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Why3
why3
Commits
abdfa014
Commit
abdfa014
authored
Dec 16, 2010
by
JeanChristophe
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doc: Einsten's puzzle as another tutorial example
parent
49f961a2
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doc/macros.tex
doc/macros.tex
+1
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doc/manpages.tex
doc/manpages.tex
+2
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doc/manual.tex
doc/manual.tex
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doc/syntax.tex
doc/syntax.tex
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examples/einstein.why
examples/einstein.why
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doc/macros.tex
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abdfa014
...
...
@@ 4,6 +4,7 @@
\newcommand
{
\eg
}{
\emph
{
e.g.
}}
% BNF grammar
\newcommand
{
\keyword
}
[1]
{
\texttt
{
#1
}}
\newcommand
{
\indextt
}
[1]
{
\index
{
#1@
\protect\keyword
{
#1
}}}
\newcommand
{
\indexttbs
}
[1]
{
\index
{
#1@
\protect\keywordbs
{
#1
}}}
\newif\ifspace
...
...
doc/manpages.tex
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abdfa014
...
...
@@ 167,7 +167,8 @@ used to provide other informations :
\item
TODO
\end{itemize}
\section
{
The
\texttt
{
why3ml
}
tool
}
% TODO (pour plus tard)
% \section{The \texttt{why3ml} tool}
\section
{
The
\texttt
{
why3ide
}
tool
}
\label
{
sec:ideref
}
...
...
doc/manual.tex
View file @
abdfa014
\documentclass
[a4paper,11pt,twoside,openright]
{
memoir
}
% rubber: module index
\usepackage
[T1]
{
fontenc
}
%\usepackage{url}
\usepackage
[a4paper,pdftex,colorlinks=true,urlcolor=blue,pdfstartview=FitH]
{
hyperref
}
...
...
@@ 18,6 +20,8 @@
\input
{
./version.tex
}
\makeindex
\begin{document}
\sloppy
\hbadness
=1100
...
...
@@ 151,6 +155,9 @@ Johannes Kanig, St\'ephane Lescuyer, Sim\~ao Melo de Sousa.
\bibliography
{
manual
}
%\input{bibliodemons}
\cleardoublepage
\printindex
\end{document}
%%% Local Variables:
...
...
doc/syntax.tex
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abdfa014
...
...
@@ 98,6 +98,117 @@ the length of a list is nonnegative.
% \section{Using and Cloning Theories} *)
\section*
{
Another Example
}
\index
{
Einstein's logic problem
}
We now consider another, slightly more complex example: to use
\why\
to solve a little puzzle known as ``Einstein's logic
problem''
\footnote
{
This was contributed by St
\'
ephane Lescuyer.
}
.
The problem is stated as follows.
\begin{itemize}
\item
TODO
\end{itemize}
The question is: what is the nationality of the fish's owner?
We start by introducing a generalpurpose theory defining the notion
of
\emph
{
bijection
}
, as two abstract types together with two functions from
one to the other and two axioms stating that these functions are
inverse of each other.
\begin{verbatim}
theory Bijection
type t
type u
logic of t : u
logic to u : t
axiom To
_
of : forall x : t. to (of x) = x
axiom Of
_
to : forall y : u. of (to y) = y
end
\end{verbatim}
We now start a new theory,
\texttt
{
Einstein
}
, which will contain all
the individuals of the problem.
\begin{verbatim}
theory Einstein "Einstein's problem"
\end{verbatim}
First we introduce enumeration types for houses, colors, persons,
drinks, cigars and pets.
\begin{verbatim}
type house = H1  H2  H3  H4  H5
type color = Blue  Green  Red  White  Yellow
type person = Dane  Englishman  German  Norwegian  Swede
type drink = Beer  Coffee  Milk  Tea  Water
type cigar = Blend  BlueMaster  Dunhill  PallMall  Prince
type pet = Birds  Cats  Dogs  Fish  Horse
\end{verbatim}
We now express that each house is associated bijectively to a color,
by cloning the
\texttt
{
Bijection
}
theory appropriately.
\begin{verbatim}
clone Bijection as Color with type t = house, type u = color
\end{verbatim}
It introduces two functions, namely
\texttt
{
Color.of
}
and
\texttt
{
Color.to
}
, from houses to colors and colors to houses,
respectively, and the two axioms relating them.
