(* This file is a part of IsarMathLib - a library of formalized mathematics for Isabelle/Isar. Copyright (C) 2008-2023 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 and order relations› theory FinOrd_ZF imports Finite_ZF func_ZF_1 NatOrder_ZF begin text‹This theory file contains properties of finite sets related to order relations. Part of this is similar to what is done in ‹Finite_ZF_1› except that the development is based on the notion of finite powerset defined in ‹Finite_ZF› rather the one defined in standard Isabelle ‹Finite› theory.› subsection‹Finite vs. bounded sets› text‹The goal of this section is to show that finite sets are bounded and have maxima and minima.› text‹For total and transitive relations nonempty finite set has a maximum.› theorem fin_has_max: assumes A1: "r {is total on} X" and A2: "trans(r)" and A3: "B ∈ FinPow(X)" and A4: "B ≠ 0" shows "HasAmaximum(r,B)" proof - have "0=0 ∨ HasAmaximum(r,0)" by simp moreover have "∀A ∈ FinPow(X). A=0 ∨ HasAmaximum(r,A) ⟶ (∀x∈X. (A ∪ {x}) = 0 ∨ HasAmaximum(r,A ∪ {x}))" proof - { fix A assume "A ∈ FinPow(X)" "A = 0 ∨ HasAmaximum(r,A)" have "∀x∈X. (A ∪ {x}) = 0 ∨ HasAmaximum(r,A ∪ {x})" proof - { fix x assume "x∈X" note ‹A = 0 ∨ HasAmaximum(r,A)› moreover { assume "A = 0" then have "A∪{x} = {x}" by simp from A1 have "refl(X,r)" using total_is_refl by simp with ‹x∈X› ‹A∪{x} = {x}› have "HasAmaximum(r,A∪{x})" using Order_ZF_4_L8 by simp } moreover { assume "HasAmaximum(r,A)" with A1 A2 ‹A ∈ FinPow(X)› ‹x∈X› have "HasAmaximum(r,A∪{x})" using FinPow_def Order_ZF_4_L9 by simp } ultimately have "A ∪ {x} = 0 ∨ HasAmaximum(r,A ∪ {x})" by auto } thus "∀x∈X. (A ∪ {x}) = 0 ∨ HasAmaximum(r,A ∪ {x})" by simp qed } thus ?thesis by simp qed moreover note A3 ultimately have "B = 0 ∨ HasAmaximum(r,B)" by (rule FinPow_induct) with A4 show "HasAmaximum(r,B)" by simp qed text‹For linearly ordered nonempty finite sets the maximum is in the set and indeed it is the greatest element of the set.› lemma linord_max_props: assumes A1: "IsLinOrder(X,r)" and A2: "A ∈ FinPow(X)" "A ≠ 0" shows "Maximum(r,A) ∈ A" "Maximum(r,A) ∈ X" "∀a∈A. ⟨a,Maximum(r,A)⟩ ∈ r" proof - from A1 A2 show "Maximum(r,A) ∈ A" and "∀a∈A. ⟨a,Maximum(r,A)⟩ ∈ r" using IsLinOrder_def fin_has_max Order_ZF_4_L3 by auto with A2 show "Maximum(r,A) ∈ X" using FinPow_def by auto qed text‹Every nonempty subset of a natural number has a maximum with expected properties.