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| 1 | +/- |
| 2 | +SPDX-License-Identifier: MPL-2.0 |
| 3 | +Copyright (c) 2026 Jonathan D.A. Jewell (hyperpolymath) <j.d.a.jewell@open.ac.uk> |
| 4 | +
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| 5 | +WokeGrammarRegular.lean — machine-checks grammar-proofs.md §7.1: the WokeLang |
| 6 | +expression language is NOT regular. Self-contained: Lean core prelude only (no |
| 7 | +Mathlib), so CI checks it with a bare `lean WokeGrammarRegular.lean`. |
| 8 | +
|
| 9 | +Method: a finite DFA cannot accept the balanced-nesting sublanguage. We |
| 10 | +- prove a bespoke finite pigeonhole from scratch (Mathlib's is unavailable here); |
| 11 | +- model a DFA over `Fin k` states; |
| 12 | +- run the fooling-set / pigeonhole argument (the combinatorial heart of the |
| 13 | + pumping lemma) on `aⁿbⁿ`: among the k+1 prefixes a⁰..aᵏ two reach the same |
| 14 | + state, so aⁱbⁱ (accepted) and aʲbⁱ (j≠i) reach the same final state, forcing |
| 15 | + the DFA to also accept aʲbⁱ ∉ L — contradiction. |
| 16 | +
|
| 17 | +Connection to the grammar: the WokeLang expression grammar derives arbitrarily |
| 18 | +deep balanced groupings `(ⁿ x )ⁿ` (the `primary = "(" expression ")"` rule, used |
| 19 | +n-deep), which is isomorphic to `aⁿbⁿ` under `( ↦ a`, `) ↦ b`. So the expression |
| 20 | +language contains a non-regular sublanguage and is itself non-regular. |
| 21 | +-/ |
| 22 | + |
| 23 | +namespace WokeGrammarRegular |
| 24 | + |
| 25 | +/-- **Finite pigeonhole** (no Mathlib): if `f` maps `[0,m)` into `[0,n)` with |
| 26 | +`n < m`, two distinct indices collide. By induction on `n`, collapsing the value |
| 27 | +at the last index out of the codomain. -/ |
| 28 | +theorem pigeon : ∀ (n m : Nat) (f : Nat → Nat), |
| 29 | + (∀ i, i < m → f i < n) → n < m → ∃ i j, i < j ∧ j < m ∧ f i = f j := by |
| 30 | + intro n |
| 31 | + induction n with |
| 32 | + | zero => intro m f hf hnm; have h0 := hf 0 (by omega); omega |
| 33 | + | succ n ih => |
| 34 | + intro m f hf hnm |
| 35 | + by_cases hcol : ∃ j, j < m - 1 ∧ f j = f (m - 1) |
| 36 | + · obtain ⟨j, hj, hfj⟩ := hcol; exact ⟨j, m - 1, by omega, by omega, by omega⟩ |
| 37 | + · have hcol' : ∀ j, j < m - 1 → f j ≠ f (m - 1) := by |
| 38 | + intro j hj hfj; exact hcol ⟨j, hj, hfj⟩ |
| 39 | + let g : Nat → Nat := fun i => if f i < f (m - 1) then f i else f i - 1 |
| 40 | + have hg : ∀ i, i < m - 1 → g i < n := by |
| 41 | + intro i hi |
| 42 | + have hbound := hf i (by omega); have hne := hcol' i hi |
| 43 | + have hlast := hf (m - 1) (by omega) |
| 44 | + simp only [g]; by_cases hlt : f i < f (m - 1) <;> simp [hlt] <;> omega |
| 45 | + obtain ⟨i, j, hij, hjm, hgij⟩ := ih (m - 1) g hg (by omega) |
| 46 | + refine ⟨i, j, hij, by omega, ?_⟩ |
| 47 | + have hnei := hcol' i (by omega); have hnej := hcol' j hjm |
| 48 | + simp only [g] at hgij |
| 49 | + by_cases hi : f i < f (m - 1) <;> by_cases hj : f j < f (m - 1) <;> |
| 50 | + simp [hi, hj] at hgij <;> omega |
| 51 | + |
| 52 | +/-- Two-symbol alphabet (`a ≙ "("`, `b ≙ ")"`). -/ |
| 53 | +inductive Sym | a | b deriving DecidableEq, Repr |
| 54 | + |
| 55 | +/-- A deterministic finite automaton with `k` states. -/ |
| 56 | +structure DFA (k : Nat) where |
| 57 | + start : Fin k |
| 58 | + step : Fin k → Sym → Fin k |
| 59 | + accept : Fin k → Bool |
| 60 | + |
| 61 | +def DFA.runFrom {k} (M : DFA k) : Fin k → List Sym → Fin k |
| 62 | + | s, [] => s |
| 63 | + | s, x :: w => M.runFrom (M.step s x) w |
| 64 | + |
| 65 | +def DFA.accepts {k} (M : DFA k) (w : List Sym) : Bool := M.accept (M.runFrom M.start w) |
| 66 | + |
