src/HOL/Partial_Function.thy
 author wenzelm Thu Mar 15 22:08:53 2012 +0100 (2012-03-15) changeset 46950 d0181abdbdac parent 46041 1e3ff542e83e child 48891 c0eafbd55de3 permissions -rw-r--r--
declare command keywords via theory header, including strict checking outside Pure;
1 (* Title:    HOL/Partial_Function.thy
2    Author:   Alexander Krauss, TU Muenchen
3 *)
5 header {* Partial Function Definitions *}
7 theory Partial_Function
8 imports Complete_Partial_Order Option
9 keywords "partial_function" :: thy_decl
10 uses
11   "Tools/Function/function_lib.ML"
12   "Tools/Function/partial_function.ML"
13 begin
15 setup Partial_Function.setup
17 subsection {* Axiomatic setup *}
19 text {* This techical locale constains the requirements for function
20   definitions with ccpo fixed points. *}
22 definition "fun_ord ord f g \<longleftrightarrow> (\<forall>x. ord (f x) (g x))"
23 definition "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
24 definition "img_ord f ord = (\<lambda>x y. ord (f x) (f y))"
25 definition "img_lub f g Lub = (\<lambda>A. g (Lub (f ` A)))"
27 lemma chain_fun:
28   assumes A: "chain (fun_ord ord) A"
29   shows "chain ord {y. \<exists>f\<in>A. y = f a}" (is "chain ord ?C")
30 proof (rule chainI)
31   fix x y assume "x \<in> ?C" "y \<in> ?C"
32   then obtain f g where fg: "f \<in> A" "g \<in> A"
33     and [simp]: "x = f a" "y = g a" by blast
34   from chainD[OF A fg]
35   show "ord x y \<or> ord y x" unfolding fun_ord_def by auto
36 qed
38 lemma call_mono[partial_function_mono]: "monotone (fun_ord ord) ord (\<lambda>f. f t)"
39 by (rule monotoneI) (auto simp: fun_ord_def)
41 lemma let_mono[partial_function_mono]:
42   "(\<And>x. monotone orda ordb (\<lambda>f. b f x))
43   \<Longrightarrow> monotone orda ordb (\<lambda>f. Let t (b f))"
46 lemma if_mono[partial_function_mono]: "monotone orda ordb F
47 \<Longrightarrow> monotone orda ordb G
48 \<Longrightarrow> monotone orda ordb (\<lambda>f. if c then F f else G f)"
49 unfolding monotone_def by simp
51 definition "mk_less R = (\<lambda>x y. R x y \<and> \<not> R y x)"
53 locale partial_function_definitions =
54   fixes leq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
55   fixes lub :: "'a set \<Rightarrow> 'a"
56   assumes leq_refl: "leq x x"
57   assumes leq_trans: "leq x y \<Longrightarrow> leq y z \<Longrightarrow> leq x z"
58   assumes leq_antisym: "leq x y \<Longrightarrow> leq y x \<Longrightarrow> x = y"
59   assumes lub_upper: "chain leq A \<Longrightarrow> x \<in> A \<Longrightarrow> leq x (lub A)"
60   assumes lub_least: "chain leq A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> leq x z) \<Longrightarrow> leq (lub A) z"
62 lemma partial_function_lift:
63   assumes "partial_function_definitions ord lb"
64   shows "partial_function_definitions (fun_ord ord) (fun_lub lb)" (is "partial_function_definitions ?ordf ?lubf")
65 proof -
66   interpret partial_function_definitions ord lb by fact
68   show ?thesis
69   proof
70     fix x show "?ordf x x"
71       unfolding fun_ord_def by (auto simp: leq_refl)
72   next
73     fix x y z assume "?ordf x y" "?ordf y z"
74     thus "?ordf x z" unfolding fun_ord_def
75       by (force dest: leq_trans)
76   next
77     fix x y assume "?ordf x y" "?ordf y x"
78     thus "x = y" unfolding fun_ord_def
79       by (force intro!: dest: leq_antisym)
80   next
81     fix A f assume f: "f \<in> A" and A: "chain ?ordf A"
82     thus "?ordf f (?lubf A)"
83       unfolding fun_lub_def fun_ord_def
