\part{Forward Proofs}

\frame[plain]{\partpage}

\begin{frame}
\frametitle{Kernel too detailed}
\begin{itemize}
\item we already discussed the HOL Logic
\item the kernel itself does not even contain basic logic operators
\item usually one uses a much higher level of abstraction
\begin{itemize}
\item many operations and datatypes are defined
\item high-level derived inference rules are used
\end{itemize}
\item let's now look at this more common abstraction level
\end{itemize}
\end{frame}


\section{Term Syntax}

\begin{frame}
\frametitle{Common Terms and Types}

\begin{tabular}{lcc}
& \emph{Unicode} & \emph{ASCII} \\
type vars & \hol{$\alpha$}, \hol{$\beta$}, \ldots & \hol{'a}, \hol{'b}, \ldots \\ 
type annotated term & \hol{term:type}  & \hol{term:type} \\ 
true & \hol{T} & \hol{T} \\
false & \hol{F} & \hol{F} \\
negation & \hol{$\neg$b} & \hol{\holNeg{}b} \\
conjunction & \hol{b1\ $\wedge$\ b2} & \hol{b1 \holAnd{} b2} \\
disjunction & \hol{b1\ $\vee$\ b2} & \hol{b1 \holOr{} b2} \\
implication & \hol{b1\ $\Longrightarrow$\ b2} & \hol{b1 \holImp{} b2} \\ 
equivalence & \hol{b1\ $\Longleftrightarrow$\ b2} & \hol{b1 \holEquiv{} b2} \\
disequation & \hol{v1\ $\neq$\ v2} & \hol{v1 <> v2} \\
all-quantification & \hol{$\forall$x.\ P x} & \hol{!x.\ P x} \\
existential quantification & \hol{$\exists$x.\ P x} & \hol{?x.\ P x} \\
Hilbert's choice operator & \hol{@x.\ P x} & \hol{@x.\ P x}
\end{tabular}
\bigskip

There are similar restrictions to constant and variable names as in SML.\\
HOL specific: don't start variable names with an underscore

\end{frame}

\begin{frame}
\frametitle{Syntax conventions}
\begin{itemize}
\item common function syntax
\begin{itemize}
\item prefix notation, \eg \hol{SUC x}
\item infix notation, \eg \hol{x + y}
\item quantifier notation, \eg \hol{$\forall$x.\ P x} means \hol{($\forall$)\ ($\lambda$x.\ P x)}
\end{itemize}
\item infix and quantifier notation can be turned into prefix notation \\
      Example: \hol{(+)\ x\ y} and \hol{\$+\ x\ y} are the same as \hol{x + y}
\item quantifiers of the same type don't need to be repeated \\
   Example:\ 
   \hol{$\forall$x\ y.\ P\ x\ y} is short for
   \hol{$\forall$x.\ $\forall$y.\ P\ x\ y}
\item there is special syntax for some functions\\
   Example:\ 
   \hol{if c then v1 else v2} is nice syntax for
   \hol{COND c v1 v2}
\item associative infix operators are usually right-associative\\
   Example:\ 
   \hol{b1 \holAnd{} b2 \holAnd{} b3} is parsed as
   \hol{b1 \holAnd{} (b2 \holAnd{} b3)}
\end{itemize}
\end{frame}




\begin{frame}
\frametitle{Creating Terms}

\begin{block}{Term Parser}
Use special quotation provided by \texttt{unquote}.
\end{block}

\begin{alertblock}{Operator Precedence}
It is easy to misjudge the binding strength of certain operators. Therefore use plenty of parenthesis.
\end{alertblock}

\begin{block}{Use Syntax Functions}
Terms are just SML values of type \texttt{term}. You can use syntax functions (usually defined in \texttt{*Syntax.sml} files) to create them.
\end{block}
\end{frame}

