(* Title: Dual_Systems.thy Author: Chelsea Edmonds *) section ‹ Dual Systems › text ‹The concept of a dual incidence system \<^cite>‹"colbournHandbookCombinatorialDesigns2007"› is an important property in design theory. It enables us to reason on the existence of several different types of design constructs through dual properties \<^cite>‹"stinsonCombinatorialDesignsConstructions2004"›› theory Dual_Systems imports Incidence_Matrices begin subsection ‹Dual Blocks › text ‹A dual design of $(\mathcal{V}, \mathcal{B})$, is the design where each block in $\mathcal{B}$ represents a point $x$, and a block in a dual design is a set of blocks which $x$ is in from the original design. It is important to note that if a block repeats in $\mathcal{B}$, each instance of the block is a distinct point. As such the definition below uses each block's list index as its identifier. The list of points would simply be the indices $0..<$length $Bs$ › definition dual_blocks :: "'a set ⇒ 'a set list ⇒ nat set multiset" where "dual_blocks 𝒱 ℬs ≡ {# {y . y < length ℬs ∧ x ∈ ℬs ! y} . x ∈# (mset_set 𝒱)#}" lemma dual_blocks_wf: "b ∈# dual_blocks V Bs ⟹ b ⊆ {0..<length Bs}" by (auto simp add: dual_blocks_def) context ordered_incidence_system begin definition dual_blocks_ordered :: "nat set list" ("ℬs*") where "dual_blocks_ordered ≡ map (λ x . {y . y < length ℬs ∧ x ∈ ℬs ! y}) 𝒱s" lemma dual_blocks_ordered_eq: "dual_blocks 𝒱 ℬs= mset (ℬs*)" by (auto simp add: distinct dual_blocks_def dual_blocks_ordered_def mset_set_set) lemma dual_blocks_len: "length ℬs* = length 𝒱s" by (simp add: dual_blocks_ordered_def) text ‹A dual system is an incidence system › sublocale dual_sys: finite_incidence_system "{0..<length ℬs}" "dual_blocks 𝒱 ℬs" using dual_blocks_wf by(unfold_locales) (auto) lemma dual_is_ordered_inc_sys: "ordered_incidence_system [0..<length ℬs] ℬs*" using inc_sys_orderedI dual_blocks_ordered_eq by (metis atLeastLessThan_upt distinct_upt dual_sys.incidence_system_axioms) interpretation ordered_dual_sys: ordered_incidence_system "[0..<length ℬs]" "ℬs*" using dual_is_ordered_inc_sys by simp subsection ‹Basic Dual Properties› lemma ord_dual_blocks_b: "ordered_dual_sys.𝖻 = 𝗏" using dual_blocks_len by (simp add: points_list_length) lemma dual_blocks_b: "dual_sys.𝖻 = 𝗏" using points_list_length by (simp add: dual_blocks_len dual_blocks_ordered_eq) lemma dual_blocks_v: "dual_sys.𝗏 = 𝖻" by fastforce lemma ord_dual_blocks_v: "ordered_dual_sys.𝗏 = 𝖻" by fastforce lemma dual_point_block: "i < 𝗏 ⟹ ℬs* ! i = {y. y < length ℬs ∧ (𝒱s ! i) ∈ ℬs ! y}" by (simp add: dual_blocks_ordered_def points_list_length) lemma dual_incidence_iff: "i < 𝗏 ⟹ j < 𝖻 ⟹ ℬs ! j = bl ⟹ 𝒱s ! i = x ⟹ (x ∈ bl ⟷ j ∈ ℬs* ! i)" using dual_point_block by (intro iffI)(simp_all) lemma dual_incidence_iff2: "i < 𝗏 ⟹ j < 𝖻 ⟹ (𝒱s ! i ∈ ℬs ! j ⟷ j ∈ ℬs* ! i)" using dual_incidence_iff by simp lemma dual_blocks_point_exists: "bl ∈# dual_blocks 𝒱 ℬs ⟹ ∃ x. x ∈ 𝒱 ∧ bl = {y . y < length ℬs ∧ x ∈ ℬs ! y}" by (auto simp add: dual_blocks_def) lemma dual_blocks_ne_index_ne: "j1 < length ℬs* ⟹ j2 < length ℬs* ⟹ ℬs* ! j1 ≠ ℬs* ! j2 ⟹ j1 ≠ j2" by auto lemma dual_blocks_list_index_img: "image_mset (λx . ℬs* ! x) (mset_set {0..