Formal Verification and Prover

Formal Verification & Embedded `OmniProof-v4` Prover

Unlike toolchains that delegate verification to gigabyte-scale external C++ solvers with unpredictable timeouts, Lume embeds its multi-theory theorem prover (OmniProof-v4) directly inside the self-hosted compiler (compiler/stage1/lume/06_typecheck.src, 06b_phase3.src, lcaas.src) and pairs it with an independent, zero-compiler-import proof certificate kernel (compiler/stage1/lcert.src).

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1. Live OmniProof-v4 Verification Metrics (lume self-prove)

Running lume self-prove --json executes live compile-time proofs and native tests across the entire self-hosted standard library and domain engine suite with zero external scripts:

Metric	Verified Result (v1.1.0)
Embedded Proof Engine	LumeCompiler-Embedded-OmniProof-v4
Domain Proof Batteries	16 / 16 certified (ok: true)
Formal Mathematical Obligations Verified	1,682 (proven + @bound budget-checked)
Runtime-Fallback or Refuted Obligations	0
Native Unit Tests Passed	372 (exit 0)
Translation-Validated lstd.src Functions	70 functions (426 proof cases)
Verified Sovereign Domains	122 domains

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2. Multi-Theory Decision Procedures

Theory / Analysis	Diagnostic Codes	Self-Hosted Decision Procedure
QF_LIA Linear Integer Arithmetic	[V0001]	Exact Fourier-Motzkin variable elimination, integer cutting planes, disjunctive or splitting, C99 truncated remainder bounds (-k + 1 <= r <= k - 1), 10^9 coefficient overflow guard, and Farkas dual certificates
QF_NIA Nonlinear Polynomials	[V0001]	Horner/Bernstein polynomial interval envelopes, native quadratic square-completion ((a*x + b)^2 >= 0), Positivstellensatz witnesses, sign-rule propagation, and 64-bit multiplication overflow guards (<= 3,037,000,499)
QF_BV Bitvectors & Shifts	[V0001]	Exact bitmask bounds (0 <= bit_and(x, m) <= m), bit_or/bit_xor algebraic identities, and bounded left/right bit-shift semantics
Relational Octagons (±x ± y <= c)	[V0001]	Difference-bound & octagon closure tracking coupled index/length invariants across branches and loops
Termination, DAG & WCET Budgets	[V0002], [K0006]	Lexicographic ranking measures for total fn, call-DAG acyclicity ([K0006]), and static instruction-cycle summation (@bound(N))
Separation Logic & Perceus Reuse	[O0001]–[O0030]	Disjoint heap frame rules (p != q), affine ownership dataflow, and alias-safe in-place cell reuse (pat_scrutinee_safe)
Constant-Time Secret Isolation	[C0009], [C0011]	Secret taint lattices forbidding secret branches, divisions, memory indices, and non-kernel calls; paired with ltv [C0010]

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3. Writing Provable Specifications (requires, ensures, invariant, law)

Linear & Nonlinear Contracts

## Quadratic non-negativity discharged by native square-completion / QF_NIA:
law shifted_parabola_floor(x: int) =
  (x * x) - (6 * x) + 14 >= 5

## Conservation-proven ledger withdrawal with static WCET bound:
@no_alloc
@bound(40)
pub fn withdraw(balance: int, amount: int, fee: int) -> int / pure
  requires balance >= 0 and amount > 0 and fee >= 0 and amount + fee <= balance
  ensures result == balance - amount - fee and result >= 0 and result < balance
= balance - amount - fee

When a contract is falsifiable, compilation aborts fail-closed (exit 3) with [V0001] before a single byte of object code is emitted.

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4. Sub-10 ms Incremental SHA-256 Proof Cache (lproof_cache.src)

To keep interactive IDE and incremental builds instantaneous on large verified modules, compiler/stage1/lproof_cache.src caches discharged obligation digests keyed by normalized function AST, contract AST, transitive callee ensures signatures, and prover version:

• Warm Cache Latency: ~6.9 ms in-process / ~7.4 ms CLI (< 10 ms, verified in G-DWARF-CACHE across 20/20 warm hits and 19/20 single-function edits).
• Sound Invalidation & Tamper Refusal: Automatically invalidates cached proofs whenever a function body, its requires/ensures, or any transitive callee's ensures changes; rejects corrupted cache entries with [PC0002] and refuses forged false-theorem cache injections with [PC0003].

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5. Standalone Zero-Compiler-Import Proof Kernel (lcert.src --check-source)

Lume enforces strict separation between proof search (lcaas.src) and proof checking (lcert.src):

1. Zero-Compiler-Import TCB: compiler/stage1/lcert.src imports zero modules from lumec or lcaas. It independently re-derives every Farkas linear combination, quadratic/Horner step, and frame condition from the emitted .cert.json artifact.
2. Source-to-Certificate AST Binding (--check-source): Running lume verify-cert --check-source <file.src> <file.cert.json> re-parses the original .src file, verifies its SHA-256 digest, enforces kernel safety laws ([K0005] symbolic W^X DMA page protection and [K0006] mutual recursion prohibition), and ensures no obligation was omitted, weakened, or forged:
   • [K0010]: Edited or mismatched source file refused.
   • [K0011]: Deleted/missing proof obligation refused.
   • [K0012]: Certificate referencing a non-existent function refused.
   • [K0013]: Weakened postcondition goal, strengthened precondition hypothesis, captured caller parameter, loosened @bound, or weakened law refused.

lume prove app.src
lume certify app.src > app.cert.json
lume verify-cert --check-source app.src app.cert.json

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6. Continuous Adversarial Soundness Fuzzing (lume prover-fuzz)

Every CI run executes tests/harness/prover_fuzz_gate.src (lume prover-fuzz) and claude_audit_gate.src:

• Generates and mutates hundreds of subtle false theorems across 6 verification domains (64-bit boundaries, division/modulo signs, records + variable shadowing, canonical arrays, separation/heap aliasing, nested quantifiers, wrapping +%/-%/*% operators, 2^40 coefficients, and lexicographic termination measures).
• Verifies that 0 false theorems are ever accepted by lumec, lcaas, or lcert, while 266 valid control theorems are independently re-derived by lcert.

For post-codegen machine-code translation validation (ltv), clean-room .smt2 solving (lsmt + z3), 0/9,618 Masking MC/DC, 118 constant-time crypto kernels, and 2oo2D lockstep qualification, continue to Binary Assurance, Crypto & Qualification.