1 theory LamEx |
|
2 imports Nominal "../Quotient" "../Quotient_List" |
|
3 begin |
|
4 |
|
5 atom_decl name |
|
6 |
|
7 datatype rlam = |
|
8 rVar "name" |
|
9 | rApp "rlam" "rlam" |
|
10 | rLam "name" "rlam" |
|
11 |
|
12 fun |
|
13 rfv :: "rlam \<Rightarrow> name set" |
|
14 where |
|
15 rfv_var: "rfv (rVar a) = {a}" |
|
16 | rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)" |
|
17 | rfv_lam: "rfv (rLam a t) = (rfv t) - {a}" |
|
18 |
|
19 overloading |
|
20 perm_rlam \<equiv> "perm :: 'x prm \<Rightarrow> rlam \<Rightarrow> rlam" (unchecked) |
|
21 begin |
|
22 |
|
23 fun |
|
24 perm_rlam |
|
25 where |
|
26 "perm_rlam pi (rVar a) = rVar (pi \<bullet> a)" |
|
27 | "perm_rlam pi (rApp t1 t2) = rApp (perm_rlam pi t1) (perm_rlam pi t2)" |
|
28 | "perm_rlam pi (rLam a t) = rLam (pi \<bullet> a) (perm_rlam pi t)" |
|
29 |
|
30 end |
|
31 |
|
32 declare perm_rlam.simps[eqvt] |
|
33 |
|
34 instance rlam::pt_name |
|
35 apply(default) |
|
36 apply(induct_tac [!] x rule: rlam.induct) |
|
37 apply(simp_all add: pt_name2 pt_name3) |
|
38 done |
|
39 |
|
40 instance rlam::fs_name |
|
41 apply(default) |
|
42 apply(induct_tac [!] x rule: rlam.induct) |
|
43 apply(simp add: supp_def) |
|
44 apply(fold supp_def) |
|
45 apply(simp add: supp_atm) |
|
46 apply(simp add: supp_def Collect_imp_eq Collect_neg_eq) |
|
47 apply(simp add: supp_def) |
|
48 apply(simp add: supp_def Collect_imp_eq Collect_neg_eq[symmetric]) |
|
49 apply(fold supp_def) |
|
50 apply(simp add: supp_atm) |
|
51 done |
|
52 |
|
53 declare set_diff_eqvt[eqvt] |
|
54 |
|
55 lemma rfv_eqvt[eqvt]: |
|
56 fixes pi::"name prm" |
|
57 shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)" |
|
58 apply(induct t) |
|
59 apply(simp_all) |
|
60 apply(simp add: perm_set_eq) |
|
61 apply(simp add: union_eqvt) |
|
62 apply(simp add: set_diff_eqvt) |
|
63 apply(simp add: perm_set_eq) |
|
64 done |
|
65 |
|
66 inductive |
|
67 alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100) |
|
68 where |
|
69 a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)" |
|
70 | a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2" |
|
71 | a3: "\<exists>pi::name prm. (rfv t - {a} = rfv s - {b} \<and> (rfv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) \<approx> s \<and> (pi \<bullet> a) = b) |
|
72 \<Longrightarrow> rLam a t \<approx> rLam b s" |
|
73 |
|
74 |
|
75 (* should be automatic with new version of eqvt-machinery *) |
|
76 lemma alpha_eqvt: |
|
77 fixes pi::"name prm" |
|
78 shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)" |
|
79 apply(induct rule: alpha.induct) |
|
80 apply(simp add: a1) |
|
81 apply(simp add: a2) |
|
82 apply(simp) |
|
83 apply(rule a3) |
|
84 apply(erule conjE) |
|
85 apply(erule exE) |
|
86 apply(erule conjE) |
|
87 apply(rule_tac x="pi \<bullet> pia" in exI) |
|
88 apply(rule conjI) |
|
89 apply(rule_tac pi1="rev pi" in perm_bij[THEN iffD1]) |
|
90 apply(perm_simp add: eqvts) |
|
91 apply(rule conjI) |
|
92 apply(rule_tac pi1="rev pi" in pt_fresh_star_bij(1)[OF pt_name_inst at_name_inst, THEN iffD1]) |
|
93 apply(perm_simp add: eqvts) |
|
94 apply(rule conjI) |
|