<length ℬs*}) = mset ℬs*" using lessThan_atLeast0 ordered_dual_sys.blocks_list_length ordered_dual_sys.blocks_mset_image by presburger lemma dual_blocks_elem_iff: assumes "j < 𝗏" shows "x ∈ (ℬs* ! j) ⟷ 𝒱s ! j ∈ ℬs ! x ∧ x < 𝖻" proof (intro iffI conjI) show "x ∈ ℬs* ! j ⟹ 𝒱s ! j ∈ ℬs ! x" using assms ordered_incidence_system.dual_point_block ordered_incidence_system_axioms by fastforce show "x ∈ ℬs* ! j ⟹ x < 𝖻" using assms dual_blocks_ordered_def dual_point_block by fastforce show "𝒱s ! j ∈ ℬs ! x ∧ x < 𝖻 ⟹ x ∈ ℬs* ! j" by (metis (full_types) assms blocks_list_length dual_incidence_iff) qed text ‹The incidence matrix of the dual of a design is just the transpose › lemma dual_incidence_mat_eq_trans: "ordered_dual_sys.N = N⇧^{T}" proof (intro eq_matI) show dimr: "dim_row ordered_dual_sys.N = dim_row N⇧^{T}" using dual_blocks_v by (simp) show dimc: "dim_col ordered_dual_sys.N = dim_col N⇧^{T}" using ord_dual_blocks_b by (simp) show "⋀i j. i < dim_row N⇧^{T}⟹ j < dim_col N⇧^{T}⟹ ordered_dual_sys.N $$ (i, j) = N⇧^{T}$$ (i, j)" proof - fix i j assume ilt: "i < dim_row N⇧^{T}" assume jlt: "j < dim_col N⇧^{T}" then have ilt2: "i < length ℬs"using dimr using blocks_list_length ord_dual_blocks_v ilt ordered_dual_sys.dim_row_is_v by linarith then have ilt3: "i < 𝖻" by simp have jlt2: "j < 𝗏" using jlt using dim_row_is_v by fastforce have "ordered_dual_sys.N $$ (i, j) = (if ([0..<length ℬs] ! i) ∈ (ℬs* ! j) then 1 else 0)" using dimr dual_blocks_len ilt jlt inc_matrix_elems_one_zero by (metis inc_mat_dim_row inc_matrix_point_in_block_iff index_transpose_mat(3) ) then have "ordered_dual_sys.N $$ (i, j) = (if 𝒱s ! j ∈ ℬs ! i then 1 else 0)" using ilt3 jlt2 dual_incidence_iff2 by simp thus "ordered_dual_sys.N $$ (i, j) = N⇧^{T}$$ (i, j)" using ilt3 jlt2 dim_row_is_v dim_col_is_b N_trans_index_val by simp qed qed lemma dual_incidence_mat_eq_trans_rev: "(ordered_dual_sys.N)⇧^{T}= N" using dual_incidence_mat_eq_trans by simp subsection ‹Incidence System Dual Properties› text ‹Many common design properties have a dual in the dual design which enables extensive reasoning Using incidence matrices and the transpose property these are easy to prove. We leave examples of counting proofs (commented out), to demonstrate how incidence matrices can significantly simplify reasoning › lemma dual_blocks_nempty: assumes "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" assumes "bl ∈# dual_blocks 𝒱 ℬs" shows "bl ≠ {}" proof - have "bl ∈# {# {y . y < length ℬs ∧ x ∈ ℬs ! y} . x ∈# (mset_set 𝒱)#}" using assms dual_blocks_def by metis then obtain x where "x ∈# (mset_set 𝒱)" and blval: "bl = {y . y < length ℬs ∧ x ∈ ℬs ! y}" by blast then obtain bl' where "bl' ∈# ℬ" and xin: "x ∈ bl'" using assms(1) using point_in_block_rep_min_iff by auto then obtain y where "y < length ℬs" and "ℬs ! y = bl'" using valid_blocks_index_cons by auto then have "y ∈ bl" by (simp add: xin blval) thus ?thesis by blast qed lemma dual_blocks_size_is_rep: "j < length ℬs* ⟹ card (ℬs* ! j) = ℬ rep (𝒱s ! j)" using dual_incidence_mat_eq_trans trans_mat_rep_block_size_sym(2) by (metis dual_blocks_len dual_is_ordered_inc_sys inc_mat_dim_row mat_rep_num_N_row ordered_incidence_system.mat_block_size_N_col points_list_length size_mset) (* Old Counting proof proof - have 1: "card (ℬs* ! j) = card {y . y < length ℬs ∧ (𝒱s ! j) ∈ ℬs ! y}" using assms dual_blocks_len dual_point_block points_list_length by force also have 2: "... = card {y ∈ {0..