# Theory Sorting

theory Sorting
imports Complex_Main Multiset
```(* Author: Tobias Nipkow *)

section "Sorting"

theory Sorting
imports
Complex_Main
"HOL-Library.Multiset"
begin

hide_const List.insort

declare Let_def [simp]

subsection "Insertion Sort"

fun insort :: "'a::linorder ⇒ 'a list ⇒ 'a list" where
"insort x [] = [x]" |
"insort x (y#ys) =
(if x ≤ y then x#y#ys else y#(insort x ys))"

fun isort :: "'a::linorder list ⇒ 'a list" where
"isort [] = []" |
"isort (x#xs) = insort x (isort xs)"

subsubsection "Functional Correctness"

lemma mset_insort: "mset (insort x xs) = add_mset x (mset xs)"
apply(induction xs)
apply auto
done

lemma mset_isort: "mset (isort xs) = mset xs"
apply(induction xs)
apply simp
apply (simp add: mset_insort)
done

lemma set_insort: "set (insort x xs) = insert x (set xs)"
by (metis mset_insort set_mset_add_mset_insert set_mset_mset)

lemma sorted_insort: "sorted (insort a xs) = sorted xs"
apply(induction xs)
apply(auto simp add: set_insort)
done

lemma sorted_isort: "sorted (isort xs)"
apply(induction xs)
apply(auto simp: sorted_insort)
done

subsubsection "Time Complexity"

text ‹We count the number of function calls.›

text‹
‹insort x [] = [x]›
‹insort x (y#ys) =
(if x ≤ y then x#y#ys else y#(insort x ys))›
›
fun t_insort :: "'a::linorder ⇒ 'a list ⇒ nat" where
"t_insort x [] = 1" |
"t_insort x (y#ys) =
(if x ≤ y then 0 else t_insort x ys) + 1"

text‹
‹isort [] = []›
‹isort (x#xs) = insort x (isort xs)›
›
fun t_isort :: "'a::linorder list ⇒ nat" where
"t_isort [] = 1" |
"t_isort (x#xs) = t_isort xs + t_insort x (isort xs) + 1"

lemma t_insort_length: "t_insort x xs ≤ length xs + 1"
apply(induction xs)
apply auto
done

lemma length_insort: "length (insort x xs) = length xs + 1"
apply(induction xs)
apply auto
done

lemma length_isort: "length (isort xs) = length xs"
apply(induction xs)
apply (auto simp: length_insort)
done

lemma t_isort_length: "t_isort xs ≤ (length xs + 1) ^ 2"
proof(induction xs)
case Nil show ?case by simp
next
case (Cons x xs)
have "t_isort (x#xs) = t_isort xs + t_insort x (isort xs) + 1" by simp
also have "… ≤ (length xs + 1) ^ 2 + t_insort x (isort xs) + 1"
using Cons.IH by simp
also have "… ≤ (length xs + 1) ^ 2 + length xs + 1 + 1"
using t_insort_length[of x "isort xs"] by (simp add: length_isort)
also have "… ≤ (length(x#xs) + 1) ^ 2"
by (simp add: power2_eq_square)
finally show ?case .
qed

subsection "Merge Sort"

fun merge :: "'a::linorder list ⇒ 'a list ⇒ 'a list" where
"merge [] ys = ys" |
"merge xs [] = xs" |
"merge (x#xs) (y#ys) = (if x ≤ y then x # merge xs (y#ys) else y # merge (x#xs) ys)"

fun msort :: "'a::linorder list ⇒ 'a list" where
"msort xs = (let n = length xs in
if n ≤ 1 then xs
else merge (msort (take (n div 2) xs)) (msort (drop (n div 2) xs)))"

declare msort.simps [simp del]

subsubsection "Functional Correctness"

lemma mset_merge: "mset(merge xs ys) = mset xs + mset ys"
by(induction xs ys rule: merge.induct) auto

lemma mset_msort: "mset (msort xs) = mset xs"
proof(induction xs rule: msort.induct)
case (1 xs)
let ?n = "length xs"
let ?ys = "take (?n div 2) xs"
let ?zs = "drop (?n div 2) xs"
show ?case
proof cases
assume "?n ≤ 1"
thus ?thesis by(simp add: msort.simps[of xs])
next
assume "¬ ?n ≤ 1"
hence "mset (msort xs) = mset (msort ?ys) + mset (msort ?zs)"
by(simp add: msort.simps[of xs] mset_merge)
also have "… = mset ?ys + mset ?zs"
using ‹¬ ?n ≤ 1› by(simp add: "1.IH")
also have "… = mset (?ys @ ?zs)" by (simp del: append_take_drop_id)
also have "… = mset xs" by simp
finally show ?thesis .