\begin{frame}
\frametitle{Creating Terms II}
\begin{tabular}{lll}
\emph{Parser} & \emph{Syntax Funs} & \\
\hol{``:bool``} & \ml{mk\_type ("bool", [])} or \ml{bool} & type of Booleans \\
\hol{``T``} & \ml{mk\_const ("T", bool)} or \ml{T} & term true \\
\hol{``\holNeg{}b``} & \hol{mk\_neg (} & negation of \\
& \hol{\ \ mk\_var ("b", bool))} & \ \ Boolean var b\\
\hol{``\ldots\ \holAnd{} \ldots``} & \hol{mk\_conj (\ldots, \ldots)} & conjunction \\
\hol{``\ldots\ \holOr{} \ldots``} & \hol{mk\_disj (\ldots, \ldots)} & disjunction \\
\hol{``\ldots\ \holImp{} \ldots``} & \hol{mk\_imp (\ldots, \ldots)} & implication \\
\hol{``\ldots\ = \ldots``} & \hol{mk\_eq (\ldots, \ldots)} & equation \\
\hol{``\ldots\ <=> \ldots``} & \hol{mk\_eq (\ldots, \ldots)} & equivalence \\
\hol{``\ldots\ <> \ldots``} & \hol{mk\_neg (mk\_eq (\ldots, \ldots))} & negated equation
\end{tabular}
\end{frame}


\section{Inference Rules}

\begin{frame}
\frametitle{Inference Rules for Equality}

\begin{columns}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right=REFL] {\ }{\entails t = t}$\\[1em]
$\inferrule*[right=ABS]{\Gamma \entails s = t\\x\ \textit{not free in}\ \Gamma}{\Gamma \entails \lambda{}x.\ s = \lambda{}x. t}$\\[1em]
$\inferrule*[right=MK\_COMB]{\Gamma \entails s = t\\\Delta \entails u = v \\\\ \textit{types fit}}{\Gamma \cup \Delta \entails s(u) = t(v)}$\\[1em]
\end{center}
\end{column}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule* [right={GSYM}] {\Gamma \entails s = t}{\Gamma \entails t = s}$\\[1em]
$\inferrule*[right=TRANS] {\Gamma \entails s = t\\\Delta \entails t = u}{\Gamma \cup \Delta \entails s = u}$\\[1em]
$\inferrule*[right=EQ\_MP]{\Gamma \entails p \Leftrightarrow q\\\Delta \entails p}{\Gamma \cup \Delta \entails q}$\\[1em]
$\inferrule*[right=BETA\_CONV]{\ }{\entails (\lambda{}x.\ t) v = t[v/x]}$\\[1em]
\end{center}
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Inference Rules for free Variables}
\begin{center}
$\inferrule*[right=INST]{\Gamma[x_1, \ldots, x_n] \entails p[x_1, \ldots, x_n]}
{\Gamma[t_1, \ldots, t_n] \entails p[t_1, \ldots, t_n]}$\\[1em]
$\inferrule*[right=INST\_TYPE]{\Gamma[\alpha_1, \ldots, \alpha_n] \entails p[\alpha_1, \ldots, \alpha_n]}
{\Gamma[\gamma_1, \ldots, \gamma_n] \entails p[\gamma_1, \ldots, \gamma_n]}$\\[1em]
\end{center}
\end{frame}

\begin{frame}
\frametitle{Inference Rules for Implication}

\begin{columns}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right={MP, MATCH\_MP}]{\Gamma \entails p \Longrightarrow q\\\Delta \entails p}{\Gamma \cup \Delta \entails q}$\\[1em]
$\inferrule*[right=EQ\_IMP\_RULE] {\Gamma \entails p = q}{\Gamma \entails p \Longrightarrow q\\\\\Gamma \entails q \Longrightarrow p}$\\[1em]
$\inferrule*[right=IMP\_ANTISYM\_RULE]{\Gamma \entails p \Longrightarrow q\\\\\Delta \entails q \Longrightarrow p}{\Gamma \cup \Delta \entails p = q}$\\[1em]
$\inferrule*[right=IMP\_TRANS] {\Gamma \entails p \Longrightarrow q\\\Delta \entails q \Longrightarrow r}{\Gamma \cup \Delta \entails p \Longrightarrow r}$\\[1em]
\end{center}
\end{column}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right=DISCH]{\Gamma \entails p}{\Gamma - \{q\} \entails q \Longrightarrow p}$\\[1em]
$\inferrule*[right=UNDISCH]{\Gamma \entails q \Longrightarrow p}{\Gamma \cup \{q\} \entails p}$\\[1em]
$\inferrule*[right=NOT\_INTRO]{\Gamma \entails p \Longrightarrow \text{F}}{\Gamma \entails \holNeg{}p}$\\[1em]
$\inferrule*[right=NOT\_ELIM]{\Gamma \entails \holNeg{}p}{\Gamma \entails p \Longrightarrow \text{F}}$\\[1em]
\end{center}
\end{column}
\end{columns}
\end{frame}


