1	# bend-ml-bpe-tokenizer: byte-level BPE with a proved roundtrip.
2	#
3	#   import bend-ml-bpe-tokenizer@0.1.2.0/main.bend as BPE
4	#
5	# A token is a raw byte (B{n}) or the result of a merge rule (M{id}).
6	# The table is a list of rules from the oldest to the newest, like GPT-2's
7	# merges.txt: the rule with id k joins the pair (a, b) into the token M{k}.
8	# `encode` applies the rules in order; `decode` expands each token back
9	# into bytes. The laws are at the end of the file, each with its proof.
10	#
11	# How to read the proofs: a proof in Bend is a function whose TYPE is the statement.
12	# `match` does case analysis, the recursive call is the induction hypothesis,
13	# `%e : P` rewrites the goal with the equality `e`, `{==}` closes when the
14	# two sides are already the same term. `+x` marks a variable that may be
15	# used more than once.
16	
17	import Base
18	import bend-ml-nat-lemmas@0.1.0.0/main.bend as NL
19	
20	# Token: a raw byte (B) or the result of a merge rule, by rule id (M).
21	type Tk is Data:
22	  B{n: Nat}
23	  M{k: Nat}
24	
25	# Merge rule: joins the pair (a, b) into the token M{id}.
26	type Rule is Data:
27	  Rule{id: Nat, a: Tk, b: Tk}
28	
29	def Tk.eq(+x: Tk, +y: Tk) -> Bool:
30	  match x y:
31	    case B{+m} B{+n}:
32	      Nat.is_eq(m, n)
33	    case M{+i} M{+j}:
34	      Nat.is_eq(i, j)
35	    case B{m} M{j}:
36	      False{}
37	    case M{i} B{n}:
38	      False{}
39	
40	# does xs start with exactly the pair (a, b)?
41	def peek(+a: Tk, +b: Tk, xs: List<&2, Tk>) -> Bool:
42	  match xs:
43	    case Nil{}:
44	      False{}
45	    case Con{+x, t}:
46	      match t:
47	        case Nil{}:
48	          False{}
49	        case Con{+y, u}:
50	          Bool.and(Tk.eq(x, a), Tk.eq(y, b))
51	
52	# Replaces each occurrence (non-overlapping, left to right) of the pair
53	# (a, b) with the token c. `hit` is always `peek(a, b, xs)`: the caller computes it,
54	# because a match can only open parameters.
55	def merge.go(xs: List<&2, Tk>, hit: Bool, +a: Tk, +b: Tk, +c: Tk) -> List<&2, Tk>:
56	  match xs hit:
57	    case Con{x, Con{y, +t}} True{}:
58	      c <> merge.go(t, peek(a, b, t), a, b, c)
59	    case Con{x, Nil{}} True{}:
60	      x <> Nil{}
61	    case Con{x, +t} False{}:
62	      x <> merge.go(t, peek(a, b, t), a, b, c)
63	    case Nil{} _:
64	      Nil{}
65	
66	def merge(+a: Tk, +b: Tk, +c: Tk, +xs: List<&2, Tk>) -> List<&2, Tk>:
67	  merge.go(xs, peek(a, b, xs), a, b, c)
68	
69	# applies the rules from the oldest to the newest; `done` keeps the ones already applied
70	# (newest first), which is exactly the table that decode needs.
71	def encode.go(todo: List<&2, Rule>, done: List<&2, Rule>, xs: List<&2, Tk>) -> List<&2, Tk>:
72	  match todo:
73	    case Nil{}:
74	      xs
75	    case Con{+r, rest}:
76	      match r:
77	        case Rule{+i, +a, +b}:
78	          encode.go(rest, Rule{i, a, b} <> done, merge(a, b, M{i}, xs))
79	
80	# The table goes from the oldest rule to the newest (like merges.txt).
81	def encode(+table: List<&2, Rule>, xs: List<&2, Tk>) -> List<&2, Tk>:
82	  encode.go(table, Nil{}, xs)
83	
84	# is the newest rule in the table the one that creates the token tk?
85	def hits(table: List<&2, Rule>, +tk: Tk) -> Bool:
86	  match table:
87	    case Nil{}:
88	      False{}
89	    case Con{Rule{+i, a, b}, rest}:
90	      match tk:
91	        case B{n}:
92	          False{}
93	        case M{+j}:
94	          Nat.is_eq(i, j)
95	
96	# Expansion of a token into bytes. The table has the newest rule first and
97	# a rule only refers to older tokens, so expanding a and b looks only at
98	# the rest of the table. `hit` is always `hits(table, tk)`.
