aboutsummaryrefslogtreecommitdiff
path: root/spartan/core/Spartan.thy
blob: 9a6d884f1754b6bc72bfb6681ed83ed8c45f6bae (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
text \<open>Spartan type theory\<close>

theory Spartan
imports
  Pure
  "HOL-Eisbach.Eisbach"
  "HOL-Eisbach.Eisbach_Tools"
keywords
  "Theorem" "Lemma" "Corollary" "Proposition" :: thy_goal_stmt and
  "focus" "\<guillemotright>" "\<^item>" "\<^enum>" "~" :: prf_script_goal % "proof" and
  "congruence" "print_coercions" :: thy_decl and
  "rhs" "derive" "prems" "vars" :: quasi_command
  

begin


section \<open>Preamble\<close>

declare [[eta_contract=false]]

syntax "_dollar" :: \<open>logic \<Rightarrow> logic \<Rightarrow> logic\<close> (infixr "$" 3)
translations "a $ b" \<rightharpoonup> "a (b)"

abbreviation (input) K where "K x \<equiv> \<lambda>_. x"


section \<open>Metatype setup\<close>

typedecl o


section \<open>Judgments\<close>

judgment has_type :: \<open>o \<Rightarrow> o \<Rightarrow> prop\<close> ("(2_:/ _)" 999)


section \<open>Universes\<close>

typedecl lvl \<comment> \<open>Universe levels\<close>

axiomatization
  O  :: \<open>lvl\<close> and
  S  :: \<open>lvl \<Rightarrow> lvl\<close> and
  lt :: \<open>lvl \<Rightarrow> lvl \<Rightarrow> prop\<close> (infix "<" 900)
  where
  O_min: "O < S i" and
  lt_S: "i < S i" and
  lt_trans: "i < j \<Longrightarrow> j < k \<Longrightarrow> i < k"

axiomatization U :: \<open>lvl \<Rightarrow> o\<close> where
  U_hierarchy: "i < j \<Longrightarrow> U i: U j" and
  U_cumulative: "A: U i \<Longrightarrow> i < j \<Longrightarrow> A: U j"

lemma U_in_U:
  "U i: U (S i)"
  by (rule U_hierarchy, rule lt_S)

lemma lift_universe:
  "A: U i \<Longrightarrow> A: U (S i)"
  by (erule U_cumulative, rule lt_S)


section \<open>\<Prod>-type\<close>

axiomatization
  Pi  :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
  lam :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
  app :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(1_ `_)" [120, 121] 120)

syntax
  "_Pi"   :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Prod>_: _./ _)" 30)
  "_lam"  :: \<open>pttrns \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<lambda>_: _./ _)" 30)
  "_lam2" :: \<open>pttrns \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
translations
  "\<Prod>x: A. B"    \<rightleftharpoons> "CONST Pi A (\<lambda>x. B)"
  "\<lambda>x xs: A. b" \<rightharpoonup> "CONST lam A (\<lambda>x. _lam2 xs A b)"
  "_lam2 x A b" \<rightharpoonup> "\<lambda>x: A. b"
  "\<lambda>x: A. b"    \<rightleftharpoons> "CONST lam A (\<lambda>x. b)"

abbreviation Fn (infixr "\<rightarrow>" 40) where "A \<rightarrow> B \<equiv> \<Prod>_: A. B"

axiomatization where
  PiF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Prod>x: A. B x: U i" and

  PiI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x: B x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x: \<Prod>x: A. B x" and

  PiE: "\<lbrakk>f: \<Prod>x: A. B x; a: A\<rbrakk> \<Longrightarrow> f `a: B a" and

  beta: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b x: B x\<rbrakk> \<Longrightarrow> (\<lambda>x: A. b x) `a \<equiv> b a" and

  eta: "f: \<Prod>x: A. B x \<Longrightarrow> \<lambda>x: A. f `x \<equiv> f" and

  Pi_cong: "\<lbrakk>
    \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
    A: U i;
    \<And>x. x: A \<Longrightarrow> B x: U i;
    \<And>x. x: A \<Longrightarrow> B' x: U i
    \<rbrakk> \<Longrightarrow> \<Prod>x: A. B x \<equiv> \<Prod>x: A. B' x" and

  lam_cong: "\<lbrakk>\<And>x. x: A \<Longrightarrow> b x \<equiv> c x; A: U i\<rbrakk> \<Longrightarrow> \<lambda>x: A. b x \<equiv> \<lambda>x: A. c x"


section \<open>\<Sum>-type\<close>

axiomatization
  Sig    :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o\<close> and
  pair   :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close> ("(2<_,/ _>)") and
  SigInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o\<close>

syntax "_Sum" :: \<open>idt \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2\<Sum>_: _./ _)" 20)

translations "\<Sum>x: A. B" \<rightleftharpoons> "CONST Sig A (\<lambda>x. B)"

abbreviation Prod (infixl "\<times>" 60)
  where "A \<times> B \<equiv> \<Sum>_: A. B"

abbreviation "and" (infixl "\<and>" 60)
  where "A \<and> B \<equiv> A \<times> B"

axiomatization where
  SigF: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; A: U i\<rbrakk> \<Longrightarrow> \<Sum>x: A. B x: U i" and

