# Theory Grid_Point

section ‹ Grid Points › theory Grid_Point imports "HOL-Analysis.Multivariate_Analysis" begin type_synonym grid_point = "(nat × int) list" definition lv :: "grid_point ⇒ nat ⇒ nat" where "lv p d = fst (p ! d)" definition ix :: "grid_point ⇒ nat ⇒ int" where "ix p d = snd (p ! d)" definition level :: "grid_point ⇒ nat" where "level p = (∑ i < length p. lv p i)" lemma level_all_eq: assumes "⋀d. d < length p ⟹ lv p d = lv p' d" and "length p = length p'" shows "level p' = level p" unfolding level_def using assms by auto datatype dir = left | right fun sgn :: "dir ⇒ int" where "sgn left = -1" | "sgn right = 1" fun inv :: "dir ⇒ dir" where "inv left = right" | "inv right = left" lemma inv_inv[simp]: "inv (inv dir) = dir" by (cases dir) simp_all lemma sgn_inv[simp]: "sgn (inv dir) = - sgn dir" by (cases dir, auto) definition child :: "grid_point ⇒ dir ⇒ nat ⇒ grid_point" where "child p dir d = p[d := (lv p d + 1, 2 * (ix p d) + sgn dir)]" lemma child_length[simp]: "length (child p dir d) = length p" unfolding child_def by simp lemma child_odd[simp]: "d < length p ⟹ odd (ix (child p dir d) d)" unfolding child_def ix_def by (cases dir, auto) lemma child_eq: "p ! d = (l, i) ⟹ ∃ j. child p dir d = p[d := (l + 1, j)]" by (auto simp add: child_def ix_def lv_def) lemma child_other: "d ≠ d' ⟹ child p dir d ! d' = p ! d'" unfolding child_def lv_def ix_def by (cases "d' < length p", auto) lemma child_invariant: assumes "d' < length p" shows "(child p dir d ! d' = p ! d') = (d ≠ d')" proof - obtain l i where "p ! d' = (l, i)" using prod.exhaust . with assms show ?thesis unfolding child_def ix_def lv_def by auto qed lemma child_single_level: "d < length p ⟹ lv (child p dir d) d > lv p d" unfolding lv_def child_def by simp lemma child_lv: "d < length p ⟹ lv (child p dir d) d = lv p d + 1" unfolding child_def lv_def by simp lemma child_lv_other: assumes "d' ≠ d" shows "lv (child p dir d') d = lv p d" using child_other[OF assms] unfolding lv_def by simp lemma child_ix_left: "d < length p ⟹ ix (child p left d) d = 2 * ix p d - 1" unfolding child_def ix_def by simp lemma child_ix_right: "d < length p ⟹ ix (child p right d) d = 2 * ix p d + 1" unfolding child_def ix_def by simp lemma child_ix: "d < length p ⟹ ix (child p dir d) d = 2 * ix p d + sgn dir" unfolding child_def ix_def by simp lemma child_level[simp]: assumes "d < length p" shows "level (child p dir d) = level p + 1" proof - have inter: "{0..<length p} ∩ {d} = {d}" using assms by auto have "level (child p dir d) = (∑ d' = 0..<length p. if d' ∈ {d} then lv p d + 1 else lv p d')" by (auto intro!: sum.cong simp add: child_lv_other child_lv level_def) moreover have "level p + 1 = (∑ d' = 0..<length p. if d' ∈ {d} then lv p d else lv p d') + 1" by (auto intro!: sum.cong simp add: child_lv_other child_lv level_def) ultimately show ?thesis unfolding sum.If_cases[OF finite_atLeastLessThan] inter using assms by auto qed lemma child_ex_neighbour: "∃ b'. child b dir d = child b' (inv dir) d" proof show "child b dir d = child (b[d := (lv b d, ix b d + sgn dir)]) (inv dir) d" unfolding child_def ix_def lv_def by (cases "d < length b", auto simp add: algebra_simps) qed lemma child_level_gt[simp]: "level (child p dir d) >= level p" by (cases "d < length p", simp, simp add: child_def) lemma child_estimate_child: assumes "d < length p" and "l ≤ lv p d" and i'_range: "ix p d < (i + 1) * 2^(lv p d - l) ∧ ix p d > (i - 1) * 2^(lv p d - l)" (is "?top p ∧ ?bottom p") and is_child: "p' = child p dir d" shows "?top p' ∧ ?bottom p'" proof from is_child and ‹d < length p› have "lv p' d = lv p d + 1" by (auto simp add: child_def ix_def lv_def) hence "lv p' d - l = lv p d - l + 1" using ‹lv p d >= l› by auto hence pow_l'': "(2^(lv p' d - l) :: int) = 2 * 2^(lv p d - l)" by auto show "?top p'" proof - from is_child and ‹d < length p› have "ix p' d ≤ 2 * (ix p d) + 1" by (cases dir, auto simp add: child_def lv_def ix_def) also have "… < (i + 1) * (2 * 2^(lv p d - l))" using i'_range by auto finally show ?thesis using pow_l'' by auto qed show "?bottom p'" proof - have "(i - 1) * 2^(lv p' d - l) = 2 * ((i - 1) * 2^(lv p d - l))" using pow_l'' by simp also have "… < 2 * ix p d - 1" using i'_range by auto finally show ?thesis using is_child and ‹d < length p› by (cases dir, auto simp add: child_def lv_def ix_def) qed qed lemma child_neighbour: assumes "child p (inv dir) d = child ps dir d" (is "?c_p = ?c_ps") shows "ps = p[d := (lv p d, ix p d - sgn dir)]" (is "_ = ?ps") proof (rule nth_equalityI) have "length ?c_ps = length ?c_p" using ‹?c_p = ?c_ps› by simp hence len_eq: "length ps = length p" by simp thus "length ps = length ?ps" by simp show "ps ! i = ?ps ! i" if "i < length ps" for i proof - have "i < length p" using that len_eq by auto show "ps ! i = ?ps ! i" proof (cases "d = i") case [simp]: True have "?c_p ! i = ?c_ps ! i" using ‹?c_p = ?c_ps› by auto hence "ix p i = ix ps d + sgn dir" and "lv p i = lv ps i" by (auto simp add: child_def nth_list_update_eq[OF ‹i < length p›] nth_list_update_eq[OF ‹i < length ps›]) thus ?thesis by (simp add: lv_def ix_def ‹i < length p›) next assume "d ≠ i" with child_other[OF this, of ps dir] child_other[OF this, of p "inv dir"] show ?thesis using assms by auto qed qed qed definition start :: "nat ⇒ grid_point" where "start dm = replicate dm (0, 1)" lemma start_lv[simp]: "d < dm ⟹ lv (start dm) d = 0" unfolding start_def lv_def by simp lemma start_ix[simp]: "d < dm ⟹ ix (start dm) d = 1" unfolding start_def ix_def by simp lemma start_length[simp]: "length (start dm) = dm" unfolding start_def by auto lemma level_start_0[simp]: "level (start dm) = 0" using level_def by auto end

# Theory Grid

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

# Theory Triangular_Function

section ‹ Hat Functions › theory Triangular_Function imports "HOL-Analysis.Equivalence_Lebesgue_Henstock_Integration" Grid begin lemma continuous_on_max[continuous_intros]: fixes f :: "_ ⇒ 'a::linorder_topology" shows "continuous_on S f ⟹ continuous_on S g ⟹ continuous_on S (λx. max (f x) (g x))" by (auto simp: continuous_on_def intro: tendsto_max) definition φ :: "(nat × int) ⇒ real ⇒ real" where "φ ≡ (λ(l,i) x. max 0 (1 - ¦ x * 2^(l + 1) - real_of_int i ¦))" definition Φ :: "(nat × int) list ⇒ (nat ⇒ real) ⇒ real" where "Φ p x = (∏d<length p. φ (p ! d) (x d))" definition l2_φ where "l2_φ p1 p2 = (∫x. φ p1 x * φ p2 x ∂lborel)" definition l2 where "l2 a b = (∫x. Φ a x * Φ b x ∂(Π⇩_{M}d∈{..<length a}. lborel))" lemma measurable_φ[measurable]: "φ p ∈ borel_measurable borel" by (cases p) (simp add: φ_def) lemma φ_nonneg: "0 ≤ φ p x" by (simp add: φ_def split: prod.split) lemma φ_zero_iff: "φ (l,i) x = 0 ⟷ x ∉ {real_of_int (i - 1) / 2^(l + 1) <..< real_of_int (i + 1) / 2^(l + 1)}" by (auto simp: φ_def field_simps split: split_max) lemma φ_zero: "x ∉ {real_of_int (i - 1) / 2^(l + 1) <..