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#|
Leftist Trees, Version 0.1
Copyright (C) 2012 by Ben Selfridge <benself@cs.utexas.edu>
License: (An MIT/X11-style license)
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to
deal in the Software without restriction, including without limitation the
rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
sell copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
IN THE SOFTWARE.
|#
(in-package "ACL2")
;;(include-book "sorting/perm" :dir :system)
;;(include-book "sorting/ordered-perms" :dir :system)
(include-book "sorting/orderedp" :dir :system)
(include-book "sorting/how-many" :dir :system)
(local (include-book "sorting/convert-perm-to-how-many" :dir :system))
(include-book "leftist-tree-defuns")
(include-book "leftist-tree-defthms")
; NOTE: Legacy doc strings defined in this file were replaced Nov. 2014 by the
; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; (defdoc leftist-tree-sort
; ":doc-section leftist-trees
; Functions and theorems about leftist tree-based heapsort.~/~/
; There are three functions related to heapsort, the most important
; being ltree-sort, which works just like the other sorting functions in
; the books/sorting contribution, except it uses leftist trees to sort
; its input. There are a number of theorems provided that prove the
; heapsort algorithm correct.
;
; ------------------------------------------------------------
; Functions and Constants
; ------------------------------------------------------------
;
; Function/Constant Name Result
; (supporting function) Type
; ---------------------- ----
; LTREE-TO-LIST list
; LTREE-SORT list
; HOW-MANY-LT natural
; ~/")
;;;;;;;;;;;;;;
;; HEAPSORT ;;
;;;;;;;;;;;;;;
;; -- At this point, heapsort is fairly simple. We define
;; "ltree-to-list" which constructs a list by calling
;; find-min and delete-min repeatedly on the tree.
;; Then to sort a list, we just need to convert it to
;; a leftist tree via build-lt, and then call
;; ltree-to-list on the result.
(defun ltree-to-list (tree)
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; ":doc-section leftist-tree-sort
; convert a leftist tree to a list~/~/
; Assuming the leftist tree in question is proper, this will result in a
; sorted list.
; ~/"
(declare
(xargs :measure (size-lt tree)))
(cond ((atom tree) nil)
(t (cons (find-min-lt tree)
(ltree-to-list (delete-min-lt tree))))))
;% (ltree-sort l)
; -- Computes a leftist-heapsort of a list.
(defun ltree-sort (l)
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; ":doc-section ltree-sort
; sort an input list using leftist tree-based heapsort~/~/
; Sorts an input list by first INSERT-LTing each element of the list into a
; leftist tree, then DELETE-MIN-LTing the min element from the tree one
; by one.
; ~/"
(ltree-to-list (build-lt l)))
(defun how-many-lt (e tree)
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; ":doc-section how-many-lt
; returns the number of times an object occurs in a leftist tree~/~/
; Counts the number of occurrences of a given object in a leftist
; tree. This function takes advantage of the heap-ordering property, and
; returns 0 if the root of the tree is larger than what we are searching
; for."
(cond ((is-empty-lt tree) 0)
((equal e (root-lt tree))
(+ 1 (how-many-lt e (left-lt tree))
(how-many-lt e (right-lt tree))))
((lexorder (root-lt tree) e)
(+ (how-many-lt e (left-lt tree))
(how-many-lt e (right-lt tree))))
(t 0)))
;;;;;;;;;;;;;;;;;;;;;;;;;
;; HEAPSORT IS CORRECT ;;
;;;;;;;;;;;;;;;;;;;;;;;;;
;; Note: This comes down to proving that the car of the
;; ltree-to-list of a proper-lt tree is equal to the
;; root of the original tree. These rewrite rules
;; are disabled in general, since neither is really
;; simpler than the other, and both are used
;; independently in proving the main theorem.
