src/HOL/Isar_Examples/Hoare.thy
author hoelzl
Tue Mar 26 12:20:58 2013 +0100 (2013-03-26)
changeset 51526 155263089e7b
parent 48891 c0eafbd55de3
child 51717 9e7d1c139569
permissions -rw-r--r--
move SEQ.thy and Lim.thy to Limits.thy
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(*  Title:      HOL/Isar_Examples/Hoare.thy
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    Author:     Markus Wenzel, TU Muenchen
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A formulation of Hoare logic suitable for Isar.
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*)
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header {* Hoare Logic *}
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theory Hoare
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imports Main
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begin
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subsection {* Abstract syntax and semantics *}
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text {* The following abstract syntax and semantics of Hoare Logic
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  over \texttt{WHILE} programs closely follows the existing tradition
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  in Isabelle/HOL of formalizing the presentation given in
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  \cite[\S6]{Winskel:1993}.  See also @{file "~~/src/HOL/Hoare"} and
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  \cite{Nipkow:1998:Winskel}. *}
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type_synonym 'a bexp = "'a set"
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type_synonym 'a assn = "'a set"
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datatype 'a com =
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    Basic "'a => 'a"
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  | Seq "'a com" "'a com"    ("(_;/ _)" [60, 61] 60)
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  | Cond "'a bexp" "'a com" "'a com"
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  | While "'a bexp" "'a assn" "'a com"
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abbreviation Skip  ("SKIP")
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  where "SKIP == Basic id"
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type_synonym 'a sem = "'a => 'a => bool"
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primrec iter :: "nat => 'a bexp => 'a sem => 'a sem"
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where
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  "iter 0 b S s s' = (s ~: b & s = s')"
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| "iter (Suc n) b S s s' = (s : b & (EX s''. S s s'' & iter n b S s'' s'))"
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primrec Sem :: "'a com => 'a sem"
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where
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  "Sem (Basic f) s s' = (s' = f s)"
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| "Sem (c1; c2) s s' = (EX s''. Sem c1 s s'' & Sem c2 s'' s')"
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| "Sem (Cond b c1 c2) s s' =
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    (if s : b then Sem c1 s s' else Sem c2 s s')"
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| "Sem (While b x c) s s' = (EX n. iter n b (Sem c) s s')"
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definition Valid :: "'a bexp => 'a com => 'a bexp => bool"
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    ("(3|- _/ (2_)/ _)" [100, 55, 100] 50)
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  where "|- P c Q \<longleftrightarrow> (\<forall>s s'. Sem c s s' --> s : P --> s' : Q)"
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notation (xsymbols) Valid  ("(3\<turnstile> _/ (2_)/ _)" [100, 55, 100] 50)
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lemma ValidI [intro?]:
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    "(!!s s'. Sem c s s' ==> s : P ==> s' : Q) ==> |- P c Q"
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  by (simp add: Valid_def)
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lemma ValidD [dest?]:
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    "|- P c Q ==> Sem c s s' ==> s : P ==> s' : Q"
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  by (simp add: Valid_def)
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subsection {* Primitive Hoare rules *}
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text {* From the semantics defined above, we derive the standard set
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  of primitive Hoare rules; e.g.\ see \cite[\S6]{Winskel:1993}.
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  Usually, variant forms of these rules are applied in actual proof,
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  see also \S\ref{sec:hoare-isar} and \S\ref{sec:hoare-vcg}.
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  \medskip The \name{basic} rule represents any kind of atomic access
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  to the state space.  This subsumes the common rules of \name{skip}
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  and \name{assign}, as formulated in \S\ref{sec:hoare-isar}. *}
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theorem basic: "|- {s. f s : P} (Basic f) P"
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proof
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  fix s s' assume s: "s : {s. f s : P}"
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  assume "Sem (Basic f) s s'"
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  then have "s' = f s" by simp
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  with s show "s' : P" by simp
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qed
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text {*
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 The rules for sequential commands and semantic consequences are
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 established in a straight forward manner as follows.
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*}
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theorem seq: "|- P c1 Q ==> |- Q c2 R ==> |- P (c1; c2) R"
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proof
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  assume cmd1: "|- P c1 Q" and cmd2: "|- Q c2 R"
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  fix s s' assume s: "s : P"
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  assume "Sem (c1; c2) s s'"
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  then obtain s'' where sem1: "Sem c1 s s''" and sem2: "Sem c2 s'' s'"
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    by auto
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  from cmd1 sem1 s have "s'' : Q" ..
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  with cmd2 sem2 show "s' : R" ..
