--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Isar_examples/Hoare.thy Tue Oct 03 22:39:49 2000 +0200
@@ -0,0 +1,437 @@
+(* Title: HOL/Isar_examples/Hoare.thy
+ ID: $Id$
+ Author: Markus Wenzel, TU Muenchen
+
+A formulation of Hoare logic suitable for Isar.
+*)
+
+header {* Hoare Logic *}
+
+theory Hoare = Main
+files ("~~/src/HOL/Hoare/Hoare.ML"):
+
+subsection {* Abstract syntax and semantics *}
+
+text {*
+ The following abstract syntax and semantics of Hoare Logic over
+ \texttt{WHILE} programs closely follows the existing tradition in
+ Isabelle/HOL of formalizing the presentation given in
+ \cite[\S6]{Winskel:1993}. See also
+ \url{http://isabelle.in.tum.de/library/Hoare/} and
+ \cite{Nipkow:1998:Winskel}.
+*}
+
+types
+ 'a bexp = "'a set"
+ 'a assn = "'a set"
+
+datatype 'a com =
+ Basic "'a => 'a"
+ | Seq "'a com" "'a com" ("(_;/ _)" [60, 61] 60)
+ | Cond "'a bexp" "'a com" "'a com"
+ | While "'a bexp" "'a assn" "'a com"
+
+syntax
+ "_skip" :: "'a com" ("SKIP")
+translations
+ "SKIP" == "Basic id"
+
+types
+ 'a sem = "'a => 'a => bool"
+
+consts
+ iter :: "nat => 'a bexp => 'a sem => 'a sem"
+primrec
+ "iter 0 b S s s' = (s ~: b & s = s')"
+ "iter (Suc n) b S s s' =
+ (s : b & (EX s''. S s s'' & iter n b S s'' s'))"
+
+consts
+ Sem :: "'a com => 'a sem"
+primrec
+ "Sem (Basic f) s s' = (s' = f s)"
+ "Sem (c1; c2) s s' = (EX s''. Sem c1 s s'' & Sem c2 s'' s')"
+ "Sem (Cond b c1 c2) s s' =
+ (if s : b then Sem c1 s s' else Sem c2 s s')"
+ "Sem (While b x c) s s' = (EX n. iter n b (Sem c) s s')"
+
+constdefs
+ Valid :: "'a bexp => 'a com => 'a bexp => bool"
+ ("(3|- _/ (2_)/ _)" [100, 55, 100] 50)
+ "|- P c Q == ALL s s'. Sem c s s' --> s : P --> s' : Q"
+
+syntax (symbols)
+ Valid :: "'a bexp => 'a com => 'a bexp => bool"
+ ("(3\<turnstile> _/ (2_)/ _)" [100, 55, 100] 50)
+
+lemma ValidI [intro?]:
+ "(!!s s'. Sem c s s' ==> s : P ==> s' : Q) ==> |- P c Q"
+ by (simp add: Valid_def)
+
+lemma ValidD [dest?]:
+ "|- P c Q ==> Sem c s s' ==> s : P ==> s' : Q"
+ by (simp add: Valid_def)
+
+
+subsection {* Primitive Hoare rules *}
+
+text {*
+ From the semantics defined above, we derive the standard set of
+ primitive Hoare rules; e.g.\ see \cite[\S6]{Winskel:1993}. Usually,
+ variant forms of these rules are applied in actual proof, see also
+ \S\ref{sec:hoare-isar} and \S\ref{sec:hoare-vcg}.
+
+ \medskip The \name{basic} rule represents any kind of atomic access
+ to the state space. This subsumes the common rules of \name{skip}
+ and \name{assign}, as formulated in \S\ref{sec:hoare-isar}.
+*}
+
+theorem basic: "|- {s. f s : P} (Basic f) P"
+proof
+ fix s s' assume s: "s : {s. f s : P}"
+ assume "Sem (Basic f) s s'"
+ hence "s' = f s" by simp
+ with s show "s' : P" by simp
+qed
+
+text {*
+ The rules for sequential commands and semantic consequences are
+ established in a straight forward manner as follows.
