--- a/src/HOL/Isar_Examples/Hoare_Ex.thy Mon Nov 02 11:43:02 2015 +0100
+++ b/src/HOL/Isar_Examples/Hoare_Ex.thy Mon Nov 02 13:58:19 2015 +0100
@@ -6,9 +6,9 @@
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>
+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
@@ -16,29 +16,28 @@
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>
+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.\<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.\<close>
-text \<open>Using the basic @{text assign} rule directly is a bit
+text \<open>Using the basic \<open>assign\<close> rule directly is a bit
cumbersome.\<close>
lemma "\<turnstile> \<lbrace>\<acute>(N_update (\<lambda>_. (2 * \<acute>N))) \<in> \<lbrace>\<acute>N = 10\<rbrace>\<rbrace> \<acute>N := 2 * \<acute>N \<lbrace>\<acute>N = 10\<rbrace>"
by (rule assign)
-text \<open>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.\<close>
+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 "\<turnstile> \<lbrace>True\<rbrace> \<acute>N := 10 \<lbrace>\<acute>N = 10\<rbrace>"
by hoare
@@ -67,8 +66,8 @@
\<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
+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>
@@ -84,9 +83,9 @@
oops
-text \<open>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.\<close>
+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 "\<turnstile> \<lbrace>\<acute>M = \<acute>N\<rbrace> \<acute>M := \<acute>M + 1 \<lbrace>\<acute>M \<noteq> \<acute>N\<rbrace>"
proof -
@@ -114,10 +113,10 @@
subsection \<open>Multiplication by addition\<close>
-text \<open>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.\<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
"\<turnstile> \<lbrace>\<acute>M = 0 \<and> \<acute>S = 0\<rbrace>
@@ -141,10 +140,10 @@
finally show ?thesis .
qed
-text \<open>The subsequent version of the proof applies the @{text 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.\<close>
+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
"\<turnstile> \<lbrace>\<acute>M = 0 \<and> \<acute>S = 0\<rbrace>
@@ -157,15 +156,15 @@
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.\<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.
-text \<open>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.\<close>
+ \<^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
"\<turnstile> \<lbrace>True\<rbrace>
@@ -203,9 +202,8 @@
finally show ?thesis .
qed
-text \<open>The next version uses the @{text hoare} method, while still
- explaining the resulting proof obligations in an abstract,
- structured manner.\<close>
+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
"\<turnstile> \<lbrace>True\<rbrace>
@@ -230,8 +228,8 @@
qed
qed
-text \<open>Certainly, this proof may be done fully automatic as well,
- provided that the invariant is given beforehand.\<close>
+text \<open>Certainly, this proof may be done fully automatic as well, provided
+ that the invariant is given beforehand.\<close>
theorem
"\<turnstile> \<lbrace>True\<rbrace>
@@ -248,8 +246,8 @@
subsection \<open>Time\<close>
-text \<open>A simple embedding of time in Hoare logic: function @{text
- timeit} inserts an extra variable to keep track of the elapsed 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