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qed
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theorem conseq: "P' <= P ==> |- P c Q ==> Q <= Q' ==> |- P' c Q'"
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proof
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  assume P'P: "P' <= P" and QQ': "Q <= Q'"
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  assume cmd: "|- P c Q"
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  fix s s' :: 'a
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  assume sem: "Sem c s s'"
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  assume "s : P'" with P'P have "s : P" ..
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  with cmd sem have "s' : Q" ..
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  with QQ' show "s' : Q'" ..
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qed
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text {* The rule for conditional commands is directly reflected by the
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  corresponding semantics; in the proof we just have to look closely
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  which cases apply. *}
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theorem cond:
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  assumes case_b: "|- (P Int b) c1 Q"
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    and case_nb: "|- (P Int -b) c2 Q"
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  shows "|- P (Cond b c1 c2) Q"
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proof
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  fix s s' assume s: "s : P"
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  assume sem: "Sem (Cond b c1 c2) s s'"
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  show "s' : Q"
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  proof cases
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    assume b: "s : b"
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    from case_b show ?thesis
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    proof
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      from sem b show "Sem c1 s s'" by simp
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      from s b show "s : P Int b" by simp
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    qed
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  next
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    assume nb: "s ~: b"
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    from case_nb show ?thesis
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    proof
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      from sem nb show "Sem c2 s s'" by simp
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      from s nb show "s : P Int -b" by simp
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    qed
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  qed
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qed
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text {* The @{text while} rule is slightly less trivial --- it is the
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  only one based on recursion, which is expressed in the semantics by
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  a Kleene-style least fixed-point construction.  The auxiliary
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  statement below, which is by induction on the number of iterations
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  is the main point to be proven; the rest is by routine application
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  of the semantics of \texttt{WHILE}. *}
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theorem while:
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  assumes body: "|- (P Int b) c P"
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  shows "|- P (While b X c) (P Int -b)"
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proof
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  fix s s' assume s: "s : P"
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  assume "Sem (While b X c) s s'"
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  then obtain n where "iter n b (Sem c) s s'" by auto
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  from this and s show "s' : P Int -b"
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  proof (induct n arbitrary: s)
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    case 0
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    then show ?case by auto
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  next
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    case (Suc n)
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    then obtain s'' where b: "s : b" and sem: "Sem c s s''"
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      and iter: "iter n b (Sem c) s'' s'" by auto
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    from Suc and b have "s : P Int b" by simp
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    with body sem have "s'' : P" ..
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    with iter show ?case by (rule Suc)
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  qed
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qed
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subsection {* Concrete syntax for assertions *}
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text {* We now introduce concrete syntax for describing commands (with
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  embedded expressions) and assertions. The basic technique is that of
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  semantic ``quote-antiquote''.  A \emph{quotation} is a syntactic
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  entity delimited by an implicit abstraction, say over the state
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  space.  An \emph{antiquotation} is a marked expression within a
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  quotation that refers the implicit argument; a typical antiquotation
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  would select (or even update) components from the state.
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  We will see some examples later in the concrete rules and
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  applications. *}
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text {* The following specification of syntax and translations is for
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  Isabelle experts only; feel free to ignore it.
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  While the first part is still a somewhat intelligible specification
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  of the concrete syntactic representation of our Hoare language, the
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  actual ``ML drivers'' is quite involved.  Just note that the we
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  re-use the basic quote/antiquote translations as already defined in
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  Isabelle/Pure (see @{ML Syntax_Trans.quote_tr}, and
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  @{ML Syntax_Trans.quote_tr'},). *}
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syntax
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  "_quote"       :: "'b => ('a => 'b)"       ("(.'(_').)" [0] 1000)
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  "_antiquote"   :: "('a => 'b) => 'b"       ("\<acute>_" [1000] 1000)
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  "_Subst"       :: "'a bexp \<Rightarrow> 'b \<Rightarrow> idt \<Rightarrow> 'a bexp"
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        ("_[_'/\<acute>_]" [1000] 999)
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  "_Assert"      :: "'a => 'a set"           ("(.{_}.)" [0] 1000)
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  "_Assign"      :: "idt => 'b => 'a com"    ("(\<acute>_ :=/ _)" [70, 65] 61)
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  "_Cond"        :: "'a bexp => 'a com => 'a com => 'a com"
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        ("(0IF _/ THEN _/ ELSE _/ FI)" [0, 0, 0] 61)
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  "_While_inv"   :: "'a bexp => 'a assn => 'a com => 'a com"
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        ("(0WHILE _/ INV _ //DO _ /OD)"  [0, 0, 0] 61)
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  "_While"       :: "'a bexp => 'a com => 'a com"
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        ("(0WHILE _ //DO _ /OD)"  [0, 0] 61)
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syntax (xsymbols)
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  "_Assert"      :: "'a => 'a set"            ("(\<lbrace>_\<rbrace>)" [0] 1000)
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translations
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  ".{b}."                   => "CONST Collect .(b)."
