
elim (function_construction
(domain f) {d in range g |
∃ a : set, a ∈ domain f ∧ g (f a) = d} (λ x, g (f x)))
=> [h [H4 [H5 H6]] | a H4].
- exists h.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) => [x y | y].
+ move: H4 H6 -> => /[swap] H6 /[swap] H7 /[dup] -> // -> // /H1 /H H8.
apply /H8; eauto using function_maps_domain_to_range.
+ rewrite H5 Specify_classification H4 => [[H7 [a [H8 <-]]]].
eauto.
- apply Specify_classification, conj;
eauto using function_maps_domain_to_range.
Qed.
Theorem card_empty : # ∅ = 0.
Proof.
rewrite -(card_of_natural 0) //.
Qed.
Theorem empty_card : ∀ S, S ~ 0 → S = ∅.
Proof.
move=> S /cardinality_sym [f] [H [<- [H0 /Surjective_classification]]].
move: H -> => H.
(apply Subset_equality_iff, conj; auto using Empty_set_is_subset) =>
x /H [y [/Empty_set_classification]] //.
Qed.
Lemma finite_subsets_precursor :
∀ E F, finite F → E ⊂ F → ∃ n : N, E ~ n ∧ n ≤ # F.
Proof.
move=>
E F /[dup]
[[n /[dup] /equivalence_to_card /[swap]
[[f [<- [H /[dup] H0 [/Injective_classification H1
/Surjective_classification H2]]]]] H3 H4 H5]].
elim (classic (E = (domain f))) =>
[-> | H6]; eauto using cardinality_sym, card_equiv, le_refl.
(elim (proper_subset_inhab E (domain f)); try split; eauto) => [z [H7 H8]].
elim (cardinality_of_subsets_of_n
{y in # (domain f) | ∃ x, x ∈ E ∧ f x = y} (# domain f)) =>
[n' [H9 [g [H10 [H11 H12]]]] | ].
- exists n'.
split; last by eapply lt_le_trans; eauto using le_refl.
apply /(cardinality_trans _ (domain g)); try now (exists g).
move: (H5) (H10).
elim (function_construction E (domain g) (λ x, f x)) =>
[f' [<- [H13 H14 H15]] | y].
+ exists f'.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) =>
[x y H17 H18 | y]; try rewrite ? H14 // => /H1 ?; auto.
move: H13 H16 H14 -> =>
-> /[swap] /Specify_classification [?] [?] [?] /[swap] <-; eauto.
+ move: H10 H3 H -> => -> <- H.
apply Specify_classification, conj; eauto.
apply /function_maps_domain_to_range; auto.
- split => [y /Specify_classification [H9 [x [H10]]] | H9] //.
contradict H8.
move: (H7) H H3 H9 =>
/function_maps_domain_to_range /[swap] ->
/[swap] <- /[swap] <-
/Specify_classification [H] [x] /and_comm [/H1] /[swap] ? <-; auto.
Qed.
Theorem subsets_of_finites_are_finite : ∀ E F, finite F → E ⊂ F → finite E.
Proof.
move=> E F H /finite_subsets_precursor [| n [H0 H1]] //.
apply /ex_intro /H0.
Qed.
Theorem infinite_subset : ∀ E F, infinite E → E ⊂ F → infinite F.
Proof.
move=> E F H /subsets_of_finites_are_finite /[apply] //.
Qed.
Theorem finite_subsets : ∀ E F, finite F → E ⊂ F → # E ≤ # F.
Proof.
move=> E F /[swap] /finite_subsets_precursor /[apply]
[[n [/equivalence_to_card <-]]] //.
Qed.
Theorem naturals_sum_diff : ∀ n m, (m + n) \ m ~ n.
Proof.
elim/Induction =>
[m | n /[swap] m /(_ m) [f [H [H0 [/Injective_classification H1
/Surjective_classification H2]]]]];
rewrite ? add_0_r ? add_succ_r -? S_is_succ;
first by apply cardinality_eq, Complement_subset, Set_is_subset.
have H3: ∀ z, z ∈ succ (m + n) \ m → z ≠ m + n → z ∈ domain f => [z | ].