95 apply(subst perm_compose[symmetric]) |
|
96 apply(simp) |
|
97 apply(subst perm_compose[symmetric]) |
|
98 apply(simp) |
|
99 done |
|
100 |
|
101 lemma alpha_refl: |
|
102 shows "t \<approx> t" |
|
103 apply(induct t rule: rlam.induct) |
|
104 apply(simp add: a1) |
|
105 apply(simp add: a2) |
|
106 apply(rule a3) |
|
107 apply(rule_tac x="[]" in exI) |
|
108 apply(simp_all add: fresh_star_def fresh_list_nil) |
|
109 done |
|
110 |
|
111 lemma alpha_sym: |
|
112 shows "t \<approx> s \<Longrightarrow> s \<approx> t" |
|
113 apply(induct rule: alpha.induct) |
|
114 apply(simp add: a1) |
|
115 apply(simp add: a2) |
|
116 apply(rule a3) |
|
117 apply(erule exE) |
|
118 apply(rule_tac x="rev pi" in exI) |
|
119 apply(simp) |
|
120 apply(simp add: fresh_star_def fresh_list_rev) |
|
121 apply(rule conjI) |
|
122 apply(erule conjE)+ |
|
123 apply(rotate_tac 3) |
|
124 apply(drule_tac pi="rev pi" in alpha_eqvt) |
|
125 apply(perm_simp) |
|
126 apply(rule pt_bij2[OF pt_name_inst at_name_inst]) |
|
127 apply(simp) |
|
128 done |
|
129 |
|
130 lemma alpha_trans: |
|
131 shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3" |
|
132 apply(induct arbitrary: t3 rule: alpha.induct) |
|
133 apply(erule alpha.cases) |
|
134 apply(simp_all) |
|
135 apply(simp add: a1) |
|
136 apply(rotate_tac 4) |
|
137 apply(erule alpha.cases) |
|
138 apply(simp_all) |
|
139 apply(simp add: a2) |
|
140 apply(rotate_tac 1) |
|
141 apply(erule alpha.cases) |
|
142 apply(simp_all) |
|
143 apply(erule conjE)+ |
|
144 apply(erule exE)+ |
|
145 apply(erule conjE)+ |
|
146 apply(rule a3) |
|
147 apply(rule_tac x="pia @ pi" in exI) |
|
148 apply(simp add: fresh_star_def fresh_list_append) |
|
149 apply(simp add: pt_name2) |
|
150 apply(drule_tac x="rev pia \<bullet> sa" in spec) |
|
151 apply(drule mp) |
|
152 apply(rotate_tac 8) |
|
153 apply(drule_tac pi="rev pia" in alpha_eqvt) |
|
154 apply(perm_simp) |
|
155 apply(rotate_tac 11) |
|
156 apply(drule_tac pi="pia" in alpha_eqvt) |
|
157 apply(perm_simp) |
|
158 done |
|
159 |
|
160 lemma alpha_equivp: |
|
161 shows "equivp alpha" |
|
162 apply(rule equivpI) |
|
163 unfolding reflp_def symp_def transp_def |
|
164 apply(auto intro: alpha_refl alpha_sym alpha_trans) |
|
165 done |
|
166 |
|
167 lemma alpha_rfv: |
|
168 shows "t \<approx> s \<Longrightarrow> rfv t = rfv s" |
|
169 apply(induct rule: alpha.induct) |
|
170 apply(simp) |
|
171 apply(simp) |
|
172 apply(simp) |
|
173 done |
|
174 |
|
175 quotient_type lam = rlam / alpha |
|
176 by (rule alpha_equivp) |
|
177 |
|
178 |
|
179 quotient_definition |
|
180 "Var :: name \<Rightarrow> lam" |
|
181 is |
|
182 "rVar" |
|
183 |
|
184 quotient_definition |
|
185 "App :: lam \<Rightarrow> lam \<Rightarrow> lam" |
|
186 is |
|
187 "rApp" |
|
188 |
|
189 quotient_definition |
|
190 "Lam :: name \<Rightarrow> lam \<Rightarrow> lam" |
|
191 is |
|
192 "rLam" |
|
193 |
|
194 quotient_definition |
|
195 "fv :: lam \<Rightarrow> name set" |
|
196 is |
|
197 "rfv" |
|
198 |
|
199 (* definition of overloaded permutation function *) |
|
200 (* for the lifted type lam *) |
|
201 overloading |
|