\begin{frame}
\frametitle{Inference Rules for Conjunction / Disjunction}

\begin{columns}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right={CONJ}]{\Gamma \entails p\\\Delta \entails q}{\Gamma \cup \Delta \entails p\ \wedge\ q}$\\[1em]
$\inferrule*[right={CONJUNCT1}]{\Gamma \entails p\ \wedge\ q}{\Gamma \entails p}$\\[1em]
$\inferrule*[right={CONJUNCT2}]{\Gamma \entails p\ \wedge\ q}{\Gamma \entails q}$\\[1em]
\end{center}
\end{column}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right={DISJ1}]{\Gamma \entails p}{\Gamma \entails p\ \vee\ q}$\\[1em]
$\inferrule*[right={DISJ2}]{\Gamma \entails q}{\Gamma \entails p\ \vee\ q}$\\[1em]
$\inferrule*[right={DISJ\_CASES}]{\Gamma \entails p \vee q\\\Delta_1 \cup \{p\} \entails r\\\Delta_2 \cup \{q\} \entails r}{\Gamma \cup \Delta_1 \cup \Delta_2 \entails r}$\\[1em]
\end{center}
\end{column}
\end{columns}
\end{frame}


\begin{frame}
\frametitle{Inference Rules for Quantifiers}

\begin{columns}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right={GEN}]{\Gamma \entails p\\x \text{\ not free in\ }\Gamma}{\Gamma \entails \forall{}x.\ p}$\\[1em]
$\inferrule*[right={SPEC}]{\Gamma \entails \forall{}x.\ p}{\Gamma \entails p[u/x]}$\\[1em]
\end{center}
\end{column}
\begin{column}{.45\textwidth}
\begin{center}
$\inferrule*[right={EXISTS}]{\Gamma \entails p[u/x]}{\Gamma \entails \exists{}x.\ p}$\\[1em]
$\inferrule*[right={CHOOSE}]{\Gamma \entails \exists{}x.\ p\\\Delta \cup \{p[u/x]\} \entails r\\
u \text{\ not free in\ } \Gamma, \Delta, p \text{ and } r}
{\Gamma \cup \Delta \entails r}$\\[1em]
\end{center}
\end{column}
\end{columns}
\end{frame}

\section{Forward Proofs}

\begin{frame}
\frametitle{Forward Proofs}

\begin{itemize}
\item axioms and inference rules are used to derive theorems
\item this method is called \emph{forward proof}
\begin{itemize}
\item one starts with basic building blocks
\item one moves step by step forward
\item finally the theorem one is interested in is derived
\end{itemize} 
\item one can also implement own proof tools
\end{itemize}
\end{frame}

\begin{frame}[fragile]
\frametitle{Forward Proofs --- Example I}

Let's prove $\forall{}p.\ p \Longrightarrow p$.
\bigskip

\begin{columns}
\begin{column}{.45\textwidth}
\begin{semiverbatim}
val IMP_REFL_THM = let
  val tm1 = ``p:bool``;
  val thm1 = ASSUME tm1;
  val thm2 = DISCH tm1 thm1;
in 
  GEN tm1 thm2

end

fun IMP_REFL t = 
  SPEC t IMP_REFL_THM;
\end{semiverbatim}
\end{column}
\begin{column}{.45\textwidth}
\begin{semiverbatim}

> val tm1 = ``p``: term
> val thm1 = [p] |- p: thm
> val thm2 = |- p ==> p: thm

> val IMP_REFL_THM = 
    |- !p. p ==> p: thm


> val IMP_REFL = 
    fn: term -> thm
\end{semiverbatim}
\end{column}
\end{columns}
\end{frame}

\begin{frame}[fragile]
\frametitle{Forward Proofs --- Example II}

Let's prove $\forall{}P\,v.\ (\exists{}x.\ (x = v) \wedge P\ x) \Longleftrightarrow P\ v$.
\bigskip

\begin{columns}
\scriptsize
\begin{column}{.45\textwidth}
\begin{semiverbatim}
val tm_v = ``v:'a``;
val tm_P = ``P:'a -> bool``;
val tm_lhs = ``?x. (x = v) \holAnd{} P x``
val tm_rhs = mk_comb (tm_P, tm_v);

val thm1 = let
  val thm1a = ASSUME tm_rhs;
  val thm1b = 
    CONJ (REFL tm_v) thm1a;
  val thm1c = 
    EXISTS (tm_lhs, tm_v) thm1b
in
  DISCH tm_rhs thm1c
end
\end{semiverbatim}
\end{column}
\begin{column}{.45\textwidth}
\begin{semiverbatim}






