Theory VEBT_MinMax
theory VEBT_MinMax imports VEBT_Member
begin
section ‹The Minimum and Maximum Operation›
fun vebt_mint :: "VEBT ⇒ nat option" where
"vebt_mint (Leaf a b) = (if a then Some 0 else if b then Some 1 else None)"|
"vebt_mint (Node None _ _ _) = None"|
"vebt_mint (Node (Some (mi,ma)) _ _ _ ) = Some mi"
fun vebt_maxt :: "VEBT ⇒ nat option" where
"vebt_maxt (Leaf a b) = (if b then Some 1 else if a then Some 0 else None)"|
"vebt_maxt (Node None _ _ _) = None"|
"vebt_maxt (Node (Some (mi,ma)) _ _ _ ) = Some ma"
context VEBT_internal begin
fun option_shift::"('a⇒'a⇒'a) ⇒'a option ⇒'a option⇒ 'a option" where
"option_shift _ None _ = None"|
"option_shift _ _ None = None"|
"option_shift f (Some a) (Some b) = Some (f a b)"
definition power::"nat option ⇒ nat option ⇒ nat option" (infixl"^⇩o" 81) where
"power= option_shift (^)"
definition add::"nat option ⇒ nat option ⇒ nat option" (infixl"+⇩o" 79) where
"add= option_shift (+)"
definition mul::"nat option ⇒ nat option ⇒ nat option" (infixl"*⇩o" 80) where
"mul = option_shift (*)"
fun option_comp_shift::"('a ⇒ 'a ⇒ bool) ⇒ 'a option ⇒ 'a option ⇒ bool" where
"option_comp_shift _ None _ = False"|
"option_comp_shift _ _ None = False"|
"option_comp_shift f (Some x) (Some y) = f x y"
fun less::"nat option ⇒ nat option ⇒ bool" (infixl"<⇩o" 80) where
"less x y= option_comp_shift (<) x y"
fun lesseq::"nat option ⇒ nat option ⇒ bool" (infixl"≤⇩o" 80) where
"lesseq x y = option_comp_shift (≤) x y"
fun greater::"nat option ⇒ nat option ⇒ bool" (infixl">⇩o" 80) where
"greater x y = option_comp_shift (>) x y"
lemma add_shift:"x+y = z ⟷ Some x +⇩o Some y = Some z"
by (simp add: add_def)
lemma mul_shift:"x*y = z ⟷ Some x *⇩o Some y = Some z" by (simp add: mul_def)
lemma power_shift:"x^y = z ⟷ Some x ^⇩o Some y = Some z" by (simp add: power_def)
lemma less_shift: "x < y ⟷ Some x <⇩o Some y" by simp
lemma lesseq_shift: "x ≤ y ⟷ Some x ≤⇩o Some y" by simp
lemma greater_shift: "x > y ⟷ Some x >⇩o Some y" by simp
definition max_in_set :: "nat set ⇒ nat ⇒ bool" where
"max_in_set xs x ⟷ (x ∈ xs ∧ (∀ y ∈ xs. y ≤ x))"
lemma maxt_member: "invar_vebt t n ⟹ vebt_maxt t = Some maxi ⟹ vebt_member t maxi"
proof(induction t n arbitrary: maxi rule: invar_vebt.induct)
case (1 a b)
then show ?case
by (metis VEBT_Member.vebt_member.simps(1) vebt_maxt.simps(1) option.distinct(1) option.inject zero_neq_one)
next
case (2 treeList n summary m deg)
then show ?case
by simp
next
case (3 treeList n summary m deg)
then show ?case
by simp
next
case (4 treeList n summary m deg mi ma)
hence "deg ≥ 2"
by (metis One_nat_def Suc_le_eq add_mono deg_not_0 numeral_2_eq_2 plus_1_eq_Suc)
then show ?case
by (metis "4.prems" VEBT_Member.vebt_member.simps(5) Suc_diff_Suc Suc_pred lessI less_le_trans vebt_maxt.simps(3) numeral_2_eq_2 option.inject zero_less_Suc)
next
case (5 treeList n summary m deg mi ma)
hence "deg ≥ 2"
by (metis Suc_leI le_add2 less_add_same_cancel2 less_le_trans not_less_iff_gr_or_eq not_one_le_zero numeral_2_eq_2 plus_1_eq_Suc set_n_deg_not_0)
then show ?case
