summaryrefslogtreecommitdiff
path: root/theories/Sorting/Heap.v
blob: 2e46312085f571f3fa56b6770e39978394d0b37b (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*i $Id$ i*)

(** 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) *)

(* G. Huet 1-9-95 uses Multiset *)

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.
  Proof.
    simpl in |- *; auto with datatypes.
  Qed.

  Lemma leA_Tree_Node :
    forall (a b:A) (G D:Tree), leA a b -> leA_Tree a (Tree_Node b G D).
  Proof.
    simpl in |- *; auto with datatypes.
  Qed.


  (** ** 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.
  Proof.
    intros; inversion H; auto with datatypes.
  Qed.

  (* This lemma ought to be generated automatically by the Inversion tools *)
  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.
  Proof.
    simple induction T; auto with datatypes.
    intros a G PG D PD PN.
    elim (invert_heap a G D); auto with datatypes.
    intros H1 H2; elim H2; intros H3 H4; elim H4; intros.
    apply X0; auto with datatypes.
  Qed.

  (* This lemma ought to be generated automatically by the Inversion tools *)
  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.
  Proof.
    simple induction T; auto with datatypes.
    intros a G PG D PD PN.
    elim (invert_heap a G D); auto with datatypes.
    intros H1 H2; elim H2; intros H3 H4; elim H4; intros.
    apply X; auto with datatypes.
  Qed.

  Lemma low_trans :
    forall (T:Tree) (a b:A), leA a b -> leA_Tree b T -> leA_Tree a T.
  Proof.
    simple induction T; auto with datatypes.
    intros; simpl in |- *; apply leA_trans with b; auto with datatypes.
  Qed.

  (** ** 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.

  Lemma merge :
    forall l1:list A, Sorted leA l1 ->
    forall l2:list A, Sorted leA l2 -> merge_lem l1 l2.
  Proof.
    simple induction 1; intros.
    apply merge_exist with l2; auto with datatypes.
    elim H2; intros.
    apply merge_exist with (a :: l); simpl in |- *; auto using cons_sort with datatypes.
    elim (leA_dec a a0); intros.

    (* 1 (leA a a0) *)
    cut (merge_lem l (a0 :: l0)); auto using cons_sort with datatypes.
    intros [l3 l3sorted l3contents Hrec].
    apply merge_exist with (a :: l3); simpl in |- *;
      auto using cons_sort, cons_leA with datatypes.
    apply meq_trans with
      (munion (singletonBag a)
	(munion (list_contents _ eqA_dec l)
          (list_contents _ eqA_dec (a0 :: l0)))).
    apply meq_right; trivial with datatypes.
    apply meq_sym; apply munion_ass.
    intros; apply cons_leA.
    apply (@HdRel_inv _ leA) with l; trivial with datatypes.

    (* 2 (leA a0 a) *)
    elim X0; simpl in |- *; intros.
    apply merge_exist with (a0 :: l3); simpl in |- *;
      auto using cons_sort, cons_leA with datatypes.
    apply meq_trans with
      (munion (singletonBag a0)
	(munion (munion (singletonBag a) (list_contents _ eqA_dec l))
          (list_contents _ eqA_dec l0))).
    apply meq_right; trivial with datatypes.
    apply munion_perm_left.
    intros; apply cons_leA; apply HdRel_inv with (l:=l0); trivial with datatypes.
  Qed.


  (** ** 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.
  Proof.
    simple induction 1; intros.
    apply insert_exist with (Tree_Node a Tree_Leaf Tree_Leaf);
      auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes.
    simpl in |- *; unfold meq, munion in |- *; auto using node_is_heap with datatypes.
    elim (leA_dec a a0); intros.
    elim (X a0); intros.
    apply insert_exist with (Tree_Node a T2 T0);
      auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes.
    simpl in |- *; apply treesort_twist1; trivial with datatypes.
    elim (X a); intros T3 HeapT3 ConT3 LeA.
    apply insert_exist with (Tree_Node a0 T2 T3);
      auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes.
    apply node_is_heap; auto using node_is_heap, nil_is_heap, leA_Tree_Leaf with datatypes.
    apply low_trans with a; auto with datatypes.
    apply LeA; auto with datatypes.
    apply low_trans with a; auto with datatypes.
    simpl in |- *; apply treesort_twist2; trivial with datatypes.
  Qed.


  (** ** 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.
  Proof.
    simple induction l.
    apply (heap_exist nil Tree_Leaf); auto with datatypes.
    simpl in |- *; unfold meq in |- *; exact nil_is_heap.
    simple induction 1.
    intros T i m; elim (insert T i a).
    intros; apply heap_exist with T1; simpl in |- *; auto with datatypes.
    apply meq_trans with (munion (contents T) (singletonBag a)).
    apply meq_trans with (munion (singletonBag a) (contents T)).
    apply meq_right; trivial with datatypes.
    apply munion_comm.
    apply meq_sym; trivial with datatypes.
  Qed.


  (** ** 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.
  Proof.
    intros T h; elim h; intros.
    apply flat_exist with (nil (A:=A)); auto with datatypes.
    elim X; intros l1 s1 i1 m1; elim X0; intros l2 s2 i2 m2.
    elim (merge _ s1 _ s2); intros.
    apply flat_exist with (a :: l); simpl in |- *; auto with datatypes.
    apply meq_trans with
      (munion (list_contents _ eqA_dec l1)
	(munion (list_contents _ eqA_dec l2) (singletonBag a))).
    apply meq_congr; auto with datatypes.
    apply meq_trans with
      (munion (singletonBag a)
	(munion (list_contents _ eqA_dec l1) (list_contents _ eqA_dec l2))).
    apply munion_rotate.
    apply meq_right; apply meq_sym; trivial with datatypes.
  Qed.


  (** * Specification of treesort *)

  Theorem treesort :
    forall l:list A,
    {m : list A | Sorted leA m & permutation _ eqA_dec l m}.
  Proof.
    intro l; unfold permutation in |- *.
    elim (list_to_heap l).
    intros.
    elim (heap_to_list T); auto with datatypes.
    intros.
    exists l0; auto with datatypes.
    apply meq_trans with (contents T); trivial with datatypes.
  Qed.

End defs.