proof: thread spec functions through stmt list seam

[Th0rgal]

Continues #2080 as the next proof-tier slice after phase 12 / #2106.

Stacked on #2106 (paloma/issue-2080-spec-functions-stmt-witness) because #2106 and the earlier #2102-#2105 chain are still open. Please review/merge this after #2106.

What changed

• Add compileStmtList_cons_eq_ok_with_internals, a narrow internal-functions-parametric cons reconstruction lemma for compileStmtList.
• Add StmtListGenericWithHelpersAndHelperIRWithInternals, a spec-functions-aware statement-list seam whose heads carry CompiledStmtStepWithHelpersAndHelperIRWithInternals.
• Add exact-scope reconstruction theorem compileStmtList_ok_of_stmtListGenericWithHelpersAndHelperIRWithInternals_exact, compiling statement lists with spec.functions.
• Compose the phase-12 ExprInternalHelperHeadStepBridgeWithInternals witness through the new list seam with cons and all-heads assemblers.
• Regenerate PrintAxioms.lean for the new theorem declarations.

Current blocker / next API

This closes exact-scope list reconstruction for the spec.functions world. Full replacement of compileStmtList_ok_of_stmtListGenericWithHelpersAndHelperIR still needs an internalFunctions-parametric version of FunctionBody.compileStmt_ok_any_scope_with_surface / compileStmtList_ok_any_scope_with_surface, because the existing scope-lift proof recursively hardcodes the default empty internal-function world.

Validation

• lake env lean Compiler/Proofs/IRGeneration/GenericInduction/ResultRelation.lean
• lake env lean Compiler/Proofs/HelperStepProofs.lean
• lean-slot lake build Compiler.Proofs.IRGeneration.GenericInduction.ResultRelation
• lean-slot lake build Compiler.Proofs.HelperStepProofs
• python3 scripts/lean_lint.py --only lean_hygiene
• python3 scripts/lean_lint.py --only proof_length
• make check

Note: lean-slot attempted Spark offload but Spark returned HTTP 403 for this isolated worktree path, so the builds ran locally and passed.

---

Note

Medium Risk
Changes sit on the formal compiler correctness proof chain (stmt-list compilation and helper IR), but are additive Lean scaffolding with no runtime or security surface.

Overview
Extends the phase-12 WithInternals helper proof path so statement lists compile and assemble under spec.functions instead of the legacy empty internal-function world.

Compiler lemmas: Adds compileStmtList_cons_eq_ok_with_internals / compileStmtList_cons_ok_inv_with_internals, generalizes compileStmt_ok_any_scope_with_surface_aux to take internalFunctions, and exposes compileStmt_*_with_internals / compileStmtList_*_with_internals scope-lift theorems.

New list seams: Introduces StmtListGenericWithHelpersAndHelperIRWithInternals and StmtListExprInternalHelperStepInterfaceWithInternals, with exact- and scope-lifted compileStmtList_ok_of_stmtListGenericWithHelpersAndHelperIRWithInternals* reconstruction and an assembly bridge from the expression-helper interface.

Composition: Wires ExprInternalHelperHeadStepBridgeWithInternals through cons / all-heads list assemblers in HelperStepProofs; updates PrintAxioms.lean.

Scope is intentionally narrow (e.g. all expression-helper heads for the whole-list bridge); mixed lists still use legacy assembly until more WithInternals interfaces exist.

^{Reviewed by Cursor Bugbot for commit 73564ef. Bugbot is set up for automated code reviews on this repo. Configure here.}

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Project	Deployment	Actions	Updated (UTC)
verity	Ready	Preview, Comment	Jul 7, 2026 6:36am

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