84       by (blast intro: lub_upper chain_fun[OF A] f)
85   next
86     fix A :: "('b \<Rightarrow> 'a) set" and g :: "'b \<Rightarrow> 'a"
87     assume A: "chain ?ordf A" and g: "\<And>f. f \<in> A \<Longrightarrow> ?ordf f g"
88     show "?ordf (?lubf A) g" unfolding fun_lub_def fun_ord_def
89       by (blast intro: lub_least chain_fun[OF A] dest: g[unfolded fun_ord_def])
90    qed
91 qed
93 lemma ccpo: assumes "partial_function_definitions ord lb"
94   shows "class.ccpo lb ord (mk_less ord)"
95 using assms unfolding partial_function_definitions_def mk_less_def
96 by unfold_locales blast+
98 lemma partial_function_image:
99   assumes "partial_function_definitions ord Lub"
100   assumes inj: "\<And>x y. f x = f y \<Longrightarrow> x = y"
101   assumes inv: "\<And>x. f (g x) = x"
102   shows "partial_function_definitions (img_ord f ord) (img_lub f g Lub)"
103 proof -
104   let ?iord = "img_ord f ord"
105   let ?ilub = "img_lub f g Lub"
107   interpret partial_function_definitions ord Lub by fact
108   show ?thesis
109   proof
110     fix A x assume "chain ?iord A" "x \<in> A"
111     then have "chain ord (f ` A)" "f x \<in> f ` A"
112       by (auto simp: img_ord_def intro: chainI dest: chainD)
113     thus "?iord x (?ilub A)"
114       unfolding inv img_lub_def img_ord_def by (rule lub_upper)
115   next
116     fix A x assume "chain ?iord A"
117       and 1: "\<And>z. z \<in> A \<Longrightarrow> ?iord z x"
118     then have "chain ord (f ` A)"
119       by (auto simp: img_ord_def intro: chainI dest: chainD)
120     thus "?iord (?ilub A) x"
121       unfolding inv img_lub_def img_ord_def
122       by (rule lub_least) (auto dest: 1[unfolded img_ord_def])
123   qed (auto simp: img_ord_def intro: leq_refl dest: leq_trans leq_antisym inj)
124 qed
126 context partial_function_definitions
127 begin
129 abbreviation "le_fun \<equiv> fun_ord leq"
130 abbreviation "lub_fun \<equiv> fun_lub lub"
131 abbreviation "fixp_fun \<equiv> ccpo.fixp lub_fun le_fun"
132 abbreviation "mono_body \<equiv> monotone le_fun leq"
135 text {* Interpret manually, to avoid flooding everything with facts about
136   orders *}
138 lemma ccpo: "class.ccpo lub_fun le_fun (mk_less le_fun)"
139 apply (rule ccpo)
140 apply (rule partial_function_lift)
141 apply (rule partial_function_definitions_axioms)
142 done
144 text {* The crucial fixed-point theorem *}
146 lemma mono_body_fixp:
147   "(\<And>x. mono_body (\<lambda>f. F f x)) \<Longrightarrow> fixp_fun F = F (fixp_fun F)"
148 by (rule ccpo.fixp_unfold[OF ccpo]) (auto simp: monotone_def fun_ord_def)
150 text {* Version with curry/uncurry combinators, to be used by package *}
152 lemma fixp_rule_uc:
153   fixes F :: "'c \<Rightarrow> 'c" and
154     U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" and
155     C :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c"
156   assumes mono: "\<And>x. mono_body (\<lambda>f. U (F (C f)) x)"
157   assumes eq: "f \<equiv> C (fixp_fun (\<lambda>f. U (F (C f))))"
158   assumes inverse: "\<And>f. C (U f) = f"
159   shows "f = F f"
160 proof -
161   have "f = C (fixp_fun (\<lambda>f. U (F (C f))))" by (simp add: eq)
162   also have "... = C (U (F (C (fixp_fun (\<lambda>f. U (F (C f)))))))"
163     by (subst mono_body_fixp[of "%f. U (F (C f))", OF mono]) (rule refl)
164   also have "... = F (C (fixp_fun (\<lambda>f. U (F (C f)))))" by (rule inverse)
165   also have "... = F f" by (simp add: eq)
166   finally show "f = F f" .
