\section{Soundness}
\label{sect:soundness}

The main result of this section is the following:

\begin{theorem}
	\label{thm:soundness}
	If $\arith^i$ proves $\forall \vec u^\normal . \forall \vec x^\safe . \exists y^\safe . A(\vec u ; \vec x , y)$ then there is a $\mubci{i-1}$ program $f(\vec u ; \vec x)$ such that $\Nat \models A(\vec u ; \vec x , f(\vec u ; \vec x))$.
\end{theorem}



The main problem for soundness is that we have predicates, for example equality, that take safe arguments in our theory but do not formally satisfy the polychecking lemma for $\mubc$ functions. 
For this we will use length-bounded witnessing, borrowing a similar idea from Bellantoni's previous work \cite{Bellantoni95}.


\begin{definition}
[Length bounded (in)equality]
%We define \emph{length-bounded equality}, $\eq(l;x,y)$ as the characteristic function of the predicate:
%\[
%x \mode l = y \mode l
%\]
%which is definable by safe recursion on $l$:
%\[
%\begin{array}{rcl}
%\eq (0 ; x,y) & \dfn & \equivfn (;\bit (0;x),\bit(0;y) ) \\
%\eq (\succ i l; x,y) & \dfn & \cond (; \eq ( u;x,y ) , 0, \equivfn (; \bit (\succ i u ; x ) , \bit (\succ i l ; y ))  )
%\end{array}
%\]
We define \emph{length-bounded inequality} as:
\[
\begin{array}{rcl}
\leqfn (0 ; x ,y) & \dfn & \cond(; \bit (0;x), 1, \bit (0;y) ) \\
\leqfn (\succ i l ; x,y) & \dfn & \orfn ( ; \cond(;\bit (\succ i l ; x) , \bit(\succ i l ; y),0 ) , \andfn (; \equivfn (\bit (\succ i l ; x) , \bit(\succ i l ; y)) , \leqfn (l;x,y ) ) )
\end{array}
\]
\end{definition}

Notice that $\leqfn (l; x,y) = 1$ just if $x \mode l \leq y \mode l$.
We can also define $\eq( l; x,y)$ as $\andfn (;\leq(l;x,y),\leq(l;y,x))$.

%\anupam{Do we need the general form of length-boundedness? E.g. the $*$ functions from Bellantoni's paper? Put above if necessary. Otherwise just add sequence manipulation functions as necessary.}
%
%Notice that $\eq$ is a polymax bounded polyomial checking function on its normal input, and so can be added to $\mubc$ without problems.

Let us say that a sorting $(\vec u ; \vec x)$ of the variables $\vec u , \vec x$ is \emph{compatible} with a formula $A$ if each variable of $\vec x$ occurs hereditarily safe with respect to the $\bc$-types of terms (see App.~\ref{appendix:arithmetic}). 
In the presence of a compatible sorting, we may easily define functions that \emph{evaluate} safe formulae in $\mubc$:


\begin{proposition}
Given a $\Sigma^\safe_i$ formula $A$ and compatible sorting $(\vec u; \vec x)$ of its variables, there is a $\mubci{i-1}$ program $\charfn{\vec u ;\vec x}{A} (l, \vec u ; \vec x)$ computing the characteristic function of $A (\vec u \mode l ; \vec x \mode l)$.
\end{proposition}


We will use the programs $\charfn{}{}$ in the witness functions we define below.
Let us write $\charfn{}{i}$ to denote the class of functions $\charfn{}{A}$ for $A \in \Pi^\safe_{i-1}$.
For the notion of bounding polynomial below we are a little informal with bounds, using `big-oh' notation, since it will suffice just to be `sufficiently large'.


