author haftmann Sun, 04 Nov 2018 15:00:30 +0000 changeset 69246 c1fe9dcc274a parent 69245 3e9f812c308c child 69249 27423819534c
concrecte sorting algorithms beyond insertion sort
```--- a/src/HOL/Library/Comparator.thy	Mon Nov 05 23:15:58 2018 +0100
+++ b/src/HOL/Library/Comparator.thy	Sun Nov 04 15:00:30 2018 +0000
@@ -76,6 +76,24 @@
"asymp (\<lambda>a b. cmp a b = Greater)"
using irreflp_greater by (auto intro!: asympI dest: asym_greater)

+lemma trans_equiv_less:
+  "cmp a c = Less" if "cmp a b = Equiv" and "cmp b c = Less"
+  using that
+  by (metis (full_types) comp.exhaust greater_iff_sym_less trans_equiv trans_less)
+
+lemma trans_less_equiv:
+  "cmp a c = Less" if "cmp a b = Less" and "cmp b c = Equiv"
+  using that
+  by (metis (full_types) comp.exhaust greater_iff_sym_less trans_equiv trans_less)
+
+lemma trans_equiv_greater:
+  "cmp a c = Greater" if "cmp a b = Equiv" and "cmp b c = Greater"
+  using that by (simp add: sym [of a b] greater_iff_sym_less trans_less_equiv)
+
+lemma trans_greater_equiv:
+  "cmp a c = Greater" if "cmp a b = Greater" and "cmp b c = Equiv"
+  using that by (simp add: sym [of b c] greater_iff_sym_less trans_equiv_less)
+
lemma transp_less:
"transp (\<lambda>a b. cmp a b = Less)"
by (rule transpI) (fact trans_less)
@@ -118,6 +136,26 @@
"transp (\<lambda>a b. cmp a b \<noteq> Greater)"
by (rule transpI) (fact trans_not_greater)

+text \<open>Substitution under equivalences\<close>
+
+lemma equiv_subst_left:
+  "cmp z y = comp \<longleftrightarrow> cmp x y = comp" if "cmp z x = Equiv" for comp
+proof -
+  from that have "cmp x z = Equiv"
+  with that show ?thesis
+    by (cases comp) (auto intro: trans_equiv trans_equiv_less trans_equiv_greater)
+qed
+
+lemma equiv_subst_right:
+  "cmp x z = comp \<longleftrightarrow> cmp x y = comp" if "cmp z y = Equiv" for comp
+proof -
+  from that have "cmp y z = Equiv"
+  with that show ?thesis
+    by (cases comp) (auto intro: trans_equiv trans_less_equiv trans_greater_equiv)
+qed
+
end

typedef 'a comparator = "{cmp :: 'a \<Rightarrow> 'a \<Rightarrow> comp. comparator cmp}"```
```--- a/src/HOL/Library/Sorting_Algorithms.thy	Mon Nov 05 23:15:58 2018 +0100
+++ b/src/HOL/Library/Sorting_Algorithms.thy	Sun Nov 04 15:00:30 2018 +0000
@@ -6,11 +6,6 @@
imports Main Multiset Comparator
begin

-text \<open>Prelude\<close>
-
-hide_const (open) sorted insort sort
-
-
section \<open>Stably sorted lists\<close>

abbreviation (input) stable_segment :: "'a comparator \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
@@ -19,17 +14,26 @@
fun sorted :: "'a comparator \<Rightarrow> 'a list \<Rightarrow> bool"
where sorted_Nil: "sorted cmp [] \<longleftrightarrow> True"
| sorted_single: "sorted cmp [x] \<longleftrightarrow> True"
-  | sorted_rec: "sorted cmp (y # x # xs) \<longleftrightarrow> compare cmp x y \<noteq> Less \<and> sorted cmp (x # xs)"
+  | sorted_rec: "sorted cmp (y # x # xs) \<longleftrightarrow> compare cmp y x \<noteq> Greater \<and> sorted cmp (x # xs)"

lemma sorted_ConsI:
"sorted cmp (x # xs)" if "sorted cmp xs"
-    and "\<And>y ys. xs = y # ys \<Longrightarrow> compare cmp y x \<noteq> Less"
+    and "\<And>y ys. xs = y # ys \<Longrightarrow> compare cmp x y \<noteq> Greater"
+  using that by (cases xs) simp_all
+
+lemma sorted_Cons_imp_sorted:
+  "sorted cmp xs" if "sorted cmp (x # xs)"
using that by (cases xs) simp_all

