src/LCF/LCF.thy
 author wenzelm Sun Nov 09 17:04:14 2014 +0100 (2014-11-09) changeset 58957 c9e744ea8a38 parent 58889 5b7a9633cfa8 child 58973 2a683fb686fd permissions -rw-r--r--
proper context for match_tac etc.;
1 (*  Title:      LCF/LCF.thy
2     Author:     Tobias Nipkow
3     Copyright   1992  University of Cambridge
4 *)
6 section {* LCF on top of First-Order Logic *}
8 theory LCF
9 imports "~~/src/FOL/FOL"
10 begin
12 text {* This theory is based on Lawrence Paulson's book Logic and Computation. *}
14 subsection {* Natural Deduction Rules for LCF *}
16 class cpo = "term"
17 default_sort cpo
19 typedecl tr
20 typedecl void
21 typedecl ('a,'b) prod  (infixl "*" 6)
22 typedecl ('a,'b) sum  (infixl "+" 5)
24 instance "fun" :: (cpo, cpo) cpo ..
25 instance prod :: (cpo, cpo) cpo ..
26 instance sum :: (cpo, cpo) cpo ..
27 instance tr :: cpo ..
28 instance void :: cpo ..
30 consts
31  UU     :: "'a"
32  TT     :: "tr"
33  FF     :: "tr"
34  FIX    :: "('a => 'a) => 'a"
35  FST    :: "'a*'b => 'a"
36  SND    :: "'a*'b => 'b"
37  INL    :: "'a => 'a+'b"
38  INR    :: "'b => 'a+'b"
39  WHEN   :: "['a=>'c, 'b=>'c, 'a+'b] => 'c"
40  adm    :: "('a => o) => o"
41  VOID   :: "void"               ("'(')")
42  PAIR   :: "['a,'b] => 'a*'b"   ("(1<_,/_>)" [0,0] 100)
43  COND   :: "[tr,'a,'a] => 'a"   ("(_ =>/ (_ |/ _))" [60,60,60] 60)
44  less   :: "['a,'a] => o"       (infixl "<<" 50)
46 axiomatization where
47   (** DOMAIN THEORY **)
49   eq_def:        "x=y == x << y & y << x" and
51   less_trans:    "[| x << y; y << z |] ==> x << z" and
53   less_ext:      "(ALL x. f(x) << g(x)) ==> f << g" and
55   mono:          "[| f << g; x << y |] ==> f(x) << g(y)" and
57   minimal:       "UU << x" and
59   FIX_eq:        "\<And>f. f(FIX(f)) = FIX(f)"
61 axiomatization where
62   (** TR **)
64   tr_cases:      "p=UU | p=TT | p=FF" and
66   not_TT_less_FF: "~ TT << FF" and
67   not_FF_less_TT: "~ FF << TT" and
68   not_TT_less_UU: "~ TT << UU" and
69   not_FF_less_UU: "~ FF << UU" and
71   COND_UU:       "UU => x | y  =  UU" and
72   COND_TT:       "TT => x | y  =  x" and
73   COND_FF:       "FF => x | y  =  y"
75 axiomatization where
76   (** PAIRS **)
78   surj_pairing:  "<FST(z),SND(z)> = z" and
80   FST:   "FST(<x,y>) = x" and
81   SND:   "SND(<x,y>) = y"
83 axiomatization where
84   (*** STRICT SUM ***)
86   INL_DEF: "~x=UU ==> ~INL(x)=UU" and
87   INR_DEF: "~x=UU ==> ~INR(x)=UU" and
89   INL_STRICT: "INL(UU) = UU" and
90   INR_STRICT: "INR(UU) = UU" and
92   WHEN_UU:  "WHEN(f,g,UU) = UU" and
93   WHEN_INL: "~x=UU ==> WHEN(f,g,INL(x)) = f(x)" and
94   WHEN_INR: "~x=UU ==> WHEN(f,g,INR(x)) = g(x)" and
96   SUM_EXHAUSTION:
97     "z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"
99 axiomatization where
100   (** VOID **)
102   void_cases:    "(x::void) = UU"
104   (** INDUCTION **)
106 axiomatization where
107   induct:        "[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"
109 axiomatization where
110   (** Admissibility / Chain Completeness **)
111   (* All rules can be found on pages 199--200 of Larry's LCF book.