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  "B [a/\<acute>x]"                => ".{\<acute>(_update_name x (\<lambda>_. a)) \<in> B}."
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  "\<acute>x := a"                 => "CONST Basic .(\<acute>(_update_name x (\<lambda>_. a)))."
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  "IF b THEN c1 ELSE c2 FI" => "CONST Cond .{b}. c1 c2"
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  "WHILE b INV i DO c OD"   => "CONST While .{b}. i c"
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  "WHILE b DO c OD"         == "WHILE b INV CONST undefined DO c OD"
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parse_translation {*
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  let
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    fun quote_tr [t] = Syntax_Trans.quote_tr @{syntax_const "_antiquote"} t
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      | quote_tr ts = raise TERM ("quote_tr", ts);
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  in [(@{syntax_const "_quote"}, quote_tr)] end
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*}
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text {* As usual in Isabelle syntax translations, the part for
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  printing is more complicated --- we cannot express parts as macro
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  rules as above.  Don't look here, unless you have to do similar
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  things for yourself. *}
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print_translation {*
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  let
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    fun quote_tr' f (t :: ts) =
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          Term.list_comb (f $ Syntax_Trans.quote_tr' @{syntax_const "_antiquote"} t, ts)
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      | quote_tr' _ _ = raise Match;
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    val assert_tr' = quote_tr' (Syntax.const @{syntax_const "_Assert"});
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    fun bexp_tr' name ((Const (@{const_syntax Collect}, _) $ t) :: ts) =
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          quote_tr' (Syntax.const name) (t :: ts)
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      | bexp_tr' _ _ = raise Match;
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    fun assign_tr' (Abs (x, _, f $ k $ Bound 0) :: ts) =
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          quote_tr' (Syntax.const @{syntax_const "_Assign"} $ Syntax_Trans.update_name_tr' f)
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            (Abs (x, dummyT, Syntax_Trans.const_abs_tr' k) :: ts)
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      | assign_tr' _ = raise Match;
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  in
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   [(@{const_syntax Collect}, assert_tr'),
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    (@{const_syntax Basic}, assign_tr'),
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    (@{const_syntax Cond}, bexp_tr' @{syntax_const "_Cond"}),
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    (@{const_syntax While}, bexp_tr' @{syntax_const "_While_inv"})]
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  end
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*}
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subsection {* Rules for single-step proof \label{sec:hoare-isar} *}
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text {* We are now ready to introduce a set of Hoare rules to be used
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  in single-step structured proofs in Isabelle/Isar.  We refer to the
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  concrete syntax introduce above.
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  \medskip Assertions of Hoare Logic may be manipulated in
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  calculational proofs, with the inclusion expressed in terms of sets
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  or predicates.  Reversed order is supported as well. *}
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lemma [trans]: "|- P c Q ==> P' <= P ==> |- P' c Q"
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  by (unfold Valid_def) blast
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lemma [trans] : "P' <= P ==> |- P c Q ==> |- P' c Q"
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  by (unfold Valid_def) blast
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lemma [trans]: "Q <= Q' ==> |- P c Q ==> |- P c Q'"
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  by (unfold Valid_def) blast
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lemma [trans]: "|- P c Q ==> Q <= Q' ==> |- P c Q'"
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  by (unfold Valid_def) blast
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lemma [trans]:
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    "|- .{\<acute>P}. c Q ==> (!!s. P' s --> P s) ==> |- .{\<acute>P'}. c Q"
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  by (simp add: Valid_def)
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lemma [trans]:
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    "(!!s. P' s --> P s) ==> |- .{\<acute>P}. c Q ==> |- .{\<acute>P'}. c Q"
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  by (simp add: Valid_def)
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lemma [trans]:
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    "|- P c .{\<acute>Q}. ==> (!!s. Q s --> Q' s) ==> |- P c .{\<acute>Q'}."
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  by (simp add: Valid_def)
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lemma [trans]:
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    "(!!s. Q s --> Q' s) ==> |- P c .{\<acute>Q}. ==> |- P c .{\<acute>Q'}."