(local
(defthmd root-equals-car-list-lt
(implies (and (consp tree)
(proper-lt tree))
(equal (cadr tree)
(car (ltree-to-list tree))))))
(local
(defthmd car-list-equals-root-lt
(implies (and (consp tree)
(proper-lt tree))
(equal (car (ltree-to-list tree))
(cadr tree)))))
(local
(defthmd orderedp-proper-ltree-to-list-ltree-L1
(implies (and (consp left_tree)
(consp right_tree)
(proper-lt left_tree)
(proper-lt right_tree)
(lexorder x (root-lt left_tree))
(lexorder x (root-lt right_tree)))
(lexorder x
(root-lt (merge-lt left_tree right_tree))))
:hints (("Goal"
:induct (merge-lt left_tree right_tree)))))
(defthmd orderedp-proper-ltree-to-list
(implies (proper-lt tree)
(orderedp (ltree-to-list tree)))
:hints (("Subgoal *1/3.3'"
:in-theory (enable car-list-equals-root-lt))
("Subgoal *1/3.2'"
:in-theory (enable root-equals-car-list-lt))
("Subgoal *1/3.1"
:in-theory (enable car-list-equals-root-lt))
("Subgoal *1/3.1'"
:use ((:instance orderedp-proper-ltree-to-list-ltree-L1
(x (cadr tree))
(left_tree (caddr tree))
(right_tree (cadddr tree)))))))
(defthm orderedp-ltree-sort
(orderedp (ltree-sort l))
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; :doc ":doc-section leftist-tree-sort
; ltree-sort produces an ordered list~/~/~/"
)
(defthm true-listp-ltree-sort
(true-listp (ltree-sort l))
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; :doc ":doc-section leftist-tree-sort
; ltree-sort produces a true-listp~/~/~/"
)
;(defthm how-many-lt-zero
; (implies (not (lexorder (root-lt tree) e))
; (equal (how-many-lt e tree) 0)))
; root(tree1) > root(tree2)
; root(tree2) > e
; ==> (how-many-lt e tree1) = 0
(local
(defthmd how-many-merge-lt-L1
(implies (and (consp tree1)
(consp tree2)
(proper-lt tree1)
(proper-lt tree2)
(not (lexorder (root-lt tree1)
(root-lt tree2)))
(not (lexorder (root-lt tree2)
e)))
(equal (how-many-lt e tree1) 0))))
; root(tree1) <= root(tree2)
; e < root(tree1)
; ==> (how-many-lt e tree2) = 0
(local
(defthmd how-many-merge-lt-L2
(implies (and (consp tree1)
(consp tree2)
(proper-lt tree1)
(proper-lt tree2)
(lexorder (root-lt tree1)
(root-lt tree2))
(not (lexorder (root-lt tree1) e)))
(equal (how-many-lt e tree2) 0))))
(defthm how-many-merge-lt
(implies (and (proper-lt tree1)
(proper-lt tree2))
(equal (how-many-lt e (merge-lt tree1 tree2))
(+ (how-many-lt e tree1)
(how-many-lt e tree2))))
:hints (("Goal"
:in-theory (enable how-many-merge-lt-L1
how-many-merge-lt-L2)
:induct (merge-lt tree1 tree2))))
(defthm how-many-insert-lt
(implies (proper-lt tree)
(equal (how-many-lt e (insert-lt e tree))
(+ 1 (how-many-lt e tree)))))
(defthm how-many-delete-min-lt
(implies (and (proper-lt tree)
(consp tree))
(equal (how-many-lt e (delete-min-lt tree))
(+ (how-many-lt e (left-lt tree))
(how-many-lt e (right-lt tree))))))
(defthm how-many-build-lt
(equal (how-many-lt e (build-lt l))
(how-many e l)))
(local
(defthm how-many-ltree-to-list-L1
(implies (and (proper-lt tree)
(not (lexorder x e))
(lexorder x (root-lt tree)))
(equal (how-many-lt e tree) 0))))
(defthm how-many-ltree-to-list
(implies (proper-lt tree)
(equal (how-many e (ltree-to-list tree))
(how-many-lt e tree))))
(defthm how-many-ltree-sort
(equal (how-many e (ltree-sort x))
(how-many e x))
;;; This legacy doc string was replaced Nov. 2014 by a corresponding
;;; auto-generated defxdoc form in file leftist-trees-xdoc.lisp.
; :doc ":doc-section leftist-tree-sort
; ltree-sort preserves how-many~/~/
; This is needed to prove that ltree-sort is equivalent to isort.~/"
)
|