qed
qed

text ‹Via the previous lemma or directly:›

lemma set_merge: "set(merge xs ys) = set xs ∪ set ys"
by (metis mset_merge set_mset_mset set_mset_union)

lemma "set(merge xs ys) = set xs ∪ set ys"
by(induction xs ys rule: merge.induct) (auto)

lemma sorted_merge: "sorted (merge xs ys) ⟷ (sorted xs ∧ sorted ys)"
by(induction xs ys rule: merge.induct) (auto simp: set_merge)

lemma sorted_msort: "sorted (msort xs)"
proof(induction xs rule: msort.induct)
case (1 xs)
let ?n = "length xs"
show ?case
proof cases
assume "?n ≤ 1"
thus ?thesis by(simp add: msort.simps[of xs] sorted01)
next
assume "¬ ?n ≤ 1"
thus ?thesis using "1.IH"
by(simp add: sorted_merge msort.simps[of xs])
qed
qed

subsubsection "Time Complexity"

text ‹We only count the number of comparisons between list elements.›

fun c_merge :: "'a::linorder list ⇒ 'a list ⇒ nat" where
"c_merge [] ys = 0" |
"c_merge xs [] = 0" |
"c_merge (x#xs) (y#ys) = 1 + (if x ≤ y then c_merge xs (y#ys) else c_merge (x#xs) ys)"

lemma c_merge_ub: "c_merge xs ys ≤ length xs + length ys"
by (induction xs ys rule: c_merge.induct) auto

fun c_msort :: "'a::linorder list ⇒ nat" where
"c_msort xs =
(let n = length xs;
ys = take (n div 2) xs;
zs = drop (n div 2) xs
in if n ≤ 1 then 0
else c_msort ys + c_msort zs + c_merge (msort ys) (msort zs))"

declare c_msort.simps [simp del]

lemma length_merge: "length(merge xs ys) = length xs + length ys"
apply (induction xs ys rule: merge.induct)
apply auto
done

lemma length_msort: "length(msort xs) = length xs"
proof (induction xs rule: msort.induct)
case (1 xs)
thus ?case by (auto simp: msort.simps[of xs] length_merge)
qed
text ‹Why structured proof?
To have the name "xs" to specialize msort.simps with xs
to ensure that msort.simps cannot be used recursively.
Also works without this precaution, but that is just luck.›

lemma c_msort_le: "length xs = 2^k ⟹ c_msort xs ≤ k * 2^k"
proof(induction k arbitrary: xs)
case 0 thus ?case by (simp add: c_msort.simps)
next
case (Suc k)
let ?n = "length xs"
let ?ys = "take (?n div 2) xs"
let ?zs = "drop (?n div 2) xs"
show ?case
proof (cases "?n ≤ 1")
case True
thus ?thesis by(simp add: c_msort.simps)
next
case False
have "c_msort(xs) =
c_msort ?ys + c_msort ?zs + c_merge (msort ?ys) (msort ?zs)"
by (simp add: c_msort.simps msort.simps)
also have "… ≤ c_msort ?ys + c_msort ?zs + length ?ys + length ?zs"
using c_merge_ub[of "msort ?ys" "msort ?zs"] length_msort[of ?ys] length_msort[of ?zs]
by arith
also have "… ≤ k * 2^k + c_msort ?zs + length ?ys + length ?zs"
using Suc.IH[of ?ys] Suc.prems by simp
also have "… ≤ k * 2^k + k * 2^k + length ?ys + length ?zs"
using Suc.IH[of ?zs] Suc.prems by simp
also have "… = 2 * k * 2^k + 2 * 2 ^ k"
using Suc.prems by simp
finally show ?thesis by simp
qed
qed

(* Beware of conversions: *)
lemma c_msort_log: "length xs = 2^k ⟹ c_msort xs ≤ length xs * log 2 (length xs)"
using c_msort_le[of xs k] apply (simp add: log_nat_power algebra_simps)
by (metis (mono_tags) numeral_power_eq_of_nat_cancel_iff of_nat_le_iff of_nat_mult)