99	def exp.go(table: List<&2, Rule>, hit: Bool, tk: Tk) -> List<&2, Nat>:
100	  match table hit tk:
101	    case Nil{} _ B{n}:
102	      n <> Nil{}
103	    case Nil{} _ M{j}:
104	      Nil{}
105	    case Con{r, rest} _ B{n}:
106	      n <> Nil{}
107	    case Con{Rule{i, +a, +b}, +rest} True{} M{j}:
108	      List.append(&2, Nat, exp.go(rest, hits(rest, a), a), exp.go(rest, hits(rest, b), b))
109	    case Con{r, +rest} False{} M{+j}:
110	      exp.go(rest, hits(rest, M{j}), M{j})
111	
112	def exp(+table: List<&2, Rule>, +tk: Tk) -> List<&2, Nat>:
113	  exp.go(table, hits(table, tk), tk)
114	
115	# decodes with the already reversed table (newest first)
116	def dec(+table: List<&2, Rule>, ids: List<&2, Tk>) -> List<&2, Nat>:
117	  match ids:
118	    case Nil{}:
119	      Nil{}
120	    case Con{+x, t}:
121	      List.append(&2, Nat, exp(table, x), dec(table, t))
122	
123	def decode(+table: List<&2, Rule>, ids: List<&2, Tk>) -> List<&2, Nat>:
124	  dec(List.reverse(&2, Rule, table), ids)
125	
126	# bytes -> initial tokens
127	def lift(bs: List<&2, Nat>) -> List<&2, Tk>:
128	  match bs:
129	    case Nil{}:
130	      Nil{}
131	    case Con{n, t}:
132	      B{n} <> lift(t)
133	
134	# ---- predicates used in the laws (all computable) ----
135	
136	# does some rule in the table have the id i?
137	def has(+D: List<&2, Rule>, +i: Nat) -> Bool:
138	  match D:
139	    case Nil{}:
140	      False{}
141	    case Con{Rule{+j, a, b}, t}:
142	      Bool.or(Nat.is_eq(i, j), has(t, i))
143	
144	# is the token a byte, or was it created by a rule in the table?
145	def kn(+D: List<&2, Rule>, tk: Tk) -> Bool:
146	  match tk:
147	    case B{n}:
148	      True{}
149	    case M{+j}:
150	      has(D, j)
151	
152	def knl(+D: List<&2, Rule>, xs: List<&2, Tk>) -> Bool:
153	  match xs:
154	    case Nil{}:
155	      True{}
156	    case Con{x, t}:
157	      Bool.and(kn(D, x), knl(D, t))
158	
159	# well-formed table: no id repeats
160	def wf.go(todo: List<&2, Rule>, +done: List<&2, Rule>) -> Bool:
161	  match todo:
162	    case Nil{}:
163	      True{}
164	    case Con{Rule{+i, +a, +b}, rest}:
165	      Bool.and(Bool.not(has(done, i)), wf.go(rest, Rule{i, a, b} <> done))
166	
167	def wf(table: List<&2, Rule>) -> Bool:
168	  wf.go(table, Nil{})
169	
170	# =====================================================================
171	# Training (needs no proof: it generates tables; a table is valid if wf(table))
172	# =====================================================================
173	# ---------------------------------------------------------------
174	# training: count adjacent pairs, merge the most frequent one, repeat
175	# (tie-break: the first pair that appeared, as in minbpe)
176	# ---------------------------------------------------------------
177	type Cnt is Data:
178	  Cnt{a: Tk, b: Tk, n: Nat}
179	
180	# is the first Cnt of cs the pair (a, b)?
181	def same_head(cs: List<&2, Cnt>, +a: Tk, +b: Tk) -> Bool:
182	  match cs:
183	    case Nil{}:
184	      False{}
185	    case Con{Cnt{+x, +y, k}, rest}:
186	      Bool.and(Tk.eq(x, a), Tk.eq(y, b))
187	
188	# adds 1 to the counter of the pair (a, b), or creates it at the end of the list
189	def bump.go(cs: List<&2, Cnt>, hit: Bool, +a: Tk, +b: Tk) -> List<&2, Cnt>:
190	  match cs hit:
191	    case Nil{} _:
192	      Cnt{a, b, 1n} <> Nil{}
193	    case Con{Cnt{x, y, +k}, rest} True{}:
194	      Cnt{x, y, 1n+k} <> rest
195	    case Con{c, +rest} False{}:
196	      c <> bump.go(rest, same_head(rest, a, b), a, b)
197	
198	def bump(+cs: List<&2, Cnt>, +a: Tk, +b: Tk) -> List<&2, Cnt>:
199	  bump.go(cs, same_head(cs, a, b), a, b)
200	
201	def count.go(xs: List<&2, Tk>, cs: List<&2, Cnt>) -> List<&2, Cnt>:
202	  match xs:
203	    case Nil{}:
204	      cs
205	    case Con{+x, +t}:
206	      match t:
207	        case Nil{}:
208	          cs
209	        case Con{+y, u}:
210	          count.go(Con{y, u}, bump(cs, x, y))
211	
212	def pick2(better: Bool, c: Cnt, cur: Cnt) -> Cnt:
213	  match better:
214	    case True{}:
215	      c
216	    case False{}:
217	      cur
