(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2008 Slawomir Kolodynski This program is free software Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. The name of the author may not be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES LOSS OF USE, DATA, OR PROFITS OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. *) section ‹Finite sets - introduction› theory Finite_ZF imports ZF1 Nat_ZF_IML Cardinal begin text{*Standard Isabelle Finite.thy contains a very useful notion of finite powerset: the set of finite subsets of a given set. The definition, however, is specific to Isabelle and based on the notion of "datatype", obviously not something that belongs to ZF set theory. This theory file devolopes the notion of finite powerset similarly as in Finite.thy, but based on standard library's Cardinal.thy. This theory file is intended to replace IsarMathLib's @{text "Finite1"} and @{text "Finite_ZF_1"} theories that are currently derived from the "datatype" approach. *} subsection{*Definition and basic properties of finite powerset*} text{*The goal of this section is to prove an induction theorem about finite powersets: if the empty set has some property and this property is preserved by adding a single element of a set, then this property is true for all finite subsets of this set. *} text{*We defined the finite powerset @{text "FinPow(X)"} as those elements of the powerset that are finite.*} definition "FinPow(X) ≡ {A ∈ Pow(X). Finite(A)}" text{*The cardinality of an element of finite powerset is a natural number.*} lemma card_fin_is_nat: assumes "A ∈ FinPow(X)" shows "|A| ∈ nat" and "A ≈ |A|" using assms FinPow_def Finite_def cardinal_cong nat_into_Card Card_cardinal_eq by auto text{*A reformulation of @{text "card_fin_is_nat"}: for a finit set $A$ there is a bijection between $|A|$ and $A$.*} lemma fin_bij_card: assumes A1: "A ∈ FinPow(X)" shows "∃b. b ∈ bij(|A|, A)" proof - from A1 have "|A| ≈ A" using card_fin_is_nat eqpoll_sym by blast then show ?thesis using eqpoll_def by auto qed text{*If a set has the same number of elements as $n \in \mathbb{N}$, then its cardinality is $n$. Recall that in set theory a natural number $n$ is a set that has $n$ elements.*} lemma card_card: assumes "A ≈ n" and "n ∈ nat" shows "|A| = n" using assms cardinal_cong nat_into_Card Card_cardinal_eq by auto text{*If we add a point to a finite set, the cardinality increases by one. To understand the second assertion $| A \cup \{ a\}| = |A| \cup \{ |A|\} $ recall that the cardinality $|A|$ of $A$ is a natural number and for natural numbers we have $n+1 = n \cup \{ n\}$. *} lemma card_fin_add_one: assumes A1: "A ∈ FinPow(X)" and A2: "a ∈ X-A" shows "|A ∪ {a}| = succ( |A| )" "|A ∪ {a}| = |A| ∪ {|A|}" proof - from A1 A2 have "cons(a,A) ≈ cons( |A|, |A| )" using card_fin_is_nat mem_not_refl cons_eqpoll_cong by auto moreover have "cons(a,A) = A ∪ {a}" by (rule consdef) moreover have "cons( |A|, |A| ) = |A| ∪ {|A|}" by (rule consdef) ultimately have "A∪{a} ≈ succ( |A| )" using succ_explained by simp with A1 show "|A ∪ {a}| = succ( |A| )" and "|A ∪ {a}| = |A| ∪ {|A|}" using card_fin_is_nat card_card by auto qed text{*We can decompose the finite powerset into collection of sets of the same natural cardinalities.