+lemma sorted_Cons_imp_not_less:
+  "compare cmp y x \<noteq> Greater" if "sorted cmp (y # xs)"
+    and "x \<in> set xs"
+  using that by (induction xs arbitrary: y) (auto dest: compare.trans_not_greater)
+
lemma sorted_induct [consumes 1, case_names Nil Cons, induct pred: sorted]:
"P xs" if "sorted cmp xs" and "P []"
and *: "\<And>x xs. sorted cmp xs \<Longrightarrow> P xs
-      \<Longrightarrow> (\<And>y. y \<in> set xs \<Longrightarrow> compare cmp y x \<noteq> Less) \<Longrightarrow> P (x # xs)"
+      \<Longrightarrow> (\<And>y. y \<in> set xs \<Longrightarrow> compare cmp x y \<noteq> Greater) \<Longrightarrow> P (x # xs)"
using \<open>sorted cmp xs\<close> proof (induction xs)
case Nil
show ?case
@@ -40,7 +44,7 @@
by (cases xs) simp_all
moreover have "P xs" using \<open>sorted cmp xs\<close>
by (rule Cons.IH)
-  moreover have "compare cmp y x \<noteq> Less" if "y \<in> set xs" for y
+  moreover have "compare cmp x y \<noteq> Greater" if "y \<in> set xs" for y
using that \<open>sorted cmp (x # xs)\<close> proof (induction xs)
case Nil
then show ?case
@@ -54,10 +58,10 @@
by simp
next
case (Cons w ws)
-      with Cons.prems have "compare cmp w z \<noteq> Less" "compare cmp z x \<noteq> Less"
+      with Cons.prems have "compare cmp z w \<noteq> Greater" "compare cmp x z \<noteq> Greater"
by auto
-      then have "compare cmp w x \<noteq> Less"
-        by (auto dest: compare.trans_not_less)
+      then have "compare cmp x w \<noteq> Greater"
+        by (auto dest: compare.trans_not_greater)
with Cons show ?thesis
using Cons.prems Cons.IH by auto
qed
@@ -69,7 +73,7 @@
lemma sorted_induct_remove1 [consumes 1, case_names Nil minimum]:
"P xs" if "sorted cmp xs" and "P []"
and *: "\<And>x xs. sorted cmp xs \<Longrightarrow> P (remove1 x xs)
-      \<Longrightarrow> x \<in> set xs \<Longrightarrow> hd (stable_segment cmp x xs) = x \<Longrightarrow> (\<And>y. y \<in> set xs \<Longrightarrow> compare cmp y x \<noteq> Less)
+      \<Longrightarrow> x \<in> set xs \<Longrightarrow> hd (stable_segment cmp x xs) = x \<Longrightarrow> (\<And>y. y \<in> set xs \<Longrightarrow> compare cmp x y \<noteq> Greater)
\<Longrightarrow> P xs"
using \<open>sorted cmp xs\<close> proof (induction xs)
case Nil
@@ -80,7 +84,7 @@
then have "sorted cmp (x # xs)"
moreover note Cons.IH
-  moreover have "\<And>y. compare cmp y x = Less \<Longrightarrow> y \<in> set xs \<Longrightarrow> False"
+  moreover have "\<And>y. compare cmp x y = Greater \<Longrightarrow> y \<in> set xs \<Longrightarrow> False"
using Cons.hyps by simp
ultimately show ?case
by (auto intro!: * [of "x # xs" x]) blast
@@ -114,7 +118,7 @@
assume "remove1 x ys = z # zs"
with \<open>x \<noteq> y\<close> have "z \<in> set ys"
using notin_set_remove1 [of z ys x] by auto
-        then show "compare cmp z y \<noteq> Less"
+        then show "compare cmp y z \<noteq> Greater"
by (rule Cons.hyps(2))
qed
with False show ?thesis
@@ -123,6 +127,20 @@
qed
qed