218	
219	def pick(c: Cnt, cur: Cnt) -> Cnt:
220	  match c cur:
221	    case Cnt{+a, +b, +k} Cnt{+x, +y, +m}:
222	      pick2(Nat.is_lt(m, k), Cnt{a, b, k}, Cnt{x, y, m})
223	
224	def best.go(cs: List<&2, Cnt>, cur: Cnt) -> Cnt:
225	  match cs:
226	    case Nil{}:
227	      cur
228	    case Con{c, rest}:
229	      best.go(rest, pick(c, cur))
230	
231	def best(cs: List<&2, Cnt>) -> Maybe<&1, Cnt>:
232	  match cs:
233	    case Nil{}:
234	      None{}
235	    case Con{c, rest}:
236	      Some{best.go(rest, c)}
237	
238	# up to n merges, ids from `next` on; m is the current best pair (best(count(xs)))
239	def train.go(n: Nat, +next: Nat, +xs: List<&2, Tk>, m: Maybe<&1, Cnt>) -> List<&2, Rule>:
240	  match n m:
241	    case 0n _:
242	      Nil{}
243	    case 1n+p None{}:
244	      Nil{}
245	    case 1n+p Some{Cnt{+a, +b, k}}:
246	      +ys = merge(a, b, M{next}, xs)
247	      Rule{next, a, b} <> train.go(p, 1n+next, ys, best(count.go(ys, Nil{})))
248	
249	# the table from the oldest rule to the newest
250	def train(n: Nat, +next: Nat, +xs: List<&2, Tk>) -> List<&2, Rule>:
251	  train.go(n, next, xs, best(count.go(xs, Nil{})))
252	
253	# =====================================================================
254	# Auxiliary lemmas for the proofs
255	# =====================================================================
256	
257	# a type chosen by a Bool: rewriting through it refutes True == False
258	def BD(b: Bool, t: Type, f: Type) -> Type:
259	  match b:
260	    case True{}:
261	      t
262	    case False{}:
263	      f
264	
265	def nat_eq_sound(a: Nat, b: Nat, e: {True{} == Nat.is_eq(a, b) : Bool}) -> {a == b : Nat}:
266	  match a b:
267	    case 0n 0n:
268	      {==}
269	    case 0n 1n+q:
270	      %e : BD(_, Unit, {0n == 1n+q : Nat})
271	      Unit{}
272	    case 1n+p 0n:
273	      %e : BD(_, Unit, {1n+p == 0n : Nat})
274	      Unit{}
275	    case 1n+p 1n+q:
276	      %nat_eq_sound(p, q, e) : {1n+p == 1n+_ : Nat}
277	      {==}
278	
279	def nat_eq_refl(a: Nat) -> {True{} == Nat.is_eq(a, a) : Bool}:
280	  match a:
281	    case 0n:
282	      {==}
283	    case 1n+p:
284	      nat_eq_refl(p)
285	
286	def and_true(p: Bool, q: Bool, e: {True{} == Bool.and(p, q) : Bool}) -> {True{} == p : Bool} & {True{} == q : Bool}:
287	  match p:
288	    case True{}:
289	      ({==}, e)
290	    case False{}:
291	      %e : BD(_, Unit, {True{} == False{} : Bool} & {True{} == q : Bool})
292	      Unit{}
293	
294	def and_intro(p: Bool, q: Bool, ep: {True{} == p : Bool}, eq: {True{} == q : Bool}) -> {True{} == Bool.and(p, q) : Bool}:
295	  match p:
296	    case True{}:
297	      eq
298	    case False{}:
299	      %ep : BD(_, Unit, {True{} == False{} : Bool})
300	      Unit{}
301	
302	def or_left(p: Bool, q: Bool, ep: {True{} == p : Bool}) -> {True{} == Bool.or(p, q) : Bool}:
303	  match p:
304	    case True{}:
305	      {==}
306	    case False{}:
307	      %ep : BD(_, Unit, {True{} == q : Bool})
308	      Unit{}
309	
310	def or_right(p: Bool, q: Bool, eq: {True{} == q : Bool}) -> {True{} == Bool.or(p, q) : Bool}:
311	  match p:
312	    case True{}:
313	      {==}
314	    case False{}:
315	      eq
316	
317	# not p is true  ->  p is false
318	def not_true(p: Bool, e: {True{} == Bool.not(p) : Bool}) -> {False{} == p : Bool}:
319	  match p:
320	    case True{}:
321	      %e : BD(_, Unit, {False{} == True{} : Bool})
322	      Unit{}
323	    case False{}:
324	      {==}
325	
326	def tk_eq_sound(+x: Tk, +y: Tk, e: {True{} == Tk.eq(x, y) : Bool}) -> {x == y : Tk}:
327	  match x y:
328	    case B{+m} B{+n}:
329	      %nat_eq_sound(m, n, e) : {B{m} == B{_} : Tk}
330	      {==}
331	    case M{+i} M{+j}:
332	      %nat_eq_sound(i, j, e) : {M{i} == M{_} : Tk}
333	      {==}
334	    case B{m} M{j}:
335	      %e : BD(_, Unit, {B{m} == M{j} : Tk})
336	      Unit{}
337	    case M{i} B{n}:
338	      %e : BD(_, Unit, {M{i} == B{n} : Tk})
339	      Unit{}
340	
341	# Adding the (new) rule i on top of the table does not change the expansion of a
342	# token M{j} that the old table already knows (j != i, because i is new).