167 qed
169 text {* Fixpoint induction rule *}
171 lemma fixp_induct_uc:
172   fixes F :: "'c \<Rightarrow> 'c" and
173     U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" and
174     C :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c" and
175     P :: "('b \<Rightarrow> 'a) \<Rightarrow> bool"
176   assumes mono: "\<And>x. mono_body (\<lambda>f. U (F (C f)) x)"
177   assumes eq: "f \<equiv> C (fixp_fun (\<lambda>f. U (F (C f))))"
178   assumes inverse: "\<And>f. U (C f) = f"
180   assumes step: "\<And>f. P (U f) \<Longrightarrow> P (U (F f))"
181   shows "P (U f)"
182 unfolding eq inverse
183 apply (rule ccpo.fixp_induct[OF ccpo adm])
184 apply (insert mono, auto simp: monotone_def fun_ord_def)[1]
185 by (rule_tac f="C x" in step, simp add: inverse)
188 text {* Rules for @{term mono_body}: *}
190 lemma const_mono[partial_function_mono]: "monotone ord leq (\<lambda>f. c)"
191 by (rule monotoneI) (rule leq_refl)
193 end
196 subsection {* Flat interpretation: tailrec and option *}
198 definition
199   "flat_ord b x y \<longleftrightarrow> x = b \<or> x = y"
201 definition
202   "flat_lub b A = (if A \<subseteq> {b} then b else (THE x. x \<in> A - {b}))"
204 lemma flat_interpretation:
205   "partial_function_definitions (flat_ord b) (flat_lub b)"
206 proof
207   fix A x assume 1: "chain (flat_ord b) A" "x \<in> A"
208   show "flat_ord b x (flat_lub b A)"
209   proof cases
210     assume "x = b"
211     thus ?thesis by (simp add: flat_ord_def)
212   next
213     assume "x \<noteq> b"
214     with 1 have "A - {b} = {x}"
215       by (auto elim: chainE simp: flat_ord_def)
216     then have "flat_lub b A = x"
217       by (auto simp: flat_lub_def)
218     thus ?thesis by (auto simp: flat_ord_def)
219   qed
220 next
221   fix A z assume A: "chain (flat_ord b) A"
222     and z: "\<And>x. x \<in> A \<Longrightarrow> flat_ord b x z"
223   show "flat_ord b (flat_lub b A) z"
224   proof cases
225     assume "A \<subseteq> {b}"
226     thus ?thesis
227       by (auto simp: flat_lub_def flat_ord_def)
228   next
229     assume nb: "\<not> A \<subseteq> {b}"
230     then obtain y where y: "y \<in> A" "y \<noteq> b" by auto
231     with A have "A - {b} = {y}"
232       by (auto elim: chainE simp: flat_ord_def)
233     with nb have "flat_lub b A = y"
234       by (auto simp: flat_lub_def)
235     with z y show ?thesis by auto
236   qed
237 qed (auto simp: flat_ord_def)
239 interpretation tailrec!:
240   partial_function_definitions "flat_ord undefined" "flat_lub undefined"
241 by (rule flat_interpretation)
243 interpretation option!:
244   partial_function_definitions "flat_ord None" "flat_lub None"
245 by (rule flat_interpretation)
248 abbreviation "option_ord \<equiv> flat_ord None"
249 abbreviation "mono_option \<equiv> monotone (fun_ord option_ord) option_ord"
251 lemma bind_mono[partial_function_mono]:
252 assumes mf: "mono_option B" and mg: "\<And>y. mono_option (\<lambda>f. C y f)"
253 shows "mono_option (\<lambda>f. Option.bind (B f) (\<lambda>y. C y f))"
254 proof (rule monotoneI)
255   fix f g :: "'a \<Rightarrow> 'b option" assume fg: "fun_ord option_ord f g"
256   with mf
257   have "option_ord (B f) (B g)" by (rule monotoneD[of _ _ _ f g])
258   then have "option_ord (Option.bind (B f) (\<lambda>y. C y f)) (Option.bind (B g) (\<lambda>y. C y f))"
259     unfolding flat_ord_def by auto
260   also from mg
261   have "\<And>y'. option_ord (C y' f) (C y' g)"
262     by (rule monotoneD) (rule fg)
263   then have "option_ord (Option.bind (B g) (\<lambda>y'. C y' f)) (Option.bind (B g) (\<lambda>y'. C y' g))"
264     unfolding flat_ord_def by (cases "B g") auto
265   finally (option.leq_trans)
266   show "option_ord (Option.bind (B f) (\<lambda>y. C y f)) (Option.bind (B g) (\<lambda>y'. C y' g))" .