202 perm_lam \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" (unchecked) |
|
203 begin |
|
204 |
|
205 quotient_definition |
|
206 "perm_lam :: 'x prm \<Rightarrow> lam \<Rightarrow> lam" |
|
207 is |
|
208 "perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam" |
|
209 |
|
210 end |
|
211 |
|
212 lemma perm_rsp[quot_respect]: |
|
213 "(op = ===> alpha ===> alpha) op \<bullet> (op \<bullet> :: (name \<times> name) list \<Rightarrow> rlam \<Rightarrow> rlam)" |
|
214 apply auto |
|
215 apply(erule alpha_eqvt) |
|
216 done |
|
217 |
|
218 lemma rVar_rsp[quot_respect]: |
|
219 "(op = ===> alpha) rVar rVar" |
|
220 by (auto intro: a1) |
|
221 |
|
222 lemma rApp_rsp[quot_respect]: "(alpha ===> alpha ===> alpha) rApp rApp" |
|
223 by (auto intro: a2) |
|
224 |
|
225 lemma rLam_rsp[quot_respect]: "(op = ===> alpha ===> alpha) rLam rLam" |
|
226 apply(auto) |
|
227 apply(rule a3) |
|
228 apply(rule_tac x="[]" in exI) |
|
229 unfolding fresh_star_def |
|
230 apply(simp add: fresh_list_nil) |
|
231 apply(simp add: alpha_rfv) |
|
232 done |
|
233 |
|
234 lemma rfv_rsp[quot_respect]: |
|
235 "(alpha ===> op =) rfv rfv" |
|
236 apply(simp add: alpha_rfv) |
|
237 done |
|
238 |
|
239 section {* lifted theorems *} |
|
240 |
|
241 lemma lam_induct: |
|
242 "\<lbrakk>\<And>name. P (Var name); |
|
243 \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2); |
|
244 \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> |
|
245 \<Longrightarrow> P lam" |
|
246 by (lifting rlam.induct) |
|
247 |
|
248 lemma perm_lam [simp]: |
|
249 fixes pi::"name prm" |
|
250 shows "pi \<bullet> Var a = Var (pi \<bullet> a)" |
|
251 and "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)" |
|
252 and "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)" |
|
253 by (lifting perm_rlam.simps[where 'a="name"]) |
|
254 |
|
255 instance lam::pt_name |
|
256 apply(default) |
|
257 apply(induct_tac [!] x rule: lam_induct) |
|
258 apply(simp_all add: pt_name2 pt_name3) |
|
259 done |
|
260 |
|
261 lemma fv_lam [simp]: |
|
262 shows "fv (Var a) = {a}" |
|
263 and "fv (App t1 t2) = fv t1 \<union> fv t2" |
|
264 and "fv (Lam a t) = fv t - {a}" |
|
265 by(lifting rfv_var rfv_app rfv_lam) |
|
266 |
|
267 lemma a1: |
|
268 "a = b \<Longrightarrow> Var a = Var b" |
|
269 by (lifting a1) |
|
270 |
|
271 lemma a2: |
|
272 "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc" |
|
273 by (lifting a2) |
|
274 |
|
275 lemma a3: |
|
276 "\<lbrakk>\<exists>pi::name prm. (fv t - {a} = fv s - {b} \<and> (fv t - {a})\<sharp>* pi \<and> (pi \<bullet> t) = s \<and> (pi \<bullet> a) = b)\<rbrakk> |
|
277 \<Longrightarrow> Lam a t = Lam b s" |
|
278 by (lifting a3) |
|
279 |
|
280 lemma alpha_cases: |
|
281 "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P; |
|
282 \<And>x xa xb xc. \<lbrakk>a1 = App x xb; a2 = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P; |
|
283 \<And>t a s b. \<lbrakk>a1 = Lam a t; a2 = Lam b s; |
|
284 \<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> (fv t - {a}) \<sharp>* pi \<and> (pi \<bullet> t) = s \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> P\<rbrakk> |
|
285 \<Longrightarrow> P" |