by (metis "5.prems" VEBT_Member.vebt_member.simps(5) add_2_eq_Suc le_add_diff_inverse vebt_maxt.simps(3) option.inject)
qed
lemma maxt_corr_help: "invar_vebt t n ⟹ vebt_maxt t = Some maxi ⟹ vebt_member t x ⟹ maxi ≥ x "
by (smt VEBT_Member.vebt_member.simps(1) le_less vebt_maxt.elims member_inv mi_ma_2_deg option.simps(1) option.simps(3) zero_le_one)
lemma maxt_corr_help_empty: "invar_vebt t n ⟹ vebt_maxt t = None ⟹ set_vebt' t = {}"
by (metis (full_types) VEBT_Member.vebt_member.simps(1) empty_Collect_eq vebt_maxt.elims minNull.simps(4) min_Null_member option.distinct(1) set_vebt'_def)
theorem maxt_corr:assumes "invar_vebt t n" and "vebt_maxt t = Some x" shows "max_in_set (set_vebt' t) x"
unfolding set_vebt'_def Max_def max_in_set_def
using assms(1) assms(2) maxt_corr_help maxt_member by blast
theorem maxt_sound:assumes "invar_vebt t n" and "max_in_set (set_vebt' t) x" shows "vebt_maxt t = Some x"
by (metis (no_types, opaque_lifting) assms(1) assms(2) empty_Collect_eq le_less max_in_set_def
maxt_corr_help maxt_corr_help_empty maxt_member mem_Collect_eq not_le option.exhaust set_vebt'_def)
definition min_in_set :: "nat set ⇒ nat ⇒ bool" where
"min_in_set xs x ⟷ (x ∈ xs ∧ (∀ y ∈ xs. y ≥ x))"
lemma mint_member: "invar_vebt t n ⟹ vebt_mint t = Some maxi ⟹ vebt_member t maxi"
proof(induction t n arbitrary: maxi rule: invar_vebt.induct)
case (1 a b)
then show ?case
by (metis VEBT_Member.vebt_member.simps(1) vebt_mint.simps(1) option.distinct(1) option.inject zero_neq_one)
next
case (2 treeList n summary m deg)
then show ?case
by simp
next
case (3 treeList n summary m deg)
then show ?case
by simp
next
case (4 treeList n summary m deg mi ma)
hence "deg ≥ 2"
by (metis One_nat_def Suc_le_eq add_mono deg_not_0 numeral_2_eq_2 plus_1_eq_Suc)
then show ?case
by (metis "4.prems" VEBT_Member.vebt_member.simps(5) One_nat_def Suc_diff_Suc Suc_pred dual_order.strict_trans1 le_imp_less_Suc le_numeral_extra(4) vebt_mint.simps(3) numeral_2_eq_2 option.inject zero_le_one)
next
case (5 treeList n summary m deg mi ma)
hence "deg ≥ 2"
by (metis Suc_leI le_add2 less_add_same_cancel2 less_le_trans not_less_iff_gr_or_eq not_one_le_zero numeral_2_eq_2 plus_1_eq_Suc set_n_deg_not_0)
then show ?case using "5.prems" VEBT_Member.vebt_member.simps(5) add_2_eq_Suc le_add_diff_inverse vebt_mint.simps(3)
by (metis option.inject)
qed
lemma mint_corr_help: "invar_vebt t n ⟹ vebt_mint t = Some mini ⟹ vebt_member t x ⟹ mini ≤ x "
by (smt VEBT_Member.vebt_member.simps(1) eq_iff option.inject less_imp_le_nat member_inv mi_ma_2_deg vebt_mint.elims of_nat_0 of_nat_0_le_iff of_nat_le_iff option.simps(3))
lemma mint_corr_help_empty: "invar_vebt t n ⟹ vebt_mint t = None ⟹ set_vebt' t = {}"
by (metis VEBT_internal.maxt_corr_help_empty option.distinct(1) vebt_maxt.simps(1) vebt_maxt.simps(2) vebt_mint.elims)
theorem mint_corr:assumes "invar_vebt t n" and "vebt_mint t = Some x" shows "min_in_set (set_vebt' t) x"
using assms(1) assms(2) min_in_set_def mint_corr_help mint_member set_vebt'_def by auto
theorem mint_sound:assumes "invar_vebt t n" and "min_in_set (set_vebt' t) x" shows "vebt_mint t = Some x"