*} lemma finpow_decomp: shows "FinPow(X) = (⋃n ∈ nat. {A ∈ Pow(X). A ≈ n})" using Finite_def FinPow_def by auto text{*Finite powerset is the union of sets of cardinality bounded by natural numbers.*} lemma finpow_union_card_nat: shows "FinPow(X) = (⋃n ∈ nat. {A ∈ Pow(X). A ≲ n})" proof - have "FinPow(X) ⊆ (⋃n ∈ nat. {A ∈ Pow(X). A ≲ n})" using finpow_decomp FinPow_def eqpoll_imp_lepoll by auto moreover have "(⋃n ∈ nat. {A ∈ Pow(X). A ≲ n}) ⊆ FinPow(X)" using lepoll_nat_imp_Finite FinPow_def by auto ultimately show ?thesis by auto qed text{*A different form of @{text "finpow_union_card_nat"} (see above) - a subset that has not more elements than a given natural number is in the finite powerset.*} lemma lepoll_nat_in_finpow: assumes "n ∈ nat" "A ⊆ X" "A ≲ n" shows "A ∈ FinPow(X)" using assms finpow_union_card_nat by auto text{*Natural numbers are finite subsets of the set of natural numbers.*} lemma nat_finpow_nat: assumes "n ∈ nat" shows "n ∈ FinPow(nat)" using assms nat_into_Finite nat_subset_nat FinPow_def by simp text{*A finite subset is a finite subset of itself.*} lemma fin_finpow_self: assumes "A ∈ FinPow(X)" shows "A ∈ FinPow(A)" using assms FinPow_def by auto text{*If we remove an element and put it back we get the set back. *} lemma rem_add_eq: assumes "a∈A" shows "(A-{a}) ∪ {a} = A" using assms by auto text{*Induction for finite powerset. This is smilar to the standard Isabelle's @{text "Fin_induct"}. *} theorem FinPow_induct: assumes A1: "P(0)" and A2: "∀A ∈ FinPow(X). P(A) ⟶ (∀a∈X. P(A ∪ {a}))" and A3: "B ∈ FinPow(X)" shows "P(B)" proof - { fix n assume "n ∈ nat" moreover from A1 have I: "∀B∈Pow(X). B ≲ 0 ⟶ P(B)" using lepoll_0_is_0 by auto moreover have "∀ k ∈ nat. (∀B ∈ Pow(X). (B ≲ k ⟶ P(B))) ⟶ (∀B ∈ Pow(X). (B ≲ succ(k) ⟶ P(B)))" proof - { fix k assume A4: "k ∈ nat" assume A5: "∀ B ∈ Pow(X). (B ≲ k ⟶ P(B))" fix B assume A6: "B ∈ Pow(X)" "B ≲ succ(k)" have "P(B)" proof - have "B = 0 ⟶ P(B)" proof - { assume "B = 0" then have "B ≲ 0" using lepoll_0_iff by simp with I A6 have "P(B)" by simp } thus "B = 0 ⟶ P(B)" by simp qed moreover have "B≠0 ⟶ P(B)" proof - { assume "B ≠ 0" then obtain a where II: "a∈B" by auto let ?A = "B - {a}" from A6 II have "?A ⊆ X" and "?A ≲ k" using Diff_sing_lepoll by auto with A4 A5 have "?A ∈ FinPow(X)" and "P(?A)" using lepoll_nat_in_finpow finpow_decomp by auto with A2 A6 II have " P(?A ∪ {a})" by auto moreover from II have "?A ∪ {a} = B" by auto ultimately have "P(B)" by simp } thus "B≠0 ⟶ P(B)" by simp qed ultimately show "P(B)" by auto qed } thus ?thesis by blast qed ultimately have "∀B ∈ Pow(X). (B ≲ n ⟶ P(B))" by (rule ind_on_nat) } then have "∀n ∈ nat. ∀B ∈ Pow(X). (B ≲ n ⟶ P(B))" by auto with A3 show "P(B)" using finpow_union_card_nat by auto qed text{*A subset of a finites subset is a finite subset.