< real_of_int (i + 1) / 2^(l + 1)} ⟹ φ (l,i) x = 0" unfolding φ_zero_iff by simp lemma φ_eq_0: assumes x: "x < 0 ∨ 1 < x" and i: "0 < i" "i < 2^Suc l" shows "φ (l, i) x = 0" using x proof assume "x < 0" also have "0 ≤ real_of_int (i - 1) / 2^(l + 1)" using i by (auto simp: field_simps) finally show ?thesis by (auto intro!: φ_zero simp: field_simps) next have "real_of_int (i + 1) / 2^(l + 1) ≤ 1" using i by (subst divide_le_eq_1_pos) (auto simp del: of_int_add power_Suc) also assume "1 < x" finally show ?thesis by (auto intro!: φ_zero simp: field_simps) qed lemma ix_lt: "p ∈ sparsegrid dm lm ⟹ d < dm ⟹ ix p d < 2^(lv p d + 1)" unfolding sparsegrid_def lgrid_def using grid_estimate[of d "start dm" p "{0 ..< dm}"] by auto lemma ix_gt: "p ∈ sparsegrid dm lm ⟹ d < dm ⟹ 0 < ix p d" unfolding sparsegrid_def lgrid_def using grid_estimate[of d "start dm" p "{0 ..< dm}"] by auto lemma Φ_eq_0: assumes x: "∃d<length p. x d < 0 ∨ 1 < x d" and p: "p ∈ sparsegrid dm lm" shows "Φ p x = 0" unfolding Φ_def proof (rule prod_zero) from x guess d .. with p[THEN ix_lt, of d] p[THEN ix_gt, of d] p show "∃a∈{..<length p}. φ (p ! a) (x a) = 0" apply (cases "p!d") apply (intro bexI[of _ d]) apply (auto intro!: φ_eq_0 simp: sparsegrid_length ix_def lv_def) done qed simp lemma φ_left_support': "x ∈ {real_of_int (i - 1) / 2^(l + 1) .. real_of_int i / 2^(l + 1)} ⟹ φ (l,i) x = 1 + x * 2^(l + 1) - real_of_int i" by (auto simp: φ_def field_simps split: split_max) lemma φ_left_support: "x ∈ {-1 .. 0::real} ⟹ φ (l,i) ((x + real_of_int i) / 2^(l + 1)) = 1 + x" by (auto simp: φ_def field_simps split: split_max) lemma φ_right_support': "x ∈ {real_of_int i / 2^(l + 1) .. real_of_int (i + 1) / 2^(l + 1)} ⟹ φ (l,i) x = 1 - x * 2^(l + 1) + real_of_int i" by (auto simp: φ_def field_simps split: split_max) lemma φ_right_support: "x ∈ {0 .. 1::real} ⟹ φ (l,i) ((x + real i) / 2^(l + 1)) = 1 - x" by (auto simp: φ_def field_simps split: split_max) lemma integrable_φ: "integrable lborel (φ p)" proof (induct p) case (Pair l i) have "integrable lborel (λx. indicator {real_of_int (i - 1) / 2^(l + 1) .. real_of_int (i + 1) / 2^(l + 1)} x *⇩_{R}φ (l, i) x)" unfolding φ_def by (intro borel_integrable_compact) (auto intro!: continuous_intros) then show ?case by (rule integrable_cong[THEN iffD1, rotated -1]) (auto simp: φ_zero_iff) qed lemma integrable_φ2: "integrable lborel (λx. φ p x * φ q x)" proof (cases p q rule: prod.exhaust[case_product prod.exhaust]) case (Pair_Pair l i l' i') have "integrable lborel (λx. indicator {real_of_int (i - 1) / 2^(l + 1) .. real_of_int (i + 1) / 2^(l + 1)} x *⇩_{R}(φ (l, i) x * φ (l', i') x))" unfolding φ_def by (intro borel_integrable_compact) (auto intro!: continuous_intros) then show ?thesis unfolding Pair_Pair by (rule integrable_cong[THEN iffD1, rotated -1]) (auto simp: φ_zero_iff) qed lemma l2_φI_DERIV: assumes n: "⋀ x. x ∈ { (real_of_int i' - 1) / 2^(l' + 1) .. real_of_int i' / 2^(l' + 1) } ⟹ DERIV Φ_n x :> (φ (l', i') x * φ (l, i) x)" (is "⋀ x. x ∈ {?a..?b} ⟹ DERIV _ _ :> ?P x") and p: "⋀ x. x ∈ { real_of_int i' / 2^(l' + 1) .. (real_of_int i' + 1) / 2^(l' + 1) } ⟹ DERIV Φ_p x :> (φ (l', i') x * φ (l, i) x)" (is "⋀ x. x ∈ {?b..?c} ⟹ _") shows "l2_φ (l', i') (l, i) = (Φ_n ?b - Φ_n ?a) + (Φ_p ?c - Φ_p ?b)" proof - have "has_bochner_integral lborel (λx. ?P x * indicator {?a..?b} x + ?P x * indicator {?b..?c} x) ((Φ_n ?b - Φ_n ?a) + (Φ_p ?c - Φ_p ?b))" by (intro has_bochner_integral_add has_bochner_integral_FTC_Icc_nonneg n p) (auto simp: φ_nonneg field_simps) then have "has_bochner_integral lborel?P ((Φ_n ?b - Φ_n ?a) + (Φ_p ?c - Φ_p ?b))" by (rule has_bochner_integral_discrete_difference[where X="{?b}", THEN iffD1, rotated -1]) (auto simp: power_add intro!: φ_zero integral_cong split: split_indicator) then show ?thesis by (simp add: has_bochner_integral_iff l2_φ_def) qed lemma l2_eq: "length a = length b ⟹ l2 a b = (∏d<length a. l2_φ (a!d) (b!d))" unfolding l2_def l2_φ_def Φ_def apply (simp add: prod.distrib[symmetric]) proof (rule product_sigma_finite.product_integral_prod) show "product_sigma_finite (λd. lborel)" .. qed (auto intro: integrable_φ2) lemma l2_when_disjoint: assumes "l ≤ l'" defines "d == l' - l" assumes "(i + 1) * 2^d < i' ∨ i' < (i - 1) * 2^d" (is "?right ∨ ?left") shows "l2_φ (l', i') (l, i) = 0" proof - let ?sup = "λl i. {real_of_int (i - 1) / 2^(l + 1) <..< real_of_int (i + 1) / 2^(l + 1)}" have l': "l' = l + d" using assms by simp have *: "⋀i l. 2 ^ l = real_of_int (2 ^ l::int)" by simp have [arith]: "0 < (2^d::int)" by simp from ‹?right ∨ ?left› ‹l ≤ l'› have empty_support: "?sup l i ∩ ?sup l' i' = {}" by (auto simp add: min_def max_def divide_simps l' power_add * of_int_mult[symmetric] simp del: of_int_diff of_int_add of_int_mult of_int_power) (simp_all add: field_simps) then have "⋀x. φ (l', i') x * φ (l, i) x = 0" unfolding φ_zero_iff mult_eq_0_iff by blast then show ?thesis by (simp add: l2_φ_def del: mult_eq_0_iff vector_space_over_itself.scale_eq_0_iff) qed lemma l2_commutative: "l2_φ p q = l2_φ q p" by (simp add: l2_φ_def mult.commute) lemma l2_when_same: "l2_φ (l, i) (l, i) = 1/3 / 2^l" proof (subst l2_φI_DERIV) let ?l = "(2 :: real)^(l + 1)" let ?in = "real_of_int i - 1" let ?ip = "real_of_int i + 1" let ?φ = "φ (l,i)" let ?φ2 = "λx. ?φ x * ?φ x" { fix x assume "x ∈ {?in / ?l .. real_of_int i / ?l}" hence φ_eq: "?φ x = ?l * x - ?in" using φ_left_support' by auto show "DERIV (λx. x^3 / 3 * ?l^2 + x * ?in^2 - x^2/2 * 2 * ?l * ?in) x :> ?φ2 x" by (auto intro!: derivative_eq_intros simp add: power2_eq_square field_simps φ_eq) } { fix x assume "x ∈ {real_of_int i / ?l .. ?ip / ?l}" hence φ_eq: "?φ x = ?ip - ?l * x" using φ_right_support' by auto show "DERIV (λx. x^3 / 3 * ?l^2 + x * ?ip^2 - x^2/2 * 2 * ?l * ?ip) x :> ?φ2 x" by (auto intro!: derivative_eq_intros simp add: power2_eq_square field_simps φ_eq) } qed (simp_all add: field_simps power_eq_if[of _ 2] power_eq_if[of _ 3]) lemma l2_when_left_child: assumes "l < l'" and i'_bot: "i' > (i - 1) * 2^(l' - l)" and i'_top: "i' < i * 2^(l' - l)" shows "l2_φ (l', i') (l, i) = (1 + real_of_int i' / 2^(l' - l) - real_of_int i) / 2^(l' + 1)" proof (subst l2_φI_DERIV) let ?l' = "(2 :: real)^(l' + 1)" let ?in' = "real_of_int i' - 1" let ?ip' = "real_of_int i' + 1" let ?l = "(2 :: real)^(l + 1)" let ?i = "real_of_int i - 1" let ?φ' = "φ (l',i')" let ?φ = "φ (l, i)" let "?φ2 x" = "?φ' x * ?φ x" define Φ_n where "Φ_n x = x^3 / 3 * ?l' * ?l + x * ?i * ?in' - x^2 / 2 * (?in' * ?l + ?i * ?l')" for x define Φ_p where "Φ_p x = x^2 / 2 * (?ip' * ?l + ?i * ?l') - x^3 / 3 * ?l' * ?l - x * ?i * ?ip'" for x have level_diff: "2^(l' - l) = 2^l' / (2^l :: real)" using power_diff[of "2::real" l l'] ‹l < l'› by auto { fix x assume x: "x ∈ {?in' / ?l' .. ?ip' / ?l'}" have "?i * 2^(l' - l) ≤ ?in'" using i'_bot int_less_real_le by auto hence "?i / ?l ≤ ?in' / ?l'" using level_diff by (auto simp: field_simps) hence "?i / ?l ≤ x" using x by auto moreover have "?ip' ≤ real_of_int i * 2^(l' - l)" using i'_top int_less_real_le by auto hence ip'_le_i: "?ip' / ?l' ≤ real_of_int i / ?l" using level_diff by (auto simp: field_simps) hence "x ≤ real_of_int i / ?l" using x by auto ultimately have "?