+*}
+
+theorem seq: "|- P c1 Q ==> |- Q c2 R ==> |- P (c1; c2) R"
+proof
+ assume cmd1: "|- P c1 Q" and cmd2: "|- Q c2 R"
+ fix s s' assume s: "s : P"
+ assume "Sem (c1; c2) s s'"
+ then obtain s'' where sem1: "Sem c1 s s''" and sem2: "Sem c2 s'' s'"
+ by auto
+ from cmd1 sem1 s have "s'' : Q" ..
+ with cmd2 sem2 show "s' : R" ..
+qed
+
+theorem conseq: "P' <= P ==> |- P c Q ==> Q <= Q' ==> |- P' c Q'"
+proof
+ assume P'P: "P' <= P" and QQ': "Q <= Q'"
+ assume cmd: "|- P c Q"
+ fix s s' :: 'a
+ assume sem: "Sem c s s'"
+ assume "s : P'" with P'P have "s : P" ..
+ with cmd sem have "s' : Q" ..
+ with QQ' show "s' : Q'" ..
+qed
+
+text {*
+ The rule for conditional commands is directly reflected by the
+ corresponding semantics; in the proof we just have to look closely
+ which cases apply.
+*}
+
+theorem cond:
+ "|- (P Int b) c1 Q ==> |- (P Int -b) c2 Q ==> |- P (Cond b c1 c2) Q"
+proof
+ assume case_b: "|- (P Int b) c1 Q" and case_nb: "|- (P Int -b) c2 Q"
+ fix s s' assume s: "s : P"
+ assume sem: "Sem (Cond b c1 c2) s s'"
+ show "s' : Q"
+ proof cases
+ assume b: "s : b"
+ from case_b show ?thesis
+ proof
+ from sem b show "Sem c1 s s'" by simp
+ from s b show "s : P Int b" by simp
+ qed
+ next
+ assume nb: "s ~: b"
+ from case_nb show ?thesis
+ proof
+ from sem nb show "Sem c2 s s'" by simp
+ from s nb show "s : P Int -b" by simp
+ qed
+ qed
+qed
+
+text {*
+ The \name{while} rule is slightly less trivial --- it is the only one
+ based on recursion, which is expressed in the semantics by a
+ Kleene-style least fixed-point construction. The auxiliary statement
+ below, which is by induction on the number of iterations is the main
+ point to be proven; the rest is by routine application of the
+ semantics of \texttt{WHILE}.
+*}
+
+theorem while: "|- (P Int b) c P ==> |- P (While b X c) (P Int -b)"
+proof
+ assume body: "|- (P Int b) c P"
+ fix s s' assume s: "s : P"
+ assume "Sem (While b X c) s s'"
+ then obtain n where iter: "iter n b (Sem c) s s'" by auto
+ show "s' : P Int -b"
+ proof -
+ have "ALL s s'. iter n b (Sem c) s s' --> s : P --> s' : P Int -b"
+ (is "?P n")
+ proof (induct (stripped) n)
+ fix s s' assume s: "s : P"
+ {
+ assume "iter 0 b (Sem c) s s'"
+ with s show "s' : P Int -b" by auto
+ next
+ fix n assume hyp: "?P n"
+ assume "iter (Suc n) b (Sem c) s s'"
+ then obtain s'' where b: "s : b" and sem: "Sem c s s''"
+ and iter: "iter n b (Sem c) s'' s'"
+ by auto
+ from s b have "s : P Int b" by simp
+ with body sem have "s'' : P" ..
+ with hyp iter show "s' : P Int -b" by simp
+ }
+ qed
+ with iter s show ?thesis by simp
+ qed
+qed
+
+
+subsection {* Concrete syntax for assertions *}
+
+text {*
+ We now introduce concrete syntax for describing commands (with
+ embedded expressions) and assertions. The basic technique is that of
+ semantic ``quote-antiquote''. A \emph{quotation} is a syntactic
+ entity delimited by an implicit abstraction, say over the state
+ space. An \emph{antiquotation} is a marked expression within a
+ quotation that refers the implicit argument; a typical antiquotation
+ would select (or even update) components from the state.