+lemma sorted_stable_segment:
+  "sorted cmp (stable_segment cmp x xs)"
+proof (induction xs)
+  case Nil
+  show ?case
+    by simp
+next
+  case (Cons y ys)
+  then show ?case
+    by (auto intro!: sorted_ConsI simp add: filter_eq_Cons_iff compare.sym)
+      (auto dest: compare.trans_equiv simp add: compare.sym compare.greater_iff_sym_less)
+
+qed
+
primrec insort :: "'a comparator \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "insort cmp y [] = [y]"
| "insort cmp y (x # xs) = (if compare cmp y x \<noteq> Greater
@@ -175,7 +193,7 @@

lemma insort_eq_ConsI:
"insort cmp x xs = x # xs"
-    if "sorted cmp xs" "\<And>y. y \<in> set xs \<Longrightarrow> compare cmp y x \<noteq> Less"
+    if "sorted cmp xs" "\<And>y. y \<in> set xs \<Longrightarrow> compare cmp x y \<noteq> Greater"
using that by (induction xs) (simp_all add: compare.greater_iff_sym_less)

lemma remove1_insort_not_same_eq [simp]:
@@ -188,14 +206,15 @@
next
case (Cons z zs)
show ?case
-  proof (cases "compare cmp z x = Less")
+  proof (cases "compare cmp x z = Greater")
case True
with Cons show ?thesis
+      by simp
next
case False
-    have "compare cmp y x \<noteq> Less" if "y \<in> set zs" for y
-      using that False Cons.hyps by (auto dest: compare.trans_not_less)
+    then have "compare cmp x y \<noteq> Greater" if "y \<in> set zs" for y
+      using that Cons.hyps
+      by (auto dest: compare.trans_not_greater)
with Cons show ?thesis
qed
@@ -211,7 +230,7 @@
next
case (Cons y ys)
then have "compare cmp x y \<noteq> Less"
-    by auto
+    by (auto simp add: compare.greater_iff_sym_less)
then consider "compare cmp x y = Greater" | "compare cmp x y = Equiv"
by (cases "compare cmp x y") auto
then show ?case proof cases
@@ -227,6 +246,31 @@
qed
qed

+lemma sorted_append_iff:
+  "sorted cmp (xs @ ys) \<longleftrightarrow> sorted cmp xs \<and> sorted cmp ys
+     \<and> (\<forall>x \<in> set xs. \<forall>y \<in> set ys. compare cmp x y \<noteq> Greater)" (is "?P \<longleftrightarrow> ?R \<and> ?S \<and> ?Q")
+proof
+  assume ?P
+  have ?R
+    using \<open>?P\<close> by (induction xs)
+        auto simp add: sorted_Cons_imp_sorted intro: sorted_ConsI)
+  moreover have ?S
+    using \<open>?P\<close> by (induction xs) (auto dest: sorted_Cons_imp_sorted)
+  moreover have ?Q
+    using \<open>?P\<close> by (induction xs) (auto simp add: sorted_Cons_imp_not_less,
+  ultimately show "?R \<and> ?S \<and> ?Q"
+    by simp
+next
+  assume "?R \<and> ?S \<and> ?Q"
+  then have ?R ?S ?Q
+    by simp_all
+  then show ?P
+    by (induction xs)
+      (auto simp add: append_eq_Cons_conv intro!: sorted_ConsI)
+qed
+
definition sort :: "'a comparator \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "sort cmp xs = foldr (insort cmp) xs []"

@@ -299,7 +343,7 @@
case (minimum x xs)
from \<open>mset ys = mset xs\<close> have ys: "set ys = set xs"
by (rule mset_eq_setD)
-    then have "compare cmp y x \<noteq> Less" if "y \<in> set ys" for y
+    then have "compare cmp x y \<noteq> Greater" if "y \<in> set ys" for y
using that minimum.hyps by simp
from minimum.prems have stable: "stable_segment cmp x ys = stable_segment cmp x xs"
by simp
@@ -318,4 +362,113 @@
qed
qed