<length ℬs} . (𝒱s ! j) ∈ ℬs ! y}" by simp also have "... = size (mset_set {y ∈ {0..<length ℬs} . (𝒱s ! j) ∈ ℬs ! y})" by simp also have "... = size {# y ∈# mset_set {0..< length ℬs} . (𝒱s ! j) ∈ ℬs ! y #}" using filter_mset_mset_set by simp finally have "card (ℬs* ! j) = size {# bl ∈# ℬ . (𝒱s ! j) ∈ bl #}" by (metis 1 2 filter_size_blocks_eq_card_indexes lessThan_atLeast0 size_mset) thus ?thesis by (simp add: point_replication_number_def) qed *) lemma dual_blocks_size_is_rep_obtain: assumes "bl ∈# dual_blocks 𝒱 ℬs" obtains x where "x ∈ 𝒱" and "card bl = ℬ rep x" proof - obtain j where jlt1: "j < length ℬs*" and bleq: "ℬs* ! j = bl" by (metis assms dual_blocks_ordered_eq in_mset_conv_nth) then have jlt: "j < 𝗏" by (simp add: dual_blocks_len points_list_length) let ?x = "𝒱s ! j" have xin: "?x ∈ 𝒱" using jlt by (simp add: valid_points_index) have "card bl = ℬ rep ?x" using dual_blocks_size_is_rep jlt1 bleq by auto thus ?thesis using xin that by auto qed lemma dual_blocks_rep_is_size: assumes "i < length ℬs" shows "(mset ℬs*) rep i = card (ℬs ! i)" proof - have "[0..<length ℬs] ! i = i" using assms by simp then have "(mset ℬs*) rep i = mat_rep_num ordered_dual_sys.N i" using ordered_dual_sys.mat_rep_num_N_row assms length_upt minus_nat.diff_0 ordered_dual_sys.points_list_length by presburger also have "... = mat_block_size (ordered_dual_sys.N)⇧^{T}i" using dual_incidence_mat_eq_trans trans_mat_rep_block_size_sym(2) by (metis assms inc_mat_dim_col index_transpose_mat(2)) finally show ?thesis using dual_incidence_mat_eq_trans_rev by (metis assms blocks_list_length mat_block_size_N_col) qed (* Counting Proof proof - have "(mset ℬs* ) rep i = size {# bl ∈# (mset ℬs* ) . i ∈ bl #}" by (simp add: point_replication_number_def) also have 1: "... = size {# bl ∈# {# {y . y < length ℬs ∧ x ∈ ℬs ! y} . x ∈# (mset_set 𝒱)#} . i ∈ bl #}" using dual_blocks_ordered_eq dual_blocks_def by metis also have "... = size (filter_mset (λ bl . i ∈ bl) (image_mset (λ x . {y . y < length ℬs ∧ x ∈ ℬs ! y}) (mset_set 𝒱)))" by simp finally have "(mset ℬs* ) rep i = size (image_mset (λ x . {y . y < length ℬs ∧ x ∈ ℬs ! y}) (filter_mset (λ bl . i ∈ {y . y < length ℬs ∧ bl ∈ ℬs ! y}) (mset_set 𝒱)))" using filter_mset_image_mset by (metis 1 ordered_dual_sys.point_rep_number_alt_def) then have "(mset ℬs* ) rep i = size (filter_mset (λ bl . i ∈ {y . y < length ℬs ∧ bl ∈ ℬs ! y}) (mset_set 𝒱))" by fastforce then have "(mset ℬs* ) rep i = size (filter_mset (λ bl . bl ∈ ℬs ! i) (mset_set 𝒱))" using assms by simp then have "(mset ℬs* ) rep i = card {x ∈ 𝒱 . x ∈ (ℬs ! i)}" by simp thus ?thesis using assms block_size_alt by auto qed *) lemma dual_blocks_inter_index: assumes "j1 < length ℬs*" "j2 < length ℬs*" shows "(ℬs* ! j1) |∩| (ℬs* ! j2) = points_index ℬ {𝒱s ! j1, 𝒱s ! j2}" proof - have assms2: "j1 < 𝗏" "j2 < 𝗏" using assms by (simp_all add: dual_blocks_len points_list_length) have "(ℬs* ! j1) |∩| (ℬs* ! j2) = mat_inter_num (ordered_dual_sys.N) j1 j2" by (simp add: assms(1) assms(2) ordered_dual_sys.mat_inter_num_conv) also have "... = mat_point_index N {j1, j2}" using dual_incidence_mat_eq_trans_rev trans_mat_point_index_inter_sym(2) by (metis assms inc_mat_dim_col) finally show ?thesis using assms2 incidence_mat_two_index by presburger qed (* Counting Proof have fin: "finite {0..