{ rewrite H ? Complement_classification =>
[[/Pairwise_union_classification
[H3 | /Singleton_classification] H4] H5] //. }
elim (function_construction
(succ (m + n) \ m) (succ n) (λ x, If x = m + n then n else f x))
=> [f' [H4 [H5 H6]] | a H4].
- exists f'.
move: (H4) (H3) <- => H7.
rewrite /bijective Injective_classification Surjective_classification H5.
(repeat split; auto) => [x y H8 H9 | y /Pairwise_union_classification
[ | /Singleton_classification ->]].
+ rewrite ? H6; try congruence.
repeat case excluded_middle_informative; try congruence; auto
=> H10 H11 H12; contradiction (no_quines n);
[ rewrite {1}H12 -H0 | rewrite -{1}H12 -H0 ];
auto using function_maps_domain_to_range.
+ rewrite -H0 => /H2 [x].
rewrite H H4 =>
[[/Complement_classification [/[dup] H8 /subset_succ H9] H10 H11]].
exists x.
rewrite ? H6 ? Complement_classification //.
repeat split => //.
case excluded_middle_informative => //.
move: H8 => /[swap] -> /no_quines //.
+ exists (m+n).
have H8: m+n ∈ domain f'.
{ rewrite H4 Complement_classification.
split.
- apply /Pairwise_union_classification /or_intror.
rewrite Singleton_classification //.
- elim (classic (n = 0)) =>
[-> | ]; rewrite ? add_0_r -? lt_is_in -? le_not_gt 1 ? add_comm;
auto using le_refl, le_add_l. }
rewrite H6 -? H4; (repeat split) => //.
case excluded_middle_informative => //.
- case excluded_middle_informative; auto using in_succ => H5.
rewrite -H0.
apply /subset_succ /function_maps_domain_to_range /H3 => //.
Qed.
Theorem finite_union_equinumerosity : ∀ E F,
finite E → finite F → E ∩ F = ∅ → (E ∪ F) ~ # E + # F.
Proof.
move=> E F [n /[dup] /equivalence_to_card -> /cardinality_sym]
/[swap] [[m /[dup] /equivalence_to_card -> /cardinality_sym]].
move: (naturals_sum_diff m n)
=> /[dup] H1 [h [H11 [H12 [/Injective_classification
H13 /Surjective_classification H14]]]]
/[dup] H0 [g [H7 [H8 [/Injective_classification
H9 /Surjective_classification H10]]]]
/[dup] H [f [H3 [H4 [/Injective_classification H5
/Surjective_classification H6]]]] H2.
elim (function_construction
(n + m) (E ∪ F) (λ x, If x ∈ n then f x else g (h x))) =>
[f' [/[dup] H15 <- [H16 H17]] | a].
- symmetry.
exists f'.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) => [x y H18 H19 | y].
+ rewrite ? H17 //.
wlog: x y H17 H18 H19 / x ∈ n ∧ y ∉ n => [x0 | ].
* (repeat case excluded_middle_informative)
=> *; [ apply H5 | apply x0 | apply eq_sym, x0 | ];
repeat case excluded_middle_informative; intuition try congruence.
apply H13; try rewrite H11 Complement_classification -H15 //.
apply H9; auto; rewrite H7 -H12 ; apply function_maps_domain_to_range;
rewrite H11 Complement_classification -H15 //.
* (repeat case excluded_middle_informative; try tauto) => H20 H21 _ H22.
contradiction (Empty_set_classification (f x)).
rewrite -H2 -H4 -H8 Pairwise_intersection_classification.
move: H3 H21 <- => /function_maps_domain_to_range.
(have: y ∈ domain h by rewrite H11 -H15 Complement_classification //)
=> /function_maps_domain_to_range.
move: H12 H7 H22 -> => <- {3}-> /function_maps_domain_to_range //.
+ rewrite H16 -H4 -H8 Pairwise_union_classification =>
[[/H6 [x] /and_comm [<-] /[dup] /[dup] | /H10 [z [H18 H19]]]].
* have: n ⊂ n + m by apply le_is_subset, le_add.
move: H3 H15 H17 => {1 2}-> => -> H17 /[apply].
repeat esplit; eauto.
rewrite H17 //.
case excluded_middle_informative => //.