*} lemma subset_finpow: assumes "A ∈ FinPow(X)" and "B ⊆ A" shows "B ∈ FinPow(X)" using assms FinPow_def subset_Finite by auto text{*If we subtract anything from a finite set, the resulting set is finite.*} lemma diff_finpow: assumes "A ∈ FinPow(X)" shows "A-B ∈ FinPow(X)" using assms subset_finpow by blast text{*If we remove a point from a finites subset, we get a finite subset.*} corollary fin_rem_point_fin: assumes "A ∈ FinPow(X)" shows "A - {a} ∈ FinPow(X)" using assms diff_finpow by simp text{*Cardinality of a nonempty finite set is a successsor of some natural number.*} lemma card_non_empty_succ: assumes A1: "A ∈ FinPow(X)" and A2: "A ≠ 0" shows "∃n ∈ nat. |A| = succ(n)" proof - from A2 obtain a where "a ∈ A" by auto let ?B = "A - {a}" from A1 `a ∈ A` have "?B ∈ FinPow(X)" and "a ∈ X - ?B" using FinPow_def fin_rem_point_fin by auto then have "|?B ∪ {a}| = succ( |?B| )" using card_fin_add_one by auto moreover from `a ∈ A` `?B ∈ FinPow(X)` have "A = ?B ∪ {a}" and "|?B| ∈ nat" using card_fin_is_nat by auto ultimately show "∃n ∈ nat. |A| = succ(n)" by auto qed text{*Nonempty set has non-zero cardinality. This is probably true without the assumption that the set is finite, but I couldn't derive it from standard Isabelle theorems. *} lemma card_non_empty_non_zero: assumes "A ∈ FinPow(X)" and "A ≠ 0" shows "|A| ≠ 0" proof - from assms obtain n where "|A| = succ(n)" using card_non_empty_succ by auto then show "|A| ≠ 0" using succ_not_0 by simp qed text{*Another variation on the induction theme: If we can show something holds for the empty set and if it holds for all finite sets with at most $k$ elements then it holds for all finite sets with at most $k+1$ elements, the it holds for all finite sets.*} theorem FinPow_card_ind: assumes A1: "P(0)" and A2: "∀k∈nat. (∀A ∈ FinPow(X). A ≲ k ⟶ P(A)) ⟶ (∀A ∈ FinPow(X). A ≲ succ(k) ⟶ P(A))" and A3: "A ∈ FinPow(X)" shows "P(A)" proof - from A3 have "|A| ∈ nat" and "A ∈ FinPow(X)" and "A ≲ |A|" using card_fin_is_nat eqpoll_imp_lepoll by auto moreover have "∀n ∈ nat. (∀A ∈ FinPow(X). A ≲ n ⟶ P(A))" proof fix n assume "n ∈ nat" moreover from A1 have "∀A ∈ FinPow(X). A ≲ 0 ⟶ P(A)" using lepoll_0_is_0 by auto moreover note A2 ultimately show "∀A ∈ FinPow(X). A ≲ n ⟶ P(A)" by (rule ind_on_nat) qed ultimately show "P(A)" by simp qed text{*Another type of induction (or, maybe recursion). The induction step we try to find a point in the set that if we remove it, the fact that the property holds for the smaller set implies that the property holds for the whole set. *} lemma FinPow_ind_rem_one: assumes A1: "P(0)" and A2: "∀ A ∈ FinPow(X). A ≠ 0 ⟶ (∃a∈A. P(A-{a}) ⟶ P(A))" and A3: "B ∈ FinPow(X)" shows "P(B)" proof - note A1 moreover have "∀k∈nat. (∀B ∈ FinPow(X). B ≲ k ⟶ P(B)) ⟶ (∀C ∈ FinPow(X). C ≲ succ(k) ⟶ P(C))" proof - { fix k assume "k ∈ nat" assume A4: "∀B ∈ FinPow(X). B ≲ k ⟶ P(B)" have "∀C ∈ FinPow(X). C ≲ succ(k) ⟶ P(C)" proof - { fix C assume "C ∈ FinPow(X)" assume "C ≲ succ(k)" note A1 moreover { assume "C ≠ 0" with A2 `C ∈ FinPow(X)` obtain a where "a∈C" and "P(C-{a}) ⟶ P(C)" by auto with A4 `C ∈ FinPow(X)` `C ≲ succ(k)` have "P(C)" using Diff_sing_lepoll fin_rem_point_fin by simp } ultimately have "P(C)" by auto } thus ?thesis by simp qed } thus ?thesis by blast qed moreover note A3 ultimately show "P(B)" by (rule FinPow_card_ind) qed text{* Yet another induction theorem. This is similar, but slightly more complicated than @{text "FinPow_ind_rem_one"}. The difference is in the treatment of the empty set to allow to show properties that are not true for empty set. *} lemma FinPow_rem_ind: assumes A1: "∀A ∈ FinPow(X). A = 0 ∨ (∃a∈A. A = {a} ∨ P(A-{a}) ⟶ P(A))" and A2: "A ∈ FinPow(X)" and A3: "A≠0" shows "P(A)" proof - have "0 = 0 ∨ P(0)" by simp moreover have "∀k∈nat. (∀B ∈ FinPow(X). B ≲ k ⟶ (B=0 ∨ P(B))) ⟶ (∀A ∈ FinPow(X). A ≲ succ(k) ⟶ (A=0 ∨ P(A)))" proof - { fix k assume "k ∈ nat" assume A4: "∀B ∈ FinPow(X). B ≲ k ⟶ (B=0 ∨ P(B))" have "∀A ∈ FinPow(X). A ≲ succ(k) ⟶ (A=0 ∨ P(A))" proof - { fix A assume "A ∈ FinPow(X)" assume "A ≲ succ(k)" "A≠0" from A1 `A ∈ FinPow(X)` `A≠0` obtain a where "a∈A" and "A = {a} ∨ P(A-{a}) ⟶ P(A)" by auto let ?B = "A-{a}" from A4 `A ∈ FinPow(X)` `A ≲ succ(k)` `a∈A` have "?B = 0 ∨ P(?B)" using Diff_sing_lepoll fin_rem_point_fin by simp with `a∈A` `A = {a} ∨ P(A-{a}) ⟶ P(A)` have "P(A)" by auto } thus ?thesis by auto qed } thus ?thesis by blast qed moreover note A2 ultimately have "A=0 ∨ P(A)" by (rule FinPow_card_ind) with A3 show "P(A)" by simp qed text{*If a family of sets is closed with respect to taking intersections of two sets then it is closed with respect to taking intersections of any nonempty finite collection.*} lemma inter_two_inter_fin: assumes A1: "∀V∈T. ∀W∈T. V ∩ W ∈ T" and A2: "N ≠ 0" and A3: "N ∈ FinPow(T)" shows "(⋂N ∈ T)" proof - have "0 = 0 ∨ (⋂0 ∈ T)" by simp moreover have "∀M ∈ FinPow(T). (M = 0 ∨ ⋂M ∈ T) ⟶ (∀W ∈ T. M∪{W} = 0 ∨ ⋂(M ∪ {W}) ∈ T)" proof - { fix M assume "M ∈ FinPow(T)" assume A4: "M = 0 ∨ ⋂M ∈ T" { assume "M = 0" hence "∀W ∈ T. M∪{W} = 0 ∨ ⋂(M ∪ {W}) ∈ T" by auto } moreover { assume "M ≠ 0" with A4 have "⋂M ∈ T" by simp { fix W assume "W ∈ T" from `M ≠ 0` have "⋂(M ∪ {W}) = (⋂M) ∩ W" by auto with A1 `⋂M ∈ T` `W ∈ T` have "⋂(M ∪ {W}) ∈ T" by simp } hence "∀W ∈ T. M∪{W} = 0 ∨ ⋂(M ∪ {W}) ∈ T" by simp } ultimately have "∀W ∈ T. M∪{W} = 0 ∨ ⋂(M ∪ {W}) ∈ T" by blast } thus ?thesis by simp qed moreover note `N ∈ FinPow(T)` ultimately have "N = 0 ∨ (⋂N ∈ T)" by (rule FinPow_induct) with A2 show "(⋂N ∈ T)" by simp qed text{*If a family of sets contains the empty set and is closed with respect to taking unions of two sets then it is closed with respect to taking unions of any finite collection.