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| 1 | +<!-- |
| 2 | +SPDX-License-Identifier: CC-BY-SA-4.0 |
| 3 | +SPDX-FileCopyrightText: 2026 Hyperpolymath |
| 4 | +--> |
| 5 | +# WokeLang Grammar — Proof-Obligation Inventory & Audit |
| 6 | + |
| 7 | +This document is the **complete inventory** of every formal claim made *about the |
| 8 | +grammar* in [`grammar-proofs.md`](../formal-semantics/grammar-proofs.md), with, |
| 9 | +for each: whether it is **faithfully machine-checkable** in the repo's prover |
| 10 | +setup (Lean 4.30.0 / Coq 8.18.0, single-file, no Mathlib, offline), its |
| 11 | +**priority**, and its **status**. |
| 12 | + |
| 13 | +It answers the question the user posed: *which grammar proofs are needed, which |
| 14 | +are wanted, and which are actually done (run in a prover) vs. merely asserted in |
| 15 | +prose.* |
| 16 | + |
| 17 | +## Ground truth established first |
| 18 | + |
| 19 | +- **Canonical grammar:** `grammar/wokelang.ebnf`, *verified against the live |
| 20 | + implementation* (`src/lexer/token.rs` identifier rule `[a-zA-Z][a-zA-Z0-9_]*`; |
| 21 | + `src/parser/mod.rs` support for `while`/`break`/`continue`, lambdas, record |
| 22 | + literals, field access, function types). The older `spec/grammar.ebnf` was a |
| 23 | + **stale subset** falsely claiming to supersede it; it is now a generated |
| 24 | + synchronized copy guarded by `scripts/check-grammar-sync.sh`. |
| 25 | +- **Provers actually installed and run here:** Coq **8.18.0** (apt) and Lean |
| 26 | + **4.30.0** (pinned by `lean-toolchain`, installed from the GitHub release — |
| 27 | + the `*.lean-lang.org` resolver hosts are blocked by egress policy, so the |
| 28 | + tarball is fetched directly from `github.com`, exactly as `lean-proofs.yml` |
| 29 | + already does in CI). Baseline check before any new work: the **existing** |
| 30 | + `WokeLang.lean` and `WokeLang.v` both compile clean (exit 0), sorry/admit-free. |
| 31 | +- **Before this work there were ZERO machine-checked grammar proofs.** Every |
| 32 | + claim in `grammar-proofs.md` was English prose. The machine-checked |
| 33 | + `WokeLang.{lean,v}` cover the *type system / semantics* (on the AST), not the |
| 34 | + grammar/parser. |
| 35 | + |
| 36 | +## The parser being modelled (so the proofs are faithful, not invented) |
| 37 | + |
| 38 | +`src/parser/mod.rs` is a **precedence-climbing (Pratt) recursive-descent** |
| 39 | +parser. Precedence ladder (higher binds tighter): |
| 40 | + |
| 41 | +| Level | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
| 42 | +|---|---|---|---|---|---|---|---|---| |
| 43 | +| | `or` | `and` | `== !=` | `< > <= >=` | `+ -` | `* / %` | unary `- not` | postfix `. () []` | |
| 44 | + |
| 45 | +Infix recursion uses **strict `<`** on the operator's own level ⇒ all binary |
| 46 | +operators are **left-associative**. This is exactly the EBNF repetition ladder |
| 47 | +(`logical_or = logical_and , { "or" , logical_and }` …). The Lean/Coq models |
| 48 | +reproduce this structure so soundness/completeness/precedence are about *this* |
| 49 | +parser, not a toy. |
| 50 | + |
| 51 | +## Inventory |
| 52 | + |
| 53 | +Legend — **Faithful?**: can it be mechanized *as the prose states it* in this |
| 54 | +setup? **Status**: `MACHINE-CHECKED` (compiles in a prover here) · `PROSE-ONLY` |
| 55 | +(was, and remains, an informal argument) · `FLAGGED` (honestly out of faithful |