subsection "Bottom-Up Merge Sort"

(* Exercise: make tail recursive *)
fun merge_adj :: "('a::linorder) list list ⇒ 'a list list" where
"merge_adj [] = []" |
"merge_adj [xs] = [xs]" |
"merge_adj (xs # ys # zss) = merge xs ys # merge_adj zss"

text ‹For the termination proof of ‹merge_all› below.›
lemma length_merge_adjacent[simp]: "length (merge_adj xs) = (length xs + 1) div 2"
by (induction xs rule: merge_adj.induct) auto

fun merge_all :: "('a::linorder) list list ⇒ 'a list" where
"merge_all [] = []" |
"merge_all [xs] = xs" |
"merge_all xss = merge_all (merge_adj xss)"

definition msort_bu :: "('a::linorder) list ⇒ 'a list" where
"msort_bu xs = merge_all (map (λx. [x]) xs)"

subsubsection "Functional Correctness"

"⋃# (image_mset mset (mset (merge_adj xss))) = ⋃# (image_mset mset (mset xss))"
by(induction xss rule: merge_adj.induct) (auto simp: mset_merge)

lemma mset_merge_all:
"mset (merge_all xss) = (⋃# (mset (map mset xss)))"
by(induction xss rule: merge_all.induct) (auto simp: mset_merge mset_merge_adj)

lemma mset_msort_bu: "mset (msort_bu xs) = mset xs"
by(simp add: msort_bu_def mset_merge_all comp_def)

"∀xs ∈ set xss. sorted xs ⟹ ∀xs ∈ set (merge_adj xss). sorted xs"
by(induction xss rule: merge_adj.induct) (auto simp: sorted_merge)

lemma sorted_merge_all:
"∀xs ∈ set xss. sorted xs ⟹ sorted (merge_all xss)"
apply(induction xss rule: merge_all.induct)
using [[simp_depth_limit=3]] by (auto simp add: sorted_merge_adj)

lemma sorted_msort_bu: "sorted (msort_bu xs)"
by(simp add: msort_bu_def sorted_merge_all)

subsubsection "Time Complexity"

fun c_merge_adj :: "('a::linorder) list list ⇒ nat" where
"c_merge_adj [] = 0" |
"c_merge_adj [xs] = 0" |
"c_merge_adj (xs # ys # zss) = c_merge xs ys + c_merge_adj zss"

fun c_merge_all :: "('a::linorder) list list ⇒ nat" where
"c_merge_all [] = 0" |
"c_merge_all [xs] = 0" |
"c_merge_all xss = c_merge_adj xss + c_merge_all (merge_adj xss)"

definition c_msort_bu :: "('a::linorder) list ⇒ nat" where
"c_msort_bu xs = c_merge_all (map (λx. [x]) xs)"

"⟦ even(length xss); ∀xs ∈ set xss. length xs = m ⟧
⟹ ∀xs ∈ set (merge_adj xss). length xs = 2*m"
by(induction xss rule: merge_adj.induct) (auto simp: length_merge)

lemma c_merge_adj: "∀xs ∈ set xss. length xs = m ⟹ c_merge_adj xss ≤ m * length xss"
proof(induction xss rule: c_merge_adj.induct)
case 1 thus ?case by simp
next
case 2 thus ?case by simp
next
case (3 x y) thus ?case using c_merge_ub[of x y] by (simp add: algebra_simps)
qed

lemma c_merge_all: "⟦ ∀xs ∈ set xss. length xs = m; length xss = 2^k ⟧
⟹ c_merge_all xss ≤ m * k * 2^k"
proof (induction xss arbitrary: k m rule: c_merge_all.induct)
case 1 thus ?case by simp
next
case 2 thus ?case by simp
next
case (3 xs ys xss)
let ?xss = "xs # ys # xss"
let ?xss2 = "merge_adj ?xss"
obtain k' where k': "k = Suc k'" using "3.prems"(2)
by (metis length_Cons nat.inject nat_power_eq_Suc_0_iff nat.exhaust)
have "even (length ?xss)" using "3.prems"(2) k' by auto
from length_merge_adj[OF this "3.prems"(1)]
have *: "∀x ∈ set(merge_adj ?xss). length x = 2*m" .
