section \<open>Using Hoare Logic\<close>
theory Hoare_Ex
imports Hoare
begin
subsection \<open>State spaces\<close>
text \<open>
First of all we provide a store of program variables that occur in any of
the programs considered later. Slightly unexpected things may happen when
attempting to work with undeclared variables.
\<close>
record vars =
I :: nat
M :: nat
N :: nat
S :: nat
text \<open>
While all of our variables happen to have the same type, nothing would
prevent us from working with many-sorted programs as well, or even
polymorphic ones. Also note that Isabelle/HOL's extensible record types even
provides simple means to extend the state space later.
\<close>
subsection \<open>Basic examples\<close>
text \<open>
We look at few trivialities involving assignment and sequential composition,
in order to get an idea of how to work with our formulation of Hoare Logic.
\<^medskip>
Using the basic \<open>assign\<close> rule directly is a bit cumbersome.
\<close>
lemma "\ \\(N_update (\_. (2 * \N))) \ \\N = 10\\ \N := 2 * \N \\N = 10\"
by (rule assign)
text \<open>
Certainly we want the state modification already done, e.g.\ by
simplification. The \<open>hoare\<close> method performs the basic state update for us;
we may apply the Simplifier afterwards to achieve ``obvious'' consequences
as well.
\<close>
lemma "\ \True\ \N := 10 \\N = 10\"
by hoare
lemma "\ \2 * \N = 10\ \N := 2 * \N \\N = 10\"
by hoare
lemma "\ \\N = 5\ \N := 2 * \N \\N = 10\"
by hoare simp
lemma "\ \\N + 1 = a + 1\ \N := \N + 1 \\N = a + 1\"
by hoare
lemma "\ \\N = a\ \N := \N + 1 \\N = a + 1\"
by hoare simp
lemma "\ \a = a \ b = b\ \M := a; \N := b \\M = a \ \N = b\"
by hoare
lemma "\ \True\ \M := a; \N := b \\M = a \ \N = b\"
by hoare
lemma
"\ \\M = a \ \N = b\
\<acute>I := \<acute>M; \<acute>M := \<acute>N; \<acute>N := \<acute>I
\<lbrace>\<acute>M = b \<and> \<acute>N = a\<rbrace>"
by hoare simp
text \<open>
It is important to note that statements like the following one can only be
proven for each individual program variable. Due to the extra-logical nature
of record fields, we cannot formulate a theorem relating record selectors
and updates schematically.
\<close>
lemma "\ \\N = a\ \N := \N \\N = a\"
by hoare
lemma "\ \\x = a\ \x := \x \\x = a\"
oops
lemma
"Valid {s. x s = a} (Basic (\s. x_update (x s) s)) {s. x s = n}"
\<comment> \<open>same statement without concrete syntax\<close>
oops
text \<open>
In the following assignments we make use of the consequence rule in order to
achieve the intended precondition. Certainly, the \<open>hoare\<close> method is able to
handle this case, too.
\<close>
lemma "\ \\M = \N\ \M := \M + 1 \\M \ \N\"
proof -
have "\\M = \N\ \ \\M + 1 \ \N\"
by auto
also have "\ \ \M := \M + 1 \\M \ \N\"
by hoare
finally show ?thesis .
qed
lemma "\ \\M = \N\ \M := \M + 1 \\M \ \N\"
proof -
have "m = n \ m + 1 \ n" for m n :: nat
\<comment> \<open>inclusion of assertions expressed in ``pure'' logic,\<close>
\<comment> \<open>without mentioning the state space\<close>
by simp
also have "\ \\M + 1 \ \N\ \M := \M + 1 \\M \ \N\"
by hoare
finally show ?thesis .
qed
lemma "\ \\M = \N\ \M := \M + 1 \\M \ \N\"
by hoare simp
subsection \<open>Multiplication by addition\<close>
text \<open>
We now do some basic examples of actual \<^verbatim>\<open>WHILE\<close> programs. This one is a
loop for calculating the product of two natural numbers, by iterated
addition. We first give detailed structured proof based on single-step Hoare
rules.
\<close>
lemma
"\ \\M = 0 \ \S = 0\
WHILE \<acute>M \<noteq> a
DO \<acute>S := \<acute>S + b; \<acute>M := \<acute>M + 1 OD
\<lbrace>\<acute>S = a * b\<rbrace>"
proof -
let "\ _ ?while _" = ?thesis
let "\\?inv\" = "\\S = \M * b\"
have "\\M = 0 \ \S = 0\ \ \\?inv\" by auto
also have "\ \ ?while \\?inv \ \ (\M \ a)\"
proof
let ?c = "\S := \S + b; \M := \M + 1"
have "\\?inv \ \M \ a\ \ \\S + b = (\M + 1) * b\"
by auto
also have "\ \ ?c \\?inv\" by hoare
finally show "\ \\?inv \ \M \ a\ ?c \\?inv\" .
qed
also have "\ \ \\S = a * b\" by auto
finally show ?thesis .
qed
text \<open>
The subsequent version of the proof applies the \<open>hoare\<close> method to reduce the
Hoare statement to a purely logical problem that can be solved fully
automatically. Note that we have to specify the \<^verbatim>\<open>WHILE\<close> loop invariant in
the original statement.