by (metis assms(1) assms(2) empty_Collect_eq eq_iff mem_Collect_eq min_in_set_def
mint_corr_help mint_corr_help_empty mint_member option.exhaust set_vebt'_def)
lemma summaxma:assumes "invar_vebt (Node (Some (mi, ma)) deg treeList summary) deg" and "mi ≠ ma"
shows "the (vebt_maxt summary) = high ma (deg div 2)"
proof-
from assms(1) show ?thesis
proof(cases)
case (4 n m)
have "both_member_options summary (high ma n)"
using "4"(10) "4"(2) "4"(4) "4"(5) "4"(6) "4"(9) assms(2) deg_not_0 exp_split_high_low(1) by blast
have "high ma n ≤ the (vebt_maxt summary)" using "4"(2) ‹both_member_options summary
(high ma n)› empty_Collect_eq option.inject maxt_corr_help maxt_corr_help_empty
not_None_eq set_vebt'_def valid_member_both_member_options
by (metis option.exhaust_sel)
have "high ma n < the (vebt_maxt summary) ⟹ False"
proof-
assume "high ma n < the (vebt_maxt summary)"
obtain maxs where "Some maxs = vebt_maxt summary"
by (metis "4"(2) ‹both_member_options summary (high ma n)› empty_Collect_eq maxt_corr_help_empty
not_None_eq set_vebt'_def valid_member_both_member_options)
hence "∃ x. both_member_options (treeList ! maxs) x"
by (metis "4"(2) "4"(6) both_member_options_equiv_member maxt_member member_bound)
then obtain x where "both_member_options (treeList ! maxs) x"
by auto
hence "vebt_member (treeList ! maxs) x"
by (metis "4"(1) "4"(2) "4"(3) ‹Some maxs = vebt_maxt summary› maxt_member member_bound nth_mem valid_member_both_member_options)
have "maxs < 2^m"
by (metis "4"(2) ‹Some maxs = vebt_maxt summary› maxt_member member_bound)
have "invar_vebt (treeList ! maxs) n"
by (metis "4"(1) "4"(3) ‹maxs < 2 ^ m› inthall member_def)
hence "x < 2^n"
using ‹vebt_member (treeList ! maxs) x› member_bound by auto
let ?X = "2^n*maxs + x"
have "high ?X n = maxs"
by (simp add: ‹x < 2 ^ n› high_inv mult.commute)
hence "both_member_options (Node (Some (mi, ma)) deg treeList summary) (2^n*maxs + x)"
by (metis "4"(3) "4"(4) "4"(5) One_nat_def Suc_leI ‹both_member_options (treeList ! maxs) x› ‹maxs < 2 ^ m› ‹x < 2 ^ n› add_self_div_2 assms(1) both_member_options_from_chilf_to_complete_tree deg_not_0 low_inv mult.commute)
hence "vebt_member (Node (Some (mi, ma)) deg treeList summary) ?X"
using assms(1) both_member_options_equiv_member by auto
have "high ?X n> high ma n"
by (metis ‹Some maxs = vebt_maxt summary› ‹high (2 ^ n * maxs + x) n = maxs› ‹high ma n < the (vebt_maxt summary)› option.exhaust_sel option.inject option.simps(3))
hence "?X > ma"
by (metis div_le_mono high_def not_le)
then show ?thesis
by (metis "4"(8) ‹vebt_member (Node (Some (mi, ma)) deg treeList summary) (2 ^ n * maxs + x)› leD member_inv not_less_iff_gr_or_eq)
qed
then show ?thesis
using "4"(4) "4"(5) ‹high ma n ≤ the (vebt_maxt summary)› by fastforce
next
case (5 n m)
have "both_member_options summary (high ma n)"
by (metis "5"(10) "5"(5) "5"(6) "5"(9) div_eq_0_iff assms(2) div_exp_eq high_def nat.simps(3) numerals(2) power_not_zero)
have "high ma n ≤ the (vebt_maxt summary)"
