Cure Programming Language - SMT Solver and Proof Assistant

The SMT (Satisfiability Modulo Theories) solver provides constraint solving capabilities for Cure's dependent type system. It handles arithmetic constraints, logical reasoning, and proof generation for dependent type verification.

Features

Constraint Solving

• Arithmetic Constraints: Linear and non-linear arithmetic over integers/reals
• Equality Reasoning: Equality and disequality constraints
• Boolean Logic: Propositional logic with AND, OR, NOT, IMPLIES
• Quantified Formulas: Existential and universal quantification

Pattern Matching Support

• Length Inference: Infers list length constraints from pattern matching
• Structural Constraints: Handles tuple/record structure constraints
• Exhaustiveness Checking: Verifies pattern match completeness
• Reachability Analysis: Determines unreachable pattern branches

Proof Generation

• Constructive Proofs: Generates proof terms for valid constraints
• Counter-examples: Provides counter-examples for invalid constraints
• Proof Verification: Checks independently generated proofs
• Automated Reasoning: Combines multiple proof strategies

Dependent Type Integration

• Type-level Arithmetic: Solves constraints involving type-level values
• Refinement Types: Verifies refinement type predicates
• Constraint Propagation: Propagates constraints through type inference
• Vector Types: Handles length-indexed vectors and arrays

Core Operations

Constraint Solving

%% Create arithmetic constraints
Constraint1 = cure_smt_solver:arithmetic_constraint(X, '+', Y),
Constraint2 = cure_smt_solver:equality_constraint(Result, Z),

%% Solve constraints
case cure_smt_solver:solve_constraints([Constraint1, Constraint2]) of
    {sat, Solution} -> 
        cure_utils:debug("Solution: ~p~n", [Solution]);
    unsat -> 
        cure_utils:debug("No solution exists~n");
    unknown -> 
        cure_utils:debug("Cannot determine satisfiability~n")
end.

Pattern Length Inference

%% Infer length constraints from list patterns
Pattern = {list_pattern, [a, b], {identifier_pattern, tail}, Location},
LengthVar = cure_smt_solver:variable_term(list_length),
Constraints = cure_smt_solver:infer_pattern_length(Pattern, LengthVar).
%% Generates: list_length = 2 + tail_length

Proof Generation

%% Prove that a constraint follows from assumptions
Assumptions = [X_greater_than_zero, Y_positive],
Goal = cure_smt_solver:arithmetic_constraint(X, '+', Y, '>', zero),

case cure_smt_solver:prove_constraint(Assumptions, Goal) of
    {proved, Proof} -> 
        cure_utils:debug("Proof: ~p~n", [Proof]);
    {disproved, CounterExample} -> 
        cure_utils:debug("Counter-example: ~p~n", [CounterExample]);
    unknown -> 
        cure_utils:debug("Cannot prove or disprove~n")
end.

Constraint Types

Arithmetic Constraints

• Linear: ax + by = c, ax + by ≤ c
• Non-linear: xy = z, x² + y² ≤ r²
• Modular: x ≡ y (mod m)
• Divisibility: x divides y

Logical Constraints

• Propositional: P ∧ Q, P ∨ Q, ¬P
• Implications: P → Q, P ↔ Q
• Quantified: ∀x. P(x), ∃x. P(x)

Type-level Constraints

• Length Constraints: length(xs) = n
• Range Constraints: 0 ≤ index < length
• Dimension Constraints: Matrix(m, n) @ Matrix(n, p) = Matrix(m, p)

SMT Solver Interface

Constraint Representation

-record(smt_constraint, {
    type :: equality | inequality | arithmetic | logical,
    left :: smt_term(),
    op :: '=' | '<' | '>' | '<=' | '>=' | '/=' | '+' | '-' | '*',
    right :: smt_term(),
    location :: location()
}).

Term Representation

-record(smt_term, {
    type :: variable | constant | expression,
    value :: atom() | integer() | float() | smt_expression(),
    location :: location()
}).

Solving Strategies

Decision Procedures

1. Linear Arithmetic: Simplex algorithm for linear constraints
2. Congruence Closure: Equality reasoning with uninterpreted functions
3. Bit-vectors: Precise reasoning about fixed-width integers
4. Arrays: Theory of arrays with select/store operations

Proof Search

1. Resolution: Propositional resolution with conflict analysis
2. DPLL(T): Integration of propositional and theory reasoning
3. Instantiation: Quantifier instantiation with triggers
4. Model Construction: Building satisfying assignments

Performance Characteristics

• Linear Arithmetic: Polynomial time in most practical cases
• General Arithmetic: May be exponential (semi-decidable)
• Propositional Logic: NP-complete but efficient SAT solvers
• Quantified Formulas: Undecidable in general, heuristic-based

Integration with Type System

Dependent Types

type Vector(T, n: Nat) where n > 0
%% SMT solver verifies: n > 0

def safe_index(v: Vector(T, n), i: Int) -> T where 0 <= i < n
%% SMT solver verifies: 0 <= i < n

Refinement Types

type Positive = Int where x > 0
type Even = Int where x % 2 == 0
%% SMT solver verifies refinement predicates

Error Handling

The SMT solver provides structured error information:

• Unsatisfiable Core: Minimal unsatisfiable subset of constraints
• Model Generation: Concrete counter-examples for failed proofs
• Timeout Handling: Graceful degradation for complex problems
• Resource Limits: Configurable limits on solving time/memory

External Solver Integration

Supports integration with external SMT solvers:

• Z3: Microsoft Research SMT solver
• CVC4: Stanford/NYU SMT solver
• Yices: SRI SMT solver
• MathSAT: FBKIRST SMT solver

Thread Safety

The SMT solver is stateless and thread-safe. Multiple constraint solving operations can run concurrently without interference.