* move: H7 H12 H18 H11 H15 H17 -> => <- /H14 /[swap] -> /[swap] -> =>
[[x [/[dup] H20 /Complement_classification [H21 H22] H23]] H17].
repeat esplit; eauto.
rewrite H17 //.
case excluded_middle_informative; intuition congruence.
- case excluded_middle_informative =>
H15 H16; apply /Pairwise_union_classification.
+ move: H3 H15 H4 <- => /function_maps_domain_to_range /[swap] ->; auto.
+ (have: a ∈ domain h by rewrite H11 Complement_classification) =>
/function_maps_domain_to_range.
move: H12 H7 H8 -> =>
<- /[swap] /function_maps_domain_to_range /[swap] ->; auto.
Qed.
Theorem finite_union_cardinality : ∀ E F,
finite E → finite F → E ∩ F = ∅ → # (E ∪ F) = # E + # F.
Proof.
eauto using equivalence_to_card, finite_union_equinumerosity.
Qed.
Theorem finite_disjoint_unions_are_finite : ∀ E F,
finite E → finite F → E ∩ F = ∅ → finite (E ∪ F).
Proof.
move=> E F H H0 H1.
apply /ex_intro /finite_union_equinumerosity => //.
Qed.
Theorem singleton_card_1 : ∀ x, {x,x} ~ 1.
Proof.
move=> x.
elim (function_construction {x,x} 1 (λ x, 0)) =>
[f [H [H0 H1]] | a H]; last by apply /in_succ.
(eapply ex_intro, conj, conj, conj; eauto;
rewrite ? Injective_classification ? Surjective_classification) => [a b | y].
- move: H -> => /Singleton_classification -> /Singleton_classification -> //.
- rewrite H H0 /one -in_S_succ => /Pairwise_union_classification =>
[[/Empty_set_classification | /Singleton_classification ->]] //.
(have: x ∈ {x,x} by apply /Singleton_classification); eauto.
Qed.
Theorem singletons_are_finite : ∀ x, finite {x,x}.
Proof.
move=> x.
apply /ex_intro /singleton_card_1.
Qed.
Lemma singleton_times_natural_equiv_natural : ∀ n m : N, ({n,n} × m ~ m).
Proof.
move=> n m.
symmetry.
elim (function_construction m ({n,n} × m) (λ x, (n,x))) =>
[f [H [H0 H1]] | a H].
2: { apply /Product_classification.
repeat esplit; rewrite ? Singleton_classification //. }
exists f.
rewrite /bijective ? Injective_classification ? Surjective_classification.
(repeat split; auto) => [x y | y].
- move: H (H1) -> =>
/[swap] ? /[dup] -> // /[swap] ? -> // /Ordered_pair_iff [?] //.
- move: H H0 -> =>
-> /Product_classification
[a [b [/Singleton_classification -> [/[dup] H /H1 H0 ->]]]]; eauto.
Qed.
Lemma singleton_times_natural_is_finite : ∀ n m : N, finite ({n,n} × m).
Proof.
move=> n m.
apply /ex_intro /singleton_times_natural_equiv_natural.
Qed.
Lemma singleton_times_m_card : ∀ n m : N, # ({n,n} × m) = m.
Proof.
move=> n m.
auto using equivalence_to_card, singleton_times_natural_equiv_natural.
Qed.
Theorem natural_products_are_finite : ∀ n m : N, finite (n × m).
Proof.
elim/Induction => [m | n IH m].
- exists 0.
now rewrite Empty_product_left /=.
- rewrite -S_is_succ /succ Product_union_distr_l.
apply /finite_disjoint_unions_are_finite;
auto using singleton_times_natural_is_finite.
rewrite -Product_intersection disjoint_succ Empty_product_left //.
Qed.
Theorem finite_empty : ∀ S, finite S → # S = 0 → S = ∅.
Proof.
move=> S /card_equiv /[swap] -> /empty_card H //.
Qed.
Theorem singleton_card : ∀ n, # {n,n} = 1.
Proof.
move=> n.
apply /equivalence_to_card /singleton_card_1.
Qed.
Theorem natural_prod_card : ∀ n m : N, # (n × m) = (# n) * (# m).