φ x = ?l * x - ?i" using φ_left_support' by auto } note φ_eq = this { fix x assume x: "x ∈ {?in' / ?l' .. real_of_int i' / ?l'}" hence φ'_eq: "?φ' x = ?l' * x - ?in'" using φ_left_support' by auto from x have x': "x ∈ {?in' / ?l' .. ?ip' / ?l'}" by (auto simp add: field_simps) show "DERIV Φ_n x :> ?φ2 x" unfolding φ_eq[OF x'] φ'_eq Φ_n_def by (auto intro!: derivative_eq_intros simp add: power2_eq_square algebra_simps) } { fix x assume x: "x ∈ {real_of_int i' / ?l' .. ?ip' / ?l'}" hence φ'_eq: "?φ' x = ?ip' - ?l' * x" using φ_right_support' by auto from x have x': "x ∈ {?in' / ?l' .. ?ip' / ?l'}" by (simp add: field_simps) show "DERIV Φ_p x :> ?φ2 x" unfolding φ_eq[OF x'] φ'_eq Φ_p_def by (auto intro!: derivative_eq_intros simp add: power2_eq_square algebra_simps) } qed (simp_all add: field_simps power_eq_if[of _ 2] power_eq_if[of _ 3] power_diff[of "2::real", OF _ ‹l < l'›[THEN less_imp_le]] ) lemma l2_when_right_child: assumes "l < l'" and i'_bot: "i' > i * 2^(l' - l)" and i'_top: "i' < (i + 1) * 2^(l' - l)" shows "l2_φ (l', i') (l, i) = (1 - real_of_int i' / 2^(l' - l) + real_of_int i) / 2^(l' + 1)" proof (subst l2_φI_DERIV) let ?l' = "(2 :: real)^(l' + 1)" let ?in' = "real_of_int i' - 1" let ?ip' = "real_of_int i' + 1" let ?l = "(2 :: real)^(l + 1)" let ?i = "real_of_int i + 1" let ?φ' = "φ (l',i')" let ?φ = "φ (l, i)" let ?φ2 = "λx. ?φ' x * ?φ x" define Φ_n where "Φ_n x = x^2 / 2 * (?in' * ?l + ?i * ?l') - x^3 / 3 * ?l' * ?l - x * ?i * ?in'" for x define Φ_p where "Φ_p x = x^3 / 3 * ?l' * ?l + x * ?i * ?ip' - x^2 / 2 * (?ip' * ?l + ?i * ?l')" for x have level_diff: "2^(l' - l) = 2^l' / (2^l :: real)" using power_diff[of "2::real" l l'] ‹l < l'› by auto { fix x assume x: "x ∈ {?in' / ?l' .. ?ip' / ?l'}" have "real_of_int i * 2^(l' - l) ≤ ?in'" using i'_bot int_less_real_le by auto hence "real_of_int i / ?l ≤ ?in' / ?l'" using level_diff by (auto simp: field_simps) hence "real_of_int i / ?l ≤ x" using x by auto moreover have "?ip' ≤ ?i * 2^(l' - l)" using i'_top int_less_real_le by auto hence ip'_le_i: "?ip' / ?l' ≤ ?i / ?l" using level_diff by (auto simp: field_simps) hence "x ≤ ?i / ?l" using x by auto ultimately have "?φ x = ?i - ?l * x" using φ_right_support' by auto } note φ_eq = this { fix x assume x: "x ∈ {?in' / ?l' .. real_of_int i' / ?l'}" hence φ'_eq: "?φ' x = ?l' * x - ?in'" using φ_left_support' by auto from x have x': "x ∈ {?in' / ?l' .. ?ip' / ?l'}" by (simp add: field_simps) show "DERIV Φ_n x :> ?φ2 x" unfolding Φ_n_def φ_eq[OF x'] φ'_eq by (auto intro!: derivative_eq_intros simp add: simp add: power2_eq_square algebra_simps) } { fix x assume x: "x ∈ {real_of_int i' / ?l' .. ?ip' / ?l'}" hence φ'_eq: "?φ' x = ?ip' - ?l' * x" using φ_right_support' by auto from x have x': "x ∈ {?in' / ?l' .. ?ip' / ?l'}" by (auto simp: field_simps) show "DERIV Φ_p x :> ?φ2 x" unfolding φ_eq[OF x'] φ'_eq Φ_p_def by (auto intro!: derivative_eq_intros simp add: power2_eq_square algebra_simps) } qed (simp_all add: field_simps power_eq_if[of _ 2] power_eq_if[of _ 3] power_diff[of "2::real", OF _ ‹l < l'›[THEN less_imp_le]] ) lemma level_shift: "lc > l ⟹ (x :: real) / 2 ^ (lc - Suc l) = x * 2 / 2 ^ (lc - l)" by (auto simp add: power_diff) lemma l2_child: assumes "d < length b" and p_grid: "p ∈ grid (child b dir d) ds" (is "p ∈ grid ?child ds") shows "l2_φ (p ! d) (b ! d) = (1 - real_of_int (sgn dir) * (real_of_int (ix p d) / 2^(lv p d - lv b d) - real_of_int (ix b d))) / 2^(lv p d + 1)" proof - have "lv ?child d ≤ lv p d" using ‹d < length b› and p_grid using grid_single_level by auto hence "lv b d < lv p d" using ‹d < length b› and p_grid using child_lv by auto let ?i_c = "ix ?child d" and ?l_c = "lv ?child d" let ?i_p = "ix p d" and ?l_p = "lv p d" let ?i_b = "ix b d" and ?l_b = "lv b d" have "(2::int) * 2^(?l_p - ?l_c) = 2^Suc (?l_p - ?l_c)" by auto also have "… = 2^(Suc ?l_p - ?l_c)" proof - have "Suc (?l_p - ?l_c) = Suc ?l_p - ?l_c" using ‹lv ?child d ≤ lv p d› by auto thus ?thesis by auto qed also have "… = 2^(?l_p - ?l_b)" using ‹d < length b› and ‹lv b d < lv p d› by (auto simp add: child_def lv_def) finally have level: "2^(?l_p - ?l_b) = (2::int) * 2^(?l_p - ?l_c)" .. from ‹d < length b› and p_grid have range_left: "?i_p > (?i_c - 1) * 2^(?l_p - ?l_c)" and range_right: "?i_p < (?i_c + 1) * 2^(?l_p - ?l_c)" using grid_estimate by auto show ?thesis proof (cases dir) case left with child_ix_left[OF ‹d < length b›] have "(?i_b - 1) * 2^(?l_p - ?l_b) = (?i_c - 1) * 2^(?l_p - ?l_c)" and "?i_b * 2^(?l_p - ?l_b) = (?i_c + 1) * 2^(?l_p - ?l_c)" using level by auto hence "?i_p > (?i_b - 1) * 2^(?l_p - ?l_b)" and "?i_p < ?i_b * 2^(?l_p - ?l_b)" using range_left and range_right by auto with ‹?l_b < ?l_p› have "l2_φ (?l_p, ?i_p) (?l_b, ?i_b) = (1 + real_of_int ?i_p / 2^(?l_p - ?l_b) - real_of_int ?i_b) / 2^(?l_p + 1)" by (rule l2_when_left_child) thus ?thesis using left by (auto simp add: ix_def lv_def) next case right hence "?i_c = 2 * ?i_b + 1" using child_ix_right and ‹d < length b› by auto hence "?i_b * 2^(?l_p - ?l_b) = (?i_c - 1) * 2^(?l_p - ?l_c)" and "(?i_b + 1) * 2^(?l_p - ?l_b) = (?i_c + 1) * 2^(?l_p - ?l_c)" using level by auto hence "?i_p > ?i_b * 2^(?l_p - ?l_b)" and "?i_p < (?i_b + 1) * 2^(?l_p - ?l_b)" using range_left and range_right by auto with ‹?l_b < ?l_p› have "l2_φ (?l_p, ?i_p) (?l_b, ?i_b) = (1 - real_of_int ?i_p / 2^(?l_p - ?l_b) + real_of_int ?i_b) / 2^(?l_p + 1)" by (rule l2_when_right_child) thus ?thesis using right by (auto simp add: ix_def lv_def) qed qed lemma l2_same: "l2_φ (p!d) (p!d) = 1/3 / 2^(lv p d)" proof - have "l2_φ (p!d) (p!d) = l2_φ (lv p d, ix p d) (lv p d, ix p d)" by (auto simp add: lv_def ix_def) thus ?thesis using l2_when_same by auto qed lemma l2_disjoint: assumes "d < length b" and "p ∈ grid b {d}" and "p' ∈ grid b {d}" and "p' ∉ grid p {d}" and "lv p' d ≥ lv p d" shows "l2_φ (p' ! d) (p ! d) = 0" proof - have range: "ix p' d > (ix p d + 1) * 2^(lv p' d - lv p d) ∨ ix p' d < (ix p d - 1) * 2^(lv p' d - lv p d)" proof (rule ccontr) assume "¬ ?thesis" hence "ix p' d ≤ (ix p d + 1) * 2^(lv p' d - lv p d)" and "ix p' d ≥ (ix p d - 1) * 2^(lv p' d - lv p d)" by auto with ‹p' ∈ grid b {d}› and ‹p ∈ grid b {d}› and ‹lv p' d ≥ lv p d› and ‹d < length b› have "p' ∈ grid p {d}" using grid_part[where p=p and b=b and d=d and p'=p'] by auto with ‹p' ∉ grid p {d}› show False by auto qed have "l2_φ (p' ! d) (p ! d) = l2_φ (lv p' d, ix p' d) (lv p d, ix p d)" by (auto simp add: ix_def lv_def) also have "… = 0" using range and ‹lv p' d ≥ lv p d› and l2_when_disjoint by auto finally show ?thesis . qed lemma l2_down2: fixes pc pd p assumes "d < length pd" assumes pc_in_grid: "pc ∈ grid (child pd dir d) {d}" assumes pd_is_child: "pd = child p dir d" (is "pd = ?pd") shows "l2_φ (pc ! d) (pd ! d) / 2 = l2_φ (pc ! d) (p ! d)" proof - have "d < length p" using pd_is_child ‹d < length pd› by auto moreover have "pc ∈ grid ?pd {d}" using pd_is_child and grid_child and pc_in_grid by auto hence "lv p d < lv pc d" using grid_child_level and ‹d < length pd› and pd_is_child by auto moreover have "real_of_int (sgn dir) * real_of_int (sgn dir) = 1" by (cases dir, auto) ultimately show ?thesis unfolding l2_child[OF ‹d < length pd› pc_in_grid] l2_child[OF ‹d < length p› ‹pc ∈ grid ?pd {d}›] using child_lv and child_ix and pd_is_child and level_shift by (auto simp add: algebra_simps diff_divide_distrib add_divide_distrib) qed lemma l2_zigzag: assumes "d < length p" and p_child: "p = child p_p dir d" and p'_grid: "p' ∈ grid (child p (inv dir) d) {d}" and ps_intro: "child p (inv dir) d = child ps dir d" (is "?c_p = ?c_ps") shows "l2_φ (p' ! d) (p_p ! d) = l2_φ (p' ! d) (ps ! d) + l2_φ (p' ! d) (p ! d) / 2" proof - have "length p = length ?c_p" by auto also have "… = length ?c_ps" using ps_intro by auto finally have "length p = length ps" using ps_intro by auto hence "d < length p_p" using p_child and ‹d < length p› by auto moreover from ps_intro have "ps = p[d := (lv p d, ix p d - sgn dir)]" by (rule child_neighbour) hence "lv ps d = lv p d" and "real_of_int (ix ps d) = real_of_int (ix p d) - real_of_int (sgn dir)" using lv_def and ix_def and ‹length p = length ps› and ‹d < length p› by auto moreover have "d < length ps" and *: "p' ∈ grid (child ps dir d) {d}" using p'_grid ps_intro ‹length p = length ps› ‹d < length p› by auto have "p' ∈ grid p {d}" using p'_grid and grid_child by auto hence p_p_grid: "p' ∈ grid (child p_p dir d) {d}" using p_child by auto hence "lv p' d > lv p_p d" using grid_child_level and ‹d < length p_p› by auto moreover have "real_of_int (sgn dir) * real_of_int (sgn dir) = 1" by (cases dir, auto) ultimately show ?thesis unfolding l2_child[OF ‹d < length p› p'_grid] l2_child[OF ‹d < length ps› *] l2_child[OF ‹d < length p_p› p_p_grid] using child_lv and child_ix and p_child level_shift by (auto simp add: add_divide_distrib algebra_simps diff_divide_distrib) qed end

# Theory UpDown_Scheme

section ‹ UpDown Scheme › theory UpDown_Scheme imports Grid begin fun down' :: "nat ⇒ nat ⇒ grid_point ⇒ real ⇒ real ⇒ vector ⇒ vector" where "down' d 0 p f⇩_{l}f⇩_{r}α = α" | "down' d (Suc l) p f⇩_{l}f⇩_{r}α = (let f⇩_{m}= (f⇩_{l}+ f⇩_{r}) / 2 + (α p); α = α(p := ((f⇩_{l}+ f⇩_{r}) / 4 + (1 / 3) * (α p)) / 2 ^ (lv p d)); α = down' d l (child p left d) f⇩_{l}f⇩_{m}α; α = down' d l (child p right d) f⇩_{m}f⇩_{r}α in α)" definition down :: "nat ⇒ nat ⇒ nat ⇒ vector ⇒ vector" where "down = lift (λ d l p. down' d l p 0 0)" fun up' :: "nat ⇒ nat ⇒ grid_point ⇒ vector ⇒ (real * real) * vector" where "up' d 0 p α = ((0, 0), α)" | "up' d (Suc l) p α = (let ((f⇩_{l}, f⇩_{m}⇩_{l}), α) = up' d l (child p left d) α; ((f⇩_{m}⇩_{r}, f⇩_{r}), α) = up' d l (child p right d) α; result = (f⇩_{m}⇩_{l}+ f⇩_{m}⇩_{r}+ (α p) / 2 ^ (lv p d) / 2) / 2 in ((f⇩_{l}+ result, f⇩_{r}+ result), α(p := f⇩_{m}⇩_{l}+ f⇩_{m}⇩_{r})))" definition up :: "nat ⇒ nat ⇒ nat ⇒ vector ⇒ vector" where "up = lift (λ d lm p α. snd (up' d lm p α))" fun updown' :: "nat ⇒ nat ⇒ nat ⇒ vector ⇒ vector" where "updown' dm lm 0 α = α" | "updown' dm lm (Suc d) α = (sum_vector (updown' dm lm d (up dm lm d α)) (down dm lm d (updown' dm lm d α)))" definition updown :: "nat ⇒ nat ⇒ vector ⇒ vector" where "updown dm lm α = updown' dm lm dm α" end

# Theory Up

section ‹ Up Part › theory Up imports UpDown_Scheme Triangular_Function begin lemma up'_inplace: assumes p'_in: "p' ∉ grid p ds" and "d ∈ ds" shows "snd (up' d l p α) p' = α p'" using p'_in proof (induct l arbitrary: p α) case (Suc l) let "?ch dir" = "child p dir d" let "?up dir α" = "up' d l (?ch dir) α" let "?upl" = "snd (?up left α)" from contrapos_nn[OF ‹p' ∉ grid p ds› grid_child[OF ‹d ∈ ds›]] have left: "p' ∉ grid (?ch left) ds" and right: "p' ∉ grid (?ch right) ds" by auto have "p ≠ p'" using grid.Start Suc.prems by auto with Suc.hyps[OF left, of α] Suc.hyps[OF right, of ?upl] show ?case by (cases "?up left α", cases "?up right ?upl", auto simp add: Let_def) qed auto lemma up'_fl_fr: "⟦ d < length p ; p = (child p_r right d) ; p = (child p_l left d) ⟧ ⟹ fst (up' d lm p α) = (∑ p' ∈ lgrid p {d} (lm + level p). (α p') * l2_φ (p' ! d) (p_r ! d), ∑ p' ∈ lgrid p {d} (lm + level p). (α p') * l2_φ (p' ! d) (p_l ! d))" proof (induct lm arbitrary: p p_l p_r α) case (Suc lm) note ‹d < length p›[simp] from child_ex_neighbour obtain pc_r pc_l where pc_r_def: "child p right d = child pc_r (inv right) d" and pc_l_def: "child p left d = child pc_l (inv left) d" by blast define pc where "pc dir = (case dir of right ⇒ pc_r | left ⇒ pc_l)" for dir { fix dir have "child p (inv dir) d = child (pc (inv dir)) dir d" by (cases dir, auto simp add: pc_def pc_r_def pc_l_def) } note pc_child = this { fix dir have "child p dir d = child (pc dir) (inv dir) d" by (cases dir, auto simp add: pc_def pc_r_def pc_l_def) } note pc_child_inv = this hence "!! dir. length (child p dir d) = length (child (pc dir) (inv dir) d)" by auto hence "!! dir. length p = length (pc dir)" by auto hence [simp]: "!! dir. d < length (pc dir)" by auto let ?l = "λs. lm + level s" let ?C = "λp p'. (α p) * l2_φ (p ! d) (p' ! d)" let ?sum' = "λs p''. ∑ p' ∈ lgrid s {d} (Suc lm + level p). ?C p' p''" let ?sum = "λs dir p. ∑ p' ∈ lgrid (child s dir d) {d} (?l (child s dir d)). ?C p' p" let ?ch = "λdir. child p dir d" let ?f = "λdir. ?sum p dir (pc dir)" let ?fm = "λdir. ?sum p dir p" let ?result = "(?fm left + ?fm right + (α p) / 2 ^ (lv p d) / 2) / 2" let ?up = "λlm p α. up' d lm p α" define βl where "βl = snd (?up lm (?ch left) α)" define βr where "βr = snd (?up lm (?ch right) βl)" define p_d where "p_d dir = (case dir of right ⇒ p_r | left ⇒ p_l)" for dir have p_d_child: "p = child (p_d dir) dir d" for dir using Suc.prems p_d_def by (cases dir) auto hence "⋀ dir. length p = length (child (p_d dir) dir d)" by auto hence "⋀ dir. d < length (p_d dir)" by auto { fix dir { fix p' assume "p' ∈ lgrid (?ch (inv dir)) {d} (?l (?ch (inv dir))) " hence "?C p' (pc (inv dir)) + (?C p' p) / 2 = ?C p' (p_d dir)" using l2_zigzag[OF _ p_d_child[of dir] _ pc_child[of dir]] by (cases dir) (auto simp add: algebra_simps) } hence inv_dir_sum: "?sum p (inv dir) (pc (inv dir)) + (?sum p (inv dir) p) / 2 = ?sum p (inv dir) (p_d dir)" by (auto simp add: sum.distrib[symmetric] sum_divide_distrib) have "?sum p dir p / 2 = ?sum p dir (p_d dir)" using l2_down2[OF _ _ ‹p = child (p_d dir) dir d›] by (force intro!: sum.cong simp add: sum_divide_distrib) moreover have "?C p (p_d dir) = (α p) / 2 ^ (lv p d) / 4" using l2_child[OF ‹d < length (p_d dir)›, of p dir "{d}"] p_d_child[of dir] ‹d < length (p_d dir)› child_lv child_ix grid.Start[of p "{d}"] by (cases dir) (auto simp add: add_divide_distrib field_simps) ultimately have "?sum' p (p_d dir) = ?sum p (inv dir) (pc (inv dir)) + (?sum p (inv dir) p) / 2 + ?sum p dir p / 2 + (α p) / 2 ^ (lv p d) / 4" using lgrid_sum[where b=p] and child_level and inv_dir_sum by (cases dir) auto hence "?sum p (inv dir) (pc (inv dir)) + ?result = ?sum' p (p_d dir)" by (cases dir) auto } note this[of left] this[of right] moreover note eq = up'_inplace[OF