<length ℬs}" by auto have j1lt: "j1 < 𝗏" using assms using dual_blocks_len points_list_length by auto have j2lt: "j2 < 𝗏" using assms dual_blocks_len points_list_length by auto have iff: "⋀ x. (x ∈(ℬs* ! j1) ∧ x ∈ (ℬs* ! j2)) ⟷ (𝒱s ! j1 ∈ ℬs ! x ∧ x < 𝖻 ∧ 𝒱s ! j2 ∈ ℬs ! x)" by (auto simp add: dual_blocks_elem_iff j1lt j2lt) have pi: "points_index ℬ {𝒱s ! j1, 𝒱s ! j2} = size {# bl ∈# ℬ . 𝒱s !j1 ∈ bl ∧ 𝒱s ! j2 ∈ bl#}" by (auto simp add: points_index_def) have "(ℬs* ! j1) |∩| (ℬs* ! j2) = card ({x . x <length ℬs ∧ x ∈ (ℬs* ! j1) ∧ x ∈ (ℬs* ! j2)})" apply (auto simp add: intersection_number_def) by (smt (verit) Collect_cong Int_Collect blocks_list_length dual_blocks_elem_iff inf.idem inf_set_def j2lt mem_Collect_eq) then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = card ({x . x <length ℬs ∧ 𝒱s ! j1 ∈ ℬs ! x ∧ 𝒱s ! j2 ∈ ℬs ! x})" using iff size_mset by (smt (verit, best) Collect_cong) then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = size (mset_set {x ∈ {0..<length ℬs}. 𝒱s ! j1 ∈ ℬs ! x ∧ 𝒱s ! j2 ∈ ℬs ! x})" by simp then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = size ({#x ∈# mset_set {0..<length ℬs}. 𝒱s ! j1 ∈ ℬs ! x ∧ 𝒱s ! j2 ∈ ℬs ! x#})" using fin by simp then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = size (filter_mset (λ x . 𝒱s ! j1 ∈ ℬs ! x ∧ 𝒱s ! j2 ∈ ℬs ! x) (mset_set {0..<length ℬs}))" by simp then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = size (image_mset (λ i. ℬs ! i) (filter_mset (λ x . 𝒱s ! j1 ∈ ℬs ! x ∧ 𝒱s ! j2 ∈ ℬs ! x) (mset_set {0..<length ℬs})))" by simp then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = size (filter_mset (λ x . 𝒱s ! j1 ∈ x ∧ 𝒱s ! j2 ∈ x) (image_mset (λ i. ℬs ! i) (mset_set {0..<length ℬs})))" by (simp add: filter_mset_image_mset) then have "(ℬs* ! j1) |∩| (ℬs* ! j2) = size {# bl ∈# ℬ . 𝒱s !j1 ∈ bl ∧ 𝒱s ! j2 ∈ bl#}" by (metis blocks_list_length blocks_mset_image lessThan_atLeast0) thus ?thesis using pi by simp qed *) lemma dual_blocks_points_index_inter: assumes "i1 < 𝖻" "i2 < 𝖻" shows "(mset ℬs*) index {i1, i2} = (ℬs ! i1) |∩| (ℬs ! i2)" proof - have "(mset ℬs*) index {i1, i2} = mat_point_index (ordered_dual_sys.N) {i1, i2}" using assms(1) assms(2) blocks_list_length ord_dual_blocks_v ordered_dual_sys.dim_row_is_v ordered_dual_sys.incidence_mat_two_index ordered_dual_sys.mat_ord_inc_sys_point by presburger also have "... = mat_inter_num N i1 i2" using dual_incidence_mat_eq_trans trans_mat_point_index_inter_sym(1) by (metis assms(1) assms(2) dual_incidence_mat_eq_trans_rev ord_dual_blocks_v ordered_dual_sys.dim_row_is_v) finally show ?thesis using mat_inter_num_conv using assms(1) assms(2) by auto qed (* Counting Proof proof - have "⋀ j . j ∈# mset_set {0..<length ℬs*} ⟹ j < 𝗏" by (metis atLeastLessThan_iff atLeastLessThan_upt dual_blocks_len mset_upt points_list_length set_mset_mset) then have iff: "⋀ j i. j ∈# mset_set {0..<length ℬs*} ⟹ i < 𝖻 ⟹ i ∈ (ℬs* ! j) ⟷ (𝒱s ! j) ∈ (ℬs ! i)" using assms dual_incidence_iff2 by simp then have iff2: "⋀ j . j ∈# mset_set {0..