› lemma nat_max_props: assumes "n∈nat" "A⊆n" "A≠0" shows "Maximum(Le,A) ∈ A" "Maximum(Le,A) ∈ nat" "∀k∈A. k ≤ Maximum(Le,A)" proof - from assms(1,2) have "A ∈ FinPow(nat)" using nat_finpow_nat subset_finpow by blast with assms(3) show "Maximum(Le,A) ∈ A" "Maximum(Le,A) ∈ nat" using NatOrder_ZF_1_L2(4) linord_max_props(1,2) by simp_all from assms(3) ‹A ∈ FinPow(nat)› have "∀k∈A. ⟨k,Maximum(Le,A)⟩ ∈ Le" using linord_max_props NatOrder_ZF_1_L2(4) by blast then show "∀k∈A. k ≤ Maximum(Le,A)" by simp qed text‹Yet another version of induction where the induction step is valid only up to $n\in \mathbb{N}$ rather than for all natural numbers. This lemma is redundant as it is easier to prove this assertion using lemma ‹fin_nat_ind› from ‹Nat_ZF_IML› which was done in lemma ‹fin_nat_ind1› there. It is left here for now as an alternative proof based on properties of the maximum of a finite set. › lemma ind_on_nat2: assumes "n∈nat" and "P(0)" and "∀j∈n. P(j)⟶P(j #+ 1)" shows "∀j∈n #+ 1. P(j)" and "P(n)" proof - let ?A = "{k∈succ(n). ∀j∈succ(k). P(j)}" let ?M = "Maximum(Le,?A)" from assms(1,2) have I: "succ(n) ∈ nat" "?A⊆succ(n)" "?A≠0" using empty_in_every_succ by auto then have "?M ∈ ?A" by (rule nat_max_props) have "n=?M" proof - from ‹?M ∈ ?A› have "?M ∈ succ(n)" by blast with assms(1) have "?M∈n ∨ ?M=n" by auto moreover { assume "?M ∈ n" from I have "?M ∈ nat" by (rule nat_max_props) from assms(3) ‹?M∈?A› ‹?M∈n› have "P(?M #+ 1)" by blast with ‹?M ∈ nat› have "P(succ(?M))" using succ_add_one(1) by simp with ‹?M∈?A› have "∀j∈succ(succ(?M)). P(j)" by blast moreover from assms(1) ‹?M ∈ n› have "succ(?M) ∈ succ(n)" using succ_ineq1 by simp moreover from I have "∀k∈?A. k ≤ ?M" by (rule nat_max_props) ultimately have False by blast } ultimately show "n=?M" by auto qed with ‹?M ∈ ?A› have "n∈?A" by (rule eq_mem) with assms(1) show "∀j∈n #+ 1. P(j)" and "P(n)" using succ_add_one(1) by simp_all qed subsection‹Order isomorphisms of finite sets› text‹In this section we establish that if two linearly ordered finite sets have the same number of elements, then they are order-isomorphic and the isomorphism is unique. This allows us to talk about ''enumeration'' of a linearly ordered finite set. We define the enumeration as the order isomorphism between the number of elements of the set (which is a natural number $n = \{0,1,..,n-1\}$) and the set.› text‹A really weird corner case - empty set is order isomorphic with itself. › lemma empty_ord_iso: shows "ord_iso(0,r,0,R) ≠ 0" proof - have "0 ≈ 0" using eqpoll_refl by simp then obtain f where "f ∈ bij(0,0)" using eqpoll_def by blast then show ?thesis using ord_iso_def by auto qed text‹Even weirder than ‹empty_ord_iso› The order automorphism of the empty set is unique.› lemma empty_ord_iso_uniq: assumes "f ∈ ord_iso(0,r,0,R)" "g ∈ ord_iso(0,r,0,R)" shows "f = g" proof - from assms have "f : 0 → 0" and "g: 0 → 0" using ord_iso_def bij_def surj_def by auto moreover have "∀x∈0. f`(x) = g`(x)" by simp ultimately show "f = g" by (rule func_eq) qed text‹The empty set is the only order automorphism of itself.