*} lemma union_two_union_fin: assumes A1: "0 ∈ C" and A2: "∀A∈C. ∀B∈C. A∪B ∈ C" and A3: "N ∈ FinPow(C)" shows "⋃N ∈ C" proof - from `0 ∈ C` have "⋃0 ∈ C" by simp moreover have "∀M ∈ FinPow(C). ⋃M ∈ C ⟶ (∀A∈C. ⋃(M ∪ {A}) ∈ C)" proof - { fix M assume "M ∈ FinPow(C)" assume "⋃M ∈ C" fix A assume "A∈C" have "⋃(M ∪ {A}) = (⋃M) ∪ A" by auto with A2 `⋃M ∈ C` `A∈C` have "⋃(M ∪ {A}) ∈ C" by simp } thus ?thesis by simp qed moreover note `N ∈ FinPow(C)` ultimately show "⋃N ∈ C" by (rule FinPow_induct) qed text{*Empty set is in finite power set.*} lemma empty_in_finpow: shows "0 ∈ FinPow(X)" using FinPow_def by simp text{*Singleton is in the finite powerset.*} lemma singleton_in_finpow: assumes "x ∈ X" shows "{x} ∈ FinPow(X)" using assms FinPow_def by simp text{*Union of two finite subsets is a finite subset.*} lemma union_finpow: assumes "A ∈ FinPow(X)" and "B ∈ FinPow(X)" shows "A ∪ B ∈ FinPow(X)" using assms FinPow_def by auto text{*Union of finite number of finite sets is finite.*} lemma fin_union_finpow: assumes "M ∈ FinPow(FinPow(X))" shows "⋃M ∈ FinPow(X)" using assms empty_in_finpow union_finpow union_two_union_fin by simp (*text{*A subset of a finites subset is a finite subset.*} lemma subset_finpow: assumes "A ∈ FinPow(X)" and "B ⊆ A" shows "B ∈ FinPow(X)" using assms FinPow_def subset_Finite by auto;*) text{*If a set is finite after removing one element, then it is finite.*} lemma rem_point_fin_fin: assumes A1: "x ∈ X" and A2: "A - {x} ∈ FinPow(X)" shows "A ∈ FinPow(X)" proof - from assms have "(A - {x}) ∪ {x} ∈ FinPow(X)" using singleton_in_finpow union_finpow by simp moreover have "A ⊆ (A - {x}) ∪ {x}" by auto ultimately show "A ∈ FinPow(X)" using FinPow_def subset_Finite by auto qed text{*An image of a finite set is finite.*} lemma fin_image_fin: assumes "∀V∈B. K(V)∈C" and "N ∈ FinPow(B)" shows "{K(V). V∈N} ∈ FinPow(C)" proof - have "{K(V). V∈0} ∈ FinPow(C)" using FinPow_def by auto moreover have "∀A ∈ FinPow(B). {K(V). V∈A} ∈ FinPow(C) ⟶ (∀a∈B. {K(V). V ∈ (A ∪ {a})} ∈ FinPow(C))" proof - { fix A assume "A ∈ FinPow(B)" assume "{K(V). V∈A} ∈ FinPow(C)" fix a assume "a∈B" have "{K(V). V ∈ (A ∪ {a})} ∈ FinPow(C)" proof - have "{K(V). V ∈ (A ∪ {a})} = {K(V). V∈A} ∪ {K(a)}" by auto moreover note `{K(V). V∈A} ∈ FinPow(C)` moreover from `∀V∈B. K(V) ∈ C` `a∈B` have "{K(a)} ∈ FinPow(C)" using singleton_in_finpow by simp ultimately show ?thesis using union_finpow by simp qed } thus ?thesis by simp qed moreover note `N ∈ FinPow(B)` ultimately show "{K(V). V∈N} ∈ FinPow(C)" by (rule FinPow_induct) qed text{*Union of a finite indexed family of finite sets is finite.*} lemma union_fin_list_fin: assumes A1: "n ∈ nat" and A2: "∀k ∈ n. N(k) ∈ FinPow(X)" shows "{N(k). k ∈ n} ∈ FinPow(FinPow(X))" and "(⋃k ∈ n. N(k)) ∈ FinPow(X)" proof - from A1 have "n ∈ FinPow(n)" using nat_finpow_nat fin_finpow_self by auto with A2 show "{N(k). k ∈ n} ∈ FinPow(FinPow(X))" by (rule fin_image_fin) then show "(⋃k ∈ n. N(k)) ∈ FinPow(X)" using fin_union_finpow by simp qed end