Similarly, we express that each house is associated bijectively to a
person
\begin{verbatim}
clone Bijection as Owner with type t = house, type u = person
\end{verbatim}
and that drinks, cigars and pets are all associated bijectively to persons:
\begin{verbatim}
clone Bijection as Drink with type t = person, type u = drink
clone Bijection as Cigar with type t = person, type u = cigar
clone Bijection as Pet with type t = person, type u = pet
\end{verbatim}
Next we need a way to state that a person lives next to another. We
first define a predicate
\texttt
{
leftof
}
over two houses.
\begin{verbatim}
logic leftof (h1 h2 : house) =
match h1, h2 with
 H1, H2
 H2, H3
 H3, H4
 H4, H5 > true

_
> false
end
\end{verbatim}
Note how we advantageously used patternmatching, with a orpattern
for the four positive cases and a universal pattern for the remaining
21 cases. It is then immediate to define a
\texttt
{
neighbour
}
predicate over two houses, which completes theory
\texttt
{
Einstein
}
.
\begin{verbatim}
logic rightof (h1 h2 : house) =
leftof h2 h1
logic neighbour (h1 h2 : house) =
leftof h1 h2 or rightof h1 h2
end
\end{verbatim}
The next theory contains the 15 hypotheses. It starts by importing
theory
\texttt
{
Einstein
}
.
\begin{verbatim}
theory EinsteinHints "Hints"
use import Einstein
\end{verbatim}
Then each hypothesis is stated in terms of
\texttt
{
to
}
and
\texttt
{
of
}
functions. For instance, the hypothesis ``The Englishman lives in a
red house'' is declared as the following axiom.
\begin{verbatim}
axiom Hint1: Color.of (Owner.to Englishman) = Red
\end{verbatim}
And so on for all other hypotheses, up to
``The man who smokes Blends has a neighbour who drinks water'', which completes
this theory.
\begin{verbatim}
...
axiom Hint15:
neighbour (Owner.to (Cigar.to Blend)) (Owner.to (Drink.to Water))
end
\end{verbatim}
TODO
%%% Local Variables:
%%% mode: latex
...
...
examples/einstein.why
View file @
abdfa014
...
...
@@ 8,21 +8,21 @@ theory Bijection
type t
type u
logic
_
of t : u
logic
_
to u : t
logic of t : u
logic to u : t
axiom To_of : forall x : t.
_to(_of(x)
) = x
axiom Of_to : forall y : u.
_of(_to(y)
) = y
axiom To_of : forall x : t.
to (of x
) = x
axiom Of_to : forall y : u.
of (to y
) = y
end
theory Einstein "Einstein's problem"
(* Types *)
type house = H1  H2  H3  H4  H5
type color = Blue  Green  Red  White  Yellow
type house
= H1  H2  H3  H4  H5
type color
= Blue  Green  Red  White  Yellow
type person = Dane  Englishman  German  Norwegian  Swede
type drink = Beer  Coffee  Milk  Tea  Water
type cigar = Blend  BlueMaster  Dunhill  PallMall  Prince
type pet = Birds  Cats  Dogs  Fish  Horse
type drink
= Beer  Coffee  Milk  Tea  Water
type cigar
= Blend  BlueMaster  Dunhill  PallMall  Prince
type pet
= Birds  Cats  Dogs  Fish  Horse
(* Each house is associated bijectively to a color and a person *)
clone Bijection as Color with type t = house, type u = color
...
...
@@ 31,76 +31,76 @@ theory Einstein "Einstein's problem"
(* Each drink, cigar brand and pet are associated bijectively to a person *)
clone Bijection as Drink with type t = person, type u = drink
clone Bijection as Cigar with type t = person, type u = cigar
clone Bijection as Pet with type t = person, type u = pet
clone Bijection as Pet
with type t = person, type u = pet
(* Relative positions of the houses *)
logic left
_
of (h1 h2 : house) =
logic leftof (h1 h2 : house) =
match h1, h2 with
 H1, H2
> true
 H2, H3
> true
 H3, H4
> true
 H1, H2
 H2, H3
 H3, H4
 H4, H5 > true
 _
, _
> false
 _
> false
end
logic right
_
of (h1 h2 : house) =
left
_
of h2 h1
logic rightof (h1 h2 : house) =
leftof h2 h1
logic neighbour (h1 h2 : house) =
left
_of h1 h2 \/ right_
of h1 h2
left
of h1 h2 or right
of h1 h2
end
theory EinsteinHints "Hints"
use import Einstein
(* The Englishman lives in a red house *)
axiom Hint1: Color.
_of (Owner._
to Englishman) = Red
axiom Hint1: Color.
of (Owner.
to Englishman) = Red
(* The Swede has dogs *)
axiom Hint2: Pet.