› lemma empty_ord_iso_empty: shows "ord_iso(0,r,0,R) = {0}" proof - have "0 ∈ ord_iso(0,r,0,R)" proof - have "ord_iso(0,r,0,R) ≠ 0" by (rule empty_ord_iso) then obtain f where "f ∈ ord_iso(0,r,0,R)" by auto then show "0 ∈ ord_iso(0,r,0,R)" using ord_iso_def bij_def surj_def fun_subset_prod by auto qed then show "ord_iso(0,r,0,R) = {0}" using empty_ord_iso_uniq by blast qed text‹An induction (or maybe recursion?) scheme for linearly ordered sets. The induction step is that we show that if the property holds when the set is a singleton or for a set with the maximum removed, then it holds for the set. The idea is that since we can build any finite set by adding elements on the right, then if the property holds for the empty set and is invariant with respect to this operation, then it must hold for all finite sets.› lemma fin_ord_induction: assumes A1: "IsLinOrder(X,r)" and A2: "P(0)" and A3: "∀A ∈ FinPow(X). A ≠ 0 ⟶ (P(A - {Maximum(r,A)}) ⟶ P(A))" and A4: "B ∈ FinPow(X)" shows "P(B)" proof - note A2 moreover have "∀ A ∈ FinPow(X). A ≠ 0 ⟶ (∃a∈A. P(A-{a}) ⟶ P(A))" proof - { fix A assume "A ∈ FinPow(X)" and "A ≠ 0" with A1 A3 have "∃a∈A. P(A-{a}) ⟶ P(A)" using IsLinOrder_def fin_has_max IsLinOrder_def Order_ZF_4_L3 by blast } thus ?thesis by simp qed moreover note A4 ultimately show "P(B)" by (rule FinPow_ind_rem_one) qed text‹A sligltly more complicated version of ‹fin_ord_induction› that allows to prove properties that are not true for the empty set.› lemma fin_ord_ind: assumes A1: "IsLinOrder(X,r)" and A2: "∀A ∈ FinPow(X). A = 0 ∨ (A = {Maximum(r,A)} ∨ P(A - {Maximum(r,A)}) ⟶ P(A))" and A3: "B ∈ FinPow(X)" and A4: "B≠0" shows "P(B)" proof - { fix A assume "A ∈ FinPow(X)" and "A ≠ 0" with A1 A2 have "∃a∈A. A = {a} ∨ P(A-{a}) ⟶ P(A)" using IsLinOrder_def fin_has_max IsLinOrder_def Order_ZF_4_L3 by blast } then have "∀A ∈ FinPow(X). A = 0 ∨ (∃a∈A. A = {a} ∨ P(A-{a}) ⟶ P(A))" by auto with A3 A4 show "P(B)" using FinPow_rem_ind by simp qed text‹Yet another induction scheme. We build a linearly ordered set by adding elements that are greater than all elements in the set.› lemma fin_ind_add_max: assumes A1: "IsLinOrder(X,r)" and A2: "P(0)" and A3: "∀ A ∈ FinPow(X). ( ∀ x ∈ X-A. P(A) ∧ (∀a∈A. ⟨a,x⟩ ∈ r ) ⟶ P(A ∪ {x}))" and A4: "B ∈ FinPow(X)" shows "P(B)" proof - note A1 A2 moreover have "∀C ∈ FinPow(X). C ≠ 0 ⟶ (P(C - {Maximum(r,C)}) ⟶ P(C))" proof - { fix C assume "C ∈ FinPow(X)" and "C ≠ 0" let ?x = "Maximum(r,C)" let ?A = "C - {?x}" assume "P(?A)" moreover from ‹C ∈ FinPow(X)› have "?A ∈ FinPow(X)" using fin_rem_point_fin by simp moreover from A1 ‹C ∈ FinPow(X)› ‹C ≠ 0› have "?x ∈ C" and "?x ∈ X - ?A" and "∀a∈?A. ⟨a,?x⟩ ∈ r" using linord_max_props by auto moreover note A3 ultimately have "P(?A ∪ {?x})" by auto moreover from ‹?x ∈ C› have "?A ∪ {?x} = C" by auto ultimately have "P(C)" by simp } thus ?thesis by simp qed moreover note A4 ultimately show "P(B)" by (rule fin_ord_induction) qed text‹The only order automorphism of a linearly ordered finite set is the identity.