proof-driven

Proof-driven development

Prove properties from requirements before writing code. Proofs guide implementation, not the reverse. Zero unproven properties in final code.

Modern insight (2025): PBT + example tests pairing is the standard -- properties discover edge cases, example tests prevent regressions and serve as documentation. Counterexamples from shrinking should always become permanent regression tests. AI-assisted PBT (Anthropic 2025) can generate properties from docstrings, but human judgment for property selection remains essential.

See frameworks for language-specific PBT and stateful testing tools. See examples for brief property test patterns per language. See formal-tools for theorem provers and bounded model checkers.

---

Property Categories

Category	Description	Example
Postcondition	Output satisfies contract	sorted(sort(xs))
Invariant	Property preserved by operation	len(xs) == len(sort(xs))
Idempotence	f(f(x)) == f(x)	deduplicate(deduplicate(xs))
Inverse / Round-trip	g(f(x)) == x	decode(encode(x)) == x
Model-based	Implementation matches reference	my_sort(xs) == stdlib_sort(xs)
Commutativity	Order doesn't matter	a + b == b + a
Metamorphic	Relationship between outputs	sin(-x) == -sin(x)

Most effective (OOPSLA 2025): Model-based properties (~80% bug detection), postconditions (~65%). Least effective: properties that reimplement the logic under test.

Anti-pattern: Don't reimplement the function in the property. Properties should be simpler than the code they test.

---

When to Apply

• Critical algorithms (sort, search, crypto, compression)
• Financial calculations (rounding, currency conversion)
• Consensus/distributed protocols (invariants across nodes)
• Safety-critical systems (medical, automotive, aerospace)
• Data structure invariants (balanced tree, heap property)
• Serialization round-trip (encode/decode fidelity)
• Stateful systems (databases, queues, caches) -- via stateful PBT

Reasoning approach

Before attempting a proof, reason through the property — SHORT-form KEYWORDS for internal scratchwork, break down the property into hypothesis and assumptions, critically review which proof strategy fits (induction, case analysis, contradiction, construction), validate each strategy against the property structure. Work through the proof step-by-step, verifying each step against the axioms. If a step fails, diagnose why before revising the strategy. For numeric calculation arising in the proof (e.g., bound arithmetic, complexity sums), invoke fend per the baseline rule; never self-calculate. Symbolic reasoning, case enumeration, and induction structure are in-head — they are not arithmetic.

Protocol

When NOT to Apply

• UI rendering, visual layout
• Simple CRUD endpoints
• Configuration management
• Non-critical utility code
• Rapidly changing requirements (properties are expensive to maintain)

---

Anti-patterns

• Happy-path-only properties: Properties must cover edge cases -- that's their primary value
• Skipping stateful testing for stateful systems: Use model-based stateful PBT (Hypothesis RuleBasedStateMachine, jqwik stateful)
• Ignoring counterexamples: Shrunk counterexamples are gold -- always convert to permanent regression tests
• Properties that test the framework: assert fast_check works is not assert my_code works
• Permanently skipped/pending properties: Zero-skip policy -- skip = unfinished work
• Conflating PBT with unit testing: PBT explores input space; unit tests verify known examples. Use both.
• Not using shrinking: If counterexample is 500-line input, it's useless. Shrinking finds minimal failing case.
• Reimplementing logic in properties: Property should be simpler than the code. If property is as complex as implementation, it adds no confidence.

---

Shrinking

Shrinking transforms a failing complex input into the minimal input that still fails. This is the most valuable feature of PBT frameworks.

• Integrated shrinking (Hypothesis, Hedgehog): Generates shrink tree during generation. Preserves generator invariants. Superior approach.
• Type-based shrinking (QuickCheck): Separate shrinker functions. Can violate generator constraints.
• Always investigate shrunk counterexamples: They reveal the essential failure, stripped of noise.

PBT vs Fuzzing (decision guidance)

Aspect	PBT	Fuzzing
Input generation	Guided by properties	Guided by code coverage
Oracle	User-written property assertions	Crashes/exceptions/timeouts
Best for	Correctness, algorithms, contracts	Security, memory safety, crash detection
Convergence (2025)	Hybrid tools (Bolero, Antithesis) combine both approaches

---

Proof Strategies

• Simplification: Reduce by known rules, use shrinking to find minimal counterexamples
• Arithmetic: Generate numeric edge cases (0, 1, MAX, negative, overflow boundaries)
• Case analysis: Split on constructors/variants, test each branch independently
• Induction: Recursive/sequential properties via stateful testing
• Fuzzing: Empirical exploration when properties are hard to specify formally
• Metamorphic relations: When oracle is unknown, test relationships between outputs

Theorem Hierarchy

Main Property (Goal)
|-- Supporting Property 1
|   +-- Helper Property 1a
|-- Supporting Property 2
+-- Edge Case Property 3

---

Workflow (language-neutral)

1. PLAN -- Identify correctness, safety, invariant, and termination properties. Design hierarchy. Choose property categories.
2. CREATE -- Write property test files. One property per concern. Tag by category (postcondition, invariant, inverse, etc.).
3. VERIFY -- Run all properties. Count unproven (skipped/pending). Analyze counterexamples via shrinking.
4. REMEDIATE -- Fill in each skipped property using proof strategies. Convert every counterexample to a permanent regression test.

---

Constitutional Rules (Non-Negotiable)

1. CREATE First: Generate all property test artifacts from plan design before verification
2. Complete All Proofs: Zero skipped/pending properties in final code
3. Totality Required: All definitions must terminate
4. Target Mirrors Model: Implementation structure corresponds to proven model
5. Iterative Remediation: Fix proof failures, don't abandon verification

Validation Gates

Gate	Pass Criteria	Blocking
Framework	PBT framework available and configured	Yes
Properties	All property tests pass	Yes
Unproven	Zero skipped/pending properties	Yes
Coverage	>= 80% line coverage	If present

Exit Codes

Code	Meaning
0	All properties pass, zero unproven/skipped
11	Property testing framework not available
12	No property tests created
13	Property tests failed or proofs incomplete
14	Coverage gaps (properties missing)