+
+ We will see some examples later in the concrete rules and
+ applications.
+*}
+
+text {*
+ The following specification of syntax and translations is for
+ Isabelle experts only; feel free to ignore it.
+
+ While the first part is still a somewhat intelligible specification
+ of the concrete syntactic representation of our Hoare language, the
+ actual ``ML drivers'' is quite involved. Just note that the we
+ re-use the basic quote/antiquote translations as already defined in
+ Isabelle/Pure (see \verb,Syntax.quote_tr, and
+ \verb,Syntax.quote_tr',).
+*}
+
+syntax
+ "_update_name" :: idt
+ "_quote" :: "'b => ('a => 'b)" ("(.'(_').)" [0] 1000)
+ "_antiquote" :: "('a => 'b) => 'b" ("`_" [1000] 1000)
+ "_Assert" :: "'a => 'a set" ("(.{_}.)" [0] 1000)
+ "_Assign" :: "idt => 'b => 'a com" ("(`_ :=/ _)" [70, 65] 61)
+ "_Cond" :: "'a bexp => 'a com => 'a com => 'a com"
+ ("(0IF _/ THEN _/ ELSE _/ FI)" [0, 0, 0] 61)
+ "_While_inv" :: "'a bexp => 'a assn => 'a com => 'a com"
+ ("(0WHILE _/ INV _ //DO _ /OD)" [0, 0, 0] 61)
+ "_While" :: "'a bexp => 'a com => 'a com"
+ ("(0WHILE _ //DO _ /OD)" [0, 0] 61)
+
+syntax (xsymbols)
+ "_Assert" :: "'a => 'a set" ("(\<lbrace>_\<rbrace>)" [0] 1000)
+
+translations
+ ".{b}." => "Collect .(b)."
+ "`x := a" => "Basic .(`(_update_name x a))."
+ "IF b THEN c1 ELSE c2 FI" => "Cond (Collect .(b).) c1 c2"
+ "WHILE b INV i DO c OD" => "While (Collect .(b).) i c"
+ "WHILE b DO c OD" == "WHILE b INV arbitrary DO c OD"
+
+parse_translation {*
+ let
+ fun update_name_tr (Free (x, T) :: ts) =
+ Term.list_comb (Free (suffix RecordPackage.updateN x, T), ts)
+ | update_name_tr (Const (x, T) :: ts) =
+ Term.list_comb (Const (suffix RecordPackage.updateN x, T), ts)
+ | update_name_tr
+ (((c as Const ("_constrain", _)) $ t $ ty) :: ts) =
+ Term.list_comb (c $ update_name_tr [t] $
+ (Syntax.const "fun" $ ty $ Syntax.const "dummy"), ts)
+ | update_name_tr ts = raise TERM ("update_name_tr", ts);
+
+ fun quote_tr [t] = Syntax.quote_tr "_antiquote" t
+ | quote_tr ts = raise TERM ("quote_tr", ts);
+ in [("_update_name", update_name_tr), ("_quote", quote_tr)] end
+*}
+
+text {*
+ As usual in Isabelle syntax translations, the part for printing is
+ more complicated --- we cannot express parts as macro rules as above.
+ Don't look here, unless you have to do similar things for yourself.
