From c8ba7ff13770ec2099dc575470d9b0c51ba6b4e0 Mon Sep 17 00:00:00 2001
From: WeetHet <stas.ale66@gmail.com>
Date: Mon, 19 Aug 2024 15:48:27 +0300
Subject: [PATCH] 033, 088

---
 033-sort_third.dfy | 61 +++++++++++++++++++++++++++++++++++++++++
 088-sort_array.dfy | 68 ++++++++++++++++++++++++++++++++++++++++++++++
 2 files changed, 129 insertions(+)
 create mode 100644 033-sort_third.dfy
 create mode 100644 088-sort_array.dfy

diff --git a/033-sort_third.dfy b/033-sort_third.dfy
new file mode 100644
index 0000000..b99580d
--- /dev/null
+++ b/033-sort_third.dfy
@@ -0,0 +1,61 @@
+method sort_third(a: seq<int>) returns (sorted_even: seq<int>)
+  requires |a| > 0
+  ensures |sorted_even| == |a|
+  ensures forall i, j :: 0 <= i < j < |sorted_even| && i % 3 == 0 && j % 3 == 0 ==>
+      sorted_even[i] <= sorted_even[j]
+  ensures forall i :: 0 <= i < |a| && i % 3 != 0 ==> sorted_even[i] == a[i]
+  ensures multiset(a) == multiset(sorted_even)
+{
+  var p := [];
+  var i := 0;
+  while i < |a|
+    invariant 0 <= i <= |a|
+    invariant |p| == i
+    invariant forall j :: 0 <= j < i ==> p[j] == (j % 3 == 0)
+  {
+    p := p + [i % 3 == 0];
+    i := i + 1;
+  }
+
+  sorted_even := SortSeqPred(a, p);
+}
+
+method SortSeqPred(s: seq<int>, p: seq<bool>) returns (sorted: seq<int>)
+  requires |s| == |p|
+  ensures |sorted| == |s|
+  ensures forall i, j :: 0 <= i < j < |sorted| && p[i] && p[j] ==> sorted[i] <= sorted[j]
+  ensures multiset(s) == multiset(sorted)
+  ensures forall i :: 0 <= i < |s| && !p[i] ==> sorted[i] == s[i]
+{
+  sorted := s;
+  var i := 0;
+  while i < |sorted|
+    invariant 0 <= i <= |sorted|
+    invariant |sorted| == |s|
+    invariant forall j, k :: 0 <= j < k < i && p[j] && p[k] ==> sorted[j] <= sorted[k]
+    invariant multiset(s) == multiset(sorted)
+    invariant forall j :: 0 <= j < i && p[j] ==> forall k :: i <= k < |sorted| && p[k] ==> sorted[j] <= sorted[k]
+    invariant |sorted| == |s|
+    invariant forall j :: 0 <= j < |s| && !p[j] ==> sorted[j] == s[j]
+  {
+    if p[i] {
+      var minIndex := i;
+      var j := i + 1;
+      while j < |sorted|
+        invariant i <= minIndex < j <= |sorted|
+        invariant p[minIndex]
+        invariant forall k :: i <= k < j && p[k] ==> sorted[minIndex] <= sorted[k]
+      {
+        if p[j] && sorted[j] < sorted[minIndex] {
+          minIndex := j;
+        }
+        j := j + 1;
+      }
+      if minIndex != i {
+        var temp := sorted[i];
+        sorted := sorted[i := sorted[minIndex]][minIndex := temp];
+      }
+    }
+    i := i + 1;
+  }
+}
\ No newline at end of file
diff --git a/088-sort_array.dfy b/088-sort_array.dfy
new file mode 100644
index 0000000..ffc6e67
--- /dev/null
+++ b/088-sort_array.dfy
@@ -0,0 +1,68 @@
+method sort_array(s: seq<int>) returns (sorted: seq<int>)
+  ensures |sorted| == |s|
+  ensures |s| > 0 && (s[0] + s[|s| - 1]) % 2 == 0 ==>  
+    forall i, j :: 0 <= i < j < |sorted| ==> sorted[i] >= sorted[j]
+  ensures |s| > 0 && (s[0] + s[|s| - 1]) % 2 != 0 ==> 
+    forall i, j :: 0 <= i < j < |sorted| ==> sorted[i] <= sorted[j]
+{
+  if |s| == 0 {
+    sorted := [];
+  } else if (s[0] + s[|s| - 1]) % 2 == 0 {
+    var t := SortSeq(s);
+    sorted := reverse(t);
+    return;
+  } else {
+    sorted := SortSeq(s);
+    return;
+  }
+}
+
+method reverse(s: seq<int>) returns (rev: seq<int>)
+  ensures |rev| == |s|
+  ensures forall k :: 0 <= k < |s| ==> rev[k] == s[|s| - 1 - k]
+{
+  rev := [];
+  var i := 0;
+  while (i < |s|)
+    invariant i >= 0 && i <= |s|
+    invariant |rev| == i
+    invariant forall k :: 0 <= k < i ==> rev[k] == s[|s| - 1 - k]
+  {
+    rev := rev + [s[|s| - i - 1]];
+    i := i + 1;
+  }
+}
+
+
+method SortSeq(s: seq<int>) returns (sorted: seq<int>)
+  ensures forall i, j :: 0 <= i < j < |sorted| ==> sorted[i] <= sorted[j]
+  ensures |sorted| == |s|
+  ensures multiset(s) == multiset(sorted)
+{
+  sorted := s;
+  var i := 0;
+  while i < |sorted|
+    invariant 0 <= i <= |sorted|
+    invariant forall j, k :: 0 <= j < k < i ==> sorted[j] <= sorted[k]
+    invariant multiset(s) == multiset(sorted)
+    invariant forall j :: 0 <= j < i ==> forall k :: i <= k < |sorted| ==> sorted[j] <= sorted[k]
+    invariant |sorted| == |s|
+  {
+    var minIndex := i;
+    var j := i + 1;
+    while j < |sorted|
+      invariant i <= minIndex < j <= |sorted|
+      invariant forall k :: i <= k < j ==> sorted[minIndex] <= sorted[k]
+    {
+      if sorted[j] < sorted[minIndex] {
+        minIndex := j;
+      }
+      j := j + 1;
+    }
+    if minIndex != i {
+      var temp := sorted[i];
+      sorted := sorted[i := sorted[minIndex]][minIndex := temp];
+    }
+    i := i + 1;
+  }
+}
\ No newline at end of file