\begin{definition}
	[Length bounded witness function]
	For a $\Sigma^\safe_{i}$ formula $A$ with a compatible sorting $(\vec u ; \vec x)$, we define the \emph{length-bounded witness function} $\wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x , w)$ in $\bc (\charfn{}{i})$ and its \emph{bounding polynomial} $b_A (l)$ as follows:
	\begin{itemize}
		\item If $A$ is $\Pi^\safe_{i-1}$ then $\wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x , w) \dfn \charfn{\vec u ; \vec x}{A} (l, \vec u ; \vec x )$ and we set $b_A (l) = 1$.
		\item If $A$ is $B \cor C$ then we may set $b_A = O(b_B + b_C)$ and define $		\wit{\vec u ; \vec x}{A} (l, \vec u ;\vec x , w) \dfn		\orfn ( ; \wit{\vec u ; \vec x}{B} (l, \vec u ; \vec x , \beta (b_B (l) , 0 ; w ) ) , \wit{\vec u ; \vec x}{C} (l, \vec u ; \vec x , \beta (b_C (l) , 0 ; w ) )  )$.
%		\[
%		\wit{\vec u ; \vec x}{A} (l, \vec u ;\vec x , w) 
%		\quad \dfn \quad
%		\orfn ( ; \wit{\vec u ; \vec x}{B} (l, \vec u ; \vec x , \beta (b_B (l) , 0 ; w ) ) , \wit{\vec u ; \vec x}{C} (l, \vec u ; \vec x , \beta (b_C (l) , 0 ; w ) )  )
%		\]
%		and we may set $b_A = O(b_B + b_C)$.
		\item Similarly if $A $ is $B \cand C$, but with $\andfn$ in place of $\orfn$.
%		\item If $A$ is $B \cand C$ then
%			\[
%			\wit{\vec u ; \vec x}{A} (l, \vec u ;\vec x , w) 
%			\quad \dfn \quad
%			\andfn ( ; \wit{\vec u ; \vec x}{B} (l, \vec u ; \vec x , \beta (b_B (l) , 0 ; w ) ) , \wit{\vec u ; \vec x}{C} (l, \vec u ; \vec x , \beta (b_C (l) , 0 ; w ) )  )
%			\]
%			and we may set $b_A = O(b_B + b_C)$.
		\item If $A$ is $\forall u \leq |t(\vec u;)| . B(u)$ we appeal to sharply bounded closure, Lemma~\ref{lem:sharply-bounded-recursion}, to define
		\(
		\wit{\vec u ; \vec x}{A}
		 \dfn
		\forall u \leq |t|.
		\wit{u, \vec u ; \vec x}{B(u)} (l, u, \vec u ; \vec x , \beta( b_{B(t)} (l) , u ; w ) )
		\)
		%appealing to Lemma~\ref{lem:sharply-bounded-recursion}, 
		and 
		we set 
		$b_A = O(b_{B(t)}^2 )$.
		\item Similarly if $A$ is $\exists u^\normal \leq |t(\vec u;)|. A'(u)$, but with $\exists u \leq |t|$ in place of $\forall u \leq |t|$.
		\item If $A$ is $\exists x^\safe . B(x) $ then
		\(
		\wit{\vec u ; \vec x}{A}
		\dfn 
		\wit{\vec u ; \vec x , x}{B(x)} ( l, \vec u ; \vec x , \beta( b_{B} (l) , 0;w ) , \beta (q(l) , 1 ;w ))
		\)
		where $q$ is obtained by the polychecking and bounded minimisation properties, Lemmas~\ref{lem:polychecking} and \ref{lem:bounded-minimisation}, for $\wit{\vec u ; \vec x , x}{B(x)}$.
		We may set $b_A = O(b_B + q )$.
	\end{itemize}
%	\[
%	\begin{array}{rcl}
%	\wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x , w) & \dfn & \charfn{}{A} (l, \vec u ; \vec x)  \text{ if $A$ is $\Pi_i$} \\
%	\smallskip
%	\wit{\vec u ; \vec x}{A \cor B} (l,\vec u ; \vec x , \vec w^A , \vec w^B) & \dfn & \cor ( ; \wit{\vec u ; \vec x}{A} (l,\vec u ; \vec x , \vec w^A) ,\wit{\vec u ; \vec x}{B} (l,\vec u ; \vec x , \vec w^B)  )  \\
%	\smallskip
%	\wit{\vec u ; \vec x}{A \cand B} (l,\vec u ; \vec x , \vec w^A , \vec w^B) & \dfn & \cand ( ; \wit{\vec u ; \vec x}{A} (l,\vec u ; \vec x , \vec w^A) ,\wit{\vec u ; \vec x}{B} (l,\vec u ; \vec x , \vec w^B)  )  \\
%	\smallskip
%	\wit{\vec u ; \vec x}{\exists x^\safe . A(x)} (l,\vec u ; \vec x , \vec w , w) & \dfn & \wit{\vec u ; \vec x , x}{A(x)} ( l,\vec u ; \vec x , w , \vec w )
%	\\
%	\smallskip
%	\wit{\vec u ; \vec x}{\forall u \leq |t(\vec u;)| . A(x)} (l , \vec u ; \vec x, w) & \dfn & 
%	\forall u \leq |t(\vec u;)| . \wit{u , \vec u ; \vec x}{A(u)} (l, u , \vec u ; \vec x, \beta(u;w) )
%	\end{array}
%	\]
%	\anupam{need length bounding for sharply bounded quantifiers}
\end{definition}



