Type Inference Validation

Core Concept

Static type validation that rejects invalid path compositions before caching. Every Navigator path must prove type compatibility with its input structure—this skill enforces that proof.

Works alongside Möbius filtering as a second line of defense: Möbius eliminates topological impossibilities, Type Inference eliminates structural impossibilities.

Why Type Validation?

Consider this dangerous scenario:

data_numbers = [1, 2, 3, 4, 5]
data_dict = Dict("x" => 10, "y" => 20)

# This path makes sense for numbers
nav1 = @late_nav([ALL, pred(iseven)])  # Type ✓: Vector → Numbers → Numbers

# But what if someone mistakenly uses it on the dict?
nav_select(nav1, data_dict, x -> [x])
# => IndexError! (dicts don't support ALL enumeration this way)

Type Inference Validation prevents this at compile time, not runtime.

Architecture

Type System

Each Navigator carries a type signature:

Navigator{InputType, OutputType, PathSteps}

Examples:

nav_vector = Navigator{Vector, Vector, [ALL, pred(f)]}
nav_dict = Navigator{Dict, Vector, [keypath(k), ALL]}
nav_sexp = Navigator{SExpression, Symbol, [sexp_nth(2), SEXP_HEAD]}

Type Refinement Pipeline

Input Type
    ↓
@late_nav(path_expr) triggers validation
    ↓
Type each step in path:
  [ALL] : Vector{T} → T
  [keypath(k)] : Dict{K,V} → V
  [pred(f)] : T → T (preserves type)
  [INDEX(i)] : Vector{T} → T
  [SEXP_HEAD] : SExpr → Symbol
    ↓
Unify step outputs with next step inputs
    ↓
Final output type computed
    ↓
Cache with type signature
    ↓
Accept or Reject

Type Incompatibility Detection

API

Type Inference

infer_type(path_expr, input_type) :: Result{OutputType, Error} Computes the output type of a path given an input type.

infer_type([ALL, pred(iseven)], Vector{Int})
# => Result(Int)  # outputs Int values

infer_type([keypath("x"), ALL], Dict{String, Vector{Int}})
# => Result(Int)  # navigates to x, then enumerates integers

infer_type([keypath("x")], Vector{Int})
# => Result(TypeError("Cannot apply keypath to Vector"))

validate_path(navigator::Navigator, input_type) :: Bool Checks if a Navigator is compatible with a given input type.

nav = @late_nav([ALL, pred(f)])
validate_path(nav, Vector{Int})      # => true
validate_path(nav, Dict{String, Int})  # => false ✗

# Causes @late_nav to reject the Navigator before caching

Type Signatures

navigator_signature(nav::Navigator) :: TypeSignature Returns the full type signature of a cached Navigator.

nav = @late_nav([ALL, pred(iseven)])
sig = navigator_signature(nav)

# => TypeSignature(
#     input: Vector{T},
#     output: T,
#     constraints: [iseven(T)],
#     steps: [[ALL], [pred(iseven)]],
#     valid_for: Vector{Int}, Vector{BigInt}, ...
#   )

Polymorphic Inference

polymorphic_infer(path_expr) :: PolymorphicType Infers the most general type for a path (before knowing input type).

polymorphic_infer([ALL, pred(iseven)])
# => ∀T. (T ∈ Enumerable, T ∈ HasEven) → T

# This means: works for ANY type T that:
# - Can be enumerated (Vector, Set, List, etc.)
# - Has an iseven() method defined

Integration with Caching

Cache keys now include type information:

cache_key = hash((path_expr, inferred_type))

# Different input types → different cache entries
nav_vec = @late_nav([ALL, pred(f)])   # cache for Vector
nav_set = @late_nav([ALL, pred(f)])   # cache for Set (if compatible)
# => Different NavigatorObjects, despite same path!

Refinement Types

Supports refinement types for predicates:

# pred(iseven) refines Int → EvenInt
# pred(x -> x > 5) refines Int → IntGt5

nav = @late_nav([ALL, pred(x -> x > 5)])
# Type: Vector{Int} → IntGt5
# (output is proven to be > 5)

Refinement types enable:

• Automatic constraint propagation downstream
• Proof that outputs satisfy predicates without re-checking
• Composition of constraints with type safety

Type Mismatch Examples

❌ INVALID: Wrong container type

nav = @late_nav([keypath("name"), ALL])
# Type error: keypath returns a string, ALL requires enumerable

# KeyPath{Dict, String} → String
# ALL{String} → ✗ (String not enumerable in that way)

❌ INVALID: Incompatible predicate

nav = @late_nav([ALL, pred(x -> x > 5)])
# If input is Vector{String}, predicate fails
# Type error: `(String > 5)` is nonsense

✓ VALID: Polymorphic path

nav = @late_nav([ALL, pred(identity)])  # identity always works
# Type: ∀T. Vector{T} → T
# Works for ANY vector type

Error Messages

Type validation errors are clear:

TypeError: Path composition invalid
  Step 1: [ALL]
    Input: Vector{Int}
    Output: Int
    ✓ Type check passed

  Step 2: [keypath("x")]
    Input: Int
    Output: ???
    ✗ TypeError: Cannot apply keypath to Int
       keypath requires Dict or record type
       Got: Int

Suggestion: Remove [keypath("x")] or ensure input is Dict/Struct

Performance

All type checking happens at compile time (during @late_nav expansion):

Composition with Other Skills

Works with Möbius Filtering:

1. Möbius filters topological impossibilities
2. Type Inference filters structural impossibilities
3. Both must pass for path to compile

Works with Color Envelopes: Type signatures include color trit information:

nav = @late_nav([ALL, pred(f)])  # trit = -1 (filtering)
sig = navigator_signature(nav)
# => TypeSignature(..., trit: -1)

# Type system respects color: composition must balance trits AND types

Enables Constraint Generalization (next skill): Once types are proven, constraints can be composed and generalized safely.

Related Skills

• möbius-path-filtering - Topological validation (works alongside this)
• tuple-nav-composition - Uses type signatures to compose products
• constraint-generalization - Builds on type-proven constraints
• specter-navigator-gadget - Uses validated types for safe composition

References

• Gradual typing: "Gradual Typing for Functional Languages" - Siek & Taha
• Refinement types: "Liquid Haskell: Haskell as a Theorem Prover" - Jhala et al.
• Type inference: "Algorithm W: Inference of Data Types" - Damas & Milner