|
286 by (lifting alpha.cases) |
|
287 |
|
288 lemma alpha_induct: |
|
289 "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b); |
|
290 \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc); |
|
291 \<And>t a s b. |
|
292 \<lbrakk>\<exists>pi::name prm. fv t - {a} = fv s - {b} \<and> |
|
293 (fv t - {a}) \<sharp>* pi \<and> ((pi \<bullet> t) = s \<and> qxb (pi \<bullet> t) s) \<and> pi \<bullet> a = b\<rbrakk> \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk> |
|
294 \<Longrightarrow> qxb qx qxa" |
|
295 by (lifting alpha.induct) |
|
296 |
|
297 lemma lam_inject [simp]: |
|
298 shows "(Var a = Var b) = (a = b)" |
|
299 and "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)" |
|
300 apply(lifting rlam.inject(1) rlam.inject(2)) |
|
301 apply(auto) |
|
302 apply(drule alpha.cases) |
|
303 apply(simp_all) |
|
304 apply(simp add: alpha.a1) |
|
305 apply(drule alpha.cases) |
|
306 apply(simp_all) |
|
307 apply(drule alpha.cases) |
|
308 apply(simp_all) |
|
309 apply(rule alpha.a2) |
|
310 apply(simp_all) |
|
311 done |
|
312 |
|
313 lemma rlam_distinct: |
|
314 shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')" |
|
315 and "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)" |
|
316 and "\<not>(rVar nam \<approx> rLam nam' rlam')" |
|
317 and "\<not>(rLam nam' rlam' \<approx> rVar nam)" |
|
318 and "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')" |
|
319 and "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)" |
|
320 apply auto |
|
321 apply(erule alpha.cases) |
|
322 apply simp_all |
|
323 apply(erule alpha.cases) |
|
324 apply simp_all |
|
325 apply(erule alpha.cases) |
|
326 apply simp_all |
|
327 apply(erule alpha.cases) |
|
328 apply simp_all |
|
329 apply(erule alpha.cases) |
|
330 apply simp_all |
|
331 apply(erule alpha.cases) |
|
332 apply simp_all |
|
333 done |
|
334 |
|
335 lemma lam_distinct[simp]: |
|
336 shows "Var nam \<noteq> App lam1' lam2'" |
|
337 and "App lam1' lam2' \<noteq> Var nam" |
|
338 and "Var nam \<noteq> Lam nam' lam'" |
|
339 and "Lam nam' lam' \<noteq> Var nam" |
|
340 and "App lam1 lam2 \<noteq> Lam nam' lam'" |
|
341 and "Lam nam' lam' \<noteq> App lam1 lam2" |
|
342 by(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6)) |
|
343 |
|
344 lemma var_supp1: |
|
345 shows "(supp (Var a)) = ((supp a)::name set)" |
|
346 by (simp add: supp_def) |
|
347 |
|
348 lemma var_supp: |
|
349 shows "(supp (Var a)) = {a::name}" |
|
350 using var_supp1 by (simp add: supp_atm) |
|
351 |
|
352 lemma app_supp: |
|
353 shows "supp (App t1 t2) = (supp t1) \<union> ((supp t2)::name set)" |
|
354 apply(simp only: perm_lam supp_def lam_inject) |
|
355 apply(simp add: Collect_imp_eq Collect_neg_eq) |
|
356 done |
|
357 |
|
358 lemma lam_supp: |
|
359 shows "supp (Lam x t) = ((supp ([x].t))::name set)" |
|
360 apply(simp add: supp_def) |
|
361 apply(simp add: abs_perm) |
|
362 sorry |
|
363 |
|
364 instance lam::fs_name |
|
365 apply(default) |
|
366 apply(induct_tac x rule: lam_induct) |
|
367 apply(simp add: var_supp) |
|
368 apply(simp add: app_supp) |
|
369 apply(simp add: lam_supp abs_supp) |
|
370 done |
|
