section ‹ Sparse Grids › theory Grid imports Grid_Point begin subsection "Vectors" type_synonym vector = "grid_point ⇒ real" definition null_vector :: "vector" where "null_vector ≡ λ p. 0" definition sum_vector :: "vector ⇒ vector ⇒ vector" where "sum_vector α β ≡ λ p. α p + β p" subsection ‹ Inductive enumeration of all grid points › inductive_set grid :: "grid_point ⇒ nat set ⇒ grid_point set" for b :: "grid_point" and ds :: "nat set" where Start[intro!]: "b ∈ grid b ds" | Child[intro!]: "⟦ p ∈ grid b ds ; d ∈ ds ⟧ ⟹ child p dir d ∈ grid b ds" lemma grid_length[simp]: "p' ∈ grid p ds ⟹ length p' = length p" by (erule grid.induct, auto) lemma grid_union_dims: "⟦ ds ⊆ ds' ; p ∈ grid b ds ⟧ ⟹ p ∈ grid b ds'" by (erule grid.induct, auto) lemma grid_transitive: "⟦ a ∈ grid b ds ; b ∈ grid c ds' ; ds' ⊆ ds'' ; ds ⊆ ds'' ⟧ ⟹ a ∈ grid c ds''" by (erule grid.induct, auto simp add: grid_union_dims) lemma grid_child[intro?]: assumes "d ∈ ds" and p_grid: "p ∈ grid (child b dir d) ds" shows "p ∈ grid b ds" using ‹d ∈ ds› grid_transitive[OF p_grid] by auto lemma grid_single_level[simp]: assumes "p ∈ grid b ds" and "d < length b" shows "lv b d ≤ lv p d" using assms proof induct case (Child p' d' dir) thus ?case by (cases "d' = d", auto simp add: child_def ix_def lv_def) qed auto lemma grid_child_level: assumes "d < length b" and p_grid: "p ∈ grid (child b dir d) ds" shows "lv b d < lv p d" proof - have "lv b d < lv (child b dir d) d" using child_lv[OF ‹d < length b›] by auto also have "… ≤ lv p d" using p_grid assms by (intro grid_single_level) auto finally show ?thesis . qed lemma child_out: "length p ≤ d ⟹ child p dir d = p" unfolding child_def by auto lemma grid_dim_remove: assumes inset: "p ∈ grid b ({d} ∪ ds)" and eq: "d < length b ⟹ p ! d = b ! d" shows "p ∈ grid b ds" using inset eq proof induct case (Child p' d' dir) show ?case proof (cases "d' ≥ length p'") case True with child_out[OF this] show ?thesis using Child by auto next case False hence "d' < length p'" by simp show ?thesis proof (cases "d' = d") case True hence "lv b d ≤ lv p' d" and "lv p' d < lv (child p' dir d) d" using child_single_level Child ‹d' < length p'› by auto hence False using Child and ‹d' = d› and lv_def and ‹¬ d' ≥ length p'› by auto thus ?thesis .. next case False hence "d' ∈ ds" using Child by auto moreover have "d < length b ⟹ p' ! d = b ! d" proof - assume "d < length b" hence "d < length p'" using Child by auto hence "child p' dir d' ! d = p' ! d" using child_invariant and False by auto thus ?thesis using Child and ‹d < length b› by auto qed hence "p' ∈ grid b ds" using Child by auto ultimately show ?thesis using grid.Child by auto qed qed qed auto lemma gridgen_dim_restrict: assumes inset: "p ∈ grid b (ds' ∪ ds)" and eq: "∀ d ∈ ds'. d ≥ length b" shows "p ∈ grid b ds" using inset eq proof induct case (Child p' d dir) thus ?case proof (cases "d ∈ ds") case False thus ?thesis using Child and child_def by auto qed auto qed auto lemma grid_dim_remove_outer: "grid b ds = grid b {d ∈ ds. d < length b}" proof have "{d ∈ ds. d < length b} ⊆ ds" by auto from grid_union_dims[OF this] show "grid b {d ∈ ds. d < length b} ⊆ grid b ds" by auto have "ds = (ds - {d ∈ ds. d < length b}) ∪ {d ∈ ds. d < length b}" by auto moreover have "grid b ((ds - {d ∈ ds. d < length b}) ∪ {d ∈ ds. d < length b}) ⊆ grid b {d ∈ ds. d < length b}" proof fix p assume "p ∈ grid b (ds - {d ∈ ds. d < length b} ∪ {d ∈ ds. d < length b})" moreover have "∀ d ∈ (ds - {d ∈ ds. d < length b}). d ≥ length b" by auto ultimately show "p ∈ grid b {d ∈ ds. d < length b}" by (rule gridgen_dim_restrict) qed ultimately show "grid b ds ⊆ grid b {d ∈ ds. d < length b}" by auto qed lemma grid_level[intro]: assumes "p ∈ grid b ds" shows "level b ≤ level p" proof - have *: "length p = length b" using grid_length assms by auto { fix i assume "i ∈ {0 ..< length p}" hence "lv b i ≤ lv p i" using ‹p ∈ grid b ds› and grid_single_level * by auto } thus ?thesis unfolding level_def * by (auto intro!: sum_mono) qed lemma grid_empty_ds[simp]: "grid b {} = { b }" proof - have "!! z. z ∈ grid b {} ⟹ z = b" by (erule grid.induct, auto) thus ?thesis by auto qed lemma grid_Start: assumes inset: "p ∈ grid b ds" and eq: "level p = level b" shows "p = b" using inset eq proof induct case (Child p d dir) show ?case proof (cases "d < length b") case True from Child have "level p ≥ level b" by auto moreover have "level p ≤ level (child p dir d)" by (rule child_level_gt) hence "level p ≤ level b" using Child by auto ultimately have "level p = level b" by auto hence "p = b " using Child(2) by auto with Child(4) have "level (child b dir d) = level b" by auto moreover have "level (child b dir d) ≠ level b" using child_level and ‹d < length b› by auto ultimately show ?thesis by auto next case False with Child have "length p = length b" by auto with False have "child p dir d = p" using child_def by auto moreover with Child have "level p = level b" by auto with Child(2) have "p = b" by auto ultimately show ?thesis by auto qed qed auto lemma grid_estimate: assumes "d < length b" and p_grid: "p ∈ grid b ds" shows "ix p d < (ix b d + 1) * 2^(lv p d - lv b d) ∧ ix p d > (ix b d - 1) * 2^(lv p d - lv b d)" using p_grid proof induct case (Child p d' dir) show ?case proof (cases "d = d'") case False with Child show ?thesis unfolding child_def lv_def ix_def by auto next case True with child_estimate_child and Child and ‹d < length b› show ?thesis using grid_single_level by auto qed qed auto lemma grid_odd: assumes "d < length b" and p_diff: "p ! d ≠ b ! d" and p_grid: "p ∈ grid b ds" shows "odd (ix p d)" using p_grid and p_diff proof induct case (Child p d' dir) show ?case proof (cases "d = d'") case True with child_odd and ‹d < length b› and Child show ?thesis by auto next case False with Child and ‹d < length b› show ?thesis using child_def and ix_def and lv_def by auto qed qed auto lemma grid_invariant: assumes "d < length b" and "d ∉ ds" and p_grid: "p ∈ grid b ds" shows "p ! d = b ! d" using p_grid proof (induct) case (Child p d' dir) hence "d' ≠ d" using ‹d ∉ ds› by auto thus ?case using child_def and Child by auto qed auto lemma grid_part: assumes "d < length b" and p_valid: "p ∈ grid b {d}" and p'_valid: "p' ∈ grid b {d}" and level: "lv p' d ≥ lv p d" and right: "ix p' d ≤ (ix p d + 1) * 2^(lv p' d - lv p d)" (is "?right p p' d") and left: "ix p' d ≥ (ix p d - 1) * 2^(lv p' d - lv p d)" (is "?left p p' d") shows "p' ∈ grid p {d}" using p'_valid left right level and p_valid proof induct case (Child p' d' dir) hence "d = d'" by auto let ?child = "child p' dir d'" show ?case