112      Note that "easiness" of types is not taken into account
113      because it cannot be expressed schematically; flatness could be. *)
115   adm_less:      "\<And>t u. adm(%x. t(x) << u(x))" and
116   adm_not_less:  "\<And>t u. adm(%x.~ t(x) << u)" and
122   adm_all:       "\<And>P. (!!y. adm(P(y))) ==> adm(%x. ALL y. P(y,x))"
125 lemma eq_imp_less1: "x = y ==> x << y"
126   by (simp add: eq_def)
128 lemma eq_imp_less2: "x = y ==> y << x"
129   by (simp add: eq_def)
131 lemma less_refl [simp]: "x << x"
132   apply (rule eq_imp_less1)
133   apply (rule refl)
134   done
136 lemma less_anti_sym: "[| x << y; y << x |] ==> x=y"
137   by (simp add: eq_def)
139 lemma ext: "(!!x::'a::cpo. f(x)=(g(x)::'b::cpo)) ==> (%x. f(x))=(%x. g(x))"
140   apply (rule less_anti_sym)
141   apply (rule less_ext)
142   apply simp
143   apply simp
144   done
146 lemma cong: "[| f=g; x=y |] ==> f(x)=g(y)"
147   by simp
149 lemma less_ap_term: "x << y ==> f(x) << f(y)"
150   by (rule less_refl [THEN mono])
152 lemma less_ap_thm: "f << g ==> f(x) << g(x)"
153   by (rule less_refl [THEN  mono])
155 lemma ap_term: "(x::'a::cpo) = y ==> (f(x)::'b::cpo) = f(y)"
156   apply (rule cong [OF refl])
157   apply simp
158   done
160 lemma ap_thm: "f = g ==> f(x) = g(x)"
161   apply (erule cong)
162   apply (rule refl)
163   done
166 lemma UU_abs: "(%x::'a::cpo. UU) = UU"
167   apply (rule less_anti_sym)
168   prefer 2
169   apply (rule minimal)
170   apply (rule less_ext)
171   apply (rule allI)
172   apply (rule minimal)
173   done
175 lemma UU_app: "UU(x) = UU"
176   by (rule UU_abs [symmetric, THEN ap_thm])
178 lemma less_UU: "x << UU ==> x=UU"
179   apply (rule less_anti_sym)
180   apply assumption
181   apply (rule minimal)
182   done
184 lemma tr_induct: "[| P(UU); P(TT); P(FF) |] ==> ALL b. P(b)"
185   apply (rule allI)
186   apply (rule mp)
187   apply (rule_tac  p = b in tr_cases)
188   apply blast
189   done
191 lemma Contrapos: "~ B ==> (A ==> B) ==> ~A"
192   by blast
194 lemma not_less_imp_not_eq1: "~ x << y \<Longrightarrow> x \<noteq> y"
195   apply (erule Contrapos)
196   apply simp
197   done
199 lemma not_less_imp_not_eq2: "~ y << x \<Longrightarrow> x \<noteq> y"
200   apply (erule Contrapos)
201   apply simp
202   done
204 lemma not_UU_eq_TT: "UU \<noteq> TT"
205   by (rule not_less_imp_not_eq2) (rule not_TT_less_UU)
206 lemma not_UU_eq_FF: "UU \<noteq> FF"
207   by (rule not_less_imp_not_eq2) (rule not_FF_less_UU)
208 lemma not_TT_eq_UU: "TT \<noteq> UU"
209   by (rule not_less_imp_not_eq1) (rule not_TT_less_UU)
210 lemma not_TT_eq_FF: "TT \<noteq> FF"
211   by (rule not_less_imp_not_eq1) (rule not_TT_less_FF)
212 lemma not_FF_eq_UU: "FF \<noteq> UU"
213   by (rule not_less_imp_not_eq1) (rule not_FF_less_UU)
214 lemma not_FF_eq_TT: "FF \<noteq> TT"
215   by (rule not_less_imp_not_eq1) (rule not_FF_less_TT)
218 lemma COND_cases_iff [rule_format]:
219     "ALL b. P(b=>x|y) <-> (b=UU-->P(UU)) & (b=TT-->P(x)) & (b=FF-->P(y))"
220   apply (insert not_UU_eq_TT not_UU_eq_FF not_TT_eq_UU
221     not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT)
222   apply (rule tr_induct)
223   apply (simplesubst COND_UU)
224   apply blast
225   apply (simplesubst COND_TT)
226   apply blast
227   apply (simplesubst COND_FF)
228   apply blast
229   done
231 lemma COND_cases:
232   "[| x = UU --> P(UU); x = TT --> P(xa); x = FF --> P(y) |] ==> P(x => xa | y)"
233   apply (rule COND_cases_iff [THEN iffD2])
234   apply blast
235   done
237 lemmas [simp] =
238   minimal
239   UU_app
240   UU_app [THEN ap_thm]
241   UU_app [THEN ap_thm, THEN ap_thm]
242   not_TT_less_FF not_FF_less_TT not_TT_less_UU not_FF_less_UU not_UU_eq_TT
243   not_UU_eq_FF not_TT_eq_UU not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT
244   COND_UU COND_TT COND_FF
245   surj_pairing FST SND