371 |
|
372 lemma fresh_lam: |
|
373 "(a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> a \<sharp> t)" |
|
374 apply(simp add: fresh_def) |
|
375 apply(simp add: lam_supp abs_supp) |
|
376 apply(auto) |
|
377 done |
|
378 |
|
379 lemma lam_induct_strong: |
|
380 fixes a::"'a::fs_name" |
|
381 assumes a1: "\<And>name b. P b (Var name)" |
|
382 and a2: "\<And>lam1 lam2 b. \<lbrakk>\<And>c. P c lam1; \<And>c. P c lam2\<rbrakk> \<Longrightarrow> P b (App lam1 lam2)" |
|
383 and a3: "\<And>name lam b. \<lbrakk>\<And>c. P c lam; name \<sharp> b\<rbrakk> \<Longrightarrow> P b (Lam name lam)" |
|
384 shows "P a lam" |
|
385 proof - |
|
386 have "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" |
|
387 proof (induct lam rule: lam_induct) |
|
388 case (1 name pi) |
|
389 show "P a (pi \<bullet> Var name)" |
|
390 apply (simp) |
|
391 apply (rule a1) |
|
392 done |
|
393 next |
|
394 case (2 lam1 lam2 pi) |
|
395 have b1: "\<And>(pi::name prm) a. P a (pi \<bullet> lam1)" by fact |
|
396 have b2: "\<And>(pi::name prm) a. P a (pi \<bullet> lam2)" by fact |
|
397 show "P a (pi \<bullet> App lam1 lam2)" |
|
398 apply (simp) |
|
399 apply (rule a2) |
|
400 apply (rule b1) |
|
401 apply (rule b2) |
|
402 done |
|
403 next |
|
404 case (3 name lam pi a) |
|
405 have b: "\<And>(pi::name prm) a. P a (pi \<bullet> lam)" by fact |
|
406 obtain c::name where fr: "c\<sharp>(a, pi\<bullet>name, pi\<bullet>lam)" |
|
407 apply(rule exists_fresh[of "(a, pi\<bullet>name, pi\<bullet>lam)"]) |
|
408 apply(simp_all add: fs_name1) |
|
409 done |
|
410 from b fr have p: "P a (Lam c (([(c, pi\<bullet>name)]@pi)\<bullet>lam))" |
|
411 apply - |
|
412 apply(rule a3) |
|
413 apply(blast) |
|
414 apply(simp) |
|
415 done |
|
416 have eq: "[(c, pi\<bullet>name)] \<bullet> Lam (pi \<bullet> name) (pi \<bullet> lam) = Lam (pi \<bullet> name) (pi \<bullet> lam)" |
|
417 apply(rule perm_fresh_fresh) |
|
418 using fr |
|
419 apply(simp add: fresh_lam) |
|
420 apply(simp add: fresh_lam) |
|
421 done |
|
422 show "P a (pi \<bullet> Lam name lam)" |
|
423 apply (simp) |
|
424 apply(subst eq[symmetric]) |
|
425 using p |
|
426 apply(simp only: perm_lam pt_name2 swap_simps) |
|
427 done |
|
428 qed |
|
429 then have "P a (([]::name prm) \<bullet> lam)" by blast |
|
430 then show "P a lam" by simp |
|
431 qed |
|
432 |
|
433 |
|
434 lemma var_fresh: |
|
435 fixes a::"name" |
|
436 shows "(a \<sharp> (Var b)) = (a \<sharp> b)" |
|
437 apply(simp add: fresh_def) |
|
438 apply(simp add: var_supp1) |
|
439 done |
|
440 |
|
441 (* lemma hom_reg: *) |
|
442 |
|
443 lemma rlam_rec_eqvt: |
|
444 fixes pi::"name prm" |
|
445 and f1::"name \<Rightarrow> ('a::pt_name)" |
|
446 shows "(pi\<bullet>rlam_rec f1 f2 f3 t) = rlam_rec (pi\<bullet>f1) (pi\<bullet>f2) (pi\<bullet>f3) (pi\<bullet>t)" |
|
447 apply(induct t) |
|
448 apply(simp_all) |
|
449 apply(simp add: perm_fun_def) |
|
450 apply(perm_simp) |
|
451 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
452 back |
|
453 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
454 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
455 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