proof (cases "lv p d = lv ?child d") case False moreover have "lv ?child d = lv p' d + 1" using child_lv and ‹d < length b› and Child and ‹d = d'› by auto ultimately have "lv p d < lv p' d + 1" using Child by auto hence lv: "Suc (lv p' d) - lv p d = Suc (lv p' d - lv p d)" by auto have "?left p p' d ∧ ?right p p' d" proof (cases dir) case left with Child have "2 * ix p' d - 1 ≤ (ix p d + 1) * 2^(Suc (lv p' d) - lv p d)" using ‹d = d'› and ‹d < length b› by (auto simp add: child_def ix_def lv_def) also have "… = 2 * (ix p d + 1) * 2^(lv p' d - lv p d)" using lv by auto finally have "2 * ix p' d - 2 < 2 * (ix p d + 1) * 2^(lv p' d - lv p d)" by auto also have "… = 2 * ((ix p d + 1) * 2^(lv p' d - lv p d))" by auto finally have left_r: "ix p' d ≤ (ix p d + 1) * 2^(lv p' d - lv p d)" by auto have "2 * ((ix p d - 1) * 2^(lv p' d - lv p d)) = 2 * (ix p d - 1) * 2^(lv p' d - lv p d)" by auto also have "… = (ix p d - 1) * 2^(Suc (lv p' d) - lv p d)" using lv by auto also have "… ≤ 2 * ix p' d - 1" using left and Child and ‹d = d'› and ‹d < length b› by (auto simp add: child_def ix_def lv_def) finally have right_r: "((ix p d - 1) * 2^(lv p' d - lv p d)) ≤ ix p' d" by auto show ?thesis using left_r and right_r by auto next case right with Child have "2 * ix p' d + 1 ≤ (ix p d + 1) * 2^(Suc (lv p' d) - lv p d)" using ‹d = d'› and ‹d < length b› by (auto simp add: child_def ix_def lv_def) also have "… = 2 * (ix p d + 1) * 2^(lv p' d - lv p d)" using lv by auto finally have "2 * ix p' d < 2 * (ix p d + 1) * 2^(lv p' d - lv p d)" by auto also have "… = 2 * ((ix p d + 1) * 2^(lv p' d - lv p d))" by auto finally have left_r: "ix p' d ≤ (ix p d + 1) * 2^(lv p' d - lv p d)" by auto have "2 * ((ix p d - 1) * 2^(lv p' d - lv p d)) = 2 * (ix p d - 1) * 2^(lv p' d - lv p d)" by auto also have "… = (ix p d - 1) * 2^(Suc (lv p' d) - lv p d)" using lv by auto also have "… ≤ 2 * ix p' d + 1" using right and Child and ‹d = d'› and ‹d < length b› by (auto simp add: child_def ix_def lv_def) also have "… < 2 * (ix p' d + 1)" by auto finally have right_r: "((ix p d - 1) * 2^(lv p' d - lv p d)) ≤ ix p' d" by auto show ?thesis using left_r and right_r by auto qed with Child and lv have "p' ∈ grid p {d}" by auto thus ?thesis using ‹d = d'› by auto next case True moreover with Child have "?left p ?child d ∧ ?right p ?child d" by auto ultimately have range: "ix p d - 1 ≤ ix ?child d ∧ ix ?child d ≤ ix p d + 1" by auto have "p ! d ≠ b ! d" proof (rule ccontr) assume "¬ (p ! d ≠ b ! d)" with ‹lv p d = lv ?child d› have "lv b d = lv ?child d" by (auto simp add: lv_def) hence "lv b d = lv p' d + 1" using ‹d = d'› and Child and ‹d < length b› and child_lv by auto moreover have "lv b d ≤ lv p' d" using ‹d = d'› and Child and ‹d < length b› and grid_single_level by auto ultimately show False by auto qed hence "odd (ix p d)" using grid_odd and ‹p ∈ grid b {d}› and ‹d < length b› by auto hence "¬ odd (ix p d + 1)" and "¬ odd (ix p d - 1)" by auto have "d < length p'" using ‹p' ∈ grid b {d}› and ‹d < length b› by auto hence odd_child: "odd (ix ?child d)" using child_odd and ‹d = d'› by auto have "ix p d - 1 ≠ ix ?child d" proof (rule ccontr) assume "¬ (ix p d - 1 ≠ ix ?child d)" hence "odd (ix p d - 1)" using odd_child by auto thus False using ‹¬ odd (ix p d - 1)› by auto qed moreover have "ix p d + 1 ≠ ix ?child d" proof (rule ccontr) assume "¬ (ix p d + 1 ≠ ix ?child d)" hence "odd (ix p d + 1)" using odd_child by auto thus False using ‹¬ odd (ix p d + 1)› by auto qed ultimately have "ix p d = ix ?child d" using range by auto with True have d_eq: "p ! d = (?child) ! d" by (auto simp add: prod_eqI ix_def lv_def) have "length p = length ?child" using ‹p ∈ grid b {d}› and ‹p' ∈ grid b {d}› by auto moreover have "p ! d'' = ?child ! d''" if "d'' < length p" for d'' proof - have "d'' < length b" using that ‹p ∈ grid b {d}› by auto show "p ! d'' = ?child ! d''" proof (cases "d = d''") case True with d_eq show ?thesis by auto next case False hence "d'' ∉ {d}" by auto from ‹d'' < length b› and this and ‹p ∈ grid b {d}› have "p ! d'' = b ! d''" by (rule grid_invariant) also have "… = p' ! d''" using ‹d'' < length b› and ‹d'' ∉ {d}› and ‹p' ∈ grid b {d}› by (rule grid_invariant[symmetric]) also have "… = ?child ! d''" proof - have "d'' < length p'" using ‹d'' < length b› and ‹p' ∈ grid b {d}› by auto hence "?child ! d'' = p' ! d''" using child_invariant and ‹d ≠ d''› and ‹d = d'› by auto thus ?thesis by auto qed finally show ?thesis . qed qed ultimately have "p = ?child" by (rule nth_equalityI) thus "?child ∈ grid p {d}" by auto qed next case Start moreover hence "lv b d ≤ lv p d" using grid_single_level and ‹d < length b› by auto ultimately have "lv b d = lv p d" by auto have "level p = level b" proof - { fix d' assume "d' < length b" have "lv b d' = lv p d'" proof (cases "d = d'") case True with ‹lv b d = lv p d› show ?thesis by auto next case False hence "d' ∉ {d}" by auto from ‹d' < length b› and this and ‹p ∈ grid b {d}› have "p ! d' = b ! d'" by (rule grid_invariant) thus ?thesis by (auto simp add: lv_def) qed } moreover have "length b = length p" using ‹p ∈ grid b {d}› by auto ultimately show ?thesis by (rule level_all_eq) qed hence "p = b" using grid_Start and ‹p ∈ grid b {d}› by auto thus ?case by auto qed lemma grid_disjunct: assumes "d < length p" shows "grid (child p left d) ds ∩ grid (child p right d) ds = {}" (is "grid ?l ds ∩ grid ?r ds = {}") proof (intro set_eqI iffI) fix x assume "x ∈ grid ?l ds ∩ grid ?r ds" hence "ix x d < (ix ?l d + 1) * 2^(lv x d - lv ?l d)" and "ix x d > (ix ?r d - 1) * 2^(lv x d - lv ?r d)" using grid_estimate ‹d < length p› by auto thus "x ∈ {}" using ‹d < length p› and child_lv and child_ix by auto qed auto lemma grid_level_eq: assumes eq: "∀ d ∈ ds. lv p d = lv b d" and grid: "p ∈ grid b ds" shows "level p = level b" proof (rule level_all_eq) { fix i assume "i < length b" show "lv b i = lv p i" proof (cases "i ∈ ds") case True with eq show ?thesis by auto next case False with ‹i < length b› and grid show ?thesis using lv_def ix_def grid_invariant by auto qed } show "length b = length p" using grid by auto qed lemma grid_partition: "grid p {d} = {p} ∪ grid (child p left d) {d} ∪ grid (child p right d) {d}" (is "_ = _ ∪ grid ?l {d} ∪ grid ?r {d}") proof - have "!! x. ⟦ x ∈ grid p {d} ; x ≠ p ; x ∉ grid ?r {d} ⟧ ⟹ x ∈ grid ?l {d}" proof (cases "d < length p") case True fix x let "?nr_r p" = "ix x d > (ix p d + 1) * 2 ^ (lv x d - lv p d)" let "?nr_l p" = "(ix p d - 1) * 2 ^ (lv x d - lv p d) > ix x d" have ix_r_eq: "ix ?r d = 2 * ix p d + 1" using ‹d < length p› and child_ix