343	def ext_m(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, +j: Nat, h: Bool, eh: {h == Nat.is_eq(i, j) : Bool}, nf: {True{} == Bool.not(has(D, i)) : Bool}, kt: {True{} == has(D, j) : Bool}) -> {exp(Rule{i, a, b} <> D, M{j}) == exp(D, M{j}) : List<&2, Nat>}:
344	  match h:
345	    case True{}:
346	      # h = True means i == j; but j is known (has D j) and i is not (not has D i)
347	      +ij = nat_eq_sound(i, j, eh)
348	      +hij = Equal.cong(Nat, Bool, k => has(D, k), i, j, ij)
349	      +f1 = Equal.trans(Bool, False{}, has(D, i), has(D, j), not_true(has(D, i), nf), hij)
350	      +tf = Equal.trans(Bool, True{}, has(D, j), False{}, kt, Equal.sym(Bool, False{}, has(D, j), f1))
351	      %tf : BD(_, Unit, {exp.go(Rule{i, a, b} <> D, Cmp.is_eq(Nat.cmp(i, j)), M{j}) == exp.go(D, hits(D, M{j}), M{j}) : List<&2, Nat>})
352	      Unit{}
353	    case False{}:
354	      %eh : {exp.go(Rule{i, a, b} <> D, _, M{j}) == exp(D, M{j}) : List<&2, Nat>}
355	      {==}
356	
357	def and_true_l(p: Bool, q: Bool, e: {True{} == Bool.and(p, q) : Bool}) -> {True{} == p : Bool}:
358	  match p:
359	    case True{}:
360	      {==}
361	    case False{}:
362	      %e : BD(_, Unit, {True{} == False{} : Bool})
363	      Unit{}
364	
365	def and_true_r(p: Bool, q: Bool, e: {True{} == Bool.and(p, q) : Bool}) -> {True{} == q : Bool}:
366	  match p:
367	    case True{}:
368	      e
369	    case False{}:
370	      %e : BD(_, Unit, {True{} == q : Bool})
371	      Unit{}
372	
373	# the same for any known token (a byte never depends on the table)
374	def ext_tok(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, t: Tk, kt: {True{} == kn(D, t) : Bool}, nf: {True{} == Bool.not(has(D, i)) : Bool}) -> {exp(Rule{i, a, b} <> D, t) == exp(D, t) : List<&2, Nat>}:
375	  match D t:
376	    case Nil{} B{n}:
377	      {==}
378	    case Con{Rule{j, a2, b2}, rest} B{n}:
379	      {==}
380	    case _ M{+j}:
381	      ext_m(D, i, a, b, j, Nat.is_eq(i, j), {==}, nf, kt)
382	
383	def ext_dec.step(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, +x: Tk, +t: List<&2, Tk>, k1: {True{} == kn(D, x) : Bool}, nf: {True{} == Bool.not(has(D, i)) : Bool}, ih: {dec(Rule{i, a, b} <> D, t) == dec(D, t) : List<&2, Nat>}) -> {dec(Rule{i, a, b} <> D, x <> t) == dec(D, x <> t) : List<&2, Nat>}:
384	  %ext_tok(D, i, a, b, x, k1, nf) : {List.append(&2, Nat, exp(Rule{i, a, b} <> D, x), dec(Rule{i, a, b} <> D, t)) == List.append(&2, Nat, _, dec(D, t)) : List<&2, Nat>}
385	  %ih : {List.append(&2, Nat, exp(Rule{i, a, b} <> D, x), dec(Rule{i, a, b} <> D, t)) == List.append(&2, Nat, exp(Rule{i, a, b} <> D, x), _) : List<&2, Nat>}
386	  {==}
387	
388	# decoding a list of known tokens does not change when the new rule is added
389	def ext_dec(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, xs: List<&2, Tk>, +kx: {True{} == knl(D, xs) : Bool}, +nf: {True{} == Bool.not(has(D, i)) : Bool}) -> {dec(Rule{i, a, b} <> D, xs) == dec(D, xs) : List<&2, Nat>}:
390	  match xs:
391	    case Nil{}:
392	      {==}
393	    case Con{+x, +t}:
394	      ext_dec.step(D, i, a, b, x, t, and_true_l(kn(D, x), knl(D, t), kx), nf, ext_dec(D, i, a, b, t, and_true_r(kn(D, x), knl(D, t), kx), nf))
395	
396	# The new token M{i} expands to the expansion of a followed by that of b
397	# (looking only at the old table D).
398	def exp_new(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, h: Bool, eh: {h == Nat.is_eq(i, i) : Bool}) -> {exp(Rule{i, a, b} <> D, M{i}) == List.append(&2, Nat, exp(D, a), exp(D, b)) : List<&2, Nat>}:
399	  match h:
400	    case True{}:
401	      %eh : {exp.go(Rule{i, a, b} <> D, _, M{i}) == List.append(&2, Nat, exp(D, a), exp(D, b)) : List<&2, Nat>}
402	      {==}
403	    case False{}:
404	      +f = Equal.trans(Bool, True{}, Nat.is_eq(i, i), False{}, nat_eq_refl(i), Equal.sym(Bool, False{}, Nat.is_eq(i, i), eh))
405	      %f : BD(_, Unit, {exp(Rule{i, a, b} <> D, M{i}) == List.append(&2, Nat, exp(D, a), exp(D, b)) : List<&2, Nat>})
406	      Unit{}
407	
408	# Central step: in place of a b t we get M{i} followed by m, where m decodes
409	# like t. Both sides decode the same because M{i} expands to exp(a) ++ exp(b).