456 apply(simp) |
|
457 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
458 back |
|
459 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
460 apply(subst pt_fun_app_eq[OF pt_name_inst at_name_inst]) |
|
461 apply(simp) |
|
462 done |
|
463 |
|
464 |
|
465 lemma rlam_rec_respects: |
|
466 assumes f1: "f_var \<in> Respects (op= ===> op=)" |
|
467 and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)" |
|
468 and f3: "f_lam \<in> Respects (op= ===> alpha ===> op= ===> op=)" |
|
469 shows "rlam_rec f_var f_app f_lam \<in> Respects (alpha ===> op =)" |
|
470 apply(simp add: mem_def) |
|
471 apply(simp add: Respects_def) |
|
472 apply(rule allI) |
|
473 apply(rule allI) |
|
474 apply(rule impI) |
|
475 apply(erule alpha.induct) |
|
476 apply(simp) |
|
477 apply(simp) |
|
478 using f2 |
|
479 apply(simp add: mem_def) |
|
480 apply(simp add: Respects_def) |
|
481 using f3[simplified mem_def Respects_def] |
|
482 apply(simp) |
|
483 apply(case_tac "a=b") |
|
484 apply(clarify) |
|
485 apply(simp) |
|
486 (* probably true *) |
|
487 sorry |
|
488 |
|
489 function |
|
490 term1_hom :: "(name \<Rightarrow> 'a) \<Rightarrow> |
|
491 (rlam \<Rightarrow> rlam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> |
|
492 ((name \<Rightarrow> rlam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> rlam \<Rightarrow> 'a" |
|
493 where |
|
494 "term1_hom var app abs' (rVar x) = (var x)" |
|
495 | "term1_hom var app abs' (rApp t u) = |
|
496 app t u (term1_hom var app abs' t) (term1_hom var app abs' u)" |
|
497 | "term1_hom var app abs' (rLam x u) = |
|
498 abs' (\<lambda>y. [(x, y)] \<bullet> u) (\<lambda>y. term1_hom var app abs' ([(x, y)] \<bullet> u))" |
|
499 apply(pat_completeness) |
|
500 apply(auto) |
|
501 done |
|
502 |
|
503 lemma pi_size: |
|
504 fixes pi::"name prm" |
|
505 and t::"rlam" |
|
506 shows "size (pi \<bullet> t) = size t" |
|
507 apply(induct t) |
|
508 apply(auto) |
|
509 done |
|
510 |
|
511 termination term1_hom |
|
512 apply(relation "measure (\<lambda>(f1, f2, f3, t). size t)") |
|
513 apply(auto simp add: pi_size) |
|
514 done |
|
515 |
|
516 lemma lam_exhaust: |
|
517 "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; \<And>rlam1 rlam2. y = App rlam1 rlam2 \<Longrightarrow> P; \<And>name rlam. y = Lam name rlam \<Longrightarrow> P\<rbrakk> |
|
518 \<Longrightarrow> P" |
|
519 apply(lifting rlam.exhaust) |
|
520 done |
|
521 |
|
522 (* THIS IS NOT TRUE, but it lets prove the existence of the hom function *) |
|
523 lemma lam_inject': |
|
524 "(Lam a x = Lam b y) = ((\<lambda>c. [(a, c)] \<bullet> x) = (\<lambda>c. [(b, c)] \<bullet> y))" |
|
525 sorry |
|
526 |
|
527 function |
|
528 hom :: "(name \<Rightarrow> 'a) \<Rightarrow> |
|
529 (lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> |
|
530 ((name \<Rightarrow> lam) \<Rightarrow> (name \<Rightarrow> 'a) \<Rightarrow> 'a) \<Rightarrow> lam \<Rightarrow> 'a" |
|
531 where |
|
532 "hom f_var f_app f_lam (Var x) = f_var x" |
|
533 | "hom f_var f_app f_lam (App l r) = f_app l r (hom f_var f_app f_lam l) (hom f_var f_app f_lam r)" |
|
