Add n-ary equivalence URMComputable ↔ Nat.Partrec'#57
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jessealama merged 4 commits intomainfrom Jan 21, 2026
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Implement equivalence between n-ary URM computability and Mathlib's n-ary partial recursive functions (Nat.Partrec'). New file Urm/PartrecPrime.lean: - Type conversion utilities (toVectorFun, fromVectorFun) - Vector encoding is URMComputable (encodeVector_computable) - Backward direction complete (Nat.Partrec'.toURMComputable) - Forward direction structure (URMComputable.toPartrec') with one sorry - Main equivalence theorem (URMComputable_iff_Partrec') Also adds Part.sequence_some lemma to PartSequence.lean.
Complete the proof of URMComputable n ↔ Nat.Partrec' by filling three sorries: - URMComputable_zero_to_one: Lift 0-arity computable functions to 1-arity by prepending Z 0 to reset R₀. Added helper lemmas for step translation between concatenated and base programs. - decode_valid_extract: Core encoding correspondence showing decoded vectors match extractListElem element-by-element. - guardAndApply_eq_decode_bind invalid case: Handle the case when hasValidLengthNum returns a non-1 value. Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>
- Extract reusable helper lemmas (prSetupPhase_no_write_low, prLoopPrologue_no_write_high) - Replace verbose by_cases/dite patterns with split_ifs - Inline trivial let bindings as show expressions - Remove redundant type annotations and show statements - Use have instead of let for proof terms - Consolidate duplicated register bound proofs Net reduction of ~295 lines across 18 files.
Convert camelCase definitions and theorems to snake_case following Mathlib/CSLib structural naming patterns: - Definitions: maxRegister → max_register, shiftJumps → shift_jumps, isHalted → is_halted, fromInputs → of_inputs, etc. - Theorems: halted_no_step → no_step_of_halted (uses _of_ pattern), write_read_same → write_read_self (uses _self pattern), halts_unique → eq_of_halts (describes result), etc. - Module-specific renames in Composition/, Minimization/, PrimitiveRecursion/, and Simulate/ directories All 43 files updated with consistent naming. Build passes.
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