by auto have lv_r_eq: "lv ?r d = lv p d + 1" using ‹d < length p› and child_lv by auto have ix_l_eq: "ix ?l d = 2 * ix p d - 1" using ‹d < length p› and child_ix by auto have lv_l_eq: "lv ?l d = lv p d + 1" using ‹d < length p› and child_lv by auto assume "x ∈ grid p {d}" and "x ≠ p" and "x ∉ grid ?r {d}" hence "lv p d ≤ lv x d" using grid_single_level and ‹d < length p› by auto moreover have "lv p d ≠ lv x d" proof (rule ccontr) assume "¬ lv p d ≠ lv x d" hence "level x = level p" using ‹x ∈ grid p {d}› and grid_level_eq[where ds="{d}"] by auto hence "x = p" using grid_Start and ‹x ∈ grid p {d}› by auto thus False using ‹x ≠ p› by auto qed ultimately have "lv p d < lv x d" by auto hence "lv ?r d ≤ lv x d" and "?r ∈ grid p {d}" using child_lv and ‹d < length p› by auto with ‹d < length p› and ‹x ∈ grid p {d}› have r_range: "¬ ?nr_r ?r ∧ ¬ ?nr_l ?r ⟹ x ∈ grid ?r {d}" using grid_part[where p="?r" and p'=x and b=p and d=d] by auto have "x ∉ grid ?r {d} ⟹ ?nr_l ?r ∨ ?nr_r ?r" by (rule ccontr, auto simp add: r_range) hence "?nr_l ?r ∨ ?nr_r ?r" using ‹x ∉ grid ?r {d}› by auto have gt0: "lv x d - lv p d > 0" using ‹lv p d < lv x d› by auto have ix_shift: "ix ?r d = ix ?l d + 2" and lv_lr: "lv ?r d = lv ?l d" and right1: "!! x :: int. x + 2 - 1 = x + 1" using ‹d < length p› and child_ix and child_lv by auto have "lv x d - lv p d = Suc (lv x d - (lv p d + 1))" using gt0 by auto hence lv_shift: "!! y :: int. y * 2 ^ (lv x d - lv p d) = y * 2 * 2 ^ (lv x d - (lv p d + 1))" by auto have "ix x d < (ix p d + 1) * 2 ^ (lv x d - lv p d)" using ‹x ∈ grid p {d}› grid_estimate and ‹d < length p› by auto also have "… = (ix ?r d + 1) * 2 ^ (lv x d - lv ?r d)" using ‹lv p d < lv x d› and ix_r_eq and lv_r_eq lv_shift[where y="ix p d + 1"] by auto finally have "?nr_l ?r" using ‹?nr_l ?r ∨ ?nr_r ?r› by auto hence r_bound: "(ix ?l d + 1) * 2 ^ (lv x d - lv ?l d) > ix x d" unfolding ix_shift lv_lr using right1 by auto have "(ix ?l d - 1) * 2 ^ (lv x d - lv ?l d) = (ix p d - 1) * 2 * 2 ^ (lv x d - (lv p d + 1))" unfolding ix_l_eq lv_l_eq by auto also have "… = (ix p d - 1) * 2 ^ (lv x d - lv p d)" using lv_shift[where y="ix p d - 1"] by auto also have " … < ix x d" using ‹x ∈ grid p {d}› grid_estimate and ‹d < length p› by auto finally have l_bound: "(ix ?l d - 1) * 2 ^ (lv x d - lv ?l d) < ix x d" . from l_bound r_bound ‹d < length p› and ‹x ∈ grid p {d}› ‹lv ?r d ≤ lv x d› and lv_lr show "x ∈ grid ?l {d}" using grid_part[where p="?l" and p'=x and d=d] by auto qed (auto simp add: child_def) thus ?thesis by (auto intro: grid_child) qed lemma grid_change_dim: assumes grid: "p ∈ grid b ds" shows "p[d := X] ∈ grid (b[d := X]) ds" using grid proof induct case (Child p d' dir) show ?case proof (cases "d ≠ d'") case True have "(child p dir d')[d := X] = child (p[d := X]) dir d'" unfolding child_def and ix_def and lv_def unfolding list_update_swap[OF ‹d ≠ d'›] and nth_list_update_neq[OF ‹d ≠ d'›] .. thus ?thesis using Child by auto next case False hence "d = d'" by auto with Child show ?thesis unfolding child_def ‹d = d'› list_update_overwrite by auto qed qed auto lemma grid_change_dim_child: assumes grid: "p ∈ grid b ds" and "d ∉ ds" shows "child p dir d ∈ grid (child b dir d) ds" proof (cases "d < length b") case True thus ?thesis using grid_change_dim[OF grid] unfolding child_def lv_def ix_def grid_invariant[OF True ‹d ∉ ds› grid] by auto next case False hence "length b ≤ d" and "length p ≤ d" using grid by auto thus ?thesis unfolding child_def using list_update_beyond assms by auto qed lemma grid_split: assumes grid: "p ∈ grid b (ds' ∪ ds)" shows "∃ x ∈ grid b ds. p ∈ grid x ds'" using grid proof induct case (Child p d dir) show ?case proof (cases "d ∈ ds'") case True with Child show ?thesis by auto next case False hence "d ∈ ds" using Child by auto obtain x where "x ∈ grid b ds" and "p ∈ grid x ds'" using Child by auto hence "child x dir d ∈ grid b ds" using ‹d ∈ ds› by auto moreover have "child p dir d ∈ grid (child x dir d) ds'" using ‹p ∈ grid x ds'› False and grid_change_dim_child by auto ultimately show ?thesis by auto qed qed auto lemma grid_union_eq: "(⋃ p ∈ grid b ds. grid p ds') = grid b (ds' ∪ ds)" using grid_split and grid_transitive[where ds''="ds' ∪ ds" and ds=ds' and ds'=ds, OF _ _ Un_upper2 Un_upper1] by auto lemma grid_onedim_split: "grid b (ds ∪ {d}) = grid b ds ∪ grid (child b left d) (ds ∪ {d}) ∪ grid (child b right d) (ds ∪ {d})" (is "_ = ?g ∪ ?l (ds ∪ {d}) ∪ ?r (ds ∪ {d})") proof - have "?g ∪ ?l (ds ∪ {d}) ∪ ?r (ds ∪ {d}) = ?g ∪ (⋃ p ∈ ?l {d}. grid p ds) ∪ (⋃ p ∈ ?r {d}. grid p ds)" unfolding grid_union_eq .. also have "… = (⋃ p ∈ ({b} ∪ ?l {d} ∪ ?r {d}). grid p ds)" by auto also have "… = (⋃ p ∈ grid b {d}. grid p ds)" unfolding grid_partition[where p=b] .. finally show ?thesis unfolding grid_union_eq by auto qed lemma grid_child_without_parent: assumes grid: "p ∈ grid (child b dir d) ds" (is "p ∈ grid ?c ds") and "d < length b" shows "p ≠ b" proof - have "level ?c ≤ level p" using grid by (rule grid_level) hence "level b < level p" using child_level and ‹d < length b› by auto thus ?thesis by auto qed lemma grid_disjunct': assumes "p ∈ grid b ds" and "p' ∈ grid b ds" and "x ∈ grid p ds'" and "p ≠ p'" and "ds ∩ ds' = {}" shows "x ∉ grid p' ds'" proof (rule ccontr) assume "¬ x ∉ grid p' ds'" hence "x ∈ grid p' ds'" by auto have l: "length b = length p" and l': "length b = length p'" using ‹p ∈ grid b ds› and ‹p' ∈ grid b ds› by auto hence "length p' = length p" by auto moreover have "∀ d < length p'. p' ! d = p ! d" proof (rule allI, rule impI) fix d assume dl': "d < length p'" hence "d < length b" using l' by auto hence dl: "d < length p" using l by auto show "p' ! d = p ! d" proof (cases "d ∈ ds'") case True with ‹ds ∩ ds' = {}› have "d ∉ ds" by auto hence "p' ! d = b ! d" and "p ! d = b ! d" using ‹d < length b› ‹p' ∈ grid b ds› and ‹p ∈ grid b ds› and grid_invariant by auto thus ?thesis by auto next case False show ?thesis using grid_invariant[OF dl' False ‹x ∈ grid p' ds'›] and grid_invariant[OF dl False ‹x ∈ grid p ds'›] by auto qed qed ultimately have "p' = p" by (metis nth_equalityI) thus False using ‹p ≠ p'› by auto qed lemma grid_split1: assumes grid: "p ∈ grid b (ds' ∪ ds)" and "ds ∩ ds' = {}" shows "∃! x ∈ grid b ds. p ∈ grid x ds'" proof (rule ex_ex1I) obtain x where "x ∈ grid b ds" and "p ∈ grid x ds'" using grid_split[OF grid] by auto thus "∃ x. x ∈ grid b ds ∧ p ∈ grid x ds'" by auto next fix x y assume "x ∈ grid b ds ∧ p ∈ grid x ds'" and "y ∈ grid b ds ∧ p ∈ grid y ds'" hence "x ∈ grid b ds" and "p ∈ grid x ds'" and "y ∈ grid b ds" and "p ∈ grid y ds'" by