410	def mp_core(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, +t: List<&2, Tk>, +m: List<&2, Tk>, ka: {True{} == kn(D, a) : Bool}, kb: {True{} == kn(D, b) : Bool}, +nf: {True{} == Bool.not(has(D, i)) : Bool}, ihm: {dec(Rule{i, a, b} <> D, m) == dec(Rule{i, a, b} <> D, t) : List<&2, Nat>}) -> {dec(Rule{i, a, b} <> D, M{i} <> m) == dec(Rule{i, a, b} <> D, a <> b <> t) : List<&2, Nat>}:
411	  %Equal.sym(List<&2, Nat>, exp(Rule{i, a, b} <> D, M{i}), List.append(&2, Nat, exp(D, a), exp(D, b)), exp_new(D, i, a, b, Nat.is_eq(i, i), {==})) : {List.append(&2, Nat, _, dec(Rule{i, a, b} <> D, m)) == List.append(&2, Nat, exp(Rule{i, a, b} <> D, a), List.append(&2, Nat, exp(Rule{i, a, b} <> D, b), dec(Rule{i, a, b} <> D, t))) : List<&2, Nat>}
412	  %Equal.sym(List<&2, Nat>, dec(Rule{i, a, b} <> D, m), dec(Rule{i, a, b} <> D, t), ihm) : {List.append(&2, Nat, List.append(&2, Nat, exp(D, a), exp(D, b)), _) == List.append(&2, Nat, exp(Rule{i, a, b} <> D, a), List.append(&2, Nat, exp(Rule{i, a, b} <> D, b), dec(Rule{i, a, b} <> D, t))) : List<&2, Nat>}
413	  %Equal.sym(List<&2, Nat>, exp(Rule{i, a, b} <> D, a), exp(D, a), ext_tok(D, i, a, b, a, ka, nf)) : {List.append(&2, Nat, List.append(&2, Nat, exp(D, a), exp(D, b)), dec(Rule{i, a, b} <> D, t)) == List.append(&2, Nat, _, List.append(&2, Nat, exp(Rule{i, a, b} <> D, b), dec(Rule{i, a, b} <> D, t))) : List<&2, Nat>}
414	  %Equal.sym(List<&2, Nat>, exp(Rule{i, a, b} <> D, b), exp(D, b), ext_tok(D, i, a, b, b, kb, nf)) : {List.append(&2, Nat, List.append(&2, Nat, exp(D, a), exp(D, b)), dec(Rule{i, a, b} <> D, t)) == List.append(&2, Nat, exp(D, a), List.append(&2, Nat, _, dec(Rule{i, a, b} <> D, t))) : List<&2, Nat>}
415	  NL.append_assoc(Nat, exp(D, a), exp(D, b), dec(Rule{i, a, b} <> D, t))
416	
417	# if x == a and x is known, a is known too
418	def kn_sub(+D: List<&2, Rule>, +x: Tk, +a: Tk, ex: {x == a : Tk}, kx: {True{} == kn(D, x) : Bool}) -> {True{} == kn(D, a) : Bool}:
419	  %ex : {True{} == kn(D, _) : Bool}
420	  kx
421	
422	# x stays out of the rewrite: if the rest decodes the same, x <> rest does too
423	def mp_false(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, +x: Tk, +t: List<&2, Tk>, +mg: List<&2, Tk>, ih: {dec(Rule{i, a, b} <> D, mg) == dec(Rule{i, a, b} <> D, t) : List<&2, Nat>}) -> {dec(Rule{i, a, b} <> D, x <> mg) == dec(Rule{i, a, b} <> D, x <> t) : List<&2, Nat>}:
424	  %ih : {List.append(&2, Nat, exp(Rule{i, a, b} <> D, x), dec(Rule{i, a, b} <> D, mg)) == List.append(&2, Nat, exp(Rule{i, a, b} <> D, x), _) : List<&2, Nat>}
425	  {==}
426	
427	# the pair is x y: the tokens were equal to a and b
428	def mp_hit(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, +x: Tk, +y: Tk, +t: List<&2, Tk>, +m: List<&2, Tk>, +ex: {x == a : Tk}, +ey: {y == b : Tk}, +kxx: {True{} == kn(D, x) : Bool}, +kyy: {True{} == kn(D, y) : Bool}, +nf: {True{} == Bool.not(has(D, i)) : Bool}, ihm: {dec(Rule{i, a, b} <> D, m) == dec(Rule{i, a, b} <> D, t) : List<&2, Nat>}) -> {dec(Rule{i, a, b} <> D, M{i} <> m) == dec(Rule{i, a, b} <> D, x <> y <> t) : List<&2, Nat>}:
429	  %Equal.sym(Tk, x, a, ex) : {dec(Rule{i, a, b} <> D, M{i} <> m) == dec(Rule{i, a, b} <> D, _ <> y <> t) : List<&2, Nat>}
430	  %Equal.sym(Tk, y, b, ey) : {dec(Rule{i, a, b} <> D, M{i} <> m) == dec(Rule{i, a, b} <> D, a <> _ <> t) : List<&2, Nat>}
431	  mp_core(D, i, a, b, t, m, kn_sub(D, x, a, ex, kxx), kn_sub(D, y, b, ey, kyy), nf, ihm)
432	
433	# merge.go does not change what decode returns (with the new rule on top of the table)
434	def mp(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, xs: List<&2, Tk>, hit: Bool, +eh: {hit == peek(a, b, xs) : Bool}, +kx: {True{} == knl(D, xs) : Bool}, +nf: {True{} == Bool.not(has(D, i)) : Bool}) -> {dec(Rule{i, a, b} <> D, merge.go(xs, hit, a, b, M{i})) == dec(Rule{i, a, b} <> D, xs) : List<&2, Nat>}:
435	  match xs hit:
436	    case Nil{} _:
437	      {==}
438	    case Con{x, Nil{}} True{}:
439	      %eh : BD(_, Unit, {dec(Rule{i, a, b} <> D, merge.go(x <> Nil{}, True{}, a, b, M{i})) == dec(Rule{i, a, b} <> D, x <> Nil{}) : List<&2, Nat>})
440	      Unit{}
441	    case Con{+x, Con{+y, +t}} True{}:
442	      mp_hit(D, i, a, b, x, y, t, merge.go(t, peek(a, b, t), a, b, M{i}), tk_eq_sound(x, a, and_true_l(Tk.eq(x, a), Tk.eq(y, b), eh)), tk_eq_sound(y, b, and_true_r(Tk.eq(x, a), Tk.eq(y, b), eh)), and_true_l(kn(D, x), Bool.and(kn(D, y), knl(D, t)), kx), and_true_l(kn(D, y), knl(D, t), and_true_r(kn(D, x), Bool.and(kn(D, y), knl(D, t)), kx)), nf, mp(D, i, a, b, t, peek(a, b, t), {==}, and_true_r(kn(D, y), knl(D, t), and_true_r(kn(D, x), Bool.and(kn(D, y), knl(D, t)), kx)), nf))
443	    case Con{+x, +t} False{}:
444	      mp_false(D, i, a, b, x, t, merge.go(t, peek(a, b, t), a, b, M{i}), mp(D, i, a, b, t, peek(a, b, t), {==}, and_true_r(kn(D, x), knl(D, t), kx), nf))
445	
446	# ---- known tokens stay known ----
447	
448	# adding a rule to the table only increases what is known
449	def kn_mono(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, t: Tk, kt: {True{} == kn(D, t) : Bool}) -> {True{} == kn(Rule{i, a, b} <> D, t) : Bool}:
450	  match t:
451	    case B{n}:
452	      {==}
453	    case M{+j}:
454	      or_right(Nat.is_eq(j, i), has(D, j), kt)
455	
456	# the token created by the new rule is known
457	def kn_new(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk) -> {True{} == kn(Rule{i, a, b} <> D, M{i}) : Bool}:
458	  or_left(Nat.is_eq(i, i), has(D, i), nat_eq_refl(i))
459	
460	def kp(+D: List<&2, Rule>, +i: Nat, +a: Tk, +b: Tk, xs: List<&2, Tk>, hit: Bool, +eh: {hit == peek(a, b, xs) : Bool}, +kx: {True{} == knl(D, xs) : Bool}) -> {True{} == knl(Rule{i, a, b} <> D, merge.go(xs, hit, a, b, M{i})) : Bool}:
461	  match xs hit:
462	    case Nil{} _:
463	      {==}
464	    case Con{x, Nil{}} True{}:
465	      %eh : BD(_, Unit, {True{} == knl(Rule{i, a, b} <> D, merge.go(x <> Nil{}, True{}, a, b, M{i})) : Bool})
466	      Unit{}
467	    case Con{+x, Con{+y, +t}} True{}:
468	      and_intro(kn(Rule{i, a, b} <> D, M{i}), knl(Rule{i, a, b} <> D, merge.go(t, peek(a, b, t), a, b, M{i})), kn_new(D, i, a, b), kp(D, i, a, b, t, peek(a, b, t), {==}, and_true_r(kn(D, y), knl(D, t), and_true_r(kn(D, x), Bool.and(kn(D, y), knl(D, t)), kx))))
469	    case Con{+x, +t} False{}:
470	      and_intro(kn(Rule{i, a, b} <> D, x), knl(Rule{i, a, b} <> D, merge.go(t, peek(a, b, t), a, b, M{i})), kn_mono(D, i, a, b, x, and_true_l(kn(D, x), knl(D, t), kx)), kp(D, i, a, b, t, peek(a, b, t), {==}, and_true_r(kn(D, x), knl(D, t), kx)))
471	
472	# ---- induction over the rules ----
473	
474	# every emitted token is known to the complete (reversed) table
475	def enc_known(todo: List<&2, Rule>, +done: List<&2, Rule>, +xs: List<&2, Tk>, +wfe: {True{} == wf.go(todo, done) : Bool}, +kx: {True{} == knl(done, xs) : Bool}) -> {True{} == knl(List.reverse.go(&2, Rule, todo, done), encode.go(todo, done, xs)) : Bool}:
476	  match todo:
477	    case Nil{}:
478	      kx
479	    case Con{Rule{+i, +a, +b}, +rest}:
480	      enc_known(rest, Rule{i, a, b} <> done, merge.go(xs, peek(a, b, xs), a, b, M{i}), and_true_r(Bool.not(has(done, i)), wf.go(rest, Rule{i, a, b} <> done), wfe), kp(done, i, a, b, xs, peek(a, b, xs), {==}, kx))
481	
482	# applying all the rules does not change what decode returns
483	def enc_dec(todo: List<&2, Rule>, +done: List<&2, Rule>, +xs: List<&2, Tk>, +wfe: {True{} == wf.go(todo, done) : Bool}, +kx: {True{} == knl(done, xs) : Bool}) -> {dec(List.reverse.go(&2, Rule, todo, done), encode.go(todo, done, xs)) == dec(done, xs) : List<&2, Nat>}:
484	  match todo:
485	    case Nil{}:
486	      {==}
487	    case Con{Rule{+i, +a, +b}, +rest}:
488	      Equal.trans(List<&2, Nat>, dec(List.reverse.go(&2, Rule, rest, Rule{i, a, b} <> done), encode.go(rest, Rule{i, a, b} <> done, merge.go(xs, peek(a, b, xs), a, b, M{i}))), dec(Rule{i, a, b} <> done, xs), dec(done, xs), Equal.trans(List<&2, Nat>, dec(List.reverse.go(&2, Rule, rest, Rule{i, a, b} <> done), encode.go(rest, Rule{i, a, b} <> done, merge.go(xs, peek(a, b, xs), a, b, M{i}))), dec(Rule{i, a, b} <> done, merge.go(xs, peek(a, b, xs), a, b, M{i})), dec(Rule{i, a, b} <> done, xs), enc_dec(rest, Rule{i, a, b} <> done, merge.go(xs, peek(a, b, xs), a, b, M{i}), and_true_r(Bool.not(has(done, i)), wf.go(rest, Rule{i, a, b} <> done), wfe), kp(done, i, a, b, xs, peek(a, b, xs), {==}, kx)), mp(done, i, a, b, xs, peek(a, b, xs), {==}, kx, and_true_l(Bool.not(has(done, i)), wf.go(rest, Rule{i, a, b} <> done), wfe))), ext_dec(done, i, a, b, xs, kx, and_true_l(Bool.not(has(done, i)), wf.go(rest, Rule{i, a, b} <> done), wfe)))
489	
490	# ---- base case and final laws ----
491	
492	# with no rules, the bytes decode to themselves
493	def lift_dec(bs: List<&2, Nat>) -> {dec(Nil{}, lift(bs)) == bs : List<&2, Nat>}:
494	  match bs:
495	    case Nil{}:
496	      {==}
497	    case Con{+n, +t}:
498	      %lift_dec(t) : {n <> dec(Nil{}, lift(t)) == n <> _ : List<&2, Nat>}
499	      {==}
500	
501	def lift_known(bs: List<&2, Nat>) -> {True{} == knl(Nil{}, lift(bs)) : Bool}:
502	  match bs:
503	    case Nil{}:
504	      {==}
505	    case Con{n, t}:
506	      lift_known(t)
507	
508	# =====================================================================
509	# LAWS (the package laws, explained in plain language in the README)
510	# =====================================================================
511	
512	# LAW 1 (roundtrip): with a well-formed table (all rule ids
513	# distinct), decoding what was encoded returns the original bytes.
514	law roundtrip:
515	  for +table: List<&2, Rule>
516	  for +bs: List<&2, Nat>
517	  for +w: {True{} == wf(table) : Bool}
518	  {decode(table, encode(table, lift(bs))) == bs : List<&2, Nat>}
519	
520	def roundtrip(table, bs, w):
521	  Equal.trans(List<&2, Nat>, decode(table, encode(table, lift(bs))), dec(Nil{}, lift(bs)), bs, enc_dec(table, Nil{}, lift(bs), w, lift_known(bs)), lift_dec(bs))
522	
523	# LAW 2 (vocabulary): every token emitted by encode is a raw byte or was
524	# created by one of the table's rules; an unknown id never appears.
525	law vocab_bound:
526	  for +table: List<&2, Rule>
527	  for +bs: List<&2, Nat>
528	  for +w: {True{} == wf(table) : Bool}
529	  {True{} == knl(List.reverse(&2, Rule, table), encode(table, lift(bs))) : Bool}
530	
531	def vocab_bound(table, bs, w):
532	  enc_known(table, Nil{}, lift(bs), w, lift_known(bs))
533	
534	# LAW 3 (concatenation): decoding two token lists together is decoding
535	# each one and joining the bytes.
536	law dec_append:
537	  for +D: List<&2, Rule>
538	  for xs: List<&2, Tk>
539	  for +ys: List<&2, Tk>
540	  {dec(D, List.append(&2, Tk, xs, ys)) == List.append(&2, Nat, dec(D, xs), dec(D, ys)) : List<&2, Nat>}
541	
542	def dec_append(D, xs, ys):
543	  match xs:
544	    case Nil{}:
545	      {==}
546	    case Con{+x, +t}:
547	      %Equal.sym(List<&2, Nat>, dec(D, List.append(&2, Tk, t, ys)), List.append(&2, Nat, dec(D, t), dec(D, ys)), dec_append(D, t, ys)) : {List.append(&2, Nat, exp(D, x), _) == List.append(&2, Nat, List.append(&2, Nat, exp(D, x), dec(D, t)), dec(D, ys)) : List<&2, Nat>}
548	      Equal.sym(List<&2, Nat>, List.append(&2, Nat, List.append(&2, Nat, exp(D, x), dec(D, t)), dec(D, ys)), List.append(&2, Nat, exp(D, x), List.append(&2, Nat, dec(D, t), dec(D, ys))), NL.append_assoc(Nat, exp(D, x), dec(D, t), dec(D, ys)))
549	
550	# =====================================================================
551	# LAW 4: train generates well-formed tables (closes the roundtrip gap)
552	# =====================================================================
553	#
554	# train creates rules with ids next, next+1, next+2, ... None repeats, so the
555	# table satisfies wf, which is the hypothesis of the roundtrip law. Proof by induction
556	# over train.go, keeping the invariant "all ids already used are smaller
557	# than the next id".