<length ℬs*} ⟹ i1 ∈ (ℬs* ! j) ∧ i2 ∈ (ℬs* ! j) ⟷ (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)" using assms by auto have ss2: "(ℬs ! i2) ⊆ 𝒱" using wellformed assms by auto then have ss: "{x . x ∈ (ℬs ! i1) ∧ x ∈ (ℬs ! i2)} ⊆ 𝒱" by auto then have inter: "(ℬs ! i1) |∩| (ℬs ! i2) = card {x ∈ 𝒱. x ∈ (ℬs ! i1) ∧ x ∈ (ℬs ! i2)}" using intersection_number_def by (metis Collect_conj_eq Collect_mem_eq Int_absorb1) have inj: "inj_on (λ j. 𝒱s ! j) {j ∈ {0..<length 𝒱s} . (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)}" by (simp add: inj_on_nth distinct) have init: "(mset ℬs* ) index {i1, i2} = size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#}" by (simp add: points_index_def) then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = size {#j ∈# mset_set {0..<length ℬs*} . i1 ∈ (ℬs* ! j) ∧ i2 ∈ (ℬs* ! j)#}" proof - have "size {#j ∈# mset_set {0..<length ℬs*} . i1 ∈ (ℬs* ! j) ∧ i2 ∈ (ℬs* ! j)#} = size (filter_mset (λ j. i1 ∈ (ℬs* ! j) ∧ i2 ∈ (ℬs* ! j)) (mset_set {0..<length ℬs*})) " by simp also have s1: "... = size (image_mset (λx . ℬs* ! x) (filter_mset (λ j. i1 ∈ (ℬs* ! j) ∧ i2 ∈ (ℬs* ! j)) (mset_set {0..<length ℬs*})))" by fastforce also have s2: "... = size (filter_mset (λ j. i1 ∈ j ∧ i2 ∈ j) (image_mset (λx . ℬs* ! x) (mset_set {0..<length ℬs*})))" by (simp add: filter_mset_image_mset) finally have "size {#j ∈# mset_set {0..<length ℬs*} . i1 ∈ (ℬs* ! j) ∧ i2 ∈ (ℬs* ! j)#} = size (filter_mset (λ j. i1 ∈ j ∧ i2 ∈ j) (mset ℬs* ))" using dual_blocks_list_index_img s2 s1 by presburger thus ?thesis by simp qed then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = size {#j ∈# mset_set {0..<length ℬs*} . (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)#}" using iff2 by (smt (verit, ccfv_SIG) filter_mset_cong) then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = card ({j ∈ {0..<length ℬs*} . (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)})" by simp then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = card ({j ∈ {0..<length 𝒱s} . (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)})" using dual_blocks_len by presburger then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = card (image (λ j. 𝒱s ! j) {j ∈ {0..<length 𝒱s} . (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)})" using inj card_image[of "(λ j. 𝒱s ! j)" "{j ∈ {0..<length 𝒱s} . (𝒱s ! j) ∈ (ℬs ! i1) ∧ (𝒱s ! j) ∈ (ℬs ! i2)}"] by simp then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = card {j ∈ image (λ j. 𝒱s ! j) {0..<length 𝒱s}. j ∈ (ℬs ! i1) ∧ j ∈ (ℬs ! i2)}" using Compr_image_eq[of "(λ j. 𝒱s ! j)" "{0..<length 𝒱s}" "(λ j . j ∈ (ℬs ! i1) ∧ j ∈ (ℬs ! i2))"] by simp then have "size {#bl ∈# (mset ℬs* ) . i1 ∈ bl ∧ i2 ∈ bl#} = card {j ∈ 𝒱. j ∈ (ℬs ! i1) ∧ j ∈ (ℬs ! i2)}" using lessThan_atLeast0 points_list_length points_set_index_img by presburger thus ?thesis using init inter by simp qed*) end subsection ‹Dual Properties for Design sub types › context ordered_design begin lemma dual_is_design: assumes "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" ― ‹ Required to ensure no blocks are empty › shows "design {0..<length ℬs} (dual_blocks 𝒱 ℬs)" using dual_blocks_nempty assms by (unfold_locales) (simp) end context ordered_proper_design begin lemma dual_sys_b_non_zero: "dual_sys.