auto show "x = y" proof (rule ccontr) assume "x ≠ y" from grid_disjunct'[OF ‹x ∈ grid b ds› ‹y ∈ grid b ds› ‹p ∈ grid x ds'› this ‹ds ∩ ds' = {}›] show False using ‹p ∈ grid y ds'› by auto qed qed subsection ‹ Grid Restricted to a Level › definition lgrid :: "grid_point ⇒ nat set ⇒ nat ⇒ grid_point set" where "lgrid b ds lm = { p ∈ grid b ds. level p < lm }" lemma lgridI[intro]: "⟦ p ∈ grid b ds ; level p < lm ⟧ ⟹ p ∈ lgrid b ds lm" unfolding lgrid_def by simp lemma lgridE[elim]: assumes "p ∈ lgrid b ds lm" assumes "⟦ p ∈ grid b ds ; level p < lm ⟧ ⟹ P" shows P using assms unfolding lgrid_def by auto lemma lgridI_child[intro]: "d ∈ ds ⟹ p ∈ lgrid (child b dir d) ds lm ⟹ p ∈ lgrid b ds lm" by (auto intro: grid_child) lemma lgrid_empty[simp]: "lgrid p ds (level p) = {}" proof (rule equals0I) fix p' assume "p' ∈ lgrid p ds (level p)" hence "level p' < level p" and "level p ≤ level p'" by auto thus False by auto qed lemma lgrid_empty': assumes "lm ≤ level p" shows "lgrid p ds lm = {}" proof (rule equals0I) fix p' assume "p' ∈ lgrid p ds lm" hence "level p' < lm" and "level p ≤ level p'" by auto thus False using ‹lm ≤ level p› by auto qed lemma grid_not_child: assumes [simp]: "d < length p" shows "p ∉ grid (child p dir d) ds" proof (rule ccontr) assume "¬ ?thesis" have "level p < level (child p dir d)" by auto with grid_level[OF ‹¬ ?thesis›[unfolded not_not]] show False by auto qed subsection ‹ Unbounded Sparse Grid › definition sparsegrid' :: "nat ⇒ grid_point set" where "sparsegrid' dm = grid (start dm) { 0 ..< dm }" lemma grid_subset_alldim: assumes p: "p ∈ grid b ds" defines "dm ≡ length b" shows "p ∈ grid b {0..<dm}" proof - have "ds ∩ {dm..} ∪ ds ∩ {0..<dm} = ds" by auto from gridgen_dim_restrict[where ds="ds ∩ {0..<dm}" and ds'="ds ∩ {dm..}"] this have "ds ∩ {0..<dm} ⊆ {0..<dm}" and "p ∈ grid b (ds ∩ {0..<dm})" using p unfolding dm_def by auto thus ?thesis by (rule grid_union_dims) qed lemma sparsegrid'_length[simp]: "b ∈ sparsegrid' dm ⟹ length b = dm" unfolding sparsegrid'_def by auto lemma sparsegrid'I[intro]: assumes b: "b ∈ sparsegrid' dm" and p: "p ∈ grid b ds" shows "p ∈ sparsegrid' dm" using sparsegrid'_length[OF b] b grid_transitive[OF grid_subset_alldim[OF p], where c="start dm" and ds''="{0..<dm}"] unfolding sparsegrid'_def by auto lemma sparsegrid'_start: assumes "b ∈ grid (start dm) ds" shows "b ∈ sparsegrid' dm" unfolding sparsegrid'_def using grid_subset_alldim[OF assms] by simp subsection ‹ Sparse Grid › definition sparsegrid :: "nat ⇒ nat ⇒ grid_point set" where "sparsegrid dm lm = lgrid (start dm) { 0 ..< dm } lm" lemma sparsegrid_length: "p ∈ sparsegrid dm lm ⟹ length p = dm" by (auto simp: sparsegrid_def) lemma sparsegrid_subset[intro]: "p ∈ sparsegrid dm lm ⟹ p ∈ sparsegrid' dm" unfolding sparsegrid_def sparsegrid'_def lgrid_def by auto lemma sparsegridI[intro]: assumes "p ∈ sparsegrid' dm" and "level p < lm" shows "p ∈ sparsegrid dm lm" using assms unfolding sparsegrid'_def sparsegrid_def lgrid_def by auto lemma sparsegrid_start: assumes "b ∈ lgrid (start dm) ds lm" shows "b ∈ sparsegrid dm lm" proof have "b ∈ grid (start dm) ds" using assms by auto thus "b ∈ sparsegrid' dm" by (rule sparsegrid'_start) qed (insert assms, auto) lemma sparsegridE[elim]: assumes "p ∈ sparsegrid dm lm" shows "p ∈ sparsegrid' dm" and "level p < lm" using assms unfolding sparsegrid'_def sparsegrid_def lgrid_def by auto subsection ‹ Compute Sparse Grid Points › fun gridgen :: "grid_point ⇒ nat set ⇒ nat ⇒ grid_point list" where "gridgen p ds 0 = []" | "gridgen p ds (Suc l) = (let sub = λ d. gridgen (child p left d) { d' ∈ ds . d' ≤ d } l @ gridgen (child p right d) { d' ∈ ds . d' ≤ d } l in p # concat (map sub [ d ← [0 ..< length p]. d ∈ ds]))" lemma gridgen_lgrid_eq: "set (gridgen p ds l) = lgrid p ds (level p + l)" proof (induct l arbitrary: p ds) case (Suc l) let "?subg dir d" = "set (gridgen (child p dir d) { d' ∈ ds . d' ≤ d } l)" let "?sub dir d" = "lgrid (child p dir d) { d' ∈ ds . d' ≤ d } (level p + Suc l)" let "?union F dm" = "{p} ∪ (⋃ d ∈ { d ∈ ds. d < dm }. F left d ∪ F right d)" have hyp: "!! dir d. d < length p ⟹ ?subg dir d = ?sub dir d" using Suc.hyps using child_level by auto { fix dm assume "dm ≤ length p" hence "?union ?sub dm = lgrid p {d ∈ ds. d < dm} (level p + Suc l)" proof (induct dm) case (Suc dm) hence "dm ≤ length p" by auto let ?l = "child p left dm" and ?r = "child p right dm" have p_lgrid: "p ∈ lgrid p {d ∈ ds. d < dm} (level p + Suc l)" by auto show ?case proof (cases "dm ∈ ds") case True let ?ds = "{d ∈ ds. d < dm} ∪ {dm}" have ds_eq: "{d' ∈ ds. d' ≤ dm} = ?ds" using True by auto have ds_eq': "{d ∈ ds. d < Suc dm} = {d ∈ ds. d < dm } ∪ {dm}" using True by auto have "?union ?sub (Suc dm) = ?union ?sub dm ∪ ({p} ∪ ?sub left dm ∪ ?sub right dm)" unfolding ds_eq' by auto also have "… = lgrid p {d ∈ ds. d < dm} (level p + Suc l) ∪ ?sub left dm ∪ ?sub right dm" unfolding Suc.hyps[OF ‹dm ≤ length p›] using p_lgrid by auto also have "… = {p' ∈ grid p {d ∈ ds. d<dm} ∪ (grid ?l ?ds) ∪ (grid ?r ?ds). level p' < level p + Suc l}" unfolding lgrid_def ds_eq by auto also have "… = lgrid p {d ∈ ds. d < Suc dm} (level p + Suc l)" unfolding lgrid_def ds_eq' unfolding grid_onedim_split[where b=p] .. finally show ?thesis . next case False hence "{d ∈ ds. d < Suc dm} = {d ∈ ds. d < dm ∨ d = dm}" by auto hence ds_eq: "{d ∈ ds. d < Suc dm} = {d ∈ ds. d < dm}" using ‹dm ∉ ds› by auto show ?thesis unfolding ds_eq Suc.hyps[OF ‹dm ≤ length p›] .. qed next case 0 thus ?case unfolding lgrid_def by auto qed } hence "?union ?sub (length p) = lgrid p {d ∈ ds. d < length p} (level p + Suc l)" by auto hence union_lgrid_eq: "?union ?sub (length p) = lgrid p ds (level p + Suc l)" unfolding lgrid_def using grid_dim_remove_outer by auto have "set (gridgen p ds (Suc l)) = ?union ?subg (length p)" unfolding gridgen.simps and Let_def by auto hence "set (gridgen p ds (Suc l)) = ?union ?sub (length p)" using hyp by auto also have "… = lgrid p ds (level p + Suc l)" using union_lgrid_eq . finally show ?case . qed auto lemma gridgen_distinct: "distinct (gridgen p ds l)" proof (induct l arbitrary: p ds) case (Suc l) let ?ds = "[d ← [0..