by (metis "5"(2) VEBT_Member.vebt_member.simps(2) ‹both_member_options summary (high ma n)› vebt_maxt.elims maxt_corr_help minNull.simps(1) min_Null_member option.exhaust_sel option.simps(3) valid_member_both_member_options)
have "high ma n < the (vebt_maxt summary) ⟹ False"
proof-
assume "high ma n < the (vebt_maxt summary)"
obtain maxs where "Some maxs = vebt_maxt summary"
by (metis "5"(2) ‹both_member_options summary (high ma n)› empty_Collect_eq maxt_corr_help_empty
not_None_eq set_vebt'_def valid_member_both_member_options)
hence "∃ x. both_member_options (treeList ! maxs) x"
by (metis "5"(2) "5"(6) both_member_options_equiv_member maxt_member member_bound)
then obtain x where "both_member_options (treeList ! maxs) x"
by auto
hence "vebt_member (treeList ! maxs) x"
by (metis "5"(1) "5"(2) "5"(3) ‹Some maxs = vebt_maxt summary› both_member_options_equiv_member maxt_member member_bound nth_mem)
have "maxs < 2^m"
by (metis "5"(2) ‹Some maxs = vebt_maxt summary› maxt_member member_bound)
have "invar_vebt (treeList ! maxs) n"
by (metis "5"(1) "5"(3) ‹maxs < 2 ^ m› inthall member_def)
hence "x < 2^n"
using ‹vebt_member (treeList ! maxs) x› member_bound by auto
let ?X = "2^n*maxs + x"
have "high ?X n = maxs"
by (simp add: ‹x < 2 ^ n› high_inv mult.commute)
hence "both_member_options (Node (Some (mi, ma)) deg treeList summary) (2^n*maxs + x)"
by (smt (z3) "5"(3) "5"(4) "5"(5) ‹both_member_options (treeList ! maxs) x› ‹maxs < 2 ^ m› ‹x < 2 ^ n› add_Suc_right add_self_div_2 both_member_options_from_chilf_to_complete_tree even_Suc_div_two le_add1 low_inv mult.commute odd_add plus_1_eq_Suc)
hence "vebt_member (Node (Some (mi, ma)) deg treeList summary) ?X"
using assms(1) both_member_options_equiv_member by auto
have "high ?X n> high ma n"
by (metis ‹Some maxs = vebt_maxt summary› ‹high (2 ^ n * maxs + x) n = maxs› ‹high ma n < the (vebt_maxt summary)› option.sel)
hence "?X > ma"
by (metis div_le_mono high_def not_le)
then show ?thesis
by (metis "5"(8) ‹vebt_member (Node (Some (mi, ma)) deg treeList summary) (2 ^ n * maxs + x)› leD member_inv not_less_iff_gr_or_eq)
qed
then show ?thesis
using "5"(4) "5"(5) ‹high ma n ≤ the(vebt_maxt summary)› by fastforce
qed
qed
lemma maxbmo: "vebt_maxt t = Some x ⟹ both_member_options t x"
apply(induction t rule: vebt_maxt.induct)
apply auto
apply (metis both_member_options_def naive_member.simps(1) option.distinct(1) option.sel zero_neq_one)
by (metis One_nat_def Suc_le_D both_member_options_def div_by_1 div_greater_zero_iff membermima.simps(3) membermima.simps(4) not_gr0)
lemma misiz:"invar_vebt t n ⟹ Some m = vebt_mint t ⟹ m < 2^n"
by (metis member_bound mint_member)
lemma mintlistlength: assumes "invar_vebt (Node (Some (mi, ma)) deg treeList summary) n " "
mi ≠ ma " shows " ma > mi ∧ (∃ m. Some m = vebt_mint summary ∧ m < 2^(n - n div 2))"
using assms(1)
proof cases
case (4 n m)
hence "both_member_options (treeList ! high ma n) (low ma n)"
by (metis assms(2) high_bound_aux)
moreover hence "both_member_options summary (high ma n)"
using "4"(10) "4"(6) "4"(7) high_bound_aux by blast
moreover then obtain mini where "Some mini = vebt_mint summary"