Proof.
elim/Induction =>
[m | n IH m];
rewrite ? Empty_product_left ? card_empty ? card_of_natural ? mul_0_l //
- S_is_succ /succ Product_union_distr_l ? finite_union_cardinality ? IH
-? add_1_r ? mul_distr_r ? singleton_times_m_card
? card_of_natural ? mul_1_l -? Product_intersection
? disjoint_succ ? Empty_product_left;
auto using naturals_are_finite, natural_products_are_finite,
singletons_are_finite, singleton_times_natural_is_finite, disjoint_succ.
Qed.
Theorem Inclusion_Exclusion : ∀ E F,
finite E → finite F → # (E ∪ F) + # (E ∩ F) = # E + # F.
Proof.
move=> E F H H0.
rewrite disjoint_union_complement finite_union_cardinality // -? add_assoc
-1 ? finite_union_cardinality ? complement_union_intersection;
eauto using disjoint_intersection_complement, subsets_of_finites_are_finite,
complement_disjoint_intersection, complement_subset, Intersection_left.
Qed.
Theorem finite_unions_are_finite : ∀ E F, finite E → finite F → finite (E ∪ F).
Proof.
move=> E F H H0.
rewrite disjoint_union_complement.
apply finite_disjoint_unions_are_finite;
eauto using disjoint_intersection_complement, complement_subset,
subsets_of_finites_are_finite.
Qed.
Theorem finite_union_card_bound : ∀ E F, # (E ∪ F) ≤ # E + # F.
Proof.
move=> E F.
case (classic (finite E)), (classic (finite F));
try (by rewrite -Inclusion_Exclusion /le; eauto);
rewrite {1}/finite_cardinality; case excluded_middle_informative
=> *; auto using zero_le; exfalso;
eauto using subsets_of_finites_are_finite, Union_right, Union_left.
Qed.
Add Morphism product with signature
equinumerous ==> equinumerous ==> equinumerous as product_equiv.
Proof.
move=>
E n [f [<- [<- [/Injective_classification H /Surjective_classification H0]]]]
F m [g [<- [<- [/Injective_classification H1
/Surjective_classification H2]]]].
elim (function_construction
(domain f × domain g) (range f × range g)
(λ z, (f (proj1 (domain f) (domain g) z),
g (proj2 (domain f) (domain g) z)))) =>
[h [H3 [H4 H5]] | z /Product_classification [a [b [H3 [H4 ->]]]]].
- exists h.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) => [x y /[dup] H6 /[swap] /[dup] H7 | z].
+ rewrite ? H5 -? H3 // ? H3 =>
/[swap] /Product_classification [a [b [H8 [H9 ->]]]]
/Product_classification [c [d [H10 [H11 ->]]]].
rewrite ? pro1j_eval ? proj2_eval ? proj1_eval // =>
/Ordered_pair_iff [/H] /[swap] /H1; intuition congruence.
+ rewrite H3 H4 => /Product_classification
[a [b [/H0 [x [H6 <-]] [/H2 [y [H7 <-]]] ->]]].
exists (x,y).
rewrite ? H5 ? Product_classification ? proj1_eval ? proj2_eval;
repeat split; eauto.
- rewrite proj1_eval ? proj2_eval ? Product_classification //.
repeat esplit; eauto using function_maps_domain_to_range.
Qed.
Theorem product_card_comm : ∀ E F, E × F ~ F × E.
Proof.
move=> E F.
eexists; auto using swap_domain, swap_range, swap_bijective.
Qed.
Theorem finite_products_are_finite :
∀ E F, finite E → finite F → finite (E × F).
Proof.
move=> E F [n ->] [m ->].
auto using natural_products_are_finite.
Qed.
Theorem infinite_products_are_infinite :
∀ E F, E ≠ ∅ → infinite F →infinite (E × F).
Proof.
move=> E F /Nonempty_classification [e] /mkSet {}e H.
have /injective_into_image H0: injective (functionify (embed E F e)).
{ rewrite Injective_classification functionify_domain => ? ? H0 H1.
rewrite (reify H0) (reify H1) ? functionify_action /= =>
/Ordered_pair_iff [] //. }
rewrite -(functionify_range (embed E F e)).
eapply infinite_subset, image_subset_range.
rewrite /infinite -H0 functionify_domain //.