558	
559	# n < 1 + n
560	def lt_succ(n: Nat) -> {True{} == Nat.is_lt(n, 1n+n) : Bool}:
561	  match n:
562	    case 0n:
563	      {==}
564	    case 1n+p:
565	      lt_succ(p)
566	
567	# j < i  ->  i != j
568	def lt_ne(j: Nat, i: Nat, e: {True{} == Nat.is_lt(j, i) : Bool}) -> {False{} == Nat.is_eq(i, j) : Bool}:
569	  match j i:
570	    case 0n 0n:
571	      %e : BD(_, Unit, {False{} == Nat.is_eq(0n, 0n) : Bool})
572	      Unit{}
573	    case 0n 1n+q:
574	      {==}
575	    case 1n+p 0n:
576	      %e : BD(_, Unit, {False{} == Nat.is_eq(0n, 1n+p) : Bool})
577	      Unit{}
578	    case 1n+p 1n+q:
579	      lt_ne(p, q, e)
580	
581	# j < i  ->  j < 1 + i
582	def lt_up(j: Nat, i: Nat, e: {True{} == Nat.is_lt(j, i) : Bool}) -> {True{} == Nat.is_lt(j, 1n+i) : Bool}:
583	  match j i:
584	    case 0n 0n:
585	      %e : BD(_, Unit, {True{} == Nat.is_lt(0n, 1n) : Bool})
586	      Unit{}
587	    case 0n 1n+q:
588	      {==}
589	    case 1n+p 0n:
590	      %e : BD(_, Unit, {True{} == Nat.is_lt(1n+p, 1n) : Bool})
591	      Unit{}
592	    case 1n+p 1n+q:
593	      lt_up(p, q, e)
594	
595	# all ids in the rule list are smaller than i
596	def below(D: List<&2, Rule>, +i: Nat) -> Bool:
597	  match D:
598	    case Nil{}:
599	      True{}
600	    case Con{Rule{+j, a, b}, t}:
601	      Bool.and(Nat.is_lt(j, i), below(t, i))
602	
603	def below_has(D: List<&2, Rule>, +i: Nat, +e: {True{} == below(D, i) : Bool}) -> {False{} == has(D, i) : Bool}:
604	  match D:
605	    case Nil{}:
606	      {==}
607	    case Con{Rule{+j, a, b}, +t}:
608	      %lt_ne(j, i, and_true_l(Nat.is_lt(j, i), below(t, i), e)) : {False{} == Bool.or(_, has(t, i)) : Bool}
609	      below_has(t, i, and_true_r(Nat.is_lt(j, i), below(t, i), e))
610	
611	def below_up(D: List<&2, Rule>, +i: Nat, +e: {True{} == below(D, i) : Bool}) -> {True{} == below(D, 1n+i) : Bool}:
612	  match D:
613	    case Nil{}:
614	      {==}
615	    case Con{Rule{+j, a, b}, +t}:
616	      and_intro(Nat.is_lt(j, 1n+i), below(t, 1n+i), lt_up(j, i, and_true_l(Nat.is_lt(j, i), below(t, i), e)), below_up(t, i, and_true_r(Nat.is_lt(j, i), below(t, i), e)))
617	
618	# adding the rule with id `next` (greater than all those in `done`) keeps the table "below 1+next"
619	def below_cons(+a: Tk, +b: Tk, +D: List<&2, Rule>, +i: Nat, +e: {True{} == below(D, i) : Bool}) -> {True{} == below(Rule{i, a, b} <> D, 1n+i) : Bool}:
620	  and_intro(Nat.is_lt(i, 1n+i), below(D, 1n+i), lt_succ(i), below_up(D, i, e))
621	
622	# not (has D i) when all the ids in D are smaller than i
623	def below_fresh(+D: List<&2, Rule>, +i: Nat, +e: {True{} == below(D, i) : Bool}) -> {True{} == Bool.not(has(D, i)) : Bool}:
624	  %below_has(D, i, e) : {True{} == Bool.not(_) : Bool}
625	  {==}
626	
627	# train.go generates rules with ids next, next+1, ...: none repeats, whatever the corpus
628	def tw(n: Nat, +next: Nat, +xs: List<&2, Tk>, m: Maybe<&1, Cnt>, +done: List<&2, Rule>, +bl: {True{} == below(done, next) : Bool}) -> {True{} == wf.go(train.go(n, next, xs, m), done) : Bool}:
629	  match n m:
630	    case 0n _:
631	      {==}
632	    case 1n++p None{}:
633	      {==}
634	    case 1n++p Some{Cnt{+a, +b, k}}:
635	      and_intro(Bool.not(has(done, next)), wf.go(train.go(p, 1n+next, merge(a, b, M{next}, xs), best(count.go(merge(a, b, M{next}, xs), Nil{}))), Rule{next, a, b} <> done), below_fresh(done, next, bl), tw(p, 1n+next, merge(a, b, M{next}, xs), best(count.go(merge(a, b, M{next}, xs), Nil{})), Rule{next, a, b} <> done, below_cons(a, b, done, next, bl)))
636	
637	# LAW (train_wf): the table that train returns is always well formed, so the roundtrip holds for it
638	law train_wf:
639	  for n: Nat
640	  for +next: Nat
641	  for +xs: List<&2, Tk>
642	  {True{} == wf(train(n, next, xs)) : Bool}
643	
644	def train_wf(n, next, xs):
645	  tw(n, next, xs, best(count.go(xs, Nil{})), Nil{}, {==})
646	
647	# LAW 5 (roundtrip_trained): train a table on any corpus, with as many merges as you like:
648	# decoding what it encodes always returns the original bytes. No hypothesis at all.
649	law roundtrip_trained:
650	  for +n: Nat
651	  for +corpus: List<&2, Nat>
652	  for +bs: List<&2, Nat>
653	  {decode(train(n, 0n, lift(corpus)), encode(train(n, 0n, lift(corpus)), lift(bs))) == bs : List<&2, Nat>}
654	
655	def roundtrip_trained(n, corpus, bs):
656	  roundtrip(train(n, 0n, lift(corpus)), bs, train_wf(n, 0n, lift(corpus)))