grid_not_child[OF ‹d < length p›], of d "{d}" lm] { fix p' assume "p' ∈ lgrid (?ch right) {d} (lm + level (?ch right))" with grid_disjunct[of d p] up'_inplace[of p' "?ch left" "{d}" d lm α] βl_def have "βl p' = α p'" by auto } hence "fst (?up (Suc lm) p α) = (?f left + ?result, ?f right + ?result)" using βl_def pc_child_inv[of left] pc_child_inv[of right] Suc.hyps[of "?ch left" "pc left" p α] eq[of left α] Suc.hyps[of "?ch right" p "pc right" βl] eq[of right βl] by (cases "?up lm (?ch left) α", cases "?up lm (?ch right) βl") (simp add: Let_def) ultimately show ?case by (auto simp add: p_d_def) next case 0 show ?case by simp qed lemma up'_β: "⟦ d < length b ; l + level b = lm ; b ∈ sparsegrid' dm ; p ∈ sparsegrid' dm ⟧ ⟹ (snd (up' d l b α)) p = (if p ∈ lgrid b {d} lm then ∑ p' ∈ (lgrid p {d} lm) - {p}. α p' * l2_φ (p' ! d) (p ! d) else α p)" (is "⟦ _ ; _ ; _ ; _ ⟧ ⟹ (?goal l b p α)") proof (induct l arbitrary: b p α) case (Suc l) let ?l = "child b left d" and ?r = "child b right d" obtain p_l where p_l_def: "?r = child p_l left d" using child_ex_neighbour[where dir=right] by auto obtain p_r where p_r_def: "?l = child p_r right d" using child_ex_neighbour[where dir=left] by auto let ?ul = "up' d l ?l α" let ?ur = "up' d l ?r (snd ?ul)" let "?C p'" = "α p' * l2_φ (p' ! d) (p ! d)" let "?s s" = "∑ p' ∈ (lgrid s {d} lm). ?C p'" from ‹b ∈ sparsegrid' dm› have "length b = dm" unfolding sparsegrid'_def start_def by auto hence "d < dm" using ‹d < length b› by auto { fix p' assume "p' ∈ grid ?r {d}" hence "p' ∉ grid ?l {d}" using grid_disjunct[OF ‹d < length b›] by auto hence "snd ?ul p' = α p'" using up'_inplace by auto } note eq = this show "?goal (Suc l) b p α" proof (cases "p = b") case True let "?C p'" = "α p' * l2_φ (p' ! d) (b ! d)" let "?s s" = "∑ p' ∈ (lgrid s {d} lm). ?C p'" have "d < length ?l" using ‹d < length b› by auto from up'_fl_fr[OF this p_r_def] have fml: "snd (fst ?ul) = (∑ p' ∈ lgrid ?l {d} (l + level ?l). ?C p')" by simp have "d < length ?r" using ‹d < length b› by auto from up'_fl_fr[OF this _ p_l_def, where α="snd ?ul"] have fmr: "fst (fst ?ur) = (∑ p' ∈ lgrid ?r {d} (l + level ?r). ((snd ?ul) p') * l2_φ (p' ! d) (b ! d))" by simp have "level b < lm" using ‹Suc l + level b = lm› by auto hence "{ b } ⊆ lgrid b {d} lm" unfolding lgrid_def by auto from sum_diff[OF lgrid_finite this] have "(∑ p' ∈ (lgrid b {d} lm) - {b}. ?C p') = ?s b - ?C b" by simp also have "… = ?s ?l + ?s ?r" using lgrid_sum and ‹level b < lm› and ‹d < length b› by auto also have "… = snd (fst ?ul) + fst (fst ?ur)" using fml and fmr and ‹Suc l + level b = lm› and child_level[OF ‹d < length b›] using eq unfolding True lgrid_def by auto finally show ?thesis unfolding up'.simps Let_def and fun_upd_def lgrid_def using ‹p = b› and ‹level b < lm› by (cases ?ul, cases ?ur, auto) next case False have "?r ∈ sparsegrid' dm" and "?l ∈ sparsegrid' dm" using ‹b ∈ sparsegrid' dm› and ‹d < dm› unfolding sparsegrid'_def by auto from Suc.hyps[OF _ _ this(1)] Suc.hyps[OF _ _ this(2)] have "?goal l ?l p α" and "?goal l ?r p (snd ?ul)" using ‹d < length b› and ‹Suc l + level b = lm› and ‹p ∈ sparsegrid' dm› by auto show ?thesis proof (cases "p ∈ lgrid b {d} lm") case True hence "level p < lm" and "p ∈ grid b {d}" unfolding lgrid_def by auto hence "p ∈ grid ?l {d} ∨ p ∈ grid ?r {d}" unfolding grid_partition[of b] using ‹p ≠ b› by auto thus ?thesis proof (rule disjE) assume "p ∈ grid (child b left d) {d}" hence "p ∉ grid (child b right d) {d}" using grid_disjunct[OF ‹d < length b›] by auto thus ?thesis using ‹?goal l ?l p α› and ‹?goal l ?r p (snd ?ul)› using ‹p ≠ b› ‹p ∈ lgrid b {d} lm› unfolding lgrid_def grid_partition[of b] by (cases ?ul, cases ?ur, auto simp add: Let_def) next assume *: "p ∈ grid (child b right d) {d}" hence "p ∉ grid (child b left d) {d}" using grid_disjunct[OF ‹d < length b›] by auto moreover { fix p' assume "p' ∈ grid p {d}" from grid_transitive[OF this *] eq[of p'] have "snd ?ul p' = α p'" by simp } ultimately show ?thesis using ‹?goal l ?l p α› and ‹?goal l ?r p (snd ?ul)› using ‹p ≠ b› ‹p ∈ lgrid b {d} lm› * unfolding lgrid_def by (cases ?ul, cases ?ur, auto simp add: Let_def) qed next case False then have "p ∉ lgrid ?l {d} lm" and "p ∉ lgrid ?r {d} lm" unfolding lgrid_def and grid_partition[where p=b] by auto with False show ?thesis using ‹?goal l ?l p α› and ‹?goal l ?r p (snd ?ul)› using ‹p ≠ b› ‹p ∉ lgrid b {d} lm› unfolding lgrid_def by (cases ?ul, cases ?ur, auto simp add: Let_def) qed qed next case 0 then have "lgrid b {d} lm = {}" using lgrid_empty'[where p=b and lm=lm and ds="{d}"] by auto with 0 show ?case unfolding up'.simps by auto qed lemma up: assumes "d < dm" and "p ∈ sparsegrid dm lm" shows "(up dm lm d α) p = (∑ p' ∈ (lgrid p {d} lm) - {p}. α p' * l2_φ (p' ! d) (p ! d))" proof - let ?S = "λ x p p'. if p' ∈ grid p {d} - {p} then x * l2_φ (p'!d) (p!d) else 0" let ?F = "λ d lm p α. snd (up' d lm p α)" { fix p b assume "p ∈ grid b {d}" from grid_transitive[OF _ this subset_refl subset_refl] have "lgrid b {d} lm ∩ (grid p {d} - {p}) = lgrid p {d} lm - {p}" unfolding lgrid_def by auto } note lgrid_eq = this { fix l b p α assume b: "b ∈ lgrid (start dm) ({0..<dm} - {d}) lm" hence "b ∈ sparsegrid' dm" and "d < length b" using sparsegrid'_start ‹d < dm› by auto assume l: "l + level b = lm" and p: "p ∈ sparsegrid dm lm" note sparsegridE[OF p] note up' = up'_β[OF ‹d < length b› l ‹b ∈ sparsegrid' dm› ‹p ∈ sparsegrid' dm›] have "?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)" proof (cases "b = base {d} p") case True with baseE(2)[OF ‹p ∈ sparsegrid' dm›] ‹level p < lm› have "p ∈ lgrid b {d} lm" and "p ∈ grid b {d}" by auto show ?thesis using lgrid_eq[OF ‹p ∈ grid b {d}›] unfolding up' if_P[OF True] if_P[OF ‹p ∈ lgrid b {d} lm›] by (intro sum.mono_neutral_cong_left lgrid_finite) auto next case False moreover have "p ∉ lgrid b {d} lm" proof (rule ccontr) assume "¬ ?thesis" hence "base {d} p = b" using b by (auto intro!: baseI) thus False using False by auto qed ultimately show ?thesis unfolding up' by auto qed } with lift[where F = ?F, OF ‹d < dm› ‹p ∈ sparsegrid dm lm›] have lift_eq: "lift ?F dm lm d α p = (∑p'∈lgrid (base {d} p) {d} lm. ?S (α p') p p')" by auto from lgrid_eq[OF baseE(2)[OF sparsegrid_subset[OF ‹p ∈ sparsegrid dm lm›]]] show ?thesis unfolding up_def lift_eq by (intro sum.mono_neutral_cong_right lgrid_finite) auto qed end