+*}
+
+print_translation {*
+ let
+ fun quote_tr' f (t :: ts) =
+ Term.list_comb (f $ Syntax.quote_tr' "_antiquote" t, ts)
+ | quote_tr' _ _ = raise Match;
+
+ val assert_tr' = quote_tr' (Syntax.const "_Assert");
+
+ fun bexp_tr' name ((Const ("Collect", _) $ t) :: ts) =
+ quote_tr' (Syntax.const name) (t :: ts)
+ | bexp_tr' _ _ = raise Match;
+
+ fun upd_tr' (x_upd, T) =
+ (case try (unsuffix RecordPackage.updateN) x_upd of
+ Some x => (x, if T = dummyT then T else Term.domain_type T)
+ | None => raise Match);
+
+ fun update_name_tr' (Free x) = Free (upd_tr' x)
+ | update_name_tr' ((c as Const ("_free", _)) $ Free x) =
+ c $ Free (upd_tr' x)
+ | update_name_tr' (Const x) = Const (upd_tr' x)
+ | update_name_tr' _ = raise Match;
+
+ fun assign_tr' (Abs (x, _, f $ t $ Bound 0) :: ts) =
+ quote_tr' (Syntax.const "_Assign" $ update_name_tr' f)
+ (Abs (x, dummyT, t) :: ts)
+ | assign_tr' _ = raise Match;
+ in
+ [("Collect", assert_tr'), ("Basic", assign_tr'),
+ ("Cond", bexp_tr' "_Cond"), ("While", bexp_tr' "_While_inv")]
+ end
+*}
+
+
+subsection {* Rules for single-step proof \label{sec:hoare-isar} *}
+
+text {*
+ We are now ready to introduce a set of Hoare rules to be used in
+ single-step structured proofs in Isabelle/Isar. We refer to the
+ concrete syntax introduce above.
+
+ \medskip Assertions of Hoare Logic may be manipulated in
+ calculational proofs, with the inclusion expressed in terms of sets
+ or predicates. Reversed order is supported as well.
+*}
+
+lemma [trans]: "|- P c Q ==> P' <= P ==> |- P' c Q"
+ by (unfold Valid_def) blast
+lemma [trans] : "P' <= P ==> |- P c Q ==> |- P' c Q"
+ by (unfold Valid_def) blast
+
+lemma [trans]: "Q <= Q' ==> |- P c Q ==> |- P c Q'"
+ by (unfold Valid_def) blast
+lemma [trans]: "|- P c Q ==> Q <= Q' ==> |- P c Q'"
+ by (unfold Valid_def) blast
+
+lemma [trans]:
+ "|- .{`P}. c Q ==> (!!s. P' s --> P s) ==> |- .{`P'}. c Q"
+ by (simp add: Valid_def)
+lemma [trans]:
+ "(!!s. P' s --> P s) ==> |- .{`P}. c Q ==> |- .{`P'}. c Q"
+ by (simp add: Valid_def)
+
+lemma [trans]:
+ "|- P c .{`Q}. ==> (!!s. Q s --> Q' s) ==> |- P c .{`Q'}."
+ by (simp add: Valid_def)
+lemma [trans]:
+ "(!!s. Q s --> Q' s) ==> |- P c .{`Q}. ==> |- P c .{`Q'}."
+ by (simp add: Valid_def)
+
+
+text {*
+ Identity and basic assignments.\footnote{The $\idt{hoare}$ method
+ introduced in \S\ref{sec:hoare-vcg} is able to provide proper
+ instances for any number of basic assignments, without producing
+ additional verification conditions.}
+*}
+
+lemma skip [intro?]: "|- P SKIP P"
+proof -
+ have "|- {s. id s : P} SKIP P" by (rule basic)
+ thus ?thesis by simp
+qed
+
+lemma assign: "|- .{`(x_update `a) : P}. `x := `a P"
+ by (rule basic)
+
+text {*
+ Note that above formulation of assignment corresponds to our
+ preferred way to model state spaces, using (extensible) record types
+ in HOL \cite{Naraschewski-Wenzel:1998:HOOL}. For any record field
+ $x$, Isabelle/HOL provides a functions $x$ (selector) and
+ $\idt{x{\dsh}update}$ (update). Above, there is only a place-holder
+ appearing for the latter kind of function: due to concrete syntax
+ \isa{`x := `a} also contains \isa{x\_update}.\footnote{Note that due
+ to the external nature of HOL record fields, we could not even state
+ a general theorem relating selector and update functions (if this
+ were required here); this would only work for any particular instance
+ of record fields introduced so far.}
+*}
+
+text {*
+ Sequential composition --- normalizing with associativity achieves
+ proper of chunks of code verified separately.