\begin{proposition}
	\label{prop:wit-rfn}
	If, for some $w$, $\wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x,w) =1$, then $A (\vec u \mode l ; \vec x \mode l)$ is true.
Conversely, if $A (\vec u \mode l ; \vec x \mode l)$ is true then there is some $w \leq b_A(l)$ such that $\wit{\vec u ; \vec x}{A} (l, \vec u ; \vec x,w) =1$.
\end{proposition}


In order to prove Thm.~\ref{thm:soundness} we need the following lemma, essentially giving an interpretation of $\arith^i$ proofs into $\mubci{i-1}$:


\begin{lemma}
		[Proof interpretation]
		\label{lem:proof-interp}
	From a typed-variable normal form $\arith^i$ proof $\pi$ of a $\Sigma^\safe_i$ sequent $\normal(\vec u), \safe(\vec x) , \Gamma  \seqar \Delta$
	there are $\bc (\charfn{}{i})$ functions $ f^\pi_B (\vec u ; \vec x , w)$ for $B\in\Delta$ such that
	% such that, for any $l, \vec u ; \vec x  , w$, we have:
	\[
%	\vec a^\nu = \vec u ,
%	\vec b^\sigma = \vec u, 
	\bigwedge\limits_{A \in \Gamma} \wit{\vec u ; \vec x}{ A} (l, \vec u ; \vec x , w_A) =1
	\ \implies \
	\bigvee\limits_{B\in \Delta} \wit{\vec u ; \vec x}{B} (l, \vec u ; \vec x , f^\pi_B((\vec u ; \vec x )\mode l, \vec w \mode p(l))) = 1
	\]
	for some polynomial $p$.
%	\anupam{Need $\vec w \mode p(l)$ for some $p$.}
%	\anupam{$l$ may occur freely in the programs $f^\pi_B$}
\end{lemma}




Now we can prove the soundness result:
	
	\begin{proof}
		[Proof of Thm.~\ref{thm:soundness} from Lemma~\ref{lem:proof-interp}]
		Suppose $\arith^i \proves \forall \vec u^\normal . \exists x^\safe . A(\vec u ; x)$. By inversion and Thm.~\ref{thm:normal-form} there is a $\arith^i$ proof $\pi$ of $\normal (\vec u ) \seqar \exists x^\safe. A(\vec u ; x )$ in typed variable normal form.
By Lemma~\ref{lem:proof-interp}, we have a $\mubci{i-1}$ function $f^\pi$ with $\wit{\vec u ;}{\exists x^\safe . A} (l, \vec u ; f(\vec u \mode l;)) =1$.
By the definition of $\wit{}{}$ and Prop.~\ref{prop:wit-rfn} we have that $\exists x . A(\vec u \mode l; x)$ is true just if $A(\vec u \mode l ; \beta (q(l), 1 ; f(\vec u \mode l;) ))$ is true.
Now, since all $\vec u$ are normal, we may simply set $l$ to have a longer length than all of these arguments, so the function $f(\vec u;) \dfn \beta (q(\sum \vec u), 1 ; f(\vec u \mode \sum \vec u;) ))$ suffices.
	\end{proof}