+lemma filter_insort:
+  "filter P (insort cmp x xs) = insort cmp x (filter P xs)"
+    if "sorted cmp xs" and "P x"
+  using that by (induction xs)
+    (auto simp add: compare.trans_not_greater insort_eq_ConsI)
+
+lemma filter_insort_triv:
+  "filter P (insort cmp x xs) = filter P xs"
+    if "\<not> P x"
+  using that by (induction xs) simp_all
+
+lemma filter_sort:
+  "filter P (sort cmp xs) = sort cmp (filter P xs)"
+  by (induction xs) (auto simp add: filter_insort filter_insort_triv)
+
+
+section \<open>Alternative sorting algorithms\<close>
+
+subsection \<open>Quicksort\<close>
+
+definition quicksort :: "'a comparator \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  where quicksort_is_sort [simp]: "quicksort = sort"
+
+lemma sort_by_quicksort:
+  "sort = quicksort"
+  by simp
+
+lemma sort_by_quicksort_rec:
+  "sort cmp xs = sort cmp [x\<leftarrow>xs. compare cmp x (xs ! (length xs div 2)) = Less]
+    @ stable_segment cmp (xs ! (length xs div 2)) xs
+    @ sort cmp [x\<leftarrow>xs. compare cmp x (xs ! (length xs div 2)) = Greater]" (is "sort cmp ?lhs = ?rhs")
+proof (rule sort_eqI)
+  show "mset ?lhs = mset ?rhs"
+    by (rule multiset_eqI) (auto simp add: compare.sym intro: comp.exhaust)
+next
+  show "sorted cmp ?rhs"
+    by (auto simp add: sorted_append_iff sorted_stable_segment compare.equiv_subst_right dest: compare.trans_greater)
+next
+  let ?pivot = "xs ! (length xs div 2)"
+  fix l
+  assume "l \<in> set xs"
+  have "compare cmp x ?pivot = comp \<and> compare cmp l x = Equiv
+    \<longleftrightarrow> compare cmp l ?pivot = comp \<and> compare cmp l x = Equiv" for x comp
+  proof -
+    have "compare cmp x ?pivot = comp \<longleftrightarrow> compare cmp l ?pivot = comp"
+      if "compare cmp l x = Equiv"
+      using that by (simp add: compare.equiv_subst_left compare.sym)
+    then show ?thesis by blast
+  qed
+  then show "stable_segment cmp l ?lhs = stable_segment cmp l ?rhs"
+    by (simp add: stable_sort compare.sym [of _ ?pivot])
+      (cases "compare cmp l ?pivot", simp_all)
+qed
+
+context
+begin
+
+qualified definition partition :: "'a comparator \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list \<times> 'a list \<times> 'a list"
+  where "partition cmp pivot xs =
+    ([x \<leftarrow> xs. compare cmp x pivot = Less], stable_segment cmp pivot xs, [x \<leftarrow> xs. compare cmp x pivot = Greater])"
+
+qualified lemma partition_code [code]:
+  "partition cmp pivot [] = ([], [], [])"
+  "partition cmp pivot (x # xs) =
+    (let (lts, eqs, gts) = partition cmp pivot xs
+     in case compare cmp x pivot of
+       Less \<Rightarrow> (x # lts, eqs, gts)
+     | Equiv \<Rightarrow> (lts, x # eqs, gts)
+     | Greater \<Rightarrow> (lts, eqs, x # gts))"
+  using comp.exhaust by (auto simp add: partition_def Let_def compare.sym [of _ pivot])
+
+lemma quicksort_code [code]:
+  "quicksort cmp xs =
+    (case xs of
+      [] \<Rightarrow> []
+    | [x] \<Rightarrow> xs
+    | [x, y] \<Rightarrow> (if compare cmp x y \<noteq> Greater then xs else [y, x])
+    | _ \<Rightarrow>
+        let (lts, eqs, gts) = partition cmp (xs ! (length xs div 2)) xs
+        in quicksort cmp lts @ eqs @ quicksort cmp gts)"
+proof (cases "length xs \<ge> 3")
+  case False
+  then have "length xs \<le> 2"
+    by simp
+  then have "length xs = 0 \<or> length xs = 1 \<or> length xs = 2"
+    using le_neq_trans less_2_cases by auto
+  then consider "xs = []" | x where "xs = [x]" | x y where "xs = [x, y]"
+    by (auto simp add: length_Suc_conv numeral_2_eq_2)
+  then show ?thesis
+    by cases simp_all
+next
+  case True
+  then obtain x y z zs where "xs = x # y # z # zs"
+    by (metis le_0_eq length_0_conv length_Cons list.exhaust not_less_eq_eq numeral_3_eq_3)
+  moreover have "quicksort cmp xs =
+    (let (lts, eqs, gts) = partition cmp (xs ! (length xs div 2)) xs
+    in quicksort cmp lts @ eqs @ quicksort cmp gts)"
+    using sort_by_quicksort_rec [of cmp xs] by (simp add: partition_def)
+  ultimately show ?thesis
+    by simp
+qed
+
end
+
+text \<open>Evaluation example\<close>
+
+value "let cmp = key abs (reversed default)
+  in quicksort cmp [65, 1705, -2322, 734, 4, (-17::int)]"
+
+end```