You write the phrase '“total” (i.e. not Turing-complete)'. This is quite misleading; if by Turing complete you mean the language can implement a Turing machine then totality does not forbid Turing completeness as McBride explains here: https://personal.cis.strath.ac.uk/conor.mcbride/pub/Totality.pdf …
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Usually I understand "Turing-complete" to mean that the language can not only implement but also run the Turing-machine
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