Aheui Turingcompleteness proof
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It can be proven that Aheui is Turingcomplete by making a translation table from Brainfuck (a known Turingcomplete language) to Aheui. This document shows how to translate any Brainfuck program (minus input and output, which are not necessary for Turingcompleteness) to Aheui.
This program uses three stacks. An alternative using two can be made, but that can be more difficult.
Simply take the Aheui blocks and stack them vertically.
Brainfuck  Aheui  Note 

Beginning of program  부 

+ / Add 
발발나다붗 루떠떠벓벓 

 / Decrease 
밟밠밥따따받두 루떠떠벓벓벝더 
Optional. This is still Turing complete without it. 
> / Glide right 
싹순 수빠쑤 부수머 우어 
Unconfirmed 
< / Glide left 
싼숙 수빠쑤 부수머 우어 
Unconfirmed 
[ / Start loop 
빠우 우처아아아아아ㅡㅡㅡㅡㅡ아우 우어어어어어어ㅡㅡㅡㅡㅡ어우 
The number of placeholder characters must match the number of placeholder characters in the corresponding end loop instruction. 
] / End loop 
빠우 우차아아아아아ㅡㅡㅡㅡㅡ오우 우어어어어어어ㅡㅡㅡㅡㅡ어어 
The number of placeholder characters must match the number of placeholder characters in the corresponding start loop instruction. 
, / Input 
마밯밣붏 루떠떠벝 
This can keep undefined characters from being pushed into the stack. 
. / Output 
뿌 뭏 

End of program  바희 