> val thm1a = [P v] |- P v: thm
> val thm1b = 
    [P v] |- (v = v) \holAnd{} P v: thm
> val thm1c = 
    [P v] |- ?x. (x = v) \holAnd{} P x

> val thm1 = [] |- 
    P v ==> ?x. (x = v) \holAnd{} P x: thm
\end{semiverbatim}
\end{column}
\end{columns}
\end{frame}

\begin{frame}[fragile]
\frametitle{Forward Proofs --- Example II cont.}

\begin{columns}
\scriptsize
\begin{column}{.45\textwidth}
\begin{semiverbatim}
val thm2 = let
  val thm2a = 
    ASSUME ``(u:'a = v) \holAnd{} P u``
  val thm2b = AP_TERM tm_P 
    (CONJUNCT1 thm2a);
  val thm2c = EQ_MP thm2b 
    (CONJUNCT2 thm2a);
  val thm2d = 
    CHOOSE (``u:'a``, 
      ASSUME tm_lhs) thm2c
in
  DISCH tm_lhs thm2d
end


val thm3 = IMP_ANTISYM_RULE thm2 thm1

val thm4 = GENL [tm_P, tm_v] thm3

\end{semiverbatim}
\end{column}
\begin{column}{.45\textwidth}
\begin{semiverbatim}

> val thm2a = [(u = v) \holAnd{} P u] |- 
    (u = v) \holAnd{} P u: thm 
> val thm2b = [(u = v) \holAnd{} P u] |- 
    P u <=> P v
> val thm2c = [(u = v) \holAnd{} P u] |- 
    P v
> val thm2d = [?x. (x = v) \holAnd{} P x] |- 
    P v


> val thm2 = [] |- 
    ?x. (x = v) \holAnd{} P x ==> P v


> val thm3 = [] |- 
    ?x. (x = v) \holAnd{} P x <=> P v
> val thm4 = [] |- !P v. 
    ?x. (x = v) \holAnd{} P x <=> P v
\end{semiverbatim}
\end{column}
\end{columns}
\end{frame}

% \section{Rules and Conversions}
% \begin{frame}
% \frametitle{Derived Tools}
% \begin{itemize}
% \item HOL lives from implementing reasoning tools in SML
% \item \emph{rules} --- use theorems to produce new theorems\\
% \begin{itemize}  
% \item SML-type \ml{thm -> thm}
% \item functions with similar type often called rule as well
% \end{itemize}
% \item \emph{conversions} --- convert a term into an equal one\\
% \begin{itemize}
% \item SML-type \ml{term -> thm}
% \item given term \ml{t} produces theorem of form \ml{[] |- t = t'}
% \item may raise exceptions \ml{HOL\_ERR} or \ml{UNCHANGED}
% \end{itemize}
% \item \ldots
% \end{itemize}
% \end{frame}

% \begin{frame}
% \frametitle{Conversions}

% \begin{itemize}
% \item HOL has very good tool support for equality reasoning
% \item \emph{conversions} are important for HOL's automation
% \item there is a lot of infrastructure for conversions
% \begin{itemize}
% \item \ml{RAND\_CONV}, \ml{RATOR\_CONV}, \ml{ABS\_CONV}
% \item \ml{DEPTH\_CONV}
% \item \ml{THENC}, \ml{TRY\_CONV}, \ml{FIRST\_CONV}
% \item \ml{REPEAT\_CONV}
% \item \ml{CHANGED\_CONV}, \ml{QCHANGED\_CONV}
% \item \ml{NO\_CONV}, \ml{ALL\_CONV}
% \item \ldots
% \end{itemize} 
% \item important conversions
% \begin{itemize}
% \item \ml{REWR\_CONV}
% \item \ml{REWRITE\_CONV}
% \item \ldots
% \end{itemize}
% \end{itemize}
% \end{frame}


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