# Theory Down

section ‹ Down part › theory Down imports Triangular_Function UpDown_Scheme begin lemma sparsegrid'_parents: assumes b: "b ∈ sparsegrid' dm" and p': "p' ∈ parents d b p" shows "p' ∈ sparsegrid' dm" using assms parents_def sparsegrid'I by auto lemma down'_β: "⟦ d < length b ; l + level b = lm ; b ∈ sparsegrid' dm ; p ∈ sparsegrid' dm ⟧ ⟹ down' d l b fl fr α p = (if p ∈ lgrid b {d} lm then (fl + (fr - fl) / 2 * (real_of_int (ix p d) / 2^(lv p d - lv b d) - real_of_int (ix b d) + 1)) / 2 ^ (lv p d + 1) + (∑ p' ∈ parents d b p. (α p') * l2_φ (p ! d) (p' ! d)) else α p)" proof (induct l arbitrary: b α fl fr p) case (Suc l) let ?l = "child b left d" and ?r = "child b right d" let ?result = "((fl + fr) / 4 + (1 / 3) * (α b)) / 2 ^ (lv b d)" let ?fm = "(fl + fr) / 2 + (α b)" let ?down_l = "down' d l (child b left d) fl ?fm (α(b := ?result))" have "length b = dm" using ‹b ∈ sparsegrid' dm› unfolding sparsegrid'_def start_def by auto hence "d < dm" using ‹d < length b› by auto have "!!dir. d < length (child b dir d)" using ‹d < length b› by auto have "!!dir. l + level (child b dir d) = lm" using ‹d < length b› and ‹Suc l + level b = lm› and child_level by auto have "!!dir. (child b dir d) ∈ sparsegrid' dm" using ‹b ∈ sparsegrid' dm› and ‹d < dm› and sparsegrid'_def by auto note hyps = Suc.hyps[OF ‹!! dir. d < length (child b dir d)› ‹!!dir. l + level (child b dir d) = lm› ‹!!dir. (child b dir d) ∈ sparsegrid' dm›] show ?case proof (cases "p ∈ lgrid b {d} lm") case False moreover hence "p ≠ b" and "p ∉ lgrid ?l {d} lm" and "p ∉ lgrid ?r {d} lm" unfolding lgrid_def unfolding grid_partition[where p=b] using ‹Suc l + level b = lm› by auto ultimately show ?thesis unfolding down'.simps Let_def fun_upd_def hyps[OF ‹p ∈ sparsegrid' dm›] by auto next case True hence "level p < lm" and "p ∈ grid b {d}" unfolding lgrid_def by auto let ?lb = "lv b d" and ?ib = "real_of_int (ix b d)" let ?lp = "lv p d" and ?ip = "real_of_int (ix p d)" show ?thesis proof (cases "∃ dir. p ∈ grid (child b dir d){d}") case True obtain dir where p_grid: "p ∈ grid (child b dir d) {d}" using True by auto hence "p ∈ lgrid (child b dir d) {d} lm" using ‹level p < lm› unfolding lgrid_def by auto have "lv b d < lv p d" using child_lv[OF ‹d < length b›] and grid_single_level[OF p_grid ‹d < length (child b dir d)›] by auto let ?ch = "child b dir d" let ?ich = "child b (inv dir) d" show ?thesis proof (cases dir) case right hence "p ∈ lgrid ?r {d} lm" and "p ∈ grid ?r {d}" using ‹p ∈ grid ?ch {d}› and ‹level p < lm› unfolding lgrid_def by auto { fix p' fix fl fr x assume p': "p' ∈ parents d (child b right d) p" hence "p' ∈ grid (child b right d) {d}" unfolding parents_def by simp hence "p' ∉ lgrid (child b left d) {d} lm" and "p' ≠ b" unfolding lgrid_def using grid_disjunct[OF ‹d < length b›] grid_not_child by auto from hyps[OF sparsegrid'_parents[OF ‹child b right d ∈ sparsegrid' dm› p']] this have "down' d l (child b left d) fl fr (α(b := x)) p' = α p'" by auto } thus ?thesis unfolding down'.simps Let_def hyps[OF ‹p ∈ sparsegrid' dm›] parent_sum[OF ‹p ∈ grid ?r {d}› ‹d < length b›] l2_child[OF ‹d < length b› ‹p ∈ grid ?r {d}›] using child_ix child_lv ‹d < length b› level_shift[OF ‹lv b d < lv p d›] sgn.simps ‹p ∈ lgrid b {d} lm› ‹p ∈ lgrid ?r {d} lm› by (auto simp add: algebra_simps diff_divide_distrib add_divide_distrib) next case left hence "p ∈ lgrid ?l {d} lm" and "p ∈ grid ?l {d}" using ‹p ∈ grid ?ch {d}› and ‹level p < lm› unfolding lgrid_def by auto hence "¬ p ∈ lgrid ?r {d} lm" using grid_disjunct[OF ‹d < length b›] unfolding lgrid_def by auto { fix p' assume p': "p' ∈ parents d (child b left d) p" hence "p' ∈ grid (child b left d) {d}" unfolding parents_def by simp hence "p' ≠ b" using grid_not_child[OF ‹d < length b›] by auto } thus ?thesis unfolding down'.simps Let_def hyps[OF ‹p ∈ sparsegrid' dm›] parent_sum[OF ‹p ∈ grid ?l {d}› ‹d < length b›] l2_child[OF ‹d < length b› ‹p ∈ grid ?l {d}›] sgn.simps if_P[OF ‹p ∈ lgrid b {d} lm›] if_P[OF ‹p ∈ lgrid ?l {d} lm›] if_not_P[OF ‹p ∉ lgrid ?r {d} lm›] using child_ix child_lv ‹d < length b› level_shift[OF ‹lv b d < lv p d›] by (auto simp add: algebra_simps diff_divide_distrib add_divide_distrib) qed next case False hence not_child: "!! dir. ¬ p ∈ grid (child b dir d) {d}" by auto hence "p = b" using grid_onedim_split[where ds="{}" and d=d and b=b] ‹p ∈ grid b {d}› unfolding grid_empty_ds[where b=b] by auto from not_child have lnot_child: "!! dir. ¬ p ∈ lgrid (child b dir d) {d} lm" unfolding lgrid_def by auto have result: "((fl + fr) / 4 + 1 / 3 * α b) / 2 ^ lv b d = (fl + (fr - fl) / 2) / 2 ^ (lv b d + 1) + α b * l2_φ (b ! d) (b ! d)" by (auto simp: l2_same diff_divide_distrib add_divide_distrib times_divide_eq_left[symmetric] algebra_simps) show ?thesis unfolding down'.simps Let_def fun_upd_def hyps[OF ‹p ∈ sparsegrid' dm›] if_P[OF ‹p ∈ lgrid b {d} lm›] if_not_P[OF lnot_child] if_P[OF ‹p = b›] unfolding ‹p = b› parents_single unfolding result by auto qed qed next case 0 have "p ∉ lgrid b {d} lm" proof (rule ccontr) assume "¬ p ∉ lgrid b {d} lm" hence "p ∈ grid b {d}" and "level p < lm" unfolding lgrid_def by auto moreover from grid_level[OF ‹p ∈ grid b {d}›] and ‹0 + level b = lm› have "lm ≤ level p" by auto ultimately show False by auto qed thus ?case unfolding down'.simps by auto qed lemma down: assumes "d < dm" and p: "p ∈ sparsegrid dm lm" shows "(down dm lm d α) p = (∑ p' ∈ parents d (base {d} p) p. (α p') * l2_φ (p ! d) (p' ! d))" proof - let "?F d l p" = "down' d l p 0 0" let "?S x p p'" = "if p' ∈ parents d (base {d} p) p then x * l2_φ (p ! d) (p' ! d) else 0" { fix p α assume "p ∈ sparsegrid dm lm" from le_less_trans[OF grid_level sparsegridE(2)[OF this]] have "parents d (base {d} p) p ⊆ lgrid (base {d} p) {d} lm" unfolding lgrid_def parents_def by auto hence "(∑p'∈lgrid (base {d} p) {d} lm. ?S (α p') p p') = (∑p'∈parents d (base {d} p) p. α p' * l2_φ (p ! d) (p' ! d))" using lgrid_finite by (intro sum.mono_neutral_cong_right) auto } note sum_eq = this { fix l p b α assume b: "b ∈ lgrid (start dm) ({0..<dm} - {d}) lm" and "l + level b = lm" and "p ∈ sparsegrid dm lm" hence b_spg: "b ∈ sparsegrid' dm" and p_spg: "p ∈ sparsegrid' dm" and "d < length b" and "level p < lm" using sparsegrid'_start sparsegrid_subset ‹d < dm› by auto have "?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)" proof (cases "b = base {d} p") case True have "p ∈ lgrid (base {d} p) {d} lm" using baseE(2)[OF p_spg] and ‹level p < lm› unfolding lgrid_def by auto thus ?thesis unfolding if_P[OF True] unfolding True sum_eq[OF ‹p ∈ sparsegrid dm lm›] unfolding down'_β[OF ‹d < length b› ‹l + level b = lm› b_spg p_spg, unfolded True] by auto next case False have "p ∉ lgrid b {d} lm" proof (rule ccontr) assume "¬ ?thesis" hence "p ∈ grid b {d}" by auto from b this have "b = base {d} p" using baseI by auto thus False using False by simp qed thus ?thesis unfolding if_not_P[OF False] unfolding down'_β[OF ‹d < length b› ‹l + level b = lm› b_spg p_spg] by auto qed } from lift[OF ‹d < dm› ‹p ∈ sparsegrid dm lm›, where F = ?F and S = ?S, OF this] show ?thesis unfolding down_def unfolding sum_eq[OF p] by simp qed end

