Tekst źródłowy strony Algorithmic theory of natural numbers

The theory of one constant 0, one one-argument functor <math>s</math> and predicate of equality =.<br />
'''Axioms''' <br />
<math>\begin{align*}
&\tag{I}  \forall_n s(n) \neq 0 \\
&\tag{M}  \forall_n \forall_m s(n)=s(m) \implies n=m \\
&\tag{S}  \forall_n \{m:=0; \mathbf{while}\ n\neq m\ \mathbf{do}\ m:=s(m)\ \mathbf{od}  \}(n=m)
\end{align*} </math><br />
This set of formulas gives a complete specification of the structure <math>\mathfrak{N} </math>  of natural numbers with the successor operation. It means that any implementation whether hardware or software (e.g. by means of a class) that satisfies three  axioms listed above is correct. 
It means also that if an algorithmic formula is valid in the standard model of these axioms then it has a proof with the use of program calculus.<br />
One may extend this set adding five axioms that define operation of addition, subtraction predecessor and predicate < and operation of multiplication.
<math>\begin{align*}
\tag{A}\label{add} & x+y  \stackrel{df}{=} \{t:=0; w:=x; \textbf{while }t\neq y\textbf{ do }t:=s(t); w:=s(w) \textbf{ od}\}w  &  \\
\tag{L}\label{less} & x < y \stackrel{df}{\equiv} \{ w:=0; \textbf{while }w\neq y\land w\neq x \textbf{ do } w:=s(w) \textbf{ od}\}(w=x \land w\neq y) & \\
\tag{P}\label{predec}  & P(x)\stackrel{df}{=} \{\begin{array}{l}w:=0;\ 
\textbf{if }x \neq 0 \textbf{ then }  
\ \textbf{while } s(w)\neq x\textbf{ do }  w:=s(w) \textbf{ od} \
\textbf{fi}\end{array} \}w  & \\
\tag{O}\label{odejm} & x\stackrel{.}{\_\_}y \stackrel{df}{=} \{w:=x; t:=0; \mathbf{while }\ t\neq y\ \mathbf{ do }\ t:=s(t); w:=P(w)\ \mathbf{ od} \}w & \\
\tag{M}\label{mult} & x\cdot y  \stackrel{df}{=} \{t:=0; w:=0; \textbf{while }t\neq y\textbf{ do }t:=s(t); w:=w+x \textbf{ od}\}w  &  \\
\end{align*} </math><br />

Among theorems of this theory one can find theorem on correctness of Euclid's algoritm <br />
<math> \tag{Eucl} \underbrace{\forall_{n\neq 0}\,\forall_{m\neq 0}\left\{\begin{array}{l}\textbf{while }n \neq m \textbf{ do }\\ \quad \textbf{if }n\geq m   \\ 
\quad \textbf{ then } \\
\qquad  n:=n-m   \\
\quad \textbf{ else }\\
\qquad    m:=m-n \\  \quad\textbf{ fi}\\ \textbf{ od} \end{array}\right\}(n=m)}_{halting\ formula\ of\ Euclid's\ algorithm}  </math>\mathfrak{N} \models
and the theorem stating that the structure of natural numbers  enjoys the Archimedean property
<math> \tag{Archi} \underbrace{\forall_{n\neq 0}\,\forall_{m\neq 0}\left\{\begin{array}{l}
a:=n; \\
\textbf{while }\ a \leq  m\ \textbf{ do }\\ 
\quad a :=a+x   \\ 
 \textbf{ od} \end{array}\right\}(a >x)}_{ \ axiom\ of\ Archimedes\ }  </math>