# Library Coq.Sorting.Heap

This file is deprecated, for a tree on list, use Mergesort.v.
A development of Treesort on Heap trees. It has an average complexity of O(n.log n) but of O(n²) in the worst case (e.g. if the list is already sorted)
Require Import List Multiset PermutSetoid Relations Sorting.
Section defs.

# Trees and heap trees

## Definition of trees over an ordered set

Variable A : Type.
Variable leA : relation A.
Variable eqA : relation A.
Let gtA (x y:A) := ~ leA x y.
Hypothesis leA_dec : forall x y:A, { leA x y } + { leA y x }.
Hypothesis eqA_dec : forall x y:A, { eqA x y } + { ~ eqA x y }.
Hypothesis leA_refl : forall x y:A, eqA x y -> leA x y.
Hypothesis leA_trans : forall x y z:A, leA x y -> leA y z -> leA x z.
Hypothesis leA_antisym : forall x y:A, leA x y -> leA y x -> eqA x y.
Hint Resolve leA_refl.
Hint Immediate eqA_dec leA_dec leA_antisym.
Let emptyBag := EmptyBag A.
Let singletonBag := SingletonBag _ eqA_dec.
Inductive Tree :=
| Tree_Leaf : Tree
| Tree_Node : A -> Tree -> Tree -> Tree.
a is lower than a Tree T if T is a Leaf or T is a Node holding b>a
Definition leA_Tree (a:A) (t:Tree) :=
match t with
| Tree_Leaf => True
| Tree_Node b T1 T2 => leA a b
end.
Lemma leA_Tree_Leaf : forall a:A, leA_Tree a Tree_Leaf.
Lemma leA_Tree_Node :
forall (a b:A) (G D:Tree), leA a b -> leA_Tree a (Tree_Node b G D).

## The heap property

Inductive is_heap : Tree -> Prop :=
| nil_is_heap : is_heap Tree_Leaf
| node_is_heap :
forall (a:A) (T1 T2:Tree),
leA_Tree a T1 ->
leA_Tree a T2 ->
is_heap T1 -> is_heap T2 -> is_heap (Tree_Node a T1 T2).
Lemma invert_heap :
forall (a:A) (T1 T2:Tree),
is_heap (Tree_Node a T1 T2) ->
leA_Tree a T1 /\ leA_Tree a T2 /\ is_heap T1 /\ is_heap T2.
Lemma is_heap_rect :
forall P:Tree -> Type,
P Tree_Leaf ->
(forall (a:A) (T1 T2:Tree),
leA_Tree a T1 ->
leA_Tree a T2 ->
is_heap T1 -> P T1 -> is_heap T2 -> P T2 -> P (Tree_Node a T1 T2)) ->
forall T:Tree, is_heap T -> P T.
Lemma is_heap_rec :
forall P:Tree -> Set,
P Tree_Leaf ->
(forall (a:A) (T1 T2:Tree),
leA_Tree a T1 ->
leA_Tree a T2 ->
is_heap T1 -> P T1 -> is_heap T2 -> P T2 -> P (Tree_Node a T1 T2)) ->
forall T:Tree, is_heap T -> P T.
Lemma low_trans :
forall (T:Tree) (a b:A), leA a b -> leA_Tree b T -> leA_Tree a T.

## Merging two sorted lists

Inductive merge_lem (l1 l2:list A) : Type :=
merge_exist :
forall l:list A,
Sorted leA l ->
meq (list_contents _ eqA_dec l)
(munion (list_contents _ eqA_dec l1) (list_contents _ eqA_dec l2)) ->
(forall a, HdRel leA a l1 -> HdRel leA a l2 -> HdRel leA a l) ->
merge_lem l1 l2.
Require Import Morphisms.
Instance: Equivalence (@meq A).
Instance: Proper (@meq A ++> @meq _ ++> @meq _) (@munion A).
Lemma merge :
forall l1:list A, Sorted leA l1 ->
forall l2:list A, Sorted leA l2 -> merge_lem l1 l2.

## From trees to multisets

contents of a tree as a multiset
Nota Bene : In what follows the definition of SingletonBag in not used. Actually, we could just take as postulate: Parameter SingletonBag : A->multiset.
Fixpoint contents (t:Tree) : multiset A :=
match t with
| Tree_Leaf => emptyBag
| Tree_Node a t1 t2 =>
munion (contents t1) (munion (contents t2) (singletonBag a))
end.
equivalence of two trees is equality of corresponding multisets
Definition equiv_Tree (t1 t2:Tree) := meq (contents t1) (contents t2).

# From lists to sorted lists

## Specification of heap insertion

Inductive insert_spec (a:A) (T:Tree) : Type :=
insert_exist :
forall T1:Tree,
is_heap T1 ->
meq (contents T1) (munion (contents T) (singletonBag a)) ->
(forall b:A, leA b a -> leA_Tree b T -> leA_Tree b T1) ->
insert_spec a T.
Lemma insert : forall T:Tree, is_heap T -> forall a:A, insert_spec a T.

## Building a heap from a list

Inductive build_heap (l:list A) : Type :=
heap_exist :
forall T:Tree,
is_heap T ->
meq (list_contents _ eqA_dec l) (contents T) -> build_heap l.
Lemma list_to_heap : forall l:list A, build_heap l.

## Building the sorted list

Inductive flat_spec (T:Tree) : Type :=
flat_exist :
forall l:list A,
Sorted leA l ->
(forall a:A, leA_Tree a T -> HdRel leA a l) ->
meq (contents T) (list_contents _ eqA_dec l) -> flat_spec T.
Lemma heap_to_list : forall T:Tree, is_heap T -> flat_spec T.

# Specification of treesort

Theorem treesort :
forall l:list A,
{ m : list A | Sorted leA m & permutation _ eqA_dec l m }.
End defs.