248 subsection {* Ordered pairs and products *}
250 lemma expand_all_PROD: "(ALL p. P(p)) <-> (ALL x y. P(<x,y>))"
251   apply (rule iffI)
252   apply blast
253   apply (rule allI)
254   apply (rule surj_pairing [THEN subst])
255   apply blast
256   done
258 lemma PROD_less: "(p::'a*'b) << q <-> FST(p) << FST(q) & SND(p) << SND(q)"
259   apply (rule iffI)
260   apply (rule conjI)
261   apply (erule less_ap_term)
262   apply (erule less_ap_term)
263   apply (erule conjE)
264   apply (rule surj_pairing [of p, THEN subst])
265   apply (rule surj_pairing [of q, THEN subst])
266   apply (rule mono, erule less_ap_term, assumption)
267   done
269 lemma PROD_eq: "p=q <-> FST(p)=FST(q) & SND(p)=SND(q)"
270   apply (rule iffI)
271   apply simp
272   apply (unfold eq_def)
273   apply (simp add: PROD_less)
274   done
276 lemma PAIR_less [simp]: "<a,b> << <c,d> <-> a<<c & b<<d"
277   by (simp add: PROD_less)
279 lemma PAIR_eq [simp]: "<a,b> = <c,d> <-> a=c & b=d"
280   by (simp add: PROD_eq)
282 lemma UU_is_UU_UU [simp]: "<UU,UU> = UU"
283   by (rule less_UU) (simp add: PROD_less)
285 lemma FST_STRICT [simp]: "FST(UU) = UU"
286   apply (rule subst [OF UU_is_UU_UU])
287   apply (simp del: UU_is_UU_UU)
288   done
290 lemma SND_STRICT [simp]: "SND(UU) = UU"
291   apply (rule subst [OF UU_is_UU_UU])
292   apply (simp del: UU_is_UU_UU)
293   done
296 subsection {* Fixedpoint theory *}
299   apply (unfold eq_def)
301   done
304   by simp
306 lemma not_eq_TT: "ALL p. ~p=TT <-> (p=FF | p=UU)"
307   and not_eq_FF: "ALL p. ~p=FF <-> (p=TT | p=UU)"
308   and not_eq_UU: "ALL p. ~p=UU <-> (p=TT | p=FF)"
309   by (rule tr_induct, simp_all)+
311 lemma adm_not_eq_tr: "ALL p::tr. adm(%x. ~t(x)=p)"
312   apply (rule tr_induct)
313   apply (simp_all add: not_eq_TT not_eq_FF not_eq_UU)
315   done
317 lemmas adm_lemmas =
322 ML {*
323   fun induct_tac ctxt v i =
324     res_inst_tac ctxt [(("f", 0), v)] @{thm induct} i THEN
325     REPEAT (resolve_tac @{thms adm_lemmas} i)
326 *}
328 lemma least_FIX: "f(p) = p ==> FIX(f) << p"
329   apply (tactic {* induct_tac @{context} "f" 1 *})
330   apply (rule minimal)
331   apply (intro strip)
332   apply (erule subst)
333   apply (erule less_ap_term)
334   done
336 lemma lfp_is_FIX:
337   assumes 1: "f(p) = p"
338     and 2: "ALL q. f(q)=q --> p << q"
339   shows "p = FIX(f)"
340   apply (rule less_anti_sym)
341   apply (rule 2 [THEN spec, THEN mp])
342   apply (rule FIX_eq)
343   apply (rule least_FIX)
344   apply (rule 1)
345   done
348 lemma FIX_pair: "<FIX(f),FIX(g)> = FIX(%p.<f(FST(p)),g(SND(p))>)"
349   apply (rule lfp_is_FIX)
350   apply (simp add: FIX_eq [of f] FIX_eq [of g])
351   apply (intro strip)
352   apply (simp add: PROD_less)
353   apply (rule conjI)
354   apply (rule least_FIX)
355   apply (erule subst, rule FST [symmetric])
356   apply (rule least_FIX)
357   apply (erule subst, rule SND [symmetric])
358   done
360 lemma FIX1: "FIX(f) = FST(FIX(%p. <f(FST(p)),g(SND(p))>))"
361   by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct1])
363 lemma FIX2: "FIX(g) = SND(FIX(%p. <f(FST(p)),g(SND(p))>))"
364   by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct2])
366 lemma induct2:
367   assumes 1: "adm(%p. P(FST(p),SND(p)))"
368     and 2: "P(UU::'a,UU::'b)"
369     and 3: "ALL x y. P(x,y) --> P(f(x),g(y))"
370   shows "P(FIX(f),FIX(g))"
371   apply (rule FIX1 [THEN ssubst, of _ f g])
372   apply (rule FIX2 [THEN ssubst, of _ f g])
373   apply (rule induct [where ?f = "%x. <f(FST(x)),g(SND(x))>"])
374   apply (rule 1)
375   apply simp
376   apply (rule 2)
377   apply (simp add: expand_all_PROD)
378   apply (rule 3)
379   done
381 ML {*
382 fun induct2_tac ctxt (f, g) i =
383   res_inst_tac ctxt [(("f", 0), f), (("g", 0), g)] @{thm induct2} i THEN
384   REPEAT(resolve_tac @{thms adm_lemmas} i)
385 *}
387 end