# Theory Up_Down

section ‹ UpDown › (* Definition of sparse grids, hierarchical bases and the up-down algorithm. * * Based on "updown_L2-Skalarprodukt.mws" from Dirk Pflüger <pflueged@in.tum.de> * * Author: Johannes Hölzl <hoelzl@in.tum.de> *) theory Up_Down imports Up Down begin lemma updown': "⟦ d ≤ dm; p ∈ sparsegrid dm lm ⟧ ⟹ (updown' dm lm d α) p = (∑ p' ∈ lgrid (base {0 ..< d} p) {0 ..< d} lm. α p' * (∏ d' ∈ {0 ..< d}. l2_φ (p' ! d') (p ! d')))" (is "⟦ _ ; _ ⟧ ⟹ _ = (∑ p' ∈ ?subgrid d p. α p' * ?prod d p' p)") proof (induct d arbitrary: α p) case 0 hence "?subgrid 0 p = {p}" using base_empty unfolding lgrid_def and sparsegrid_def sparsegrid'_def by auto thus ?case unfolding updown'.simps by auto next case (Suc d) let "?leafs p" = "(lgrid p {d} lm) - {p}" let "?parents" = "parents d (base {d} p) p" let ?b = "base {0..<d} p" have "d < dm" using ‹Suc d ≤ dm› by auto have p_spg: "p ∈ grid (start dm) {0..<dm}" and p_spg': "p ∈ sparsegrid' dm" and "level p < lm" using ‹p ∈ sparsegrid dm lm› unfolding sparsegrid_def and sparsegrid'_def and lgrid_def by auto have p'_in_spg: "!! p'. p' ∈ ?subgrid d p ⟹ p' ∈ sparsegrid dm lm" using base_grid[OF p_spg'] unfolding sparsegrid'_def sparsegrid_def lgrid_def by auto from baseE[OF p_spg', where ds="{0..<d}"] have "?b ∈ grid (start dm) {d..<dm}" and p_bgrid: "p ∈ grid ?b {0..<d}" by auto hence "d < length ?b" using ‹Suc d ≤ dm› by auto have "p ! d = ?b ! d" using base_out[OF _ _ p_spg'] ‹Suc d ≤ dm› by auto have "length p = dm" using ‹p ∈ sparsegrid dm lm› unfolding sparsegrid_def lgrid_def by auto hence "d < length p" using ‹d < dm› by auto have "updown' dm lm d (up dm lm d α) p = (∑ p' ∈ ?subgrid d p. (up dm lm d α) p' * (?prod d p' p))" using Suc by auto also have "… = (∑ p' ∈ ?subgrid d p. (∑ p'' ∈ ?leafs p'. α p'' * ?prod (Suc d) p'' p))" proof (intro sum.cong refl) fix p' assume "p' ∈ ?subgrid d p" hence "d < length p'" unfolding lgrid_def using base_length[OF p_spg'] ‹Suc d ≤ dm› by auto have "up dm lm d α p' * ?prod d p' p = (∑ p'' ∈ ?leafs p'. α p'' * l2_φ (p'' ! d) (p' ! d)) * ?prod d p' p" using ‹p' ∈ ?subgrid d p› up ‹Suc d ≤ dm› p'_in_spg by auto also have "… = (∑ p'' ∈ ?leafs p'. α p'' * l2_φ (p'' ! d) (p' ! d) * ?prod d p' p)" using sum_distrib_right by auto also have "… = (∑ p'' ∈ ?leafs p'. α p'' * ?prod (Suc d) p'' p)" proof (intro sum.cong refl) fix p'' assume "p'' ∈ ?leafs p'" have "?prod d p' p = ?prod d p'' p" proof (intro prod.cong refl) fix d' assume "d' ∈ {0..<d}" hence d_lt_p: "d' < length p'" and d'_not_d: "d' ∉ {d}" using ‹d < length p'› by auto hence "p' ! d' = p'' ! d'" using ‹p'' ∈ ?leafs p'› grid_invariant[OF d_lt_p d'_not_d] unfolding lgrid_def by auto thus "l2_φ (p'!d') (p!d') = l2_φ (p''!d') (p!d')" by auto qed moreover have "p' ! d = p ! d" using ‹p' ∈ ?subgrid d p› and grid_invariant[OF ‹d < length ?b›, where p=p' and ds="{0..<d}"] unfolding lgrid_def ‹p ! d = ?b ! d› by auto ultimately have "l2_φ (p'' ! d) (p' ! d) * ?prod d p' p = l2_φ (p'' ! d) (p ! d) * ?prod d p'' p" by auto also have "… = ?prod (Suc d) p'' p" proof - have "insert d {0..<d} = {0..<Suc d}" by auto moreover from prod.insert have "prod (λ d'. l2_φ (p'' ! d') (p ! d')) (insert d {0..<d}) = (λ d'. l2_φ (p'' ! d') (p ! d')) d * prod (λ d'. l2_φ (p'' ! d') (p ! d')) {0..<d}" by auto ultimately show ?thesis by auto qed finally show "α p'' * l2_φ (p'' ! d) (p' ! d) * ?prod d p' p = α p'' * ?prod (Suc d) p'' p" by auto qed finally show "(up dm lm d α) p' * (?prod d p' p) = (∑ p'' ∈ ?leafs p'. α p'' * ?prod (Suc d) p'' p)" by auto qed also have "… = (∑ (p', p'') ∈ Sigma (?subgrid d p) (λp'. (?leafs p')). (α p'') * (?prod (Suc d) p'' p))" by (rule sum.Sigma, auto simp add: lgrid_finite) also have "… = (∑ p''' ∈ (⋃ p' ∈ ?subgrid d p. (⋃ p'' ∈ ?leafs p'. { (p', p'') })). (((λ p''. α p'' * ?prod (Suc d) p'' p) o snd) p''') )" unfolding Sigma_def by (rule sum.cong[OF refl], auto) also have "… = (∑ p'' ∈ snd ` (⋃ p' ∈ ?subgrid d p. (⋃ p'' ∈ ?leafs p'. { (p', p'') })). α p'' * (?prod (Suc d) p'' p))" unfolding lgrid_def by (rule sum.reindex[symmetric], rule subset_inj_on[OF grid_grid_inj_on[OF ivl_disj_int(15)[where l=0 and m="d" and u="d"], where b="?b"]]) auto also have "… = (∑ p'' ∈ (⋃ p' ∈ ?subgrid d p. (⋃ p'' ∈ ?leafs p'. snd ` { (p', p'') })). α p'' * (?prod (Suc d) p'' p))" by (auto simp only: image_UN) also have "… = (∑ p'' ∈ (⋃ p' ∈ ?subgrid d p. ?leafs p'). α p'' * (?prod (Suc d) p'' p))" by auto finally have up_part: "updown' dm lm d (up dm lm d α) p = (∑ p'' ∈ (⋃ p' ∈ ?subgrid d p. ?leafs p'). α p'' * (?prod (Suc d) p'' p))" . have "down dm lm d (updown' dm lm d α) p = (∑ p' ∈ ?parents. (updown' dm lm d α p') * l2_φ (p ! d) (p' ! d))" using ‹Suc d ≤ dm› and down and ‹p ∈ sparsegrid dm lm› by auto also have "… = (∑ p' ∈ ?parents. ∑ p'' ∈ ?subgrid d p'. α p'' * ?prod (Suc d) p'' p)" proof (rule sum.cong[OF refl]) fix p' let ?b' = "base {d} p" assume "p' ∈ ?parents" hence p_lgrid: "p' ∈ lgrid ?b' {d} (level p + 1)" using parents_subset_lgrid by auto hence "p' ∈ sparsegrid dm lm" and p'_spg': "p' ∈ sparsegrid' dm" using ‹level p < lm› base_grid[OF p_spg'] unfolding sparsegrid_def lgrid_def sparsegrid'_def by auto hence "length p' = dm" unfolding sparsegrid_def lgrid_def by auto hence "d < length p'" using ‹Suc d ≤ dm› by auto from p_lgrid have p'_grid: "p' ∈ grid ?b' {d}" unfolding lgrid_def by auto have "(updown' dm lm d α p') * l2_φ (p ! d) (p' ! d) = (∑ p'' ∈ ?subgrid d p'. α p'' * ?prod d p'' p') * l2_φ (p ! d) (p' ! d)" using ‹p' ∈ sparsegrid dm lm› Suc by auto also have "… = (∑ p'' ∈ ?subgrid d p'. α p'' * ?prod d p'' p' * l2_φ (p ! d) (p' ! d))" using sum_distrib_right by auto also have "… = (∑ p'' ∈ ?subgrid d p'. α p'' * ?prod (Suc d) p'' p)" proof (rule sum.cong[OF refl]) fix p'' assume "p'' ∈ ?subgrid d p'" have "?prod d p'' p' = ?prod d p'' p" proof (rule prod.cong, rule refl) fix d' assume "d' ∈ {0..<d}" hence "d' < dm" and "d' ∉ {d}" using ‹Suc d ≤ dm› by auto from grid_base_out[OF this p_spg' p'_grid] show "l2_φ (p''!d') (p'!d') = l2_φ (p''!d') (p!d')" by auto qed moreover have "l2_φ (p ! d) (p' ! d) = l2_φ (p'' ! d) (p ! d)" proof - have "d < dm" and "d ∉ {0..<d}" using ‹Suc d ≤ dm› base_length p'_spg' by auto from grid_base_out[OF this p'_spg'] ‹p'' ∈ ?subgrid d p'›[unfolded lgrid_def] show ?thesis using l2_commutative by auto qed moreover have "?prod d p'' p * l2_φ (p'' ! d) (p ! d) = ?prod (Suc d) p'' p" proof - have "insert d {0..<d} = {0..<Suc d}" by auto moreover from prod.insert have "(λ d'. l2_φ (p'' ! d') (p ! d')) d * prod (λ d'. l2_φ (p'' ! d') (p ! d')) {0..<d} = prod (λ d'. l2_φ (p'' ! d') (p ! d')) (insert d {0..<d})" by auto hence "(prod (λ d'. l2_φ (p'' ! d') (p ! d')) {0..<d}) * (λ d'. l2_φ (p'' ! d') (p ! d')) d = prod (λ d'. l2_φ (p'' ! d') (p ! d')) (insert d {0..<d})" by auto ultimately show ?thesis by auto qed ultimately show "α p'' * ?prod d p'' p' * l2_φ (p ! d) (p' ! d) = α p'' * ?prod (Suc d) p'' p" by auto qed finally show "(updown' dm lm d α p') * l2_φ (p ! d) (p' ! d) = (∑ p'' ∈ ?subgrid d p'. α p'' * ?prod (Suc d) p'' p)" by auto qed also have "… = (∑ (p', p'') ∈ (Sigma ?parents (?subgrid d)). α p'' * ?prod (Suc d) p'' p)" by (rule sum.Sigma, auto simp add: parents_finite lgrid_finite) also have "… = (∑ p''' ∈ (⋃ p' ∈ ?parents. (⋃ p'' ∈ ?subgrid d p'. { (p', p'') })). ( ((λ p''. α p'' * ?prod (Suc d) p'' p) o snd) p''') )" unfolding Sigma_def by (rule sum.cong[OF refl], auto) also have "… = (∑ p'' ∈ snd ` (⋃ p' ∈ ?parents. (⋃ p'' ∈ ?subgrid d p'. { (p', p'') })). α p'' * (?prod (Suc d) p'' p))" proof (rule sum.reindex[symmetric], rule inj_onI) fix x y assume "x ∈ (⋃p'∈parents d (base {d} p) p. ⋃p''∈lgrid (base {0..