by (metis "4"(3) empty_Collect_eq mint_corr_help_empty option.exhaust_sel set_vebt'_def valid_member_both_member_options)
moreover hence "mini < 2^m"
by (metis "4"(3) mint_member member_bound)
moreover have "m = (deg - deg div 2)" using 4(6) 4(5)
by auto
ultimately show ?thesis using 4(1) assms 4(9) by auto
next
case (5 n m)
hence "both_member_options (treeList ! high ma n) (low ma n)"
by (metis assms(2) high_bound_aux)
moreover hence "both_member_options summary (high ma n)"
using "5"(10) "5"(6) "5"(7) high_bound_aux by blast
moreover then obtain mini where "Some mini = vebt_mint summary"
by (metis "5"(3) empty_Collect_eq mint_corr_help_empty option.exhaust_sel set_vebt'_def valid_member_both_member_options)
moreover hence "mini < 2^m"
by (metis "5"(3) mint_member member_bound)
moreover have "m = (deg - deg div 2)" using 5(6) 5(5)
by auto
ultimately show ?thesis using 5(1) assms 5(9) by auto
qed
lemma power_minus_is_div:
"b ≤ a ⟹ (2 :: nat) ^ (a - b) = 2 ^ a div 2 ^ b"
apply (induct a arbitrary: b)
apply simp
apply (erule le_SucE)
apply (clarsimp simp:Suc_diff_le le_iff_add power_add)
apply simp
done
lemma nested_mint:assumes "invar_vebt (Node (Some (mi, ma)) deg treeList summary) n " "n = Suc (Suc va) ""
¬ ma < mi "" ma ≠ mi " shows "
high (the (vebt_mint summary) * (2 * 2 ^ (va div 2)) + the (vebt_mint (treeList ! the (vebt_mint summary)))) (Suc (va div 2))
< length treeList"
proof-
have setprop: "t ∈ set treeList ⟹ invar_vebt t (n div 2 )" for t using assms(1)
by (cases) simp+
have listlength: "length treeList = 2^(n - n div 2)" using assms(1)
by (cases) simp+
have sumprop: "invar_vebt summary (n - n div 2)" using assms(1)
by (cases) simp+
have mimaxprop: "mi ≤ ma ∧ ma ≤ 2^n" using assms(1)
by cases simp+
hence xbound: "mi ≤ x ⟹ x ≤ ma ⟹ high x (n div 2) ≤ length treeList " for x
using div_le_dividend div_le_mono high_def listlength power_minus_is_div by auto
have contcong:"i < length treeList ⟹ ∃ x. both_member_options (treeList ! i) x ⟷ both_member_options summary i " for i
using assms(1)by cases auto+
obtain m where " Some m = vebt_mint summary ∧ m < 2^(n - n div 2)"
using assms(1) assms(4) mintlistlength by blast
then obtain miny where "(vebt_mint (treeList ! the (vebt_mint summary))) =Some miny"
by (metis both_member_options_equiv_member contcong empty_Collect_eq listlength mint_corr_help_empty mint_member nth_mem option.exhaust_sel option.sel setprop sumprop set_vebt'_def)
hence "miny < 2^(n div 2)"
by (metis ‹⋀thesis. (⋀m. Some m = vebt_mint summary ∧ m < 2 ^ (n - n div 2) ⟹ thesis) ⟹ thesis› listlength misiz nth_mem option.sel setprop)
then show ?thesis
by (metis ‹⋀thesis. (⋀m. Some m = vebt_mint summary ∧ m < 2 ^ (n - n div 2) ⟹ thesis) ⟹ thesis› ‹vebt_mint (treeList ! the (vebt_mint summary)) = Some miny› assms(2) div2_Suc_Suc high_inv listlength option.sel power_Suc)
qed
lemma minminNull: "vebt_mint t = None ⟹ minNull t"
by (metis minNull.simps(1) minNull.simps(4) vebt_mint.elims option.distinct(1))
lemma minNullmin: "minNull t ⟹ vebt_mint t = None"
by (metis minNull.elims(2) vebt_mint.simps(1) vebt_mint.simps(2))
end
end