𝖻 ≠ 0" using v_non_zero dual_blocks_b by auto lemma dual_is_proper_design: assumes "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" ― ‹ Required to ensure no blocks are empty › shows "proper_design {0..<length ℬs} (dual_blocks 𝒱 ℬs)" using dual_blocks_nempty dual_sys_b_non_zero assms by (unfold_locales) (simp_all) end context ordered_block_design begin lemma dual_blocks_const_rep: "i ∈ {0..<length ℬs} ⟹ (mset ℬs*) rep i = 𝗄" using dual_blocks_rep_is_size uniform by (metis atLeastLessThan_iff nth_mem_mset) lemma dual_blocks_constant_rep_design: assumes "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" shows "constant_rep_design {0..<length ℬs} (dual_blocks 𝒱 ℬs) 𝗄" proof - interpret des: proper_design "{0..<length ℬs}" "(dual_blocks 𝒱 ℬs)" using dual_is_proper_design assms by simp show ?thesis using dual_blocks_const_rep dual_blocks_ordered_eq by (unfold_locales) (simp) qed end context ordered_constant_rep begin lemma dual_blocks_const_size: "j < length ℬs* ⟹ card (ℬs* ! j) = 𝗋" using dual_blocks_rep_is_size dual_blocks_len dual_blocks_size_is_rep by fastforce lemma dual_is_block_design: "block_design {0..<length ℬs} (dual_blocks 𝒱 ℬs) 𝗋" proof - have "𝗋 > 0" by (simp add: r_gzero) then have "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" using rep_number by simp then interpret pdes: proper_design "{0..<length ℬs}" "(dual_blocks 𝒱 ℬs)" using dual_is_proper_design by simp have "⋀ bl. bl ∈# dual_blocks 𝒱 ℬs ⟹ card bl = 𝗋" using dual_blocks_const_size by (metis dual_blocks_ordered_eq in_set_conv_nth set_mset_mset) thus ?thesis by (unfold_locales) (simp) qed end context ordered_pairwise_balance begin lemma dual_blocks_const_intersect: assumes "j1 < length ℬs*" "j2 < length ℬs*" assumes "j1 ≠ j2" shows "(ℬs* ! j1) |∩| (ℬs* ! j2) = Λ" proof - have "𝒱s ! j1 ≠ 𝒱s ! j2" using assms(3) using assms(1) assms(2) distinct dual_blocks_len nth_eq_iff_index_eq by auto then have c: "card {𝒱s ! j1, 𝒱s ! j2} = 2" using card_2_iff by blast have ss: "{𝒱s ! j1, 𝒱s ! j2} ⊆ 𝒱" using assms points_list_length using dual_blocks_len by auto have "(ℬs* ! j1) |∩| (ℬs* ! j2) = points_index ℬ {𝒱s ! j1, 𝒱s ! j2}" using dual_blocks_inter_index assms by simp thus ?thesis using ss c balanced by blast qed lemma dual_is_const_intersect_des: assumes "Λ > 0" shows "const_intersect_design {0..<(length ℬs)} (dual_blocks 𝒱 ℬs) Λ" proof - have "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x ≥ Λ)" using const_index_lt_rep by simp then have "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" using assms by (metis gr_zeroI le_zero_eq) then interpret pd: proper_design "{0..<(length ℬs)}" "(dual_blocks 𝒱 ℬs)" using dual_is_proper_design by (simp) show ?thesis proof (unfold_locales) fix b1 b2 assume b1in: "b1 ∈# dual_blocks 𝒱 ℬs" assume b2in: "b2 ∈# remove1_mset b1 (dual_blocks 𝒱 ℬs)" obtain j1 where b1eq: "b1 = ℬs* ! j1" and j1lt: "j1 < length ℬs*" using b1in by (metis dual_blocks_ordered_eq in_set_conv_nth set_mset_mset) obtain j2 where b2eq: "b2 = ℬs* ! j2" and j2lt: "j2 < length ℬs*" and "j1 ≠ j2" using b2in index_remove1_mset_ne by (metis (mono_tags) b1eq dual_blocks_ordered_eq j1lt nth_mem set_mset_mset) then show "b1 |∩| b2 = Λ" using dual_blocks_const_intersect b1eq b2eq j1lt j2lt by simp qed qed lemma dual_is_simp_const_inter_des: assumes "Λ > 0" assumes "⋀ bl. bl ∈# ℬ ⟹ incomplete_block bl" shows "simple_const_intersect_design {0..