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Why obsess over the compiler running into a loop? That's way less harmful than the compiled application getting into a loop on the end user's machine.

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I think part of the purpose of the Agda project is to see how far they can push programs as proofs, which in turn means they care a lot about termination. A language that was aimed more at industrial development would not have this restriction.

If you want to prove propositions at compile time, isn't it hampering not to have Turing-complete power?

No it's not, you can write an extremely large set of programs in non-Turing complete languages. I personally consider Turing completeness a bug, not a feature, it simply means "I cannot prove the halting problem of this language" which blocks tons of useful properties.

You can write useful programs in Agda. I wrote all kinds of programs in Agda including parsers and compilers.

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