  SigI: "\<lbrakk>\<And>x. x: A \<Longrightarrow> B x: U i; a: A; b: B a\<rbrakk> \<Longrightarrow> <a, b>: \<Sum>x: A. B x" and

  SigE: "\<lbrakk>
    p: \<Sum>x: A. B x;
    A: U i;
    \<And>x. x : A \<Longrightarrow> B x: U i;
    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>;
    \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i
    \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f p: C p" and

  Sig_comp: "\<lbrakk>
    a: A;
    b: B a;
    \<And>x. x: A \<Longrightarrow> B x: U i;
    \<And>p. p: \<Sum>x: A. B x \<Longrightarrow> C p: U i;
    \<And>x y. \<lbrakk>x: A; y: B x\<rbrakk> \<Longrightarrow> f x y: C <x, y>
    \<rbrakk> \<Longrightarrow> SigInd A (\<lambda>x. B x) (\<lambda>p. C p) f <a, b> \<equiv> f a b" and

  Sig_cong: "\<lbrakk>
    \<And>x. x: A \<Longrightarrow> B x \<equiv> B' x;
    A: U i;
    \<And>x. x : A \<Longrightarrow> B x: U i;
    \<And>x. x : A \<Longrightarrow> B' x: U i
    \<rbrakk> \<Longrightarrow> \<Sum>x: A. B x \<equiv> \<Sum>x: A. B' x"


section \<open>Proof commands\<close>

named_theorems typechk

ML_file \<open>lib/lib.ML\<close>
ML_file \<open>lib/goals.ML\<close>
ML_file \<open>lib/focus.ML\<close>


section \<open>Congruence automation\<close>

consts "rhs" :: \<open>'a\<close> ("..")

ML_file \<open>lib/congruence.ML\<close>


section \<open>Methods\<close>

ML_file \<open>lib/elimination.ML\<close> \<comment> \<open>elimination rules\<close>
ML_file \<open>lib/cases.ML\<close> \<comment> \<open>case reasoning rules\<close>

named_theorems intros and comps
lemmas
  [intros] = PiF PiI SigF SigI and
  [elims "?f"] = PiE and
  [elims "?p"] = SigE and
  [comps] = beta Sig_comp and
  [cong] = Pi_cong lam_cong Sig_cong

ML_file \<open>lib/tactics.ML\<close>

method_setup assumptions =
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
    CHANGED (TRYALL (assumptions_tac ctxt))))\<close>

method_setup known =
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
    CHANGED (TRYALL (known_tac ctxt))))\<close>

method_setup intro =
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intro_tac ctxt)))\<close>

method_setup intros =
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (intros_tac ctxt)))\<close>

method_setup elim =
  \<open>Scan.repeat Args.term >> (fn tms => fn ctxt =>
    CONTEXT_METHOD (K (elim_context_tac tms ctxt 1)))\<close>

method elims = elim+

method_setup cases =
  \<open>Args.term >> (fn tm => fn ctxt => SIMPLE_METHOD' (cases_tac tm ctxt))\<close>

(*This could possibly use additional simplification hints via a simp: modifier*)
method_setup typechk =
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (
    SIDE_CONDS (typechk_tac ctxt) ctxt))
    (* CHANGED (ALLGOALS (TRY o typechk_tac ctxt)))) *)\<close>

method_setup rule =
  \<open>Attrib.thms >> (fn ths => fn ctxt =>
    SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (rule_tac ths ctxt) ctxt)))\<close>

method_setup dest =
  \<open>Scan.lift (Scan.option (Args.parens Parse.int)) -- Attrib.thms
    >> (fn (opt_n, ths) => fn ctxt =>
        SIMPLE_METHOD (HEADGOAL (SIDE_CONDS (dest_tac opt_n ths ctxt) ctxt)))\<close>

subsection \<open>Reflexivity\<close>

named_theorems refl
method refl = (rule refl)

subsection \<open>Trivial proofs modulo typechecking\<close>

method_setup this =
  \<open>Scan.succeed (fn ctxt => METHOD (
    EVERY o map (HEADGOAL o
      (fn ths => SIDE_CONDS (resolve_tac ctxt ths) ctxt) o
      single)
    ))\<close>

subsection \<open>Rewriting\<close>

\<comment> \<open>\<open>subst\<close> method\<close>
ML_file \<open>~~/src/Tools/misc_legacy.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/isand.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/rw_inst.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/zipper.ML\<close>
ML_file \<open>lib/eqsubst.ML\<close>