\<close>
lemma
"\ \\M = 0 \ \S = 0\
WHILE \<acute>M \<noteq> a
INV \<lbrace>\<acute>S = \<acute>M * b\<rbrace>
DO \<acute>S := \<acute>S + b; \<acute>M := \<acute>M + 1 OD
\<lbrace>\<acute>S = a * b\<rbrace>"
by hoare auto
subsection \<open>Summing natural numbers\<close>
text \<open>
We verify an imperative program to sum natural numbers up to a given limit.
First some functional definition for proper specification of the problem.
\<^medskip>
The following proof is quite explicit in the individual steps taken, with
the \<open>hoare\<close> method only applied locally to take care of assignment and
sequential composition. Note that we express intermediate proof obligation
in pure logic, without referring to the state space.
\<close>
theorem
"\ \True\
\<acute>S := 0; \<acute>I := 1;
WHILE \<acute>I \<noteq> n
DO
\<acute>S := \<acute>S + \<acute>I;
\<acute>I := \<acute>I + 1
OD
\<lbrace>\<acute>S = (\<Sum>j<n. j)\<rbrace>"
(is "\ _ (_; ?while) _")
proof -
let ?sum = "\k::nat. \j
let ?inv = "\s i::nat. s = ?sum i"
have "\ \True\ \S := 0; \I := 1 \?inv \S \I\"
proof -
have "True \ 0 = ?sum 1"
by simp
also have "\ \\\ \S := 0; \I := 1 \?inv \S \I\"
by hoare
finally show ?thesis .
qed
also have "\ \ ?while \?inv \S \I \ \ \I \ n\"
proof
let ?body = "\S := \S + \I; \I := \I + 1"
have "?inv s i \ i \ n \ ?inv (s + i) (i + 1)" for s i
by simp
also have "\ \\S + \I = ?sum (\I + 1)\ ?body \?inv \S \I\"
by hoare
finally show "\ \?inv \S \I \ \I \ n\ ?body \?inv \S \I\" .
qed
also have "s = ?sum i \ \ i \ n \ s = ?sum n" for s i
by simp
finally show ?thesis .
qed
text \<open>
The next version uses the \<open>hoare\<close> method, while still explaining the
resulting proof obligations in an abstract, structured manner.
\<close>
theorem
"\ \True\
\<acute>S := 0; \<acute>I := 1;
WHILE \<acute>I \<noteq> n
INV \<lbrace>\<acute>S = (\<Sum>j<\<acute>I. j)\<rbrace>
DO
\<acute>S := \<acute>S + \<acute>I;
\<acute>I := \<acute>I + 1
OD
\<lbrace>\<acute>S = (\<Sum>j<n. j)\<rbrace>"
proof -
let ?sum = "\k::nat. \j
let ?inv = "\s i::nat. s = ?sum i"
show ?thesis
proof hoare
show "?inv 0 1" by simp
show "?inv (s + i) (i + 1)" if "?inv s i \ i \ n" for s i
using that by simp
show "s = ?sum n" if "?inv s i \ \ i \ n" for s i
using that by simp
qed
qed
text \<open>
Certainly, this proof may be done fully automatic as well, provided that the
invariant is given beforehand.
\<close>
theorem
"\ \True\
\<acute>S := 0; \<acute>I := 1;
WHILE \<acute>I \<noteq> n
INV \<lbrace>\<acute>S = (\<Sum>j<\<acute>I. j)\<rbrace>
DO
\<acute>S := \<acute>S + \<acute>I;
\<acute>I := \<acute>I + 1
OD
\<lbrace>\<acute>S = (\<Sum>j<n. j)\<rbrace>"
by hoare auto
subsection \<open>Time\<close>
text \<open>
A simple embedding of time in Hoare logic: function \<open>timeit\<close> inserts an
extra variable to keep track of the elapsed time.
\<close>
record tstate = time :: nat
type_synonym 'a time = "\time :: nat, \ :: 'a\"
primrec timeit :: "'a time com \ 'a time com"
where
"timeit (Basic f) = (Basic f; Basic(\s. s\time := Suc (time s)\))"
| "timeit (c1; c2) = (timeit c1; timeit c2)"
| "timeit (Cond b c1 c2) = Cond b (timeit c1) (timeit c2)"
| "timeit (While b iv v c) = While b iv v (timeit c)"
record tvars = tstate +
I :: nat
J :: nat
lemma lem: "(0::nat) < n \ n + n \ Suc (n * n)"
by (induct n) simp_all
lemma
"\ \i = \I \ \time = 0\
(timeit
(WHILE \<acute>I \<noteq> 0
INV \<lbrace>2 *\<acute> time + \<acute>I * \<acute>I + 5 * \<acute>I = i * i + 5 * i\<rbrace>
DO
\<acute>J := \<acute>I;
WHILE \<acute>J \<noteq> 0
INV \<lbrace>0 < \<acute>I \<and> 2 * \<acute>time + \<acute>I * \<acute>I + 3 * \<acute>I + 2 * \<acute>J - 2 = i * i + 5 * i\<rbrace>
DO \<acute>J := \<acute>J - 1 OD;
\<acute>I := \<acute>I - 1
OD))
\<lbrace>2 * \<acute>time = i * i + 5 * i\<rbrace>"
apply simp
apply hoare
apply simp
apply clarsimp
apply clarsimp
apply arith
prefer 2
apply clarsimp
apply (clarsimp simp: nat_distrib)
apply (frule lem)
apply arith
done
end
¤ Dauer der Verarbeitung: 0.20 Sekunden
(vorverarbeitet)
¤
|
Haftungshinweis
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
|