<length p]. d ∈ ds]" let "?left d" = "gridgen (child p left d) { d' ∈ ds . d' ≤ d } l" and "?right d" = "gridgen (child p right d) { d' ∈ ds . d' ≤ d } l" let "?sub d" = "?left d @ ?right d" have "distinct (concat (map ?sub ?ds))" proof (cases l) case (Suc l') have inj_on: "inj_on ?sub (set ?ds)" proof (rule inj_onI, rule ccontr) fix d d' assume "d ∈ set ?ds" and "d' ∈ set ?ds" hence "d < length p" and "d ∈ set ?ds" and "d' < length p" by auto assume *: "?sub d = ?sub d'" have in_d: "child p left d ∈ set (?sub d)" using ‹d ∈ set ?ds› Suc by (auto simp add: gridgen_lgrid_eq lgrid_def grid_Start) have in_d': "child p left d' ∈ set (?sub d')" using ‹d ∈ set ?ds› Suc by (auto simp add: gridgen_lgrid_eq lgrid_def grid_Start) { fix p' d assume "d ∈ set ?ds" and "p' ∈ set (?sub d)" hence "lv p d < lv p' d" using grid_child_level by (auto simp add: gridgen_lgrid_eq lgrid_def grid_child_level) } note level_less = this assume "d ≠ d'" show False proof (cases "d' < d") case True with in_d' ‹?sub d = ?sub d'› level_less[OF ‹d ∈ set ?ds›] have "lv p d < lv (child p left d') d" by simp thus False unfolding lv_def using child_invariant[OF ‹d < length p›, of left d'] ‹d ≠ d'› by auto next case False hence "d < d'" using ‹d ≠ d'› by auto with in_d ‹?sub d = ?sub d'› level_less[OF ‹d' ∈ set ?ds›] have "lv p d' < lv (child p left d) d'" by simp thus False unfolding lv_def using child_invariant[OF ‹d' < length p›, of left d] ‹d ≠ d'› by auto qed qed show ?thesis proof (rule distinct_concat) show "distinct (map ?sub ?ds)" unfolding distinct_map using inj_on by simp next fix ys assume "ys ∈ set (map ?sub ?ds)" then obtain d where "d ∈ ds" and "d < length p" and *: "ys = ?sub d" by auto show "distinct ys" unfolding * using grid_disjunct[OF ‹d < length p›, of "{d' ∈ ds. d' ≤ d}"] gridgen_lgrid_eq lgrid_def ‹distinct (?left d)› ‹distinct (?right d)› by auto next fix ys zs assume "ys ∈ set (map ?sub ?ds)" then obtain d where ys: "ys = ?sub d" and "d ∈ set ?ds" by auto hence "d < length p" by auto assume "zs ∈ set (map ?sub ?ds)" then obtain d' where zs: "zs = ?sub d'" and "d' ∈ set ?ds" by auto hence "d' < length p" by auto assume "ys ≠ zs" hence "d' ≠ d" unfolding ys zs by auto show "set ys ∩ set zs = {}" proof (rule ccontr) assume "¬ ?thesis" then obtain p' where "p' ∈ set (?sub d)" and "p' ∈ set (?sub d')" unfolding ys zs by auto hence "lv p d < lv p' d" "lv p d' < lv p' d'" using grid_child_level ‹d ∈ set ?ds› ‹d' ∈ set ?ds› by (auto simp add: gridgen_lgrid_eq lgrid_def grid_child_level) show False proof (cases "d < d'") case True from ‹p' ∈ set (?sub d)› have "p ! d' = p' ! d'" using grid_invariant[of d' "child p right d" "{d' ∈ ds. d' ≤ d}"] using grid_invariant[of d' "child p left d" "{d' ∈ ds. d' ≤ d}"] using child_invariant[of d' _ _ d] ‹d < d'› ‹d' < length p› using gridgen_lgrid_eq lgrid_def by auto thus False using ‹lv p d' < lv p' d'› unfolding lv_def by auto next case False hence "d' < d" using ‹d' ≠ d› by simp from ‹p' ∈ set (?sub d')› have "p ! d = p' ! d" using grid_invariant[of d "child p right d'" "{d ∈ ds. d ≤ d'}"] using grid_invariant[of d "child p left d'" "{d ∈ ds. d ≤ d'}"] using child_invariant[of d _ _ d'] ‹d' < d› ‹d < length p› using gridgen_lgrid_eq lgrid_def by auto thus False using ‹lv p d < lv p' d› unfolding lv_def by auto qed qed qed qed (simp add: map_replicate_const) moreover have "p ∉ set (concat (map ?sub ?ds))" using gridgen_lgrid_eq lgrid_def grid_not_child[of _ p] by simp ultimately show ?case unfolding gridgen.simps Let_def distinct.simps by simp qed auto lemma lgrid_finite: "finite (lgrid b ds lm)" proof (cases "level b ≤ lm") case True from iffD1[OF le_iff_add True] obtain l where l: "lm = level b + l" by auto show ?thesis unfolding l gridgen_lgrid_eq[symmetric] by auto next case False hence "!! x. x ∈ grid b ds ⟹ (¬ level x < lm)" proof - fix x assume "x ∈ grid b ds" from grid_level[OF this] show "¬ level x < lm" using False by auto qed hence "lgrid b ds lm = {}" unfolding lgrid_def by auto thus ?thesis by auto qed lemma lgrid_sum: fixes F :: "grid_point ⇒ real" assumes "d < length b" and "level b < lm" shows "(∑ p ∈ lgrid b {d} lm. F p) = (∑ p ∈ lgrid (child b left d) {d} lm. F p) + (∑ p ∈ lgrid (child b right d) {d} lm. F p) + F b" (is "(∑ p ∈ ?grid b. F p) = (∑ p ∈ ?grid ?l . F p) + (?sum (?grid ?r)) + F b") proof - have "!! dir. b ∉ ?grid (child b dir d)" using grid_child_without_parent[where ds="{d}"] and ‹d < length b› and lgrid_def by auto hence b_distinct: "b ∉ (?grid ?l ∪ ?grid ?r)" by auto have "?grid ?l ∩ ?grid ?r = {}" unfolding lgrid_def using grid_disjunct and ‹d < length b› by auto from lgrid_finite lgrid_finite and this have child_eq: "?sum ((?grid ?l) ∪ (?grid ?r)) = ?sum (?grid ?l) + ?sum (?grid ?r)" by (rule sum.union_disjoint) have "?grid b = {b} ∪ (?grid ?l) ∪ (?grid ?r)" unfolding lgrid_def grid_partition[where p=b] using assms by auto hence "?sum (?grid b) = F b + ?sum ((?grid ?l) ∪ (?grid ?r))" using b_distinct and lgrid_finite by auto thus ?thesis using child_eq by auto qed subsection ‹ Base Points › definition base :: "nat set ⇒ grid_point ⇒ grid_point" where "base ds p = (THE b. b ∈ grid (start (length p)) ({0 ..< length p} - ds) ∧ p ∈ grid b ds)" lemma baseE: assumes p_grid: "p ∈ sparsegrid' dm" shows "base ds p ∈ grid (start dm) ({0..<dm} - ds)" and "p ∈ grid (base ds p) ds" proof - from p_grid[unfolded sparsegrid'_def] have *: "∃! x ∈ grid (start dm) ({0..<dm} - ds). p ∈ grid x ds" by (intro grid_split1) (auto intro: grid_union_dims) then obtain x where x_eq: "x ∈ grid (start dm) ({0..<dm} - ds) ∧ p ∈ grid x ds" by auto with * have "base ds p = x" unfolding base_def by auto thus "base ds p ∈ grid (start dm) ({0..<dm} - ds)" and "p ∈ grid (base ds p) ds" using x_eq by auto qed lemma baseI: assumes x_grid: "x ∈ grid (start dm) ({0..<dm} - ds)" and p_xgrid: "p ∈ grid x ds" shows "base ds p = x" proof - have "p ∈ grid (start dm) (ds ∪ ({0..<dm} - ds))" using grid_transitive[OF p_xgrid x_grid, where ds''="ds ∪ ({0..<dm} - ds)"] by auto moreover have "ds ∩ ({0..<dm} - ds) = {}" by auto ultimately have "∃! x ∈ grid (start dm) ({0..<dm} - ds). p ∈ grid x ds" using grid_split1[where p=p and b="start dm" and ds'=ds and ds="{0..<dm} - ds"] by auto thus "base ds p = x" using x_grid p_xgrid unfolding base_def by auto qed lemma base_empty: assumes p_grid: "p ∈ sparsegrid' dm" shows "base {} p = p" using grid_empty_ds and p_grid and grid_split1[where ds="{0..<dm}" and ds'="{}"] unfolding base_def sparsegrid'_def by auto lemma base_start_eq: assumes p_spg: "p ∈ sparsegrid dm lm" shows "start dm = base {0..<dm} p" proof - from p_spg have "start dm ∈ grid (start dm) ({0..<dm} - {0..<dm})" and "p ∈ grid (start dm) {0..<dm}" using sparsegrid'_def by auto from baseI[OF this(1) this(2)] show ?thesis by auto qed lemma base_in_grid: assumes p_grid: "p ∈ sparsegrid' dm" shows "base ds p ∈ grid (start dm) {0..<dm}" proof - let ?ds = "ds ∪ {0..<dm}" have ds_eq: "{ d ∈ ?ds. d < length (start dm) } = { 0..