534 | "hom f_var f_app f_lam (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom f_var f_app f_lam ([(a,b)] \<bullet> x))" |
|
535 defer |
|
536 apply(simp_all add: lam_inject') (* inject, distinct *) |
|
537 apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *}) |
|
538 apply(rule refl) |
|
539 apply(rule ext) |
|
540 apply(tactic {* Cong_Tac.cong_tac @{thm cong} 1 *}) |
|
541 apply simp_all |
|
542 apply(erule conjE)+ |
|
543 apply(rule_tac x="b" in cong) |
|
544 apply simp_all |
|
545 apply auto |
|
546 apply(rule_tac y="b" in lam_exhaust) |
|
547 apply simp_all |
|
548 apply auto |
|
549 apply meson |
|
550 apply(simp_all add: lam_inject') |
|
551 apply metis |
|
552 done |
|
553 |
|
554 termination hom |
|
555 apply - |
|
556 (* |
|
557 ML_prf {* Size.size_thms @{theory} "LamEx.lam" *} |
|
558 *) |
|
559 sorry |
|
560 |
|
561 thm hom.simps |
|
562 |
|
563 lemma term1_hom_rsp: |
|
564 "\<lbrakk>(alpha ===> alpha ===> op =) f_app f_app; ((op = ===> alpha) ===> op =) f_lam f_lam\<rbrakk> |
|
565 \<Longrightarrow> (alpha ===> op =) (term1_hom f_var f_app f_lam) (term1_hom f_var f_app f_lam)" |
|
566 apply(simp) |
|
567 apply(rule allI)+ |
|
568 apply(rule impI) |
|
569 apply(erule alpha.induct) |
|
570 apply(auto)[1] |
|
571 apply(auto)[1] |
|
572 apply(simp) |
|
573 apply(erule conjE)+ |
|
574 apply(erule exE)+ |
|
575 apply(erule conjE)+ |
|
576 apply(clarify) |
|
577 sorry |
|
578 |
|
579 lemma hom: " |
|
580 \<forall>f_var. \<forall>f_app \<in> Respects(alpha ===> alpha ===> op =). |
|
581 \<forall>f_lam \<in> Respects((op = ===> alpha) ===> op =). |
|
582 \<exists>hom\<in>Respects (alpha ===> op =). |
|
583 ((\<forall>x. hom (rVar x) = f_var x) \<and> |
|
584 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and> |
|
585 (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))" |
|
586 apply(rule allI) |
|
587 apply(rule ballI)+ |
|
588 apply(rule_tac x="term1_hom f_var f_app f_lam" in bexI) |
|
589 apply(simp_all) |
|
590 apply(simp only: in_respects) |
|
591 apply(rule term1_hom_rsp) |
|
592 apply(assumption)+ |
|
593 done |
|
594 |
|
595 lemma hom': |
|
596 "\<exists>hom. |
|
597 ((\<forall>x. hom (Var x) = f_var x) \<and> |
|
598 (\<forall>l r. hom (App l r) = f_app l r (hom l) (hom r)) \<and> |
|
599 (\<forall>x a. hom (Lam a x) = f_lam (\<lambda>b. ([(a,b)] \<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))" |
|
600 apply (lifting hom) |
|
601 done |
|
602 |
|
603 (* test test |
|
604 lemma raw_hom_correct: |
|
605 assumes f1: "f_var \<in> Respects (op= ===> op=)" |
|
606 and f2: "f_app \<in> Respects (alpha ===> alpha ===> op= ===> op= ===> op=)" |
|
607 and f3: "f_lam \<in> Respects ((op= ===> alpha) ===> (op= ===> op=) ===> op=)" |
|
608 shows "\<exists>!hom\<in>Respects (alpha ===> op =). |
|
609 ((\<forall>x. hom (rVar x) = f_var x) \<and> |
|
610 (\<forall>l r. hom (rApp l r) = f_app l r (hom l) (hom r)) \<and> |
|
611 (\<forall>x a. hom (rLam a x) = f_lam (\<lambda>b. ([(a,b)]\<bullet> x)) (\<lambda>b. hom ([(a,b)] \<bullet> x))))" |
|
612 unfolding Bex1_def |
|
613 apply(rule ex1I) |
|
614 sorry |
|
615 *) |
|
616 |
|
617 |
|
618 end |
|
619 |
|