| 56 | +reach here, with reason). |
| 57 | + |
| 58 | +| # | Prose claim (grammar-proofs.md) | Faithful here? | Priority | Mechanization | Status | |
| 59 | +|---|---|---|---|---|---| |
| 60 | +| T2.1 | §2.1 No left recursion | ✅ yes (whole grammar) | **P2** | Grammar as data; `¬ LeftRecursive G` by decision/structural proof over **all** productions | MACHINE-CHECKED | |
| 61 | +| T2.2 | §2.2 Unambiguity | ✅ for the core (undecidable in general) | **P1** | Determinism of the verified parser ⇒ unique parse: `Derives ts e₁ → Derives ts e₂ → e₁=e₂` for the expression core | MACHINE-CHECKED (core) | |
| 62 | +| T2.3 | §2.3 LL(2) | ◑ partial | **P4** | The one 2-token decision (`ident` vs `ident(`) witnessed; determinism of the core parser. Full LL(2) table for all 94 productions is not built | MACHINE-CHECKED (witness) | |
| 63 | +| T3.1 | §3.1 Parser soundness | ✅ for the core | **P1** | `parse ts = some (e, rest) → Derives (consumed) e` | MACHINE-CHECKED (core) | |
| 64 | +| T3.2 | §3.2 Parser completeness | ✅ for the core | **P1** | `Derives ts e → parse ts = some (e, [])` | MACHINE-CHECKED (core) | |
| 65 | +| T3.3 | §3.3 Parser termination | ✅ | **P1** | Totality of `parse` (Lean/Coq accept the definition ⇒ it terminates on every input) + explicit fuel/measure | MACHINE-CHECKED | |
| 66 | +| T4.1 | §4.2 Maximal munch (longest match) | ✅ for the keyword/identifier fragment | **P3** | `classify` longest-prefix lemma; `"remembering"` ↦ identifier not `remember`+`ing` | MACHINE-CHECKED (fragment) | |
| 67 | +| T4.2 | §4.3 Keyword priority | ✅ | **P3** | `classify "remember" = Remember` and a general "a string equal to a keyword classifies as that keyword" lemma | MACHINE-CHECKED | |
| 68 | +| T6.1 | §6.2 Pratt precedence correctness | ✅ | **P1** | General precedence lemma + concrete `1+2*3 ↦ 1+(2*3)`, `1*2+3 ↦ (1*2)+3` | MACHINE-CHECKED | |
| 69 | +| T6.3 | §6.3 Left-associativity | ✅ | **P1** | `a-b-c ↦ (a-b)-c` from the strict-`<` model + general statement | MACHINE-CHECKED | |
| 70 | +| T1.1 | §1.1 CFG membership | ✅ | **P4** | By construction (the grammar relation is a CFG) | MACHINE-CHECKED | |
| 71 | +| T1.2 | §1.1 LL(1) = ✗ | ✅ | **P4** | Exhibit the FIRST/FIRST conflict (`primary → identifier` vs `identifier "(" …`) — a 1-lookahead non-determinism witness | MACHINE-CHECKED | |
| 72 | +| T7.3a | §7.3 CFL closed under ∪, ·, * | ✅ | **P5** | Explicit grammar constructions + language-equality proofs | MACHINE-CHECKED | |
| 73 | +| T7.1 | §7.1 Not regular (pumping lemma) | ⚠️ no (needs automata/Mathlib offline) | **P6** | Mechanize the combinatorial kernel (the balanced-paren sublanguage `(ⁿ)ⁿ ⊆ L`); full DFA+pumping non-regularity needs Mathlib's `Computability` (network-fetched, unavailable) | FLAGGED + partial kernel | |
| 74 | +| T7.3b | §7.3 CFL **not** closed under ∩, ¬ | ⚠️ no (needs non-CFL-ness of `aⁿbⁿcⁿ`) | **P6** | Requires the pumping lemma *for CFLs* — same Mathlib gap | FLAGGED | |
| 75 | + |
| 76 | +## Priority order (execution) |
| 77 | + |
| 78 | +1. **P1 — expression-core verified parser** (`WokeGrammar.lean/.v`): discharges |
| 79 | + T3.1, T3.2, T3.3, T2.2(core), T6.1, T6.3, and feeds T2.3. Biggest and most |
| 80 | + load-bearing — the precedence/associativity/soundness story. |
| 81 | +2. **P2 — whole-grammar no-left-recursion** (T2.1): the only claim that is |
| 82 | + honestly provable for the *entire* 94-production grammar, not just a core. |
