header {* Using Hoare Logic *}
theory Hoare_Ex
imports Hoare
begin
subsection {* State spaces *}
text {*
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.
*}
record vars =
I :: nat
M :: nat
N :: nat
S :: nat
text {*
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.
*}
subsection {* Basic examples *}
text {*
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.
*}
text {*
Using the basic \name{assign} rule directly is a bit cumbersome.
*}
lemma
"|- .{\<acute>(N_update (\<lambda>_. (2 * \<acute>N))) : .{\<acute>N = 10}.}. \<acute>N := 2 * \<acute>N .{\<acute>N = 10}."
by (rule assign)
text {*
Certainly we want the state modification already done, e.g.\ by
simplification. The \name{hoare} method performs the basic state
update for us; we may apply the Simplifier afterwards to achieve
``obvious'' consequences as well.
*}
lemma "|- .{True}. \<acute>N := 10 .{\<acute>N = 10}."
by hoare
lemma "|- .{2 * \<acute>N = 10}. \<acute>N := 2 * \<acute>N .{\<acute>N = 10}."
by hoare
lemma "|- .{\<acute>N = 5}. \<acute>N := 2 * \<acute>N .{\<acute>N = 10}."
by hoare simp
lemma "|- .{\<acute>N + 1 = a + 1}. \<acute>N := \<acute>N + 1 .{\<acute>N = a + 1}."
by hoare
lemma "|- .{\<acute>N = a}. \<acute>N := \<acute>N + 1 .{\<acute>N = a + 1}."
by hoare simp
lemma "|- .{a = a & b = b}. \<acute>M := a; \<acute>N := b .{\<acute>M = a & \<acute>N = b}."
by hoare
lemma "|- .{True}. \<acute>M := a; \<acute>N := b .{\<acute>M = a & \<acute>N = b}."
by hoare simp
lemma
"|- .{\<acute>M = a & \<acute>N = b}.
\<acute>I := \<acute>M; \<acute>M := \<acute>N; \<acute>N := \<acute>I
.{\<acute>M = b & \<acute>N = a}."
by hoare simp
text {*
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.
*}
lemma "|- .{\<acute>N = a}. \<acute>N := \<acute>N .{\<acute>N = a}."
by hoare
lemma "|- .{\<acute>x = a}. \<acute>x := \<acute>x .{\<acute>x = a}."
oops
lemma
"Valid {s. x s = a} (Basic (\<lambda>s. x_update (x s) s)) {s. x s = n}"
-- {* same statement without concrete syntax *}
oops
text {*
In the following assignments we make use of the consequence rule in
order to achieve the intended precondition. Certainly, the
\name{hoare} method is able to handle this case, too.
*}
lemma "|- .{\<acute>M = \<acute>N}. \<acute>M := \<acute>M + 1 .{\<acute>M ~= \<acute>N}."
proof -
have ".{\<acute>M = \<acute>N}. <= .{\<acute>M + 1 ~= \<acute>N}."
by auto
also have "|- ... \<acute>M := \<acute>M + 1 .{\<acute>M ~= \<acute>N}."
by hoare
finally show ?thesis .
qed
lemma "|- .{\<acute>M = \<acute>N}. \<acute>M := \<acute>M + 1 .{\<acute>M ~= \<acute>N}."
proof -
have "!!m n::nat. m = n --> m + 1 ~= n"
-- {* inclusion of assertions expressed in ``pure'' logic, *}
-- {* without mentioning the state space *}
by simp
also have "|- .{\<acute>M + 1 ~= \<acute>N}. \<acute>M := \<acute>M + 1 .{\<acute>M ~= \<acute>N}."
by hoare
finally show ?thesis .
qed
lemma "|- .{\<acute>M = \<acute>N}. \<acute>M := \<acute>M + 1 .{\<acute>M ~= \<acute>N}."
by hoare simp
subsection {* Multiplication by addition *}
text {*
We now do some basic examples of actual \texttt{WHILE} 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.
*}
lemma
"|- .{\<acute>M = 0 & \<acute>S = 0}.
WHILE \<acute>M ~= a
DO \<acute>S := \<acute>S + b; \<acute>M := \<acute>M + 1 OD
.{\<acute>S = a * b}."
proof -
let "|- _ ?while _" = ?thesis
let ".{\<acute>?inv}." = ".{\<acute>S = \<acute>M * b}."
have ".{\<acute>M = 0 & \<acute>S = 0}. <= .{\<acute>?inv}." by auto
also have "|- ... ?while .{\<acute>?inv & ~ (\<acute>M ~= a)}."
proof
let ?c = "\<acute>S := \<acute>S + b; \<acute>M := \<acute>M + 1"
have ".{\<acute>?inv & \<acute>M ~= a}. <= .{\<acute>S + b = (\<acute>M + 1) * b}."
by auto
also have "|- ... ?c .{\<acute>?inv}." by hoare
finally show "|- .{\<acute>?inv & \<acute>M ~= a}. ?c .{\<acute>?inv}." .
qed
also have "... <= .{\<acute>S = a * b}." by auto
finally show ?thesis .
qed
text {*
The subsequent version of the proof applies the \name{hoare} method
to reduce the Hoare statement to a purely logical problem that can be
solved fully automatically. Note that we have to specify the
\texttt{WHILE} loop invariant in the original statement.