<(length ℬs)} (dual_blocks 𝒱 ℬs) Λ" proof - interpret d: const_intersect_design "{0..<(length ℬs)}" "(dual_blocks 𝒱 ℬs)" "Λ" using assms dual_is_const_intersect_des by simp ― ‹ Show that m < block size for all blocks › have "⋀ x. x ∈ 𝒱 ⟹ Λ < ℬ rep x" using assms incomplete_index_strict_lt_rep by blast then have "⋀ bl. bl ∈# (dual_blocks 𝒱 ℬs) ⟹ Λ < card bl" by (metis dual_blocks_size_is_rep_obtain) then interpret s: simple_design "{0..<(length ℬs)}" "(dual_blocks 𝒱 ℬs)" using d.simple_const_inter_block_size by simp show ?thesis by (unfold_locales) qed end context ordered_const_intersect_design begin lemma dual_is_balanced: assumes "ps ⊆ {0..<length ℬs}" assumes "card ps = 2" shows "(dual_blocks 𝒱 ℬs) index ps = 𝗆" proof - obtain i1 i2 where psin: "ps = {i1, i2}" and neq: "i1 ≠ i2" using assms by (meson card_2_iff) then have lt: "i1 < 𝖻" using assms by (metis atLeastLessThan_iff blocks_list_length insert_subset) have lt2: "i2 < 𝖻" using assms psin by (metis atLeastLessThan_iff blocks_list_length insert_subset) then have inter: "(dual_blocks 𝒱 ℬs) index ps = (ℬs ! i1) |∩| (ℬs ! i2)" using dual_blocks_points_index_inter neq lt using dual_blocks_ordered_eq psin by presburger have inb1: "(ℬs ! i1) ∈# ℬ" using lt by auto have inb2: "(ℬs ! i2) ∈# (ℬ - {#(ℬs ! i1)#})" using lt2 neq blocks_index_ne_belong by (metis blocks_list_length lt) thus ?thesis using const_intersect inb1 inb2 inter by blast qed lemma dual_is_pbd: assumes "(⋀ x . x ∈ 𝒱 ⟹ ℬ rep x > 0)" assumes "𝖻 ≥ 2" shows "pairwise_balance {0..<(length ℬs)} (dual_blocks 𝒱 ℬs) 𝗆" proof - interpret pd: proper_design "{0..<(length ℬs)}" "(dual_blocks 𝒱 ℬs)" using dual_is_proper_design by (simp add: assms) show ?thesis proof (unfold_locales) show "(1 ::nat) ≤ 2" by simp then show "2 ≤ dual_sys.𝗏" using assms(2) by fastforce show "⋀ps. ps ⊆ {0..<length ℬs} ⟹ card ps = 2 ⟹ dual_blocks 𝒱 ℬs index ps = 𝗆" using dual_is_balanced by simp qed qed end context ordered_sym_bibd begin lemma dual_is_balanced: assumes "ps ⊆ {0..<length ℬs}" assumes "card ps = 2" shows "(dual_blocks 𝒱 ℬs) index ps = Λ" proof - obtain i1 i2 where psin: "ps = {i1, i2}" and neq: "i1 ≠ i2" using assms by (meson card_2_iff) then have lt: "i1 < 𝖻" using assms by (metis atLeastLessThan_iff blocks_list_length insert_subset) have lt2: "i2 < 𝖻" using assms psin by (metis atLeastLessThan_iff blocks_list_length insert_subset) then have inter: "(dual_blocks 𝒱 ℬs) index ps = (ℬs ! i1) |∩| (ℬs ! i2)" using dual_blocks_points_index_inter neq lt dual_blocks_ordered_eq psin by presburger have inb1: "(ℬs ! i1) ∈# ℬ" using lt by auto have inb2: "(ℬs ! i2) ∈# (ℬ - {#(ℬs ! i1)#})" using lt2 neq blocks_index_simp_unique by (metis blocks_list_length in_remove1_mset_neq lt valid_blocks_index) thus ?thesis using sym_block_intersections_index inb1 inter by blast qed lemma dual_bibd: "bibd {0..<(length ℬs)} (dual_blocks 𝒱 ℬs) 𝗋 Λ" proof - interpret block: block_design "{0..<(length ℬs)}" "(dual_blocks 𝒱 ℬs)" 𝗋 using dual_is_block_design by simp show ?thesis proof (unfold_locales) show "𝗋 < dual_sys.