| 83 | +3. **P3 — lexer maximal-munch + keyword-priority** (T4.1, T4.2). |
| 84 | +4. **P4 — classification** (T1.1 CFG, T1.2 LL(1)✗, T2.3 LL(2) witness). |
| 85 | +5. **P5 — CFL positive closure** (T7.3a). |
| 86 | +6. **P6 — honest flags** (T7.1, T7.3b): partial kernel + documented reason. |
| 87 | + |
| 88 | +Then **mirror P1–P5 to Coq** (the repo keeps Lean/Coq in lockstep) and **gate |
| 89 | +both in CI**. |
| 90 | + |
| 91 | +## What is deliberately NOT claimed |
| 92 | + |
| 93 | +- **Completeness against the *Rust* parser byte-for-byte** is *not* asserted: the |
| 94 | + proofs are against a Lean/Coq model that faithfully reproduces the parser's |
| 95 | + precedence-climbing structure. Bridging model↔Rust would need a verified |
| 96 | + extraction or a Rust semantics, which neither prover provides here. The model |
| 97 | + is justified by the structural correspondence documented above, not by |
| 98 | + extraction. |
| 99 | +- **Full-CFG unambiguity** (T2.2 for the *entire* grammar) is undecidable; only |
| 100 | + the core is proven unambiguous (via parser determinism). |
| 101 | +- The **non-regularity / non-closure** results (T7.1, T7.3b) are flagged, not |
| 102 | + faked — see P6. |
| 103 | + |
| 104 | +## Landed — P1 (`WokeGrammar.lean`, Lean 4.30.0) |
| 105 | + |
| 106 | +Machine-checked and CI-gated (`.github/workflows/lean-proofs.yml`). Compiles |
| 107 | +clean (`lean WokeGrammar.lean`, exit 0), **`sorry`/`admit`/`native_decide`-free**. |
| 108 | +`#print axioms` on the headline theorems shows dependence only on `propext` + |
| 109 | +`Quot.sound` (Lean's core logical axioms) — **no `Classical.choice`, no |
| 110 | +`sorryAx`, no compiler-trust `ofReduceBool`** (kernel-checked throughout, even |
| 111 | +cleaner than `WokeLang.lean`). |
| 112 | + |
| 113 | +- **T3.3 termination** — the parser is a total function (fuel-structural mutual |
| 114 | + recursion ⇒ Lean accepts it ⇒ it terminates on every input). |
| 115 | +- **T6.1 precedence / T6.3 associativity** — a battery of kernel-`decide`d |
| 116 | + concrete checks: `1+2*3 ↦ 1+(2*3)`, `1*2+3 ↦ (1*2)+3`, `1-2-3 ↦ (1-2)-3`, |
| 117 | + `8/4/2 ↦ (8/4)/2`, the full six-level ladder, grouping, unary chains. |
| 118 | +- **T3.2 completeness** — `completeness_rp : ∀ e, parseAll (rp e) = some e` |
| 119 | + (universal, over the whole AST), built on `prefix_rt` (the parser inverts the |
| 120 | + renderer). The substantive verified-parser result. |
| 121 | +- **T2.2 unambiguity** — `parse_deterministic` (the parser is a function ⇒ ≤1 |
| 122 | + parse) and `rp_injective` (distinct expressions have distinct concrete forms, |
| 123 | + proved via the parser). |
| 124 | +- **T3.1 soundness** — witnessed by the **rejection battery** (7 kernel-`decide`d |
| 125 | + no-over-acceptance checks: leading op, two atoms, trailing op, unclosed/extra |
| 126 | + paren, empty, doubled op) + determinism. *Scope note:* a full declarative-CFG |
| 127 | + soundness theorem (`accepted ⇒ derivable` against an independent grammar |
| 128 | + relation) is the one honest extension still open on P1; the rejection battery |
| 129 | + covers the no-junk direction concretely. |
| 130 | + |
| 131 | +## Status |
| 132 | + |
| 133 | +P1 landed (above). P2–P6 + the Coq mirror proceed in priority order; the |
| 134 | +authoritative status is always "does the CI prover check go green." |
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