< dm}" unfolding start_def by auto have "base ds p ∈ grid (start dm) ?ds" using grid_union_dims[OF _ baseE(1)[OF p_grid, where ds=ds], where ds'="?ds"] by auto thus ?thesis using grid_dim_remove_outer[where b="start dm" and ds="?ds"] unfolding ds_eq by auto qed lemma base_grid: assumes p_grid: "p ∈ sparsegrid' dm" shows "grid (base ds p) ds ⊆ sparsegrid' dm" proof fix x assume xgrid: "x ∈ grid (base ds p) ds" have ds_eq: "{ d ∈ {0..<dm} ∪ ds. d < length (start dm) } = {0..<dm}" by auto from grid_transitive[OF xgrid base_in_grid[OF p_grid], where ds''="{0..<dm} ∪ ds"] show "x ∈ sparsegrid' dm" unfolding sparsegrid'_def using grid_dim_remove_outer[where b="start dm" and ds="{0..<dm} ∪ ds"] unfolding ds_eq unfolding Un_ac(3)[of "{0..<dm}"] by auto qed lemma base_length[simp]: assumes p_grid: "p ∈ sparsegrid' dm" shows "length (base ds p) = dm" proof - from baseE[OF p_grid] have "base ds p ∈ grid (start dm) ({0..<dm} - ds)" by auto thus ?thesis by auto qed lemma base_in[simp]: assumes "d < dm" and "d ∈ ds" and p_grid: "p ∈ sparsegrid' dm" shows "base ds p ! d = start dm ! d" proof - have ds: "d ∉ {0..<dm} - ds" using ‹d ∈ ds› by auto have "d < length (start dm)" using ‹d < dm› by auto with grid_invariant[OF this ds] baseE(1)[OF p_grid] show ?thesis by auto qed lemma base_out[simp]: assumes "d < dm" and "d ∉ ds" and p_grid: "p ∈ sparsegrid' dm" shows "base ds p ! d = p ! d" proof - have "d < length (base ds p)" using base_length[OF p_grid] ‹d < dm› by auto with grid_invariant[OF this ‹d ∉ ds›] baseE(2)[OF p_grid] show ?thesis by auto qed lemma base_base: assumes p_grid: "p ∈ sparsegrid' dm" shows "base ds (base ds' p) = base (ds ∪ ds') p" proof (rule nth_equalityI) have b_spg: "base ds' p ∈ sparsegrid' dm" unfolding sparsegrid'_def using grid_union_dims[OF Diff_subset[where A="{0..<dm}" and B="ds'"] baseE(1)[OF p_grid]] . from base_length[OF b_spg] base_length[OF p_grid] show "length (base ds (base ds' p)) = length (base (ds ∪ ds') p)" by auto show "base ds (base ds' p) ! i = base (ds ∪ ds') p ! i" if "i < length (base ds (base ds' p))" for i proof - have "i < dm" using that base_length[OF b_spg] by auto show "base ds (base ds' p) ! i = base (ds ∪ ds') p ! i" proof (cases "i ∈ ds ∪ ds'") case True show ?thesis proof (cases "i ∈ ds") case True from base_in[OF ‹i < dm› ‹i ∈ ds ∪ ds'› p_grid] base_in[OF ‹i < dm› this b_spg] show ?thesis by auto next case False hence "i ∈ ds'" using ‹i ∈ ds ∪ ds'› by auto from base_in[OF ‹i < dm› ‹i ∈ ds ∪ ds'› p_grid] base_out[OF ‹i < dm› ‹i ∉ ds› b_spg] base_in[OF ‹i < dm› ‹i ∈ ds'› p_grid] show ?thesis by auto qed next case False hence "i ∉ ds" and "i ∉ ds'" by auto from base_out[OF ‹i < dm› ‹i ∉ ds ∪ ds'› p_grid] base_out[OF ‹i < dm› ‹i ∉ ds› b_spg] base_out[OF ‹i < dm› ‹i ∉ ds'› p_grid] show ?thesis by auto qed qed qed lemma grid_base_out: assumes "d < dm" and "d ∉ ds" and p_grid: "b ∈ sparsegrid' dm" and "p ∈ grid (base ds b) ds" shows "p ! d = b ! d" proof - have "base ds b ! d = b ! d" using assms by auto moreover have "d < length (base ds b)" using assms by auto from grid_invariant[OF this] have "p ! d = base ds b ! d" using assms by auto ultimately show ?thesis by auto qed lemma grid_grid_inj_on: assumes "ds ∩ ds' = {}" shows "inj_on snd (⋃p'∈grid b ds. ⋃p''∈grid p' ds'. {(p', p'')})" proof (rule inj_onI) fix x y assume "x ∈ (⋃p'∈grid b ds. ⋃p''∈grid p' ds'. {(p', p'')})" hence "snd x ∈ grid (fst x) ds'" and "fst x ∈ grid b ds" by auto assume "y ∈ (⋃p'∈grid b ds. ⋃p''∈grid p' ds'. {(p', p'')})" hence "snd y ∈ grid (fst y) ds'" and "fst y ∈ grid b ds" by auto assume "snd x = snd y" have "fst x = fst y" proof (rule ccontr) assume "fst x ≠ fst y" from grid_disjunct'[OF ‹fst x ∈ grid b ds› ‹fst y ∈ grid b ds› ‹snd x ∈ grid (fst x) ds'› this ‹ds ∩ ds' = {}›] show False using ‹snd y ∈ grid (fst y) ds'› unfolding ‹snd x = snd y› by auto qed show "x = y" using prod_eqI[OF ‹fst x = fst y› ‹snd x = snd y›] . qed lemma grid_level_d: assumes "d < length b" and p_grid: "p ∈ grid b {d}" and "p ≠ b" shows "lv p d > lv b d" proof - from p_grid[unfolded grid_partition[where p=b]] show ?thesis using grid_child_level using assms by auto qed lemma grid_base_base: assumes "b ∈ sparsegrid' dm" shows "base ds' b ∈ grid (base ds (base ds' b)) (ds ∪ ds')" proof - from base_grid[OF ‹b ∈ sparsegrid' dm›] have "base ds' b ∈ sparsegrid' dm" by auto from grid_union_dims[OF _ baseE(2)[OF this], of ds "ds ∪ ds'"] show ?thesis by auto qed lemma grid_base_union: assumes b_spg: "b ∈ sparsegrid' dm" and p_grid: "p ∈ grid (base ds b) ds" and x_grid: "x ∈ grid (base ds' p) ds'" shows "x ∈ grid (base (ds ∪ ds') b) (ds ∪ ds')" proof - have ds_union: "ds ∪ ds' = ds' ∪ (ds ∪ ds')" by auto from base_grid[OF b_spg] p_grid have p_spg: "p ∈ sparsegrid' dm" by auto with assms and grid_base_base have base_b': "base ds' p ∈ grid (base ds (base ds' p)) (ds ∪ ds')" by auto moreover have "base ds' (base ds b) = base ds' (base ds p)" (is "?b = ?p") proof (rule nth_equalityI) have bb_spg: "base ds b ∈ sparsegrid' dm" using base_grid[OF b_spg] grid.Start by auto hence "dm = length (base ds b)" by auto have bp_spg: "base ds p ∈ sparsegrid' dm" using base_grid[OF p_spg] grid.Start by auto show "length ?b = length ?p" using base_length[OF bp_spg] base_length[OF bb_spg] by auto show "?b ! i = ?p ! i" if "i < length ?b" for i proof - have "i < dm" and "i < length (base ds b)" using that base_length[OF bb_spg] ‹dm = length (base ds b)› by auto show "?b ! i = ?p ! i" proof (cases "i ∈ ds ∪ ds'") case True hence "!! x. base ds x ∈ sparsegrid' dm ⟹ x ∈ sparsegrid' dm ⟹ base ds' (base ds x) ! i = (start dm) ! i" proof - fix x assume x_spg: "x ∈ sparsegrid' dm" and xb_spg: "base ds x ∈ sparsegrid' dm" show "base ds' (base ds x) ! i = (start dm) ! i" proof (cases "i ∈ ds'") case True from base_in[OF ‹i < dm› this xb_spg] show ?thesis . next case False hence "i ∈ ds" using ‹i ∈ ds ∪ ds'› by auto from base_out[OF ‹i < dm› False xb_spg] base_in[OF ‹i < dm› this x_spg] show ?thesis by auto qed qed from this[OF bp_spg p_spg] this[OF bb_spg b_spg] show ?thesis by auto next case False hence "i ∉ ds" and "i ∉ ds'" by auto from grid_invariant[OF ‹i < length (base ds b)› ‹i ∉ ds› p_grid] base_out[OF ‹i < dm› ‹i ∉ ds'› bp_spg] base_out[OF ‹i < dm› ‹i ∉ ds› p_spg] base_out[OF ‹i < dm› ‹i ∉ ds'› bb_spg] show ?thesis by auto qed qed qed ultimately have "base ds' p ∈ grid (base (ds ∪ ds') b) (ds ∪ ds')" by (simp only: base_base[OF p_spg] base_base[OF b_spg] Un_ac(3)) from grid_transitive[OF x_grid this] show ?thesis using ds_union by auto qed lemma grid_base_dim_add: assumes "ds' ⊆ ds" and b_spg: "b ∈ sparsegrid' dm" and p_grid: "p ∈ grid (base ds' b) ds'" shows "p ∈ grid (base ds