𝗏" using dual_blocks_v incomplete symmetric_condition_1 symmetric_condition_2 by presburger show "(1 ::nat) ≤ 2" by simp have "𝗏 ≥ 2" by (simp add: t_lt_order) then have "𝖻 ≥ 2" using local.symmetric by auto then show "2 ≤ dual_sys.𝗏" by simp show "⋀ps. ps ⊆ {0..<length ℬs} ⟹ card ps = 2 ⟹ dual_blocks 𝒱 ℬs index ps = Λ" using dual_is_balanced by simp show "2 ≤ 𝗋" using r_ge_two by blast qed qed text ‹The dual of a BIBD must by symmetric › lemma dual_bibd_symmetric: "symmetric_bibd {0..<(length ℬs)} (dual_blocks 𝒱 ℬs) 𝗋 Λ" proof - interpret bibd: bibd "{0..<(length ℬs)}" "(dual_blocks 𝒱 ℬs)" 𝗋 Λ using dual_bibd by simp show ?thesis using dual_blocks_b local.symmetric by (unfold_locales) (simp) qed end subsection ‹Generalise Dual Concept › text ‹The above formalisation relies on one translation of a dual design. However, any design with an ordering of points and blocks such that the matrix is the transpose of the original is a dual. The definition below encapsulates this concept. Additionally, we prove an isomorphism exists between the generated dual from @{term "dual_blocks"} and any design satisfying the is dual definition› context ordered_incidence_system begin definition is_dual:: "'b list ⇒ 'b set list ⇒ bool" where "is_dual Vs' Bs' ≡ ordered_incidence_system Vs' Bs' ∧ (inc_mat_of Vs' Bs' = N⇧^{T})" lemma is_dualI: assumes "ordered_incidence_system Vs' Bs'" assumes "(inc_mat_of Vs' Bs' = N⇧^{T})" shows "is_dual Vs' Bs'" by (auto simp add: is_dual_def assms) lemma is_dualD1: assumes "is_dual Vs' Bs'" shows "(inc_mat_of Vs' Bs' = N⇧^{T})" using is_dual_def assms by auto lemma is_dualD2: assumes "is_dual Vs' Bs'" shows "ordered_incidence_system Vs' Bs'" using is_dual_def assms by auto lemma generated_is_dual: "is_dual [0..<(length ℬs)] ℬs*" proof - interpret osys: ordered_incidence_system "[0..<(length ℬs)]" "ℬs*" using dual_is_ordered_inc_sys by simp show ?thesis using is_dual_def by (simp add: is_dual_def dual_incidence_mat_eq_trans osys.ordered_incidence_system_axioms) qed lemma is_dual_isomorphism_generated: assumes "is_dual Vs' Bs'" shows "∃ π. incidence_system_isomorphism (set Vs') (mset Bs') ({0..<(length ℬs)}) (dual_blocks 𝒱 ℬs) π" proof - interpret os2: ordered_incidence_system "([0..<(length ℬs)])" "(ℬs*)" by (simp add: dual_is_ordered_inc_sys) interpret os1: ordered_incidence_system Vs' Bs' using assms by (simp add: is_dualD2) interpret tos: two_ordered_sys Vs' Bs' "([0..<(length ℬs)])" "(ℬs*)" using assms ordered_incidence_system_axioms two_ordered_sys.intro by (simp add: is_dualD2 two_ordered_sys.intro dual_is_ordered_inc_sys) have os2V: "os2.𝒱 = {0..<(length ℬs)}" by auto have os2B: "os2.ℬ = dual_blocks 𝒱 ℬs" by (simp add: dual_blocks_ordered_eq) have "os1.N = inc_mat_of Vs' Bs'" by simp then have "os2.N = os1.N" using assms is_dualD1 dual_incidence_mat_eq_trans by fastforce thus ?thesis using tos.equal_inc_mat_isomorphism_ex os2V os2B by auto qed interpretation ordered_dual_sys: ordered_incidence_system "[0..<length ℬs]" "ℬs*" using dual_is_ordered_inc_sys by simp text ‹Original system is dual of the dual › lemma is_dual_rev: "ordered_dual_sys.is_dual 𝒱s ℬs" by (simp add: dual_incidence_mat_eq_trans_rev ordered_dual_sys.is_dualI ordered_incidence_system_axioms) end end