_
of Swede = Dogs
axiom Hint2: Pet.of Swede = Dogs
(* The Dane drinks tea *)
axiom Hint3: Drink.
_
of Dane = Tea
axiom Hint3: Drink.of Dane = Tea
(* The green house is on the left of the white one *)
axiom Hint4: left
_of (Color._to Green) (Color._
to White)
axiom Hint4: left
of (Color.to Green) (Color.
to White)
(* The green house's owner drinks coffee *)
axiom Hint5: Drink.
_of (Owner._of (Color._
to Green)) = Coffee
axiom Hint5: Drink.
of (Owner.of (Color.
to Green)) = Coffee
(* The person who smokes Pall Mall has birds *)
axiom Hint6: Pet.
_of (Cigar._
to PallMall) = Birds
axiom Hint6: Pet.
of (Cigar.
to PallMall) = Birds
(* The yellow house's owner smokes Dunhill *)
axiom Hint7: Cigar.
_of (Owner._of (Color._
to Yellow)) = Dunhill
axiom Hint7: Cigar.
of (Owner.of (Color.
to Yellow)) = Dunhill
(* In the house in the center lives someone who drinks milk *)
axiom Hint8: Drink.
_of (Owner._
of H3) = Milk
axiom Hint8: Drink.
of (Owner.
of H3) = Milk
(* The Norwegian lives in the first house *)
axiom Hint9: Owner.
_
of H1 = Norwegian
axiom Hint9: Owner.of H1 = Norwegian
(* The man who smokes Blends lives next to the one who has cats *)
axiom Hint10: neighbour
(Owner.
_to (Cigar._to Blend)) (Owner._to (Pet._
to Cats))
(Owner.
to (Cigar.to Blend)) (Owner.to (Pet.
to Cats))
(* The man who owns a horse lives next to the one who smokes Dunhills *)
axiom Hint11: neighbour
(Owner.
_to (Pet._to Horse)) (Owner._to (Cigar._
to Dunhill))
(Owner.
to (Pet.to Horse)) (Owner.to (Cigar.
to Dunhill))
(* The man who smokes Blue Masters drinks beer *)
axiom Hint12:
Drink.
_of (Cigar._
to BlueMaster) = Beer
Drink.
of (Cigar.
to BlueMaster) = Beer
(* The German smokes Prince *)
axiom Hint13:
Cigar.
_
of German = Prince
Cigar.of German = Prince
(* The Norwegian lives next to the blue house *)
axiom Hint14:
neighbour (Owner.
_to Norwegian) (Color._
to Blue)
neighbour (Owner.
to Norwegian) (Color.
to Blue)
(* The man who smokes Blends has a neighbour who drinks water *)
axiom Hint15:
neighbour (Owner.
_to (Cigar._to Blend)) (Owner._to (Drink._
to Water))
neighbour (Owner.
to (Cigar.to Blend)) (Owner.to (Drink.
to Water))
end
...
...
@@ 108,20 +108,20 @@ theory Goals "Goals about Einstein's problem"
use import Einstein
use import EinsteinHints
(* lemma First_House_Not_White: Color.
_
of H1 <> White *)
(* lemma Last_House_Not_Green: Color.
_
of H5 <> Green *)
(* lemma First_House_Not_White: Color.of H1 <> White *)
(* lemma Last_House_Not_Green: Color.of H5 <> Green *)
(* lemma Blue_not_Red: Blue <> Red *)
(* lemma Englishman_not_H2: Owner.
_
to Englishman <> H2 *)
(* lemma Englishman_not_H1: Owner.
_
to Englishman <> H1 *)
(* lemma Englishman_not_H2: Owner.to Englishman <> H2 *)
(* lemma Englishman_not_H1: Owner.to Englishman <> H1 *)
(* lemma Second_House_Blue: Color.
_
of H2 = Blue *)
(* lemma Green_H4 : Color.
_
of H4 = Green *)
(* lemma White_H5 : Color.
_
of H5 = White *)
(* lemma Red_H3 : Color.
_
of H3 = Red *)
(* lemma Yellow_H1 : Color.
_
of H1 = Yellow *)
(* lemma Second_House_Blue: Color.of H2 = Blue *)
(* lemma Green_H4 : Color.of H4 = Green *)
(* lemma White_H5 : Color.of H5 = White *)
(* lemma Red_H3 : Color.of H3 = Red *)
(* lemma Yellow_H1 : Color.of H1 = Yellow *)
goal G: Pet.
_
to Fish = German
goal G: Pet.to Fish = German
end
...
...
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