b) ds" proof - have ds_eq: "ds' ∪ ds = ds" using assms by auto have "p ∈ sparsegrid' dm" using base_grid[OF b_spg] p_grid by auto hence "p ∈ grid (base ds p) ds" using baseE by auto from grid_base_union[OF b_spg p_grid this] show ?thesis using ds_eq by auto qed lemma grid_replace_dim: assumes "d < length b'" and "d < length b" and p_grid: "p ∈ grid b ds" and p'_grid: "p' ∈ grid b' ds" shows "p[d := p' ! d] ∈ grid (b[d := b' ! d]) ds" (is "_ ∈ grid ?b ds") using p'_grid and p_grid proof induct case (Child p'' d' dir) hence p''_grid: "p[d := p'' ! d] ∈ grid ?b ds" and "d < length p''" using assms by auto have "d < length p" using p_grid assms by auto thus ?case proof (cases "d' = d") case True from grid.Child[OF p''_grid ‹d' ∈ ds›] show ?thesis unfolding child_def ix_def lv_def list_update_overwrite ‹d' = d› nth_list_update_eq[OF ‹d < length p''›] nth_list_update_eq[OF ‹d < length p›] . next case False show ?thesis unfolding child_def nth_list_update_neq[OF False] using Child by auto qed qed (rule grid_change_dim) lemma grid_shift_base: assumes ds_dj: "ds ∩ ds' = {}" and b_spg: "b ∈ sparsegrid' dm" and p_grid: "p ∈ grid (base (ds' ∪ ds) b) (ds' ∪ ds)" shows "base ds' p ∈ grid (base (ds ∪ ds') b) ds" proof - from grid_split[OF p_grid] obtain x where x_grid: "x ∈ grid (base (ds' ∪ ds) b) ds" and p_xgrid: "p ∈ grid x ds'" by auto from grid_union_dims[OF _ this(1)] have x_spg: "x ∈ sparsegrid' dm" using base_grid[OF b_spg] by auto have b_len: "length (base (ds' ∪ ds) b) = dm" using base_length[OF b_spg] by auto define d' where "d' = dm" moreover have "d' ≤ dm ⟹ x ∈ grid (start dm) ({0..<dm} - {d ∈ ds'. d < d'})" proof (induct d') case (Suc d') with b_len have d'_b: "d' < length (base (ds' ∪ ds) b)" by auto show ?case proof (cases "d' ∈ ds'") case True hence "d' ∉ ds" and "d' ∈ ds' ∪ ds" using ds_dj by auto have "{0..<dm} - {d ∈ ds'. d < d'} = ({0..<dm} - {d ∈ ds'. d < d'}) - {d'} ∪ {d'}" using ‹Suc d' ≤ dm› by auto also have "… = ({0..<dm} - {d ∈ ds'. d < Suc d'}) ∪ {d'}" by auto finally have x_g: "x ∈ grid (start dm) ({d'} ∪ ({0..<dm} - {d ∈ ds'. d < Suc d'}))" using Suc by auto from grid_invariant[OF d'_b ‹d' ∉ ds› x_grid] base_in[OF _ ‹d' ∈ ds' ∪ ds› b_spg] ‹Suc d' ≤ dm› have "x ! d' = start dm ! d'" by auto from grid_dim_remove[OF x_g this] show ?thesis . next case False hence "{d ∈ ds'. d < Suc d'} = {d ∈ ds'. d < d' ∨ d = d'}" by auto also have "… = {d ∈ ds'. d < d'}" using False by auto finally show ?thesis using Suc by auto qed next case 0 show ?case using x_spg[unfolded sparsegrid'_def] by auto qed moreover have "{0..<dm} - ds' = {0..<dm} - {d ∈ ds'. d < dm}" by auto ultimately have "x ∈ grid (start dm) ({0..<dm} - ds')" by auto from baseI[OF this p_xgrid] and x_grid show ?thesis by (auto simp: Un_ac(3)) qed subsection ‹ Lift Operation over all Grid Points › definition lift :: "(nat ⇒ nat ⇒ grid_point ⇒ vector ⇒ vector) ⇒ nat ⇒ nat ⇒ nat ⇒ vector ⇒ vector" where "lift f dm lm d = foldr (λ p. f d (lm - level p) p) (gridgen (start dm) ({ 0 ..< dm } - { d }) lm)" lemma lift: assumes "d < dm" and "p ∈ sparsegrid dm lm" and Fintro: "⋀ l b p α. ⟦ b ∈ lgrid (start dm) ({0..<dm} - {d}) lm ; l + level b = lm ; p ∈ sparsegrid dm lm ⟧ ⟹ F d l b α p = (if b = base {d} p then (∑ p' ∈ lgrid b {d} lm. S (α p') p p') else α p)" shows "lift F dm lm d α p = (∑ p' ∈ lgrid (base {d} p) {d} lm. S (α p') p p')" (is "?lift = ?S p α") proof - let ?gridgen = "gridgen (start dm) ({0..<dm} - {d}) lm" let "?f p" = "F d (lm - level p) p" { fix bs β b assume "set bs ⊆ set ?gridgen" and "distinct bs" and "p ∈ sparsegrid dm lm" hence "foldr ?f bs β p = (if base {d} p ∈ set bs then ?S p β else β p)" proof (induct bs arbitrary: p) case (Cons b bs) hence "b ∈ lgrid (start dm) ({0..<dm} - {d}) lm" and "(lm - level b) + level b = lm" and b_grid: "b ∈ grid (start dm) ({0..<dm} - {d})" using lgrid_def gridgen_lgrid_eq by auto note F = Fintro[OF this(1,2) ‹p ∈ sparsegrid dm lm›] have "b ∉ set bs" using ‹distinct (b#bs)› by auto show ?case proof (cases "base {d} p ∈ set (b#bs)") case True note base_in_set = this show ?thesis proof (cases "b = base {d} p") case True moreover { fix p' assume "p' ∈ lgrid b {d} lm" hence "p' ∈ grid b {d}" and "level p' < lm" unfolding lgrid_def by auto from grid_transitive[OF this(1) b_grid, of "{0..<dm}"] ‹d < dm› baseI[OF b_grid ‹p' ∈ grid b {d}›] ‹b ∉ set bs› Cons.prems Cons.hyps[of p'] this(2) have "foldr ?f bs β p' = β p'" unfolding sparsegrid_def lgrid_def by auto } ultimately show ?thesis using F base_in_set by auto next case False with base_in_set have "base {d} p ∈ set bs" by auto with Cons.hyps[of p] Cons.prems have "foldr ?f bs β p = ?S p β" by auto thus ?thesis using F base_in_set False by auto qed next case False hence "b ≠ base {d} p" by auto from False Cons.hyps[of p] Cons.prems have "foldr ?f bs β p = β p" by auto thus ?thesis using False F ‹b ≠ base {d} p› by auto qed qed auto } moreover have "base {d} p ∈ set ?gridgen" proof - have "p ∈ grid (base {d} p) {d}" using ‹p ∈ sparsegrid dm lm›[THEN sparsegrid_subset] by (rule baseE) from grid_level[OF this] baseE(1)[OF sparsegrid_subset[OF ‹p ∈ sparsegrid dm lm›]] show ?thesis using ‹p ∈ sparsegrid dm lm› unfolding gridgen_lgrid_eq sparsegrid'_def lgrid_def sparsegrid_def by auto qed ultimately show ?thesis unfolding lift_def using gridgen_distinct ‹p ∈ sparsegrid dm lm› by auto qed subsection ‹ Parent Points › definition parents :: "nat ⇒ grid_point ⇒ grid_point ⇒ grid_point set" where "parents d b p = { x ∈ grid b {d}. p ∈ grid x {d} }" lemma parents_split: assumes p_grid: "p ∈ grid (child b dir d) {d}" shows "parents d b p = { b } ∪ parents d (child b dir d) p" proof (intro set_eqI iffI) let ?chd = "child b dir d" and ?chid = "child b (inv dir) d" fix x assume "x ∈ parents d b p" hence "x ∈ grid b {d}" and "p ∈ grid x {d}" unfolding parents_def by auto hence x_split: "x ∈ {b} ∪ grid ?chd {d} ∪ grid ?chid {d}" using grid_onedim_split[where ds="{}" and b=b] and grid_empty_ds by (cases dir, auto) thus "x ∈ {b} ∪ parents d (child b dir d) p" proof (cases "x = b") case False have "d < length b" proof (rule ccontr) assume "¬ d < length b" hence empty: "{d' ∈ {d}. d' < length b} = {}" by auto have "x = b" using ‹x ∈ grid b {d}› unfolding grid_dim_remove_outer[where ds="{d}" and b=b] empty using grid_empty_ds by auto thus False using ‹¬ x = b› by auto qed have "x ∉ grid ?chid {d}" proof (rule ccontr) assume "¬ x ∉ grid ?chid {d}" hence "p ∈ grid ?chid {d}" using grid_transitive[OF ‹p ∈ grid x {d}›, where ds'="{d}"] by auto hence "p ∉ grid ?chd {d}" using grid_disjunct[OF ‹d < length b›] by (cases dir, auto) thus