Qed.
Theorem infinite_product_card_right :
∀ E F, infinite F → # (E × F) = (# E) * (# F).
Proof.
move=> E F H.
(case (classic (E = ∅)) => [-> | /infinite_products_are_infinite /(_ H)]);
rewrite ? Empty_product_left ? card_empty ? {1 3}/finite_cardinality;
repeat case excluded_middle_informative => //= *; ring.
Qed.
Theorem infinite_product_card_left :
∀ E F, infinite E → # (E × F) = (# E) * (# F).
Proof.
move=> E F H.
rewrite product_card_comm infinite_product_card_right; intuition ring.
Qed.
Theorem product_card : ∀ E F, # (E × F) = (# E) * (# F).
Proof.
move=> E F.
case (classic (finite F)) => [ | /infinite_product_card_right] //.
case (classic (finite E)) => [[n ->] [m ->] | /infinite_product_card_left];
auto using natural_prod_card.
Qed.
Theorem power_union_r : ∀ A B C, B ∩ C = ∅ → A ^ (B ∪ C) ~ A ^ B × A ^ C.
Proof.
have Z:
∀ A B C z, z ∈ A ^ (B ∪ C) → {x in z | proj1 (B ∪ C) A x ∈ B} ∈ A ^ B
=> [A B C z /Specify_classification [H [H0 H1]] | A B C H].
{ rewrite Specify_classification Powerset_classification.
have: {x in z | proj1 (B ∪ C) A x ∈ B} ⊂ B × A =>
[x /Specify_classification
[/H0 /Product_classification [b [a [H2 [H3 ->]]]]] | ];
first by rewrite proj1_eval ? Product_classification; eauto.
(repeat split; auto) => a /[dup] H3 /Union_left =>
/(_ C) /H1 [y [[H4 H5] H6]].
exists y.
repeat split => // =>
[ | x' [H7]]; rewrite Specify_classification ? rewrite proj1_eval;
try apply Union_left; intuition. }
elim (function_construction
(A ^ (B ∪ C)) (A ^ B × A ^ C)
(λ z, ({x in z | proj1 (B ∪ C) A x ∈ B},
{x in z | proj1 (B ∪ C) A x ∈ C}))) =>
[f [/[dup] H0 <- [H1 H2]] | z /[dup]].
2: { rewrite {2 4}Union_comm => *.
eapply Product_classification, ex_intro, ex_intro, conj, conj; eauto. }
(do 2 esplit; repeat split; eauto;
rewrite ? Injective_classification ? Surjective_classification) => [u v | y].
- move: H0 => /[swap] /[dup] H3 /[swap] /[dup] {2}->
/[swap] /Specify_classification [/Powerset_classification H4 _]
/[swap] /[dup] H5 /[swap] -> /Specify_classification
[/Powerset_classification H6 _].
rewrite ? H2 // => /Ordered_pair_iff [H7 H8].
apply Extensionality.
move=> z; wlog: u v z H3 H4 H5 H6 H7 H8 / z ∈ u => [H0 | ].
{ split => *; [ apply (H0 u v z) | apply (H0 v u z) ]; auto. }
(split; try tauto) =>
/H4 /Product_classification
[y [x [/Pairwise_union_classification H9 [H10 H11]]]] =>
{Z H H1 H2 H3 H4 H5 H6}.
wlog: B C y H7 H8 H9 H10 H11 / y ∈ B => [ | /[dup] H /(Union_left _ C) H1].
{ elim H9 => *; [ | rewrite Union_comm in H7 H8 ]; eauto. }
have: z ∈ {x in v | proj1 (B ∪ C) A x ∈ B} =>
[ | /Specify_classification]; last by tauto.
move: H11 H0 H7 -> => H0 <-.
apply Specify_classification.
rewrite proj1_eval //.
- move: H1 -> => /Product_classification
[u [v [/Specify_classification [H3 [H4 H5]]
[/Specify_classification [H6 [H7 H8]] H9]]]].
exists (u ∪ v).
have: u ∪ v ∈ A ^ (B ∪ C).