+*}
+
+lemmas [trans, intro?] = seq
+
+lemma seq_assoc [simp]: "( |- P c1;(c2;c3) Q) = ( |- P (c1;c2);c3 Q)"
+ by (auto simp add: Valid_def)
+
+text {*
+ Conditional statements.
+*}
+
+lemmas [trans, intro?] = cond
+
+lemma [trans, intro?]:
+ "|- .{`P & `b}. c1 Q
+ ==> |- .{`P & ~ `b}. c2 Q
+ ==> |- .{`P}. IF `b THEN c1 ELSE c2 FI Q"
+ by (rule cond) (simp_all add: Valid_def)
+
+text {*
+ While statements --- with optional invariant.
+*}
+
+lemma [intro?]:
+ "|- (P Int b) c P ==> |- P (While b P c) (P Int -b)"
+ by (rule while)
+
+lemma [intro?]:
+ "|- (P Int b) c P ==> |- P (While b arbitrary c) (P Int -b)"
+ by (rule while)
+
+
+lemma [intro?]:
+ "|- .{`P & `b}. c .{`P}.
+ ==> |- .{`P}. WHILE `b INV .{`P}. DO c OD .{`P & ~ `b}."
+ by (simp add: while Collect_conj_eq Collect_neg_eq)
+
+lemma [intro?]:
+ "|- .{`P & `b}. c .{`P}.
+ ==> |- .{`P}. WHILE `b DO c OD .{`P & ~ `b}."
+ by (simp add: while Collect_conj_eq Collect_neg_eq)
+
+
+subsection {* Verification conditions \label{sec:hoare-vcg} *}
+
+text {*
+ We now load the \emph{original} ML file for proof scripts and tactic
+ definition for the Hoare Verification Condition Generator (see
+ \url{http://isabelle.in.tum.de/library/Hoare/}). As far as we are
+ concerned here, the result is a proof method \name{hoare}, which may
+ be applied to a Hoare Logic assertion to extract purely logical
+ verification conditions. It is important to note that the method
+ requires \texttt{WHILE} loops to be fully annotated with invariants
+ beforehand. Furthermore, only \emph{concrete} pieces of code are
+ handled --- the underlying tactic fails ungracefully if supplied with
+ meta-variables or parameters, for example.
+*}
+
+ML {* val Valid_def = thm "Valid_def" *}
+use "~~/src/HOL/Hoare/Hoare.ML"
+
+method_setup hoare = {*
+ Method.no_args
+ (Method.SIMPLE_METHOD' HEADGOAL (hoare_tac (K all_tac))) *}
+ "verification condition generator for Hoare logic"
+
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Isar_examples/HoareEx.thy Tue Oct 03 22:39:49 2000 +0200
@@ -0,0 +1,287 @@
+
+header {* Using Hoare Logic *}
+
+theory HoareEx = Hoare:
+
+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
+ "|- .{`(N_update (2 * `N)) : .{`N = #10}.}. `N := 2 * `N .{`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}. `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 simp
+
+lemma
+"|- .{`M = a & `N = b}.
+ `I := `M; `M := `N; `N := `I
+ .{`M = b & `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 "|- .{`N = a}. `N := `N .{`N = a}."
+ by hoare
+
+lemma "|- .{`x = a}. `x := `x .{`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 "|- .{`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 = n --> m + 1 ~= n"
+ -- {* inclusion of assertions expressed in ``pure'' logic, *}
+ -- {* without mentioning the state space *}
+ 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 {* 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
+ "|- .{`M = 0 & `S = 0}.
+ WHILE `M ~= a
+ DO `S := `S + b; `M := `M + 1 OD
+ .{`S = a * b}."
+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 {*
+ 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
+ "|- .{`M = 0 & `S = 0}.
+ WHILE `M ~= a
+ INV .{`S = `M * b}.
+ DO `S := `S + b; `M := `M + 1 OD
+ .{`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.