False using ‹p ∈ grid ?chd {d}› .. qed with False and x_split have "x ∈ grid ?chd {d}" by auto thus ?thesis unfolding parents_def using ‹p ∈ grid x {d}› by auto qed auto next let ?chd = "child b dir d" and ?chid = "child b (inv dir) d" fix x assume x_in: "x ∈ {b} ∪ parents d ?chd p" thus "x ∈ parents d b p" proof (cases "x = b") case False hence "x ∈ parents d ?chd p" using x_in by auto thus ?thesis unfolding parents_def using grid_child[where b=b] by auto next from p_grid have "p ∈ grid b {d}" using grid_child[where b=b] by auto case True thus ?thesis unfolding parents_def using ‹p ∈ grid b {d}› by auto qed qed lemma parents_no_parent: assumes "d < length b" shows "b ∉ parents d (child b dir d) p" (is "_ ∉ parents _ ?ch _") proof assume "b ∈ parents d ?ch p" hence "b ∈ grid ?ch {d}" unfolding parents_def by auto from grid_level[OF this] have "level b + 1 ≤ level b" unfolding child_level[OF ‹d < length b›] . thus False by auto qed lemma parents_subset_lgrid: "parents d b p ⊆ lgrid b {d} (level p + 1)" proof fix x assume "x ∈ parents d b p" hence "x ∈ grid b {d}" and "p ∈ grid x {d}" unfolding parents_def by auto moreover hence "level x ≤ level p" using grid_level by auto hence "level x < level p + 1" by auto ultimately show "x ∈ lgrid b {d} (level p + 1)" unfolding lgrid_def by auto qed lemma parents_finite: "finite (parents d b p)" using finite_subset[OF parents_subset_lgrid lgrid_finite] . lemma parent_sum: assumes p_grid: "p ∈ grid (child b dir d) {d}" and "d < length b" shows "(∑ x ∈ parents d b p. F x) = F b + (∑ x ∈ parents d (child b dir d) p. F x)" unfolding parents_split[OF p_grid] using parents_no_parent[OF ‹d < length b›, where dir=dir and p=p] using parents_finite by auto lemma parents_single: "parents d b b = { b }" proof have "parents d b b ⊆ lgrid b {d} (level b + (Suc 0))" using parents_subset_lgrid by auto also have "… = {b}" unfolding gridgen_lgrid_eq[symmetric] gridgen.simps Let_def by auto finally show "parents d b b ⊆ { b }" . next have "b ∈ parents d b b" unfolding parents_def by auto thus "{ b } ⊆ parents d b b" by auto qed lemma grid_single_dimensional_specification: assumes "d < length b" and "odd i" and "lv b d + l' = l" and "i < (ix b d + 1) * 2^l'" and "i > (ix b d - 1) * 2^l'" shows "b[d := (l,i)] ∈ grid b {d}" using assms proof (induct l' arbitrary: b) case 0 hence "i = ix b d" and "l = lv b d" by auto thus ?case unfolding ix_def lv_def by auto next case (Suc l') have "d ∈ {d}" by auto show ?case proof (rule linorder_cases) assume "i = ix b d * 2^(Suc l')" hence "even i" by auto thus ?thesis using ‹odd i› by blast next assume *: "i < ix b d * 2^(Suc l')" let ?b = "child b left d" have "d < length ?b" using Suc by auto moreover note ‹odd i› moreover have "lv ?b d + l' = l" and "i < (ix ?b d + 1) * 2^l'" and "(ix ?b d - 1) * 2^l' < i" unfolding child_ix_left[OF Suc.prems(1)] using Suc.prems * child_lv by (auto simp add: field_simps) ultimately have "?b[d := (l,i)] ∈ grid ?b {d}" by (rule Suc.hyps) thus ?thesis by (auto intro!: grid_child[OF ‹d ∈ {d}›, of _ b left] simp add: child_def) next assume *: "ix b d * 2^(Suc l') < i" let ?b = "child b right d" have "d < length ?b" using Suc by auto moreover note ‹odd i› moreover have "lv ?b d + l' = l" and "i < (ix ?b d + 1) * 2^l'" and "(ix ?b d - 1) * 2^l' < i" unfolding child_ix_right[OF Suc.prems(1)] using Suc.prems * child_lv by (auto simp add: field_simps) ultimately have "?b[d := (l,i)] ∈ grid ?b {d}" by (rule Suc.hyps) thus ?thesis by (auto intro!: grid_child[OF ‹d ∈ {d}›, of _ b right] simp add: child_def) qed qed lemma grid_multi_dimensional_specification: assumes "dm ≤ length b" and "length p = length b" and "⋀ d. d < dm ⟹ odd (ix p d) ∧ lv b d ≤ lv p d ∧ ix p d < (ix b d + 1) * 2^(lv p d - lv b d) ∧ ix p d > (ix b d - 1) * 2^(lv p d - lv b d)" (is "⋀ d. d < dm ⟹ ?bounded p d") and "⋀ d. ⟦ dm ≤ d ; d < length b ⟧ ⟹ p ! d = b ! d" shows "p ∈ grid b {0..<dm}" using assms proof (induct dm arbitrary: p) case 0 hence "p = b" by (auto intro!: nth_equalityI) thus ?case by auto next case (Suc dm) hence "dm ≤ length b" and "dm < length p" by auto let ?p = "p[dm := b ! dm]" note ‹dm ≤ length b› moreover have "length ?p = length b" using ‹length p = length b› by simp moreover { fix d assume "d < dm" hence *: "d < Suc dm" and "dm ≠ d" by auto have "?p ! d = p ! d" by (rule nth_list_update_neq[OF ‹dm ≠ d›]) hence "?bounded ?p d" using Suc.prems(3)[OF *] lv_def ix_def by simp } moreover { fix d assume "dm ≤ d" and "d < length b" have "?p ! d = b ! d" proof (cases "d = dm") case True thus ?thesis using ‹d < length b› ‹length p = length b› by auto next case False hence "Suc dm ≤ d" using ‹dm ≤ d› by auto thus ?thesis using Suc.prems(4) ‹d < length b› by auto qed } ultimately have *: "?p ∈ grid b {0..<dm}" by (auto intro!: Suc.hyps) have "lv b dm ≤ lv p dm" using Suc.prems(3)[OF lessI] by simp have [simp]: "lv ?p dm = lv b dm" using lv_def ‹dm < length p› by auto have [simp]: "ix ?p dm = ix b dm" using ix_def ‹dm < length p› by auto have [simp]: "p[dm := (lv p dm, ix p dm)] = p" using lv_def ix_def ‹dm < length p› by auto have "dm < length ?p" and [simp]: "lv b dm + (lv p dm - lv b dm) = lv p dm" using ‹dm < length p› ‹lv b dm ≤ lv p dm› by auto from grid_single_dimensional_specification[OF this(1), where l="lv p dm" and i="ix p dm" and l'="lv p dm - lv b dm", simplified] have "p ∈ grid ?p {dm}" using Suc.prems(3)[OF lessI] by blast from grid_transitive[OF this *] show ?case by auto qed lemma sparsegrid: "sparsegrid dm lm = {p. length p = dm ∧ level p < lm ∧ (∀ d < dm. odd (ix p d) ∧ 0 < ix p d ∧ ix p d < 2^(lv p d + 1))}" (is "_ = ?set") proof (rule equalityI[OF subsetI subsetI]) fix p assume *: "p ∈ sparsegrid dm lm" hence "length p = dm" and "level p < lm" unfolding sparsegrid_def by auto moreover { fix d assume "d < dm" hence **: "p ∈ grid (start dm) {0..<dm}" and "d < length (start dm)" using * unfolding sparsegrid_def by auto have "odd (ix p d)" proof (cases "p ! d = start dm ! d") case True thus ?thesis unfolding start_def using ‹d < dm› ix_def by auto next case False from grid_odd[OF _ this **] show ?thesis using ‹d < dm› by auto qed hence "odd (ix p d) ∧ 0 < ix p d ∧ ix p d < 2^(lv p d + 1)" using grid_estimate[OF ‹d < length (start dm)› **] unfolding ix_def lv_def start_def using ‹d < dm› by auto } ultimately show "p ∈ ?set" using sparsegrid_def lgrid_def by auto next fix p assume "p ∈ ?set" with grid_multi_dimensional_specification[of dm "start dm" p] have "p ∈ grid (start dm) {0..<dm}" and "level p < lm" by auto thus "p ∈ sparsegrid dm lm" unfolding sparsegrid_def lgrid_def by auto qed end