User blog comment:Rpakr/A Formal Definition of Pair Sequence System using BM1 Rules/@comment-35470197-20180814221408/@comment-35470197-20180814235838

> so it is technically not multiply defined

You can overload it in this way, but I guess that it conflicts what you intended in the formula "\(P \in \textrm{Pair} \Leftrightarrow \exists A \exists B (P = \textrm{Pair}(A,B))\)".

Well, you do not have to write logical formulae in order to write down formal proofs, if it is easier for you to interprete it into natural language. I guess that it makes your formal proof harder for non-Japanese readers to understand. (I also like this way, though.)

Additional comments:


 * Adding a number to a pair
 * Undefined reference to \(S\).


 * Adding a number to a sequence
 * Undefined reference to \(P(n,S,k))+n\)
 * \(P(n,S,k+1)\) refers to the undefined value when \(S\) is not a pair or \(k > 0\).
 * \(\textrm{AddSeq}_n(S)\) might be a typo of \(\textrm{AddSeq}(n,S)\).


 * Adding a number to a sequence and repeat
 * \(\textrm{AddSeqCon}(S,d,1)\) is multiply-defined.
 * \(\textrm{AddSeqCon}(S,d,r)\) might be a typo of \(\textrm{AddSeqCon}(S,d,r+1)\).


 * Bad Root
 * \(R\) is not declared or quantified.
 * Exacltly, it uniquely exists by the codition which you wrote, but you need \(\forall R \in \mathbb{N}\) at the begining of the defining formula of \(\textrm{BR}\) in order to formally declare \(R\). Or something like \(\textrm{BR}(E) = \min \sim\) might be prefered.


 * Good Part
 * Undefined reference to \(\textrm{Pair}(1,0,\textrm{Seq}(E))\)
 * \(\textrm{Good}(E)\) refers to the undefined value in the case \(\textrm{BR}(E) = 1\).