{ rewrite Specify_classification Powerset_classification.
have: u ∪ v ⊂ (B ∪ C) × A => [z /Pairwise_union_classification H10 | H1].
{ rewrite Product_union_distr_l Pairwise_union_classification.
intuition eauto. }
repeat split; auto => a /Pairwise_union_classification H10.
clear Z f H0 H2 H3 H6 H9 H1.
wlog: A B C a u v H H4 H5 H7 H8 H10 y / a ∈ B =>
[ | {H10} /[dup] H10 /H5 [z [[H0 /[dup] /H4 /Product_in_left
H1 /(Union_left _ v) H2] H3]]].
{ elim H10; [ | rewrite ? (Union_comm B C) ? (Union_comm u v)
1 ? Intersection_comm in H y |-* ]; eauto. }
exists z.
repeat split; auto =>
x' [H6 /Pairwise_union_classification [H9 | /H7 /Product_in_left H9]];
apply H3 => //.
contradiction (Empty_set_classification a).
rewrite -H Pairwise_intersection_classification //. }
move: H0 => /[swap] /[dup] H0 /[swap] <- H1.
split; auto.
rewrite H2 //.
suff H10: (∀ X Y W a b : set, Y ∩ W = ∅ → a ⊂ Y × X → b ⊂ W × X →
a = {x in a ∪ b | proj1 (Y ∪ W) X x ∈ Y}).
{ suff <- : u = {x in u ∪ v | proj1 (B ∪ C) A x ∈ B}; auto.
suff <- : v = {x in u ∪ v | proj1 (B ∪ C) A x ∈ C};
rewrite (Union_comm B C) Union_comm
Intersection_comm in H0 H |-*; auto. }
clear => X Y W a b H H0 H1.
apply Extensionality => z.
split =>
[/[dup] /H0 /Product_classification
[x [y [/[dup] H2 /(Union_left _ W) H3 [H4 ->]]]] /(Union_left _ b) H5 |
/Specify_classification
[/Pairwise_union_classification
[H2 | /H1 /Product_classification
[x [y [/[dup] H2 /(Union_right Y) H3 [H4 ->]]]]] H5]]; auto.
+ rewrite Specify_classification proj1_eval //.
+ contradiction (Empty_set_classification (proj1 (Y ∪ W) X (x, y))).
rewrite -H Pairwise_intersection_classification {2}proj1_eval //.
Qed.
Theorem power_1_r : ∀ m n, m^{n,n} ~ m.
Proof.
move=> m n.
symmetry.
elim (function_construction m (m^{n,n}) (λ x, {(n,x),(n,x)}))
=> [f [H [H0 H1]] | a H].
- exists f.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) => [x y | y].
+ move: H H1 -> => /[swap] H2 /[swap] H3 /[dup] -> // -> //.
move: (in_singleton (n,x)) =>
/[swap] -> /Singleton_classification /Ordered_pair_iff [] //.
+ move: (H) H0 (in_singleton n) -> => -> =>
/[swap] /Specify_classification [? [H2 /(_ n)]] /[apply] [[a [[? ?] H3]]].
exists a.
rewrite H1; repeat split; auto.
apply Extensionality => z.
split => [/Singleton_classification -> |
/[dup] /H2 /Product_classification
[c [d [/Singleton_classification -> [? ->] ?]]]] //.
apply Singleton_classification, Ordered_pair_iff,
conj, eq_sym, H3, conj => //.
- rewrite Specify_classification Powerset_classification.
have H0: {(n,a),(n,a)} ⊂ {n,n} × m => [x | ].
{ rewrite Singleton_classification Product_classification.
exists n, a.
rewrite Singleton_classification //. }
(repeat split; auto) => a' /Singleton_classification ->.
exists a.
((repeat split; auto) => [ | x']);
rewrite ? Singleton_classification // ? Ordered_pair_iff => [[]] _ [] //.
Qed.
Theorem natural_powers_are_finite : ∀ m n : N, finite (m^n).
Proof.
move /[swap].
elim/Induction
=> *; rewrite ? power_0_r -? S_is_succ /succ ? power_union_r ? power_1_r;
auto using finite_products_are_finite, naturals_are_finite,
disjoint_succ, (naturals_are_finite 1).
Qed.
