feat(MultiTapeTM): halting-time, output-length, and visited-position lemmas for multi-tape TMs - #816
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…lti-tape TMs General lemmas independent of any particular construction: - exists_minimal_halting_config: a padded halting run has a least halting time reaching the same configuration - outputString_length_le / output_length_le_time: the output is no longer than the elapsed time - spaceUsedByTape_le_of_positions: space comparison between two machines via inclusion of visited head positions
SamuelSchlesinger
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chenson2018 and
fmontesi
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August 18, 2026 16:20
crei
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Aug 18, 2026
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| /-- Every padded halting computation has a least halting time, and the configuration there is the | ||
| same as at the padded time. -/ | ||
| lemma exists_minimal_halting_config (tm : MultiTapeTM k Symbol State) |
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I think this lemma could be reworked a little. It seems to be related to haltsAtStep and halting_step_unique, so should maybe move there. Also the conjunction is rather complicated, maybe it should be split?
Maybe we could also prove Monotone fun t => (tm.configs cfg t).state.isNone?
crei
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Aug 18, 2026
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| have hone : (tm.outputSymbol (tm.configs cfg t)).toList.length ≤ 1 := by | ||
| cases tm.outputSymbol (tm.configs cfg t) <;> simp | ||
| simp only [List.length_append] | ||
| omega |
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There does not seem to be a lemma Option.toList_length (x : Option α) : x.toList.length ≤ 1 (which would he hone). But maybe this one suffices (I did not test it):
Suggested change
| have hone : (tm.outputSymbol (tm.configs cfg t)).toList.length ≤ 1 := by | |
| cases tm.outputSymbol (tm.configs cfg t) <;> simp | |
| simp only [List.length_append] | |
| omega | |
| grind [Option.toList] |
crei
reviewed
Aug 18, 2026
| _ = k * t + k := by simp [Nat.mul_succ] | ||
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| /-- | ||
| The space used by one tape is monotone under inclusion of its visited head positions in the |
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I think this lemma deserves a bit more explanation, in particular, I don't understand the meaning of hpos.
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Heads up: This might conflict with #819 |
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This is in advance of a PR for Turing machine composition.