have **: "length ?xss2 = 2 ^ k'" using "3.prems"(2) k' by auto
have "c_merge_all ?xss = c_merge_adj ?xss + c_merge_all ?xss2" by simp
also have "… ≤ m * 2^k + c_merge_all ?xss2"
using "3.prems"(2) c_merge_adj[OF "3.prems"(1)] by (auto simp: algebra_simps)
also have "… ≤ m * 2^k + (2*m) * k' * 2^k'"
using "3.IH"[OF * **] by simp
also have "… = m * k * 2^k"
using k' by (simp add: algebra_simps)
finally show ?case .
qed

corollary c_msort_bu: "length xs = 2 ^ k ⟹ c_msort_bu xs ≤ k * 2 ^ k"
using c_merge_all[of "map (λx. [x]) xs" 1] by (simp add: c_msort_bu_def)

subsection "Quicksort"

fun quicksort :: "('a::linorder) list ⇒ 'a list" where
"quicksort []     = []" |
"quicksort (x#xs) = quicksort (filter (λy. y < x) xs) @ [x] @ quicksort (filter (λy. x ≤ y) xs)"

lemma mset_quicksort: "mset (quicksort xs) = mset xs"
apply (induction xs rule: quicksort.induct)
apply (auto simp: not_le)
done

lemma set_quicksort: "set (quicksort xs) = set xs"
by(rule mset_eq_setD[OF mset_quicksort])

lemma sorted_quicksort: "sorted (quicksort xs)"
apply (induction xs rule: quicksort.induct)
apply (auto simp add: sorted_append set_quicksort)
done

subsection "Insertion Sort w.r.t. Keys and Stability"

text ‹Note that \<^const>‹insort_key› is already defined in theory \<^theory>‹HOL.List›.
Thus some of the lemmas are already present as well.›

fun isort_key :: "('a ⇒ 'k::linorder) ⇒ 'a list ⇒ 'a list" where
"isort_key f [] = []" |
"isort_key f (x # xs) = insort_key f x (isort_key f xs)"

subsubsection "Standard functional correctness"

lemma mset_insort_key: "mset (insort_key f x xs) = add_mset x (mset xs)"
by(induction xs) simp_all

lemma mset_isort_key: "mset (isort_key f xs) = mset xs"
by(induction xs) (simp_all add: mset_insort_key)

lemma set_isort_key: "set (isort_key f xs) = set xs"
by (rule mset_eq_setD[OF mset_isort_key])

lemma sorted_insort_key: "sorted (map f (insort_key f a xs)) = sorted (map f xs)"
by(induction xs)(auto simp: set_insort_key)

lemma sorted_isort_key: "sorted (map f (isort_key f xs))"
by(induction xs)(simp_all add: sorted_insort_key)

subsubsection "Stability"

lemma insort_is_Cons: "∀x∈set xs. f a ≤ f x ⟹ insort_key f a xs = a # xs"
by (cases xs) auto

lemma filter_insort_key_neg:
"¬ P x ⟹ filter P (insort_key f x xs) = filter P xs"
by (induction xs) simp_all

lemma filter_insort_key_pos:
"sorted (map f xs) ⟹ P x ⟹ filter P (insort_key f x xs) = insort_key f x (filter P xs)"
by (induction xs) (auto, subst insort_is_Cons, auto)

lemma sort_key_stable: "filter (λy. f y = k) (isort_key f xs) = filter (λy. f y = k) xs"
proof (induction xs)
case Nil thus ?case by simp
next
case (Cons a xs)
thus ?case
proof (cases "f a = k")
case False thus ?thesis  by (simp add: Cons.IH filter_insort_key_neg)
next
case True
have "filter (λy. f y = k) (isort_key f (a # xs))
= filter (λy. f y = k) (insort_key f a (isort_key f xs))"  by simp
also have "… = insort_key f a (filter (λy. f y = k) (isort_key f xs))"
by (simp add: True filter_insort_key_pos sorted_isort_key)
also have "… = insort_key f a (filter (λy. f y = k) xs)"  by (simp add: Cons.IH)
also have "… = a # (filter (λy. f y = k) xs)"  by(simp add: True insort_is_Cons)
also have "… = filter (λy. f y = k) (a # xs)" by (simp add: True)
finally show ?thesis .
qed
qed

end
```