\<comment> \<open>\<open>rewrite\<close> method\<close>
consts rewrite_HOLE :: "'a::{}"  ("\<hole>")

lemma eta_expand:
  fixes f :: "'a::{} \<Rightarrow> 'b::{}"
  shows "f \<equiv> \<lambda>x. f x" .

lemma rewr_imp:
  assumes "PROP A \<equiv> PROP B"
  shows "(PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B \<Longrightarrow> PROP C)"
  apply (rule Pure.equal_intr_rule)
  apply (drule equal_elim_rule2[OF assms]; assumption)
  apply (drule equal_elim_rule1[OF assms]; assumption)
  done

lemma imp_cong_eq:
  "(PROP A \<Longrightarrow> (PROP B \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP C')) \<equiv>
    ((PROP B \<Longrightarrow> PROP A \<Longrightarrow> PROP C) \<equiv> (PROP B' \<Longrightarrow> PROP A \<Longrightarrow> PROP C'))"
  apply (Pure.intro Pure.equal_intr_rule)
    apply (drule (1) cut_rl; drule Pure.equal_elim_rule1 Pure.equal_elim_rule2;
      assumption)+
    apply (drule Pure.equal_elim_rule1 Pure.equal_elim_rule2; assumption)+
  done

ML_file \<open>~~/src/HOL/Library/cconv.ML\<close>
ML_file \<open>lib/rewrite.ML\<close>

\<comment> \<open>\<open>reduce\<close> computes terms via judgmental equalities\<close>
setup \<open>map_theory_simpset (fn ctxt => ctxt addSolver (mk_solver "" typechk_tac))\<close>

method reduce uses add = (simp add: comps add | subst comps)+


section \<open>Implicit notations\<close>

text \<open>
  \<open>?\<close> is used to mark implicit arguments in definitions, while \<open>{}\<close> is expanded
  immediately for elaboration in statements.
\<close>

consts
  iarg :: \<open>'a\<close> ("?")
  hole :: \<open>'b\<close> ("{}")

ML_file \<open>lib/implicits.ML\<close>

attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close>

ML \<open>
val _ = Context.>>
  (Syntax_Phases.term_check 1 "" (fn ctxt => map (Implicits.make_holes ctxt)))
\<close>

text \<open>Automatically insert inhabitation judgments where needed:\<close>

syntax inhabited :: \<open>o \<Rightarrow> prop\<close> ("(_)")
translations "inhabited A" \<rightharpoonup> "CONST has_type {} A"


section \<open>Lambda coercion\<close>

\<comment> \<open>Coerce object lambdas to meta-lambdas\<close>
abbreviation (input) lambda :: \<open>o \<Rightarrow> o \<Rightarrow> o\<close>
  where "lambda f \<equiv> \<lambda>x. f `x"

ML_file \<open>~~/src/Tools/subtyping.ML\<close>
declare [[coercion_enabled, coercion lambda]]

translations "f x" \<leftharpoondown> "f `x"


section \<open>Functions\<close>

lemma eta_exp:
  assumes "f: \<Prod>x: A. B x"
  shows "f \<equiv> \<lambda>x: A. f x"
  by (rule eta[symmetric])

lemma lift_universe_codomain:
  assumes "A: U i" "f: A \<rightarrow> U j"
  shows "f: A \<rightarrow> U (S j)"
  apply (sub eta_exp)
    apply known
    apply (Pure.rule intros; rule lift_universe)
  done

subsection \<open>Function composition\<close>

definition "funcomp A g f \<equiv> \<lambda>x: A. g `(f `x)"

syntax
  "_funcomp" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ \<circ>\<^bsub>_\<^esub>/ _)" [111, 0, 110] 110)
translations
  "g \<circ>\<^bsub>A\<^esub> f" \<rightleftharpoons> "CONST funcomp A g f"

lemma funcompI [typechk]:
  assumes
    "A: U i"
    "B: U i"
    "\<And>x. x: B \<Longrightarrow> C x: U i"
    "f: A \<rightarrow> B"
    "g: \<Prod>x: B. C x"
  shows
    "g \<circ>\<^bsub>A\<^esub> f: \<Prod>x: A. C (f x)"
  unfolding funcomp_def by typechk

lemma funcomp_assoc [comps]:
  assumes
    "f: A \<rightarrow> B"
    "g: B \<rightarrow> C"
    "h: \<Prod>x: C. D x"
    "A: U i"
  shows
    "(h \<circ>\<^bsub>B\<^esub> g) \<circ>\<^bsub>A\<^esub> f \<equiv> h \<circ>\<^bsub>A\<^esub> g \<circ>\<^bsub>A\<^esub> f"
  unfolding funcomp_def by reduce