*}
lemma
"|- .{\<acute>M = 0 & \<acute>S = 0}.
WHILE \<acute>M ~= a
INV .{\<acute>S = \<acute>M * b}.
DO \<acute>S := \<acute>S + b; \<acute>M := \<acute>M + 1 OD
.{\<acute>S = a * b}."
by hoare auto
subsection {* Summing natural numbers *}
text {*
We verify an imperative program to sum natural numbers up to a given
limit. First some functional definition for proper specification of
the problem.
*}
text {*
The following proof is quite explicit in the individual steps taken,
with the \name{hoare} 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.
*}
declare atLeast0LessThan[symmetric,simp]
theorem
"|- .{True}.
\<acute>S := 0; \<acute>I := 1;
WHILE \<acute>I ~= n
DO
\<acute>S := \<acute>S + \<acute>I;
\<acute>I := \<acute>I + 1
OD
.{\<acute>S = (SUM j<n. j)}."
(is "|- _ (_; ?while) _")
proof -
let ?sum = "\<lambda>k::nat. SUM j<k. j"
let ?inv = "\<lambda>s i::nat. s = ?sum i"
have "|- .{True}. \<acute>S := 0; \<acute>I := 1 .{?inv \<acute>S \<acute>I}."
proof -
have "True --> 0 = ?sum 1"
by simp
also have "|- .{...}. \<acute>S := 0; \<acute>I := 1 .{?inv \<acute>S \<acute>I}."
by hoare
finally show ?thesis .
qed
also have "|- ... ?while .{?inv \<acute>S \<acute>I & ~ \<acute>I ~= n}."
proof
let ?body = "\<acute>S := \<acute>S + \<acute>I; \<acute>I := \<acute>I + 1"
have "!!s i. ?inv s i & i ~= n --> ?inv (s + i) (i + 1)"
by simp
also have "|- .{\<acute>S + \<acute>I = ?sum (\<acute>I + 1)}. ?body .{?inv \<acute>S \<acute>I}."
by hoare
finally show "|- .{?inv \<acute>S \<acute>I & \<acute>I ~= n}. ?body .{?inv \<acute>S \<acute>I}." .
qed
also have "!!s i. s = ?sum i & ~ i ~= n --> s = ?sum n"
by simp
finally show ?thesis .
qed
text {*
The next version uses the \name{hoare} method, while still explaining
the resulting proof obligations in an abstract, structured manner.
*}
theorem
"|- .{True}.
\<acute>S := 0; \<acute>I := 1;
WHILE \<acute>I ~= n
INV .{\<acute>S = (SUM j<\<acute>I. j)}.
DO
\<acute>S := \<acute>S + \<acute>I;
\<acute>I := \<acute>I + 1
OD
.{\<acute>S = (SUM j<n. j)}."
proof -
let ?sum = "\<lambda>k::nat. SUM j<k. j"
let ?inv = "\<lambda>s i::nat. s = ?sum i"
show ?thesis
proof hoare
show "?inv 0 1" by simp
next
fix s i assume "?inv s i & i ~= n"
thus "?inv (s + i) (i + 1)" by simp
next
fix s i assume "?inv s i & ~ i ~= n"
thus "s = ?sum n" by simp
qed
qed
text {*
Certainly, this proof may be done fully automatic as well, provided
that the invariant is given beforehand.
*}
theorem
"|- .{True}.
\<acute>S := 0; \<acute>I := 1;
WHILE \<acute>I ~= n
INV .{\<acute>S = (SUM j<\<acute>I. j)}.
DO
\<acute>S := \<acute>S + \<acute>I;
\<acute>I := \<acute>I + 1
OD
.{\<acute>S = (SUM j<n. j)}."
by hoare auto
subsection{* Time *}
text{*
A simple embedding of time in Hoare logic: function @{text timeit}
inserts an extra variable to keep track of the elapsed time.
*}
record tstate = time :: nat
types 'a time = "\<lparr>time :: nat, \<dots> :: 'a\<rparr>"
consts timeit :: "'a time com \<Rightarrow> 'a time com"
primrec
"timeit (Basic f) = (Basic f; Basic(\<lambda>s. s\<lparr>time := Suc (time s)\<rparr>))"
"timeit (c1; c2) = (timeit c1; timeit c2)"
"timeit (Cond b c1 c2) = Cond b (timeit c1) (timeit c2)"
"timeit (While b iv c) = While b iv (timeit c)"
record tvars = tstate +
I :: nat
J :: nat
lemma lem: "(0::nat) < n \<Longrightarrow> n + n \<le> Suc (n * n)"
by (induct n) simp_all
lemma "|- .{i = \<acute>I & \<acute>time = 0}.
timeit(
WHILE \<acute>I \<noteq> 0
INV .{2*\<acute>time + \<acute>I*\<acute>I + 5*\<acute>I = i*i + 5*i}.
DO
\<acute>J := \<acute>I;
WHILE \<acute>J \<noteq> 0
INV .{0 < \<acute>I & 2*\<acute>time + \<acute>I*\<acute>I + 3*\<acute>I + 2*\<acute>J - 2 = i*i + 5*i}.
DO \<acute>J := \<acute>J - 1 OD;
\<acute>I := \<acute>I - 1
OD
) .{2*\<acute>time = i*i + 5*i}."
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