| 67 | +theorem runFrom_append {k} (M : DFA k) (s : Fin k) (w1 w2 : List Sym) : |
| 68 | + M.runFrom s (w1 ++ w2) = M.runFrom (M.runFrom s w1) w2 := by |
| 69 | + induction w1 generalizing s with |
| 70 | + | nil => rfl |
| 71 | + | cons x w ih => simp only [List.cons_append, DFA.runFrom, ih] |
| 72 | + |
| 73 | +/-- The language `aⁿbⁿ`. -/ |
| 74 | +def isAnBn (w : List Sym) : Prop := |
| 75 | + ∃ n, w = List.replicate n Sym.a ++ List.replicate n Sym.b |
| 76 | + |
| 77 | +def countA : List Sym → Nat |
| 78 | + | [] => 0 |
| 79 | + | Sym.a :: w => countA w + 1 |
| 80 | + | Sym.b :: w => countA w |
| 81 | + |
| 82 | +theorem countA_append (u v : List Sym) : countA (u ++ v) = countA u + countA v := by |
| 83 | + induction u with |
| 84 | + | nil => simp [countA] |
| 85 | + | cons x w ih => cases x <;> simp [countA, ih] <;> omega |
| 86 | + |
| 87 | +theorem countA_repl_a (n : Nat) : countA (List.replicate n Sym.a) = n := by |
| 88 | + induction n with |
| 89 | + | zero => rfl |
| 90 | + | succ n ih => simp [List.replicate, countA, ih] |
| 91 | + |
| 92 | +theorem countA_repl_b (n : Nat) : countA (List.replicate n Sym.b) = 0 := by |
| 93 | + induction n with |
| 94 | + | zero => rfl |
| 95 | + | succ n ih => simp [List.replicate, countA, ih] |
| 96 | + |
| 97 | +/-- If `aʲbⁱ = aⁿbⁿ` then `j = n` and `i = n` (count `a`s, then lengths). -/ |
| 98 | +theorem anbn_inj {i j n : Nat} |
| 99 | + (h : List.replicate j Sym.a ++ List.replicate i Sym.b |
| 100 | + = List.replicate n Sym.a ++ List.replicate n Sym.b) : j = n ∧ i = n := by |
| 101 | + have hc : countA (List.replicate j Sym.a ++ List.replicate i Sym.b) |
| 102 | + = countA (List.replicate n Sym.a ++ List.replicate n Sym.b) := by rw [h] |
| 103 | + rw [countA_append, countA_append, countA_repl_a, countA_repl_a, |
| 104 | + countA_repl_b, countA_repl_b] at hc |
| 105 | + have hlen : (List.replicate j Sym.a ++ List.replicate i Sym.b).length |
| 106 | + = (List.replicate n Sym.a ++ List.replicate n Sym.b).length := by rw [h] |
| 107 | + simp only [List.length_append, List.length_replicate] at hlen |
| 108 | + omega |
| 109 | + |
| 110 | +/-- **T7.1 — not regular.** No finite DFA accepts exactly `aⁿbⁿ`. Via the grammar's |
| 111 | +balanced-nesting sublanguage `(ⁿ x )ⁿ` (isomorphic to `aⁿbⁿ`), the WokeLang |
| 112 | +expression language is therefore not regular. -/ |
| 113 | +theorem anbn_not_regular (k : Nat) (M : DFA k) : |
| 114 | + ¬ (∀ w, M.accepts w = true ↔ isAnBn w) := by |
| 115 | + intro hM |
| 116 | + let f : Nat → Nat := fun i => (M.runFrom M.start (List.replicate i Sym.a)).val |
| 117 | + have hf : ∀ i, i < k + 1 → f i < k := fun i _ => |
| 118 | + (M.runFrom M.start (List.replicate i Sym.a)).isLt |
| 119 | + obtain ⟨i, j, hij, _, hfij⟩ := pigeon k (k + 1) f hf (by omega) |
| 120 | + have hstate : M.runFrom M.start (List.replicate i Sym.a) |
| 121 | + = M.runFrom M.start (List.replicate j Sym.a) := Fin.eq_of_val_eq hfij |
| 122 | + have hacc_i : M.accepts (List.replicate i Sym.a ++ List.replicate i Sym.b) = true := |
| 123 | + (hM _).mpr ⟨i, rfl⟩ |
| 124 | + have hsame : M.accepts (List.replicate j Sym.a ++ List.replicate i Sym.b) |
| 125 | + = M.accepts (List.replicate i Sym.a ++ List.replicate i Sym.b) := by |
| 126 | + simp only [DFA.accepts, runFrom_append, ← hstate] |
| 127 | + have hacc_j : M.accepts (List.replicate j Sym.a ++ List.replicate i Sym.b) = true := by |
| 128 | + rw [hsame]; exact hacc_i |
| 129 | + obtain ⟨n, hn⟩ := (hM _).mp hacc_j |
| 130 | + obtain ⟨hj, hi⟩ := anbn_inj hn |
| 131 | + omega |
| 132 | + |
| 133 | +end WokeGrammarRegular |
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