lemma funcomp_lambda_comp [comps]:
  assumes
    "A: U i"
    "\<And>x. x: A \<Longrightarrow> b x: B"
    "\<And>x. x: B \<Longrightarrow> c x: C x"
  shows
    "(\<lambda>x: B. c x) \<circ>\<^bsub>A\<^esub> (\<lambda>x: A. b x) \<equiv> \<lambda>x: A. c (b x)"
  unfolding funcomp_def by reduce

lemma funcomp_apply_comp [comps]:
  assumes
    "f: A \<rightarrow> B" "g: \<Prod>x: B. C x"
    "x: A"
    "A: U i" "B: U i"
    "\<And>x y. x: B \<Longrightarrow> C x: U i"
  shows "(g \<circ>\<^bsub>A\<^esub> f) x \<equiv> g (f x)"
  unfolding funcomp_def by reduce

text \<open>Notation:\<close>

definition funcomp_i (infixr "\<circ>" 120)
  where [implicit]: "funcomp_i g f \<equiv> g \<circ>\<^bsub>?\<^esub> f"

translations "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f"

subsection \<open>Identity function\<close>

abbreviation id where "id A \<equiv> \<lambda>x: A. x"

lemma
  id_type[typechk]: "A: U i \<Longrightarrow> id A: A \<rightarrow> A" and
  id_comp [comps]: "x: A \<Longrightarrow> (id A) x \<equiv> x" \<comment> \<open>for the occasional manual rewrite\<close>
  by reduce

lemma id_left [comps]:
  assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
  shows "(id B) \<circ>\<^bsub>A\<^esub> f \<equiv> f"
  by (subst eta_exp[of f]) (reduce, rule eta)

lemma id_right [comps]:
  assumes "f: A \<rightarrow> B" "A: U i" "B: U i"
  shows "f \<circ>\<^bsub>A\<^esub> (id A) \<equiv> f"
  by (subst eta_exp[of f]) (reduce, rule eta)

lemma id_U [typechk]:
  "id (U i): U i \<rightarrow> U i"
  by typechk (fact U_in_U)


section \<open>Pairs\<close>

definition "fst A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>_. A) (\<lambda>x y. x) p"
definition "snd A B \<equiv> \<lambda>p: \<Sum>x: A. B x. SigInd A B (\<lambda>p. B (fst A B p)) (\<lambda>x y. y) p"

lemma fst_type [typechk]:
  assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
  shows "fst A B: (\<Sum>x: A. B x) \<rightarrow> A"
  unfolding fst_def by typechk

lemma fst_comp [comps]:
  assumes
    "a: A"
    "b: B a"
    "A: U i"
    "\<And>x. x: A \<Longrightarrow> B x: U i"
  shows "fst A B <a, b> \<equiv> a"
  unfolding fst_def by reduce

lemma snd_type [typechk]:
  assumes "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
  shows "snd A B: \<Prod>p: \<Sum>x: A. B x. B (fst A B p)"
  unfolding snd_def by typechk reduce

lemma snd_comp [comps]:
  assumes "a: A" "b: B a" "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
  shows "snd A B <a, b> \<equiv> b"
  unfolding snd_def by reduce

subsection \<open>Notation\<close>

definition fst_i ("fst")
  where [implicit]: "fst \<equiv> Spartan.fst ? ?"

definition snd_i ("snd")
  where [implicit]: "snd \<equiv> Spartan.snd ? ?"

translations
  "fst" \<leftharpoondown> "CONST Spartan.fst A B"
  "snd" \<leftharpoondown> "CONST Spartan.snd A B"

subsection \<open>Projections\<close>

Lemma fst [typechk]:
  assumes
    "p: \<Sum>x: A. B x"
    "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
  shows "fst p: A"
  by typechk

Lemma snd [typechk]:
  assumes
    "p: \<Sum>x: A. B x"
    "A: U i" "\<And>x. x: A \<Longrightarrow> B x: U i"
  shows "snd p: B (fst p)"
  by typechk

method fst for p::o = rule fst[of p]
method snd for p::o = rule snd[of p]

subsection \<open>Properties of \<Sigma>\<close>

Lemma (derive) Sig_dist_exp:
  assumes
    "p: \<Sum>x: A. B x \<times> C x"
    "A: U i"
    "\<And>x. x: A \<Longrightarrow> B x: U i"
    "\<And>x. x: A \<Longrightarrow> C x: U i"
  shows "(\<Sum>x: A. B x) \<times> (\<Sum>x: A. C x)"
  apply (elim p)
    focus vars x y
      apply intro
        \<guillemotright> apply intro
            apply assumption
            apply (fst y)
          done
        \<guillemotright> apply intro
            apply assumption
            apply (snd y)
          done
    done
  done


end