# Type a Boolean identity application

KL-FCS-004 · Type systems · version 1.0.0

## Problem

Reconstruct the type of (λx:Bool. x) true in the simply typed lambda fragment.

## Context

Reconstruct a typing judgment from syntax, annotations, and context. Separate the validity of one judgment from the soundness of a language.

## Definitions

- **Context**: A map assigning types to variables currently in scope.
- **Arrow type**: A function domain and codomain, written A → B.
- **Application rule**: A function may be applied only to an argument matching its domain type.

## Checked result

Reconstructed type: Bool.

The abstraction has type Bool → Bool. Its argument has type Bool, so application produces Bool. The checker reconstructs these types from syntax and an initially empty context.

## Checker reasoning

1. Walk the term recursively from an empty context.
2. Extend the context when entering an annotated lambda.
3. Construct arrow types for abstractions.
4. At each application, check the argument type and return the codomain.

## Dataset construction

{
  "family": "type-systems",
  "task": "Reconstruct the type of (λx:Bool. x) true in the simply typed lambda fragment.",
  "input_encoding": "Structured JSON; field meanings are stated in the specification.",
  "coverage": "Exact type reconstruction · 1 judgment",
  "acceptance": [
    "Walk the term recursively from an empty context.",
    "Extend the context when entering an annotated lambda.",
    "Construct arrow types for abstractions.",
    "At each application, check the argument type and return the codomain."
  ],
  "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
  "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
}

## Formal payload

```json
{
  "specification": {
    "term": [
      "app",
      [
        "lam",
        "x",
        "Bool",
        [
          "var",
          "x"
        ]
      ],
      [
        "bool",
        true
      ]
    ],
    "rules": "Boolean literals have Bool; variables use the context; abstractions form arrows; applications require an exact domain match."
  },
  "claim": {
    "type": "Bool"
  },
  "witness": {
    "method": "syntax-directed reconstruction"
  }
}
```

## Complexity

The syntax-directed checker visits each term node; context lookup and type comparison add implementation-dependent cost.

## Limits

A single typing judgment; no proof of type soundness, normalization, or full language implementation.

## Common error and further work

Being well typed is not, by itself, a proof of preservation or progress.

Add independently checked derivation trees and a broader type grammar.

## Verification

Exact type reconstruction · 1 judgment. 1 checker units.
Replay with `python3 tools/verify.py`. Mechanical status: checked; human review
has not yet been recorded. Custom Python verification, not a proof-assistant
claim. Checker 1.0.0 and exact source hashes are in `verification.json`.

## Provenance and references

Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.
No external dataset or model-generation experiment. Reuse-license selection
remains pending.

- [Conceptual reference](https://softwarefoundations.cis.upenn.edu/plf-current/Stlc.html)