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  by (simp add: Valid_def)
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text {* Identity and basic assignments.\footnote{The $\idt{hoare}$
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  method introduced in \S\ref{sec:hoare-vcg} is able to provide proper
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  instances for any number of basic assignments, without producing
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  additional verification conditions.} *}
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lemma skip [intro?]: "|- P SKIP P"
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proof -
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  have "|- {s. id s : P} SKIP P" by (rule basic)
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  then show ?thesis by simp
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qed
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lemma assign: "|- P [\<acute>a/\<acute>x::'a] \<acute>x := \<acute>a P"
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  by (rule basic)
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text {* Note that above formulation of assignment corresponds to our
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  preferred way to model state spaces, using (extensible) record types
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  in HOL \cite{Naraschewski-Wenzel:1998:HOOL}.  For any record field
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  $x$, Isabelle/HOL provides a functions $x$ (selector) and
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  $\idt{x{\dsh}update}$ (update).  Above, there is only a place-holder
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  appearing for the latter kind of function: due to concrete syntax
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  \isa{\'x := \'a} also contains \isa{x\_update}.\footnote{Note that
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  due to the external nature of HOL record fields, we could not even
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  state a general theorem relating selector and update functions (if
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  this were required here); this would only work for any particular
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  instance of record fields introduced so far.} *}
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text {* Sequential composition --- normalizing with associativity
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  achieves proper of chunks of code verified separately. *}
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lemmas [trans, intro?] = seq
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lemma seq_assoc [simp]: "( |- P c1;(c2;c3) Q) = ( |- P (c1;c2);c3 Q)"
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  by (auto simp add: Valid_def)
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text {* Conditional statements. *}
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lemmas [trans, intro?] = cond
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lemma [trans, intro?]:
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  "|- .{\<acute>P & \<acute>b}. c1 Q
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      ==> |- .{\<acute>P & ~ \<acute>b}. c2 Q
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      ==> |- .{\<acute>P}. IF \<acute>b THEN c1 ELSE c2 FI Q"
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    by (rule cond) (simp_all add: Valid_def)
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text {* While statements --- with optional invariant. *}
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lemma [intro?]:
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    "|- (P Int b) c P ==> |- P (While b P c) (P Int -b)"
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  by (rule while)
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lemma [intro?]:
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    "|- (P Int b) c P ==> |- P (While b undefined c) (P Int -b)"
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  by (rule while)
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lemma [intro?]:
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  "|- .{\<acute>P & \<acute>b}. c .{\<acute>P}.
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    ==> |- .{\<acute>P}. WHILE \<acute>b INV .{\<acute>P}. DO c OD .{\<acute>P & ~ \<acute>b}."
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  by (simp add: while Collect_conj_eq Collect_neg_eq)
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lemma [intro?]:
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  "|- .{\<acute>P & \<acute>b}. c .{\<acute>P}.
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    ==> |- .{\<acute>P}. WHILE \<acute>b DO c OD .{\<acute>P & ~ \<acute>b}."
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  by (simp add: while Collect_conj_eq Collect_neg_eq)
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subsection {* Verification conditions \label{sec:hoare-vcg} *}
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text {* We now load the \emph{original} ML file for proof scripts and
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  tactic definition for the Hoare Verification Condition Generator
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  (see @{file "~~/src/HOL/Hoare/"}).  As far as we
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  are concerned here, the result is a proof method \name{hoare}, which
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  may be applied to a Hoare Logic assertion to extract purely logical
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  verification conditions.  It is important to note that the method
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  requires \texttt{WHILE} loops to be fully annotated with invariants
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  beforehand.  Furthermore, only \emph{concrete} pieces of code are
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  handled --- the underlying tactic fails ungracefully if supplied
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  with meta-variables or parameters, for example. *}
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lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q"
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  by (auto simp add: Valid_def)
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lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q"
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  by (auto simp: Valid_def)
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lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R"
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  by (auto simp: Valid_def)
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lemma CondRule:
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  "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}
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    \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q"
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  by (auto simp: Valid_def)
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lemma iter_aux:
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  "\<forall>s s'. Sem c s s' --> s : I & s : b --> s' : I ==>
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       (\<And>s s'. s : I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' : I & s' ~: b)"
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  apply(induct n)
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   apply clarsimp
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   apply (simp (no_asm_use))
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   apply blast
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  done
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lemma WhileRule:
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    "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q"
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  apply (clarsimp simp: Valid_def)
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  apply (drule iter_aux)
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    prefer 2
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    apply assumption
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   apply blast
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  apply blast
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  done
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lemma Compl_Collect: "- Collect b = {x. \<not> b x}"
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  by blast
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lemmas AbortRule = SkipRule  -- "dummy version"
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ML_file "~~/src/HOL/Hoare/hoare_tac.ML"
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method_setup hoare = {*
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  Scan.succeed (fn ctxt =>
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    (SIMPLE_METHOD'
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       (hoare_tac ctxt (simp_tac (HOL_basic_ss addsimps [@{thm "Record.K_record_comp"}] ))))) *}
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  "verification condition generator for Hoare logic"
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end