CAB Certification

VERIFIED

The Certified Abstract Backend (CAB) is a formally verified compilation pipeline that preserves mathematical correctness from high-level theorem provers to executable machine code.

What is CAB?

The Certified Abstract Backend (CAB) represents a paradigm shift in how we bridge the gap between formal mathematical proofs and executable code. Traditional compilation pipelines can introduce bugs at any stage, undermining the guarantees provided by formal verification. CAB solves this by making every transformation step itself formally verified.

Built on the foundation of Lean 4's dependent type system, CAB extends the “Proving Lean with Lean” bootstrap approach to create an end-to-end verified compilation chain. This means that if a theorem is proven correct in Lean, the generated executable code inherits that correctness guarantee with mathematical certainty.

“The kernel of Lean 4 is designed to be small and auditable. By verifying the kernel in Lean itself, we create a self-certifying system where trust is established through mathematical proof rather than code review alone.”

- Proving Lean with Lean, 2024

The Verification Pipeline

STAGE 1

Lean 4

Source theorems and proofs written in Lean 4's dependent type theory. Type-checked by Lean's verified kernel.

2,900+ proofs verified

STAGE 2

LeanCore v2

Certified subset of Lean with verified extraction rules. Preserves type-theoretic semantics.

Type-preserving transformation

STAGE 3

Lambda IR

Simply-typed lambda calculus intermediate representation with verified semantics.

39 proofs at this stage

STAGE 4

MiniC

Verified subset of C with formal semantics. Memory-safe by construction.

50 proofs compiled to MiniC

STAGE 5

Verified C / End-to-End

Fully verified executable C code. Correctness proven from Lean source to machine output.

42 end-to-end verified programs

Why CAB Matters

Trustless Verification

Traditional smart contracts and critical systems require trust in auditors and developers. CAB eliminates this trust requirement by providing machine-checkable proofs that the code behaves exactly as specified. No trust in humans required - only in mathematics.

Proof-Carrying Code

Every piece of compiled code carries with it a certificate of correctness. This certificate can be independently verified by any party, enabling a new paradigm of “verify once, trust forever” for critical software infrastructure.

Categorical Semantics

CAB is grounded in categorical semantics, treating program transformations as functors that preserve essential structure. This mathematical foundation ensures that correctness properties compose and are preserved across abstraction boundaries.

Neural-Symbolic Integration

The CAB framework extends to neural networks through proof-carrying neural dynamics. Nucleus bottlenecks in neural architectures can be formally verified, enabling AI systems with mathematically guaranteed behavior in critical components.

Technical Foundation

Dependent Type Theory

Lean 4 uses the Calculus of Inductive Constructions (CIC), where types can depend on values. This enables proofs and programs to be unified in a single framework, with type checking equivalent to proof verification.

Nucleus Operators

Inspired by Lawvere-Tierney topologies, nucleus operators act as “ratchets” that preserve correctness invariants through program transformations. Once a property is established, it cannot be lost.

Laws of Form

Spencer-Brown's calculus of indications provides a minimal foundation for logical reasoning. CAB uses this to establish base-level semantic invariants that propagate through the compilation pipeline.

-- Example: End-to-End Verified Function

theorem

succTwice_correct : forall (n : Nat), compile(succTwice) n = n + 2 :=
-- Proof verified by Lean 4
-- Compilation verified through Lambda IR
-- C output verified by MiniC semantics
rfl

Research Foundation

Proving Lean with Lean

Establishes the foundational bootstrap where Lean 4's kernel is verified in Lean itself, creating a self-certifying type checker that forms the root of trust for CAB.

Formal Verification | Type Theory | Self-Reference

Proof-Carrying Program Synthesis via Categorical Modalities

Unifies Laws of Form, synthetic computability, and neuro-symbolic architectures through categorical modalities. Provides the theoretical framework for proof-preserving program synthesis.

Category Theory | Program Synthesis | Modalities

Proof-Carrying Neural Dynamics

Extends CAB to neural networks through Lean-verified nucleus bottlenecks for Koopman autoencoders. Enables formally verified machine learning components.

Neural Networks | Dynamical Systems | Verified ML

Explore the Proof Corpus

Visualize and navigate the formally verified proofs in the HeytingLean corpus.

Explore Full Proof Corpus→

CAB Certification | HeytingLean Project | Apoth3osis

All proofs machine-verified using Lean 4