267 qed
269 lemma flat_lub_in_chain:
270   assumes ch: "chain (flat_ord b) A "
271   assumes lub: "flat_lub b A = a"
272   shows "a = b \<or> a \<in> A"
273 proof (cases "A \<subseteq> {b}")
274   case True
275   then have "flat_lub b A = b" unfolding flat_lub_def by simp
276   with lub show ?thesis by simp
277 next
278   case False
279   then obtain c where "c \<in> A" and "c \<noteq> b" by auto
280   { fix z assume "z \<in> A"
281     from chainD[OF ch `c \<in> A` this] have "z = c \<or> z = b"
282       unfolding flat_ord_def using `c \<noteq> b` by auto }
283   with False have "A - {b} = {c}" by auto
284   with False have "flat_lub b A = c" by (auto simp: flat_lub_def)
285   with `c \<in> A` lub show ?thesis by simp
286 qed
289   (\<forall>x y. f x = Some y \<longrightarrow> P x y))"
291   fix A :: "('a \<Rightarrow> 'b option) set"
292   assume ch: "chain option.le_fun A"
293     and IH: "\<forall>f\<in>A. \<forall>x y. f x = Some y \<longrightarrow> P x y"
294   from ch have ch': "\<And>x. chain option_ord {y. \<exists>f\<in>A. y = f x}" by (rule chain_fun)
295   show "\<forall>x y. option.lub_fun A x = Some y \<longrightarrow> P x y"
296   proof (intro allI impI)
297     fix x y assume "option.lub_fun A x = Some y"
298     from flat_lub_in_chain[OF ch' this[unfolded fun_lub_def]]
299     have "Some y \<in> {y. \<exists>f\<in>A. y = f x}" by simp
300     then have "\<exists>f\<in>A. f x = Some y" by auto
301     with IH show "P x y" by auto
302   qed
303 qed
305 lemma fixp_induct_option:
306   fixes F :: "'c \<Rightarrow> 'c" and
307     U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a option" and
308     C :: "('b \<Rightarrow> 'a option) \<Rightarrow> 'c" and
309     P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"
310   assumes mono: "\<And>x. mono_option (\<lambda>f. U (F (C f)) x)"
311   assumes eq: "f \<equiv> C (ccpo.fixp (fun_lub (flat_lub None)) (fun_ord option_ord) (\<lambda>f. U (F (C f))))"
312   assumes inverse2: "\<And>f. U (C f) = f"
313   assumes step: "\<And>f x y. (\<And>x y. U f x = Some y \<Longrightarrow> P x y) \<Longrightarrow> U (F f) x = Some y \<Longrightarrow> P x y"
314   assumes defined: "U f x = Some y"
315   shows "P x y"
316   using step defined option.fixp_induct_uc[of U F C, OF mono eq inverse2 option_admissible]
317   by blast
319 declaration {* Partial_Function.init "tailrec" @{term tailrec.fixp_fun}
320   @{term tailrec.mono_body} @{thm tailrec.fixp_rule_uc} NONE *}
322 declaration {* Partial_Function.init "option" @{term option.fixp_fun}
323   @{term option.mono_body} @{thm option.fixp_rule_uc}
324   (SOME @{thm fixp_induct_option}) *}
327 hide_const (open) chain
329 end