Theorem natural_powers_card : ∀ m n : N, # (m^n) = ((# m)^(# n))%N.
Proof.
move /[swap].
elim/Induction => [m | n IH m].
- rewrite ? card_of_natural power_0_r pow_0_r -(card_of_natural 1) //.
- rewrite
-S_is_succ /succ power_union_r ? disjoint_succ // finite_union_cardinality
? disjoint_succ // ? singleton_card ? pow_add_r ? pow_1_r
? power_1_r ? product_card ? IH //;
auto using naturals_are_finite, singletons_are_finite,
natural_powers_are_finite.
Qed.
Add Morphism power with signature
equinumerous ==> equinumerous ==> equinumerous as power_equiv.
Proof.
move=> A C [f [<- [<- [/Injective_classification H
/Surjective_classification H0]]]] B D
[g [<- [<- [/Injective_classification H1
/Surjective_classification H2]]]].
elim (function_construction
((domain f)^(domain g)) ((range f)^(range g))
(λ S, {z in (range g) × (range f) |
∃ x, x ∈ S ∧ g (proj1 (domain g) (domain f) x) =
proj1 (range g) (range f) z ∧
f (proj2 (domain g) (domain f) x) =
proj2 (range g) (range f) z})) =>
[h [H3 [H4 H5]] | φ /Specify_classification [H3 [H4 H5]]].
- exists h.
rewrite /bijective Injective_classification Surjective_classification.
(repeat split; auto) => [x y H6 H7 | y].
+ move: H3 H6 H7 H5 (H5) -> =>
/[dup] H3 /Specify_classification [_ [H6 H7]] /[dup] H8
/Specify_classification [_ [H9 H10]] -> // -> // H11.
apply Extensionality => z {A B C D H0 H2 h H3 H4 H8}.
wlog: x y z H H1 H6 H7 H9 H10 H11 / z ∈ x => [H0 | ].
* split => *; [ apply (H0 x y z) | apply (H0 y x z) ]; auto.
* split => [/[dup] /[swap] /H6 /Product_classification
[a [b [H2 [H3 ->]]] H4 {z H0}] | H2] //.
have: (g a, f b) ∈ {z in range g × range f | ∃ x0 : set,
x0 ∈ x ∧ g (proj1 (domain g) (domain f) x0) =
proj1 (range g) (range f) z
∧ f (proj2 (domain g) (domain f) x0) =
proj2 (range g) (range f) z}.
{ apply Specify_classification, conj.
- apply Product_classification.
exists (g a), (f b).
repeat split; auto; now apply function_maps_domain_to_range.
- exists (a, b).
rewrite ? proj1_eval ? proj2_eval;
auto using function_maps_domain_to_range. }
rewrite H11 Specify_classification =>
[[H0 [z [/[dup] /H9 /Product_classification [a' [b' [? [? ->] ?]]]]]]].
rewrite ? proj1_eval ? proj2_eval;
auto using function_maps_domain_to_range =>
[[]] /H1 /[swap] /H; intuition congruence.
+ move: (H3) (H4) -> => -> => /Specify_classification [_ [H6 H7]].
exists {z in domain g × domain f |
(g (proj1 (domain g) (domain f) z),
f (proj2 (domain g) (domain f) z)) ∈ y}.
have H8: {z in domain g × domain f |
(g (proj1 (domain g) (domain f) z),
f (proj2 (domain g) (domain f) z)) ∈ y} ∈ domain f ^ domain g.
{ (apply Specify_classification, conj, conj;
rewrite ? Powerset_classification) =>
[z /Specify_classification [] | z /Specify_classification [] |
z /[dup] H8 /function_maps_domain_to_range /H7
[x [[/H0 [w [] /[swap] <- H9] H11] H12]]] //.
exists w.
(repeat split; auto;
first by rewrite Specify_classification Product_classification
proj1_eval ? proj2_eval; intuition eauto)
=> x' [H13 /Specify_classification [H14]].
rewrite proj1_eval ? proj2_eval //.
auto using function_maps_domain_to_range. }
split; rewrite ? H5; auto.
apply Extensionality => z.
split => [/Specify_classification
[/Product_classification [a [b [H9 [H10 ->]]]]