<d} p') {0..<d} lm. {(p', p'')})" hence x_snd: "snd x ∈ grid (base {0..<d} (fst x)) {0..<d}" and "fst x ∈ grid (base {d} p) {d}" and "p ∈ grid (fst x) {d}" unfolding parents_def lgrid_def by auto hence x_spg: "fst x ∈ sparsegrid' dm" using base_grid[OF p_spg'] by auto assume "y ∈ (⋃p'∈parents d (base {d} p) p. ⋃p''∈lgrid (base {0..<d} p') {0..<d} lm. {(p', p'')})" hence y_snd: "snd y ∈ grid (base {0..<d} (fst y)) {0..<d}" and "fst y ∈ grid (base {d} p) {d}" and "p ∈ grid (fst y) {d}" unfolding parents_def lgrid_def by auto hence y_spg: "fst y ∈ sparsegrid' dm" using base_grid[OF p_spg'] by auto hence "length (fst y) = dm" unfolding sparsegrid'_def by auto assume "snd x = snd y" have "fst x = fst y" proof (rule nth_equalityI) show l_eq: "length (fst x) = length (fst y)" using grid_length[OF ‹p ∈ grid (fst y) {d}›] grid_length[OF ‹p ∈ grid (fst x) {d}›] by auto show "fst x ! i = fst y ! i" if "i < length (fst x)" for i proof - have "i < length (fst y)" and "i < dm" using that l_eq and ‹length (fst y) = dm› by auto show "fst x ! i = fst y ! i" proof (cases "i = d") case False hence "i ∉ {d}" by auto with grid_invariant[OF ‹i < length (fst x)› this ‹p ∈ grid (fst x) {d}›] grid_invariant[OF ‹i < length (fst y)› this ‹p ∈ grid (fst y) {d}›] show ?thesis by auto next case True with grid_base_out[OF ‹i < dm› _ y_spg y_snd] grid_base_out[OF ‹i < dm› _ x_spg x_snd] show ?thesis using ‹snd x = snd y› by auto qed qed qed show "x = y" using prod_eqI[OF ‹fst x = fst y› ‹snd x = snd y›] . qed also have "… = (∑ p'' ∈ (⋃ p' ∈ ?parents. (⋃ p'' ∈ ?subgrid d p'. snd ` { (p', p'') })). α p'' * (?prod (Suc d) p'' p))" by (auto simp only: image_UN) also have "… = (∑ p'' ∈ (⋃ p' ∈ ?parents. ?subgrid d p'). α p'' * ?prod (Suc d) p'' p)" by auto finally have down_part: "down dm lm d (updown' dm lm d α) p = (∑ p'' ∈ (⋃ p' ∈ ?parents. ?subgrid d p'). α p'' * ?prod (Suc d) p'' p)" . have "updown' dm lm (Suc d) α p = (∑ p'' ∈ (⋃ p' ∈ ?subgrid d p. ?leafs p'). α p'' * ?prod (Suc d) p'' p) + (∑ p'' ∈ (⋃ p' ∈ ?parents. ?subgrid d p'). α p'' * ?prod (Suc d) p'' p)" unfolding sum_vector_def updown'.simps down_part and up_part .. also have "… = (∑ p'' ∈ (⋃ p' ∈ ?subgrid d p. ?leafs p') ∪ (⋃ p' ∈ ?parents. ?subgrid d p'). α p'' * ?prod (Suc d) p'' p)" proof (rule sum.union_disjoint[symmetric], simp add: lgrid_finite, simp add: lgrid_finite parents_finite, rule iffD2[OF disjoint_iff_not_equal], rule ballI, rule ballI) fix x y assume "x ∈ (⋃ p' ∈ ?subgrid d p. ?leafs p')" then obtain px where "px ∈ grid (base {0..<d} p) {0..<d}" and "x ∈ grid px {d}" and "x ≠ px" unfolding lgrid_def by auto with grid_base_out[OF _ _ p_spg' this(1)] ‹Suc d ≤ dm› base_length[OF p_spg'] grid_level_d have "lv px d < lv x d" and "px ! d = p ! d" by auto hence "lv p d < lv x d" unfolding lv_def by auto moreover assume "y ∈ (⋃ p' ∈ ?parents. ?subgrid d p')" then obtain py where y_grid: "y ∈ grid (base {0..<d} py) {0..<d}" and "py ∈ ?parents" unfolding lgrid_def by auto hence "py ∈ grid (base {d} p) {d}" and "p ∈ grid py {d}" unfolding parents_def by auto hence py_spg: "py ∈ sparsegrid' dm" using base_grid[OF p_spg'] by auto have "y ! d = py ! d" using grid_base_out[OF _ _ py_spg y_grid] ‹Suc d ≤ dm› by auto hence "lv y d ≤ lv p d" using grid_single_level[OF ‹p ∈ grid py {d}›] ‹Suc d ≤ dm› and sparsegrid'_length[OF py_spg] unfolding lv_def by auto ultimately show "x ≠ y" by auto qed also have "… = (∑ p' ∈ ?subgrid (Suc d) p. α p' * ?prod (Suc d) p' p)" (is "(∑ x ∈ ?in. ?F x) = (∑ x ∈ ?out. ?F x)") proof (rule sum.mono_neutral_left, simp add: lgrid_finite) show "?in ⊆ ?out" (is "?children ∪ ?siblings ⊆ _") proof (rule subsetI) fix x assume "x ∈ ?in" show "x ∈ ?out" proof (cases "x ∈ ?children") case False hence "x ∈ ?siblings" using ‹x ∈ ?in› by auto then obtain px where "px ∈ parents d (base {d} p) p" and "x ∈ lgrid (base {0..<d} px) {0..<d} lm" by auto hence "level x < lm" and "px ∈ grid (base {d} p) {d}" and "x ∈ grid (base {0..<d} px) {0..<d}" and "{d} ∪ {0..<d} = {0..<Suc d}" unfolding lgrid_def parents_def by auto with grid_base_union[OF p_spg' this(2) this(3)] show ?thesis unfolding lgrid_def by auto next have d_eq: "{0..<Suc d} ∪ {d} = {0..<Suc d}" by auto case True then obtain px where "px ∈ ?subgrid d p" and "x ∈ lgrid px {d} lm" and "x ≠ px" by auto hence "px ∈ grid (base {0..<d} p) {0..<d}" and "x ∈ grid px {d}" and "level x < lm" and "{d} ∪ {0..<d} = {0..<Suc d}" unfolding lgrid_def by auto from grid_base_dim_add[OF _ p_spg' this(1)] have "px ∈ grid (base {0..<Suc d} p) {0..<Suc d}" by auto from grid_transitive[OF ‹x ∈ grid px {d}› this] show ?thesis unfolding lgrid_def using ‹level x < lm› d_eq by auto qed qed show "∀ x ∈ ?out - ?in. ?F x = 0" proof fix x assume "x ∈ ?out - ?in" hence "x ∈ ?out" and up_ps': "!! p'. p' ∈ ?subgrid d p ⟹ x ∉ lgrid p' {d} lm - {p'}" and down_ps': "!! p'. p' ∈ ?parents ⟹ x ∉ ?subgrid d p'" by auto hence x_eq: "x ∈ grid (base {0..<Suc d} p) {0..<Suc d}" and "level x < lm" unfolding lgrid_def by auto hence up_ps: "!! p'. p' ∈ ?subgrid d p ⟹ x ∉ grid p' {d} - {p'}" and down_ps: "!! p'. p' ∈ ?parents ⟹ x ∉ grid (base {0..<d} p') {0..<d}" using up_ps' down_ps' unfolding lgrid_def by auto have ds_eq: "{0..<Suc d} = {0..<d} ∪ {d}" by auto have "x ∉ grid (base {0..<d} p) {0..<Suc d} - grid (base {0..<d} p) {0..<d}" proof assume "x ∈ grid (base {0..<d} p) {0..<Suc d} - grid (base {0..<d} p) {0..<d}" hence "x ∈ grid (base {0..<d} p) ({d} ∪ {0..<d})" and x_ngrid: "x ∉ grid (base {0..<d} p) {0..<d}" using ds_eq by auto from grid_split[OF this(1)] obtain px where px_grid: "px ∈ grid (base {0..<d} p) {0..<d}" and "x ∈ grid px {d}" by auto from grid_level[OF this(2)] ‹level x < lm› have "level px < lm" by auto hence "px ∈ ?subgrid d p" using px_grid unfolding lgrid_def by auto hence "x ∉ grid px {d} - {px}" using up_ps by auto moreover have "x ≠ px" proof (rule ccontr) assume "¬ x ≠ px" with px_grid and x_ngrid show False by auto qed ultimately show False using ‹x ∈ grid px {d}› by auto qed moreover have "p ∈ ?parents" unfolding parents_def using baseE(2)[OF p_spg'] by auto hence "x ∉ grid (base {0..<d} p) {0..<d}" by (rule down_ps) ultimately have x_ngrid: "x ∉ grid (base {0..<d} p) {0..<Suc d}" by auto have x_spg: "x ∈ sparsegrid' dm" using base_grid[OF p_spg'] x_eq by auto hence "length x = dm" using grid_length by auto let ?bx = "base {0..<d} x" and ?bp = "base {0..<d} p" and ?bx1 = "base {d} x" and ?bp1 = "base {d} p" and ?px = "p[d := x ! d]" have x_nochild_p: "?bx ∉ grid ?bp {d}" proof (rule ccontr) assume "¬ base {0..<d} x ∉ grid (base {0..<d} p) {d}" hence "base {0..<d} x ∈ grid (base {0..<d} p) {d}" by auto from grid_transitive[OF baseE(2)[OF x_spg] this] have "x ∈ grid (base {0..<d} p) {0..<Suc d}" using ds_eq by auto thus False using x_ngrid by auto qed have "d < length ?bx" and "d < length ?bp" and "d < length ?bx1" and "d < length ?bp1"