+*}
+
+consts
+ sum :: "(nat => nat) => nat => nat"
+primrec
+ "sum f 0 = 0"
+ "sum f (Suc n) = f n + sum f n"
+
+syntax
+ "_sum" :: "idt => nat => nat => nat"
+ ("SUM _<_. _" [0, 0, 10] 10)
+translations
+ "SUM j<k. b" == "sum (\<lambda>j. b) k"
+
+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.
+*}
+
+theorem
+ "|- .{True}.
+ `S := 0; `I := 1;
+ WHILE `I ~= n
+ DO
+ `S := `S + `I;
+ `I := `I + 1
+ OD
+ .{`S = (SUM j<n. j)}."
+ (is "|- _ (_; ?while) _")
+proof -
+ let ?sum = "\<lambda>k. SUM j<k. j"
+ let ?inv = "\<lambda>s i. 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 "!!s i. ?inv s i & i ~= n --> ?inv (s + i) (i + 1)"
+ 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 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}.
+ `S := 0; `I := 1;
+ WHILE `I ~= n
+ INV .{`S = (SUM j<`I. j)}.
+ DO
+ `S := `S + `I;
+ `I := `I + 1
+ OD
+ .{`S = (SUM j<n. j)}."
+proof -
+ let ?sum = "\<lambda>k. SUM j<k. j"
+ let ?inv = "\<lambda>s i. 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}.
+ `S := 0; `I := 1;
+ WHILE `I ~= n
+ INV .{`S = (SUM j<`I. j)}.
+ DO
+ `S := `S + `I;
+ `I := `I + 1
+ OD
+ .{`S = (SUM j<n. j)}."
+ by hoare auto
+
+end
\ No newline at end of file
--- a/src/HOL/Isar_examples/document/root.bib Tue Oct 03 22:37:16 2000 +0200
+++ b/src/HOL/Isar_examples/document/root.bib Tue Oct 03 22:39:49 2000 +0200
@@ -4,7 +4,6 @@
@string{Springer="Springer-Verlag"}
@string{TUM="TU Munich"}
-
@Book{Concrete-Math,
author = {R. L. Graham and D. E. Knuth and O. Patashnik},
title = {Concrete Mathematics},
@@ -12,11 +11,33 @@
year = 1989
}
+@InProceedings{Naraschewski-Wenzel:1998:HOOL,
+ author = {Wolfgang Naraschewski and Markus Wenzel},
+ title = {Object-Oriented Verification based on Record Subtyping in
+ {H}igher-{O}rder {L}ogic},
+ crossref = {tphols98}}
+
+@Article{Nipkow:1998:Winskel,
+ author = {Tobias Nipkow},
+ title = {Winskel is (almost) Right: Towards a Mechanized Semantics Textbook},
+ journal = {Formal Aspects of Computing},
+ year = 1998,
+ volume = 10,
+ pages = {171--186}
+}
+
@InProceedings{Wenzel:1999:TPHOL,
author = {Markus Wenzel},
title = {{Isar} --- a Generic Interpretative Approach to Readable Formal Proof Documents},
crossref = {tphols99}}
+@Book{Winskel:1993,
+ author = {G. Winskel},
+ title = {The Formal Semantics of Programming Languages},
+ publisher = {MIT Press},
+ year = 1993
+}
+
@Book{davey-priestley,
author = {B. A. Davey and H. A. Priestley},
title = {Introduction to Lattices and Order},
@@ -53,6 +74,14 @@
note = {\url{http://www.ftp.cl.cam.ac.uk/ftp/papers/reports/}}
}
+@Proceedings{tphols98,
+ title = {Theorem Proving in Higher Order Logics: {TPHOLs} '98},
+ booktitle = {Theorem Proving in Higher Order Logics: {TPHOLs} '98},
+ editor = {Jim Grundy and Malcom Newey},
+ series = {LNCS},
+ volume = 1479,
+ year = 1998}
+
@Proceedings{tphols99,
title = {Theorem Proving in Higher Order Logics: {TPHOLs} '99},
booktitle = {Theorem Proving in Higher Order Logics: {TPHOLs} '99},