src/HOL/Bali/Basis.thy
 author skalberg Thu Mar 03 12:43:01 2005 +0100 (2005-03-03) changeset 15570 8d8c70b41bab parent 15531 08c8dad8e399 child 16121 a80aa66d2271 permissions -rw-r--r--
Move towards standard functions.
1 (*  Title:      HOL/Bali/Basis.thy
2     ID:         \$Id\$
3     Author:     David von Oheimb
5 *)
6 header {* Definitions extending HOL as logical basis of Bali *}
8 theory Basis = Main:
10 ML_setup {*
11 Unify.search_bound := 40;
12 Unify.trace_bound  := 40;
13 *}
14 (*print_depth 100;*)
15 (*Syntax.ambiguity_level := 1;*)
17 section "misc"
19 declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
21 ML {*
22 fun cond_simproc name pat pred thm = Simplifier.simproc (Thm.sign_of_thm thm) name [pat]
23   (fn _ => fn _ => fn t => if pred t then NONE else SOME (mk_meta_eq thm));
24 *}
26 declare split_if_asm  [split] option.split [split] option.split_asm [split]
27 ML {*
28 simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
29 *}
30 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
31 declare length_Suc_conv [iff];
33 (*###to be phased out *)
34 ML {*
35 bind_thm ("make_imp", rearrange_prems [1,0] mp)
36 *}
38 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
39 apply auto
40 done
42 lemma subset_insertD:
43   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
44 apply (case_tac "x:A")
45 apply (rule disjI2)
46 apply (rule_tac x = "A-{x}" in exI)
47 apply fast+
48 done
50 syntax
51   "3" :: nat   ("3")
52   "4" :: nat   ("4")
53 translations
54  "3" == "Suc 2"
55  "4" == "Suc 3"
57 (*unused*)
58 lemma range_bool_domain: "range f = {f True, f False}"
59 apply auto
60 apply (case_tac "xa")
61 apply auto
62 done
64 (* irrefl_tranclI in Transitive_Closure.thy is more general *)
65 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
66 by(blast elim: tranclE dest: trancl_into_rtrancl)
69 lemma trancl_rtrancl_trancl:
70 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
71 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
73 lemma rtrancl_into_trancl3:
74 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+"
75 apply (drule rtranclD)
76 apply auto
77 done
79 lemma rtrancl_into_rtrancl2:
80   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
81 by (auto intro: r_into_rtrancl rtrancl_trans)
83 lemma triangle_lemma:
84  "\<lbrakk> \<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c; (a,x)\<in>r\<^sup>*; (a,y)\<in>r\<^sup>*\<rbrakk>
85  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
86 proof -
87   note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
88   note converse_rtranclE = converse_rtranclE [consumes 1]
89   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
90   assume "(a,x)\<in>r\<^sup>*"
91   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
92   proof (induct rule: converse_rtrancl_induct)
93     assume "(x,y)\<in>r\<^sup>*"
94     then show ?thesis
95       by blast
96   next
97     fix a v
98     assume a_v_r: "(a, v) \<in> r" and
99           v_x_rt: "(v, x) \<in> r\<^sup>*" and
100           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
101              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
102     from a_y_rt
103     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
104     proof (cases rule: converse_rtranclE)
105       assume "a=y"
106       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
107 	by (auto intro: r_into_rtrancl rtrancl_trans)
108       then show ?thesis
109 	by blast
110     next
111       fix w
112       assume a_w_r: "(a, w) \<in> r" and
113             w_y_rt: "(w, y) \<in> r\<^sup>*"
114       from a_v_r a_w_r unique
115       have "v=w"
116 	by auto
117       with w_y_rt hyp
118       show ?thesis
119 	by blast
120     qed
121   qed
122 qed
125 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
126  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
127 apply (erule rtranclE)
128 apply (auto dest: rtrancl_into_trancl1)
129 done
131 (* ### To Transitive_Closure *)
132 theorems converse_rtrancl_induct
133  = converse_rtrancl_induct [consumes 1,case_names Id Step]
135 theorems converse_trancl_induct
136          = converse_trancl_induct [consumes 1,case_names Single Step]
138 (* context (theory "Set") *)
139 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
140 by auto
142 (* context (theory "Finite") *)
143 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>
144   finite {f y x |x y. P y}"
145 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
146 prefer 2 apply  fast
147 apply (erule ssubst)
148 apply (erule finite_UN_I)
149 apply fast
150 done
153 (* ### TO theory "List" *)
154 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
155  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
156 apply (induct_tac "xs1")
157 apply simp
158 apply (rule allI)
159 apply (induct_tac "xs2")
160 apply simp
161 apply (rule allI)
162 apply (induct_tac "xs3")
163 apply auto
164 done
167 section "pairs"
169 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
170   snd (snd (snd (snd p))))"
171 apply auto
172 done
174 lemma fst_splitE [elim!]:
175 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
176 apply (cut_tac p = "s'" in surjective_pairing)
177 apply auto
178 done
180 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
181 apply (induct_tac "l")
182 apply  auto
183 done
186 section "quantifiers"
188 (*###to be phased out *)
189 ML {*
190 fun noAll_simpset () = simpset() setmksimps
191 	mksimps (List.filter (fn (x,_) => x<>"All") mksimps_pairs)
192 *}
194 lemma All_Ex_refl_eq2 [simp]:
195  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
196 apply auto
197 done
199 lemma ex_ex_miniscope1 [simp]:
200   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
201 apply auto
202 done
204 lemma ex_miniscope2 [simp]:
205   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))"
206 apply auto
207 done
209 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
210 apply auto
211 done
213 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
214 apply auto
215 done
218 section "sums"
220 hide const In0 In1
222 syntax
223   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
224 translations
225  "fun_sum" == "sum_case"
227 consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
228           the_Inr  :: "'a + 'b \<Rightarrow> 'b"
229 primrec  "the_Inl (Inl a) = a"
230 primrec  "the_Inr (Inr b) = b"
232 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
234 consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
235           the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
236           the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
237 primrec  "the_In1 (In1 a) = a"
238 primrec  "the_In2 (In2 b) = b"
239 primrec  "the_In3 (In3 c) = c"
241 syntax
242 	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
243 	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
244 translations
245 	"In1l e" == "In1 (Inl e)"
246 	"In1r c" == "In1 (Inr c)"
248 syntax the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
249        the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
250 translations
251    "the_In1l" == "the_Inl \<circ> the_In1"
252    "the_In1r" == "the_Inr \<circ> the_In1"
254 ML {*
255 fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
256  (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
257 *}
258 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
260 translations
261   "option"<= (type) "Datatype.option"
262   "list"  <= (type) "List.list"
263   "sum3"  <= (type) "Basis.sum3"
266 section "quantifiers for option type"
268 syntax
269   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
270   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
272 syntax (symbols)
273   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
274   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
276 translations
277   "! x:A: P"    == "! x:o2s A. P"
278   "? x:A: P"    == "? x:o2s A. P"
281 section "unique association lists"
283 constdefs
284   unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
285  "unique \<equiv> distinct \<circ> map fst"
287 lemma uniqueD [rule_format (no_asm)]:
288 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
289 apply (unfold unique_def o_def)
290 apply (induct_tac "l")
291 apply  (auto dest: fst_in_set_lemma)
292 done
294 lemma unique_Nil [simp]: "unique []"
295 apply (unfold unique_def)
296 apply (simp (no_asm))
297 done
299 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
300 apply (unfold unique_def)
301 apply  (auto dest: fst_in_set_lemma)
302 done
304 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
306 lemma unique_single [simp]: "!!p. unique [p]"
307 apply auto
308 done
310 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
312 done
314 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->
315   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
316 apply (induct_tac "l")
317 apply  (auto dest: fst_in_set_lemma)
318 done
320 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
321 apply (induct_tac "l")
322 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
323 done
325 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
326 apply (induct_tac "l")
327 apply auto
328 done
331 section "list patterns"
333 consts
334   lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
335 defs
336   lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
337 (*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
338 syntax
339   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
340 translations
341   "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
342   "%x#xs  . b"  == "lsplit (%x xs  . b)"
344 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
345 apply (unfold lsplit_def)
346 apply (simp (no_asm))
347 done
349 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
350 apply (unfold lsplit_def)
351 apply simp
352 done
355 section "dummy pattern for quantifiers, let, etc."
357 syntax
358   "@dummy_pat"   :: pttrn    ("'_")
360 parse_translation {*
361 let fun dummy_pat_tr [] = Free ("_",dummyT)
362   | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
363 in [("@dummy_pat", dummy_pat_tr)]
364 end
365 *}
367 end