Brainfuck with while 0 loops

This page is proof that brainfuck with unbounded tape and while 0 loops is Turing complete, which is proven by simulation of Cyclic Tag.

Initialise data

put +>> at the start of a program then initialise program with 0 being >+> and 1 being >>. After initialising data append >+<<[<<]>>[

Simulating Cyclic Tag

Replace every ; with +>>, 0 with +>[[>>]->+>+<<[<<]->+]<[+>-<]- and 1 with +>[[>>]->>+<<[<<]->+]<[+>-<]-. End program with ]. It also proves that it is Turing-complete with nesting depth of 3

Explanation

To simulate Cyclic Tag in this version of brainfuck we will use following layout:

 10 xy xy xy ... 1

x is 1 if bit is deleted and 0 otherwise. y is 1 if bit is 0 and 0 if bit is 1.