› theorem fin_ord_auto_id: assumes A1: "IsLinOrder(X,r)" and A2: "B ∈ FinPow(X)" and A3: "B≠0" shows "ord_iso(B,r,B,r) = {id(B)}" proof - note A1 moreover { fix A assume "A ∈ FinPow(X)" "A≠0" let ?M = "Maximum(r,A)" let ?A⇩_{0}= "A - {?M}" assume "A = {?M} ∨ ord_iso(?A⇩_{0},r,?A⇩_{0},r) = {id(?A⇩_{0})}" moreover { assume "A = {?M}" have "ord_iso({?M},r,{?M},r) = {id({?M})}" using id_ord_auto_singleton by simp with ‹A = {?M}› have "ord_iso(A,r,A,r) = {id(A)}" by simp } moreover { assume "ord_iso(?A⇩_{0},r,?A⇩_{0},r) = {id(?A⇩_{0})}" have "ord_iso(A,r,A,r) = {id(A)}" proof show "{id(A)} ⊆ ord_iso(A,r,A,r)" using id_ord_iso by simp { fix f assume "f ∈ ord_iso(A,r,A,r)" with A1 ‹A ∈ FinPow(X)› ‹A≠0› have "restrict(f,?A⇩_{0}) ∈ ord_iso(?A⇩_{0}, r, A-{f`(?M)},r)" using IsLinOrder_def fin_has_max ord_iso_rem_max by auto with A1 ‹A ∈ FinPow(X)› ‹A≠0› ‹f ∈ ord_iso(A,r,A,r)› ‹ord_iso(?A⇩_{0},r,?A⇩_{0},r) = {id(?A⇩_{0})}› have "restrict(f,?A⇩_{0}) = id(?A⇩_{0})" using IsLinOrder_def fin_has_max max_auto_fixpoint by auto moreover from A1 ‹f ∈ ord_iso(A,r,A,r)› ‹A ∈ FinPow(X)› ‹A≠0› have "f : A → A" and "?M ∈ A" and "f`(?M) = ?M" using ord_iso_def bij_is_fun IsLinOrder_def fin_has_max Order_ZF_4_L3 max_auto_fixpoint by auto ultimately have "f = id(A)" using id_fixpoint_rem by simp } then show "ord_iso(A,r,A,r) ⊆ {id(A)}" by auto qed } ultimately have "ord_iso(A,r,A,r) = {id(A)}" by auto } then have "∀A ∈ FinPow(X). A = 0 ∨ (A = {Maximum(r,A)} ∨ ord_iso(A-{Maximum(r,A)},r,A-{Maximum(r,A)},r) = {id(A-{Maximum(r,A)})} ⟶ ord_iso(A,r,A,r) = {id(A)})" by auto moreover note A2 A3 ultimately show "ord_iso(B,r,B,r) = {id(B)}" by (rule fin_ord_ind) qed text‹Every two finite linearly ordered sets are order isomorphic. The statement is formulated to make the proof by induction on the size of the set easier, see ‹fin_ord_iso_ex› for an alternative formulation. › lemma fin_order_iso: assumes A1: "IsLinOrder(X,r)" "IsLinOrder(Y,R)" and A2: "n ∈ nat" shows "∀A ∈ FinPow(X). ∀B ∈ FinPow(Y). A ≈ n ∧ B ≈ n ⟶ ord_iso(A,r,B,R) ≠ 0" proof - note A2 moreover have "∀A ∈ FinPow(X). ∀B ∈ FinPow(Y). A ≈ 0 ∧ B ≈ 0 ⟶ ord_iso(A,r,B,R) ≠ 0" using eqpoll_0_is_0 empty_ord_iso by blast moreover have "∀k ∈ nat. (∀A ∈ FinPow(X). ∀B ∈ FinPow(Y). A ≈ k ∧ B ≈ k ⟶ ord_iso(A,r,B,R) ≠ 0) ⟶ (∀C ∈ FinPow(X). ∀D ∈ FinPow(Y). C ≈ succ(k) ∧ D ≈ succ(k) ⟶ ord_iso(C,r,D,R) ≠ 0)" proof - { fix k assume "k ∈ nat" assume A3: "∀A ∈ FinPow(X). ∀B ∈ FinPow(Y). A ≈ k ∧ B ≈ k ⟶ ord_iso(A,r,B,R) ≠ 0" have "∀C ∈ FinPow(X). ∀D ∈ FinPow(Y). C ≈ succ(k) ∧ D ≈ succ(k) ⟶ ord_iso(C,r,D,R) ≠ 0" proof - { fix C assume "C ∈ FinPow(X)" fix D assume "D ∈ FinPow(Y)" assume "C ≈ succ(k)" "D ≈ succ(k)" then have "C ≠ 0" and "D≠ 0" using eqpoll_succ_imp_not_empty by auto let ?M⇩_{C}= "Maximum(r,C)" let ?M⇩_{D}= "Maximum(R,D)" let ?C⇩_{0}= "C - {?M⇩_{C}}" let ?D⇩_{0}= "D - {?M⇩_{D}}" from ‹C ∈ FinPow(X)› have "C ⊆ X" using FinPow_def by simp with A1 have "IsLinOrder(C,r)" using ord_linear_subset by blast from ‹D ∈ FinPow(Y)› have "D ⊆ Y" using FinPow_def by simp with A1 have "IsLinOrder(D,R)" using ord_linear_subset by blast from A1 ‹C ∈ FinPow(X)› ‹D ∈ FinPow(Y)› ‹C ≠ 0› ‹D≠ 0› have "HasAmaximum(r,C)" and "HasAmaximum(R,D)" using IsLinOrder_def fin_has_max by auto with A1 have "?M⇩_{C}∈ C" and "?M⇩_{D}∈ D" using IsLinOrder_def Order_ZF_4_L3 by auto with ‹C ≈ succ(k)› ‹D ≈ succ(k)› have "?C⇩_{0}≈ k" and "?D⇩_{0}≈ k" using Diff_sing_eqpoll by auto from ‹C ∈ FinPow(X)› ‹D ∈ FinPow(Y)› have "?C⇩_{0}∈ FinPow(X)" and "?D⇩_{0}∈ FinPow(Y)" using fin_rem_point_fin by auto with A3 ‹?C⇩_{0}≈ k› ‹?D⇩_{0}≈ k› have "ord_iso(?C⇩_{0},r,?D⇩_{0},R) ≠ 0" by simp with ‹IsLinOrder(C,r)› ‹IsLinOrder(D,R)› ‹HasAmaximum(r,C)› ‹HasAmaximum(R,D)› have "ord_iso(C,r,D,R) ≠ 0" by (rule rem_max_ord_iso) } thus ?thesis by simp qed } thus ?thesis by blast qed ultimately show ?thesis by (rule ind_on_nat) qed text‹Every two finite linearly ordered sets are order isomorphic.› lemma fin_ord_iso_ex: assumes A1: "IsLinOrder(X,r)" "IsLinOrder(Y,R)" and A2: "A ∈ FinPow(X)" "B ∈ FinPow(Y)" and A3: "B ≈ A" shows "ord_iso(A,r,B,R) ≠ 0" proof - from A2 obtain n where "n ∈ nat" and "A ≈ n" using finpow_decomp by auto from A3 ‹A ≈ n› have "B ≈ n" by (rule eqpoll_trans) with A1 A2 ‹A ≈ n› ‹n ∈ nat› show "ord_iso(A,r,B,R) ≠ 0" using fin_order_iso by simp qed text‹Existence and uniqueness of order isomorphism for two linearly ordered sets with the same number of elements.› theorem fin_ord_iso_ex_uniq: assumes A1: "IsLinOrder(X,r)" "IsLinOrder(Y,R)" and A2: "A ∈ FinPow(X)" "B ∈ FinPow(Y)" and A3: "B ≈ A" shows "∃!f. f ∈ ord_iso(A,r,B,R)" proof from assms show "∃f. f ∈ ord_iso(A,r,B,R)" using fin_ord_iso_ex by blast fix f g assume A4: "f ∈ ord_iso(A,r,B,R)" "g ∈ ord_iso(A,r,B,R)" then have "converse(g) ∈ ord_iso(B,R,A,r)" using ord_iso_sym by simp with ‹f ∈ ord_iso(A,r,B,R)› have I: "converse(g) O f ∈ ord_iso(A,r,A,r)" by (rule ord_iso_trans) { assume "A ≠ 0" with A1 A2 I have "converse(g) O f = id(A)" using fin_ord_auto_id by auto with A4 have "f = g" using ord_iso_def comp_inv_id_eq_bij by auto } moreover { assume "A = 0" then have "A ≈ 0" using eqpoll_0_iff by simp with A3 have "B ≈ 0" by (rule eqpoll_trans) with A4 ‹A = 0› have "f ∈ ord_iso(0,r,0,R)" and "g ∈ ord_iso(0,r,0,R)" using eqpoll_0_iff by auto then have "f = g" by (rule empty_ord_iso_uniq) } ultimately show "f = g" using ord_iso_def comp_inv_id_eq_bij by auto qed end