src/HOL/Complete_Partial_Order.thy
 author hoelzl Thu Jan 31 11:31:27 2013 +0100 (2013-01-31) changeset 50999 3de230ed0547 parent 46041 1e3ff542e83e child 53361 1cb7d3c0cf31 permissions -rw-r--r--
introduce order topology
1 (* Title:    HOL/Complete_Partial_Order.thy
2    Author:   Brian Huffman, Portland State University
3    Author:   Alexander Krauss, TU Muenchen
4 *)
6 header {* Chain-complete partial orders and their fixpoints *}
8 theory Complete_Partial_Order
9 imports Product_Type
10 begin
12 subsection {* Monotone functions *}
14 text {* Dictionary-passing version of @{const Orderings.mono}. *}
16 definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
17 where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
19 lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y))
20  \<Longrightarrow> monotone orda ordb f"
21 unfolding monotone_def by iprover
23 lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
24 unfolding monotone_def by iprover
27 subsection {* Chains *}
29 text {* A chain is a totally-ordered set. Chains are parameterized over
30   the order for maximal flexibility, since type classes are not enough.
31 *}
33 definition
34   chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
35 where
36   "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
38 lemma chainI:
39   assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x"
40   shows "chain ord S"
41 using assms unfolding chain_def by fast
43 lemma chainD:
44   assumes "chain ord S" and "x \<in> S" and "y \<in> S"
45   shows "ord x y \<or> ord y x"
46 using assms unfolding chain_def by fast
48 lemma chainE:
49   assumes "chain ord S" and "x \<in> S" and "y \<in> S"
50   obtains "ord x y" | "ord y x"
51 using assms unfolding chain_def by fast
53 subsection {* Chain-complete partial orders *}
55 text {*
56   A ccpo has a least upper bound for any chain.  In particular, the
57   empty set is a chain, so every ccpo must have a bottom element.
58 *}
60 class ccpo = order + Sup +
61   assumes ccpo_Sup_upper: "\<lbrakk>chain (op \<le>) A; x \<in> A\<rbrakk> \<Longrightarrow> x \<le> Sup A"
62   assumes ccpo_Sup_least: "\<lbrakk>chain (op \<le>) A; \<And>x. x \<in> A \<Longrightarrow> x \<le> z\<rbrakk> \<Longrightarrow> Sup A \<le> z"
63 begin
65 subsection {* Transfinite iteration of a function *}
67 inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"
68 for f :: "'a \<Rightarrow> 'a"
69 where
70   step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
71 | Sup: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
73 lemma iterates_le_f:
74   "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
75 by (induct x rule: iterates.induct)
76   (force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+
78 lemma chain_iterates:
79   assumes f: "monotone (op \<le>) (op \<le>) f"
80   shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")
81 proof (rule chainI)
82   fix x y assume "x \<in> ?C" "y \<in> ?C"
83   then show "x \<le> y \<or> y \<le> x"
84   proof (induct x arbitrary: y rule: iterates.induct)
85     fix x y assume y: "y \<in> ?C"
86     and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
87     from y show "f x \<le> y \<or> y \<le> f x"
88     proof (induct y rule: iterates.induct)
89       case (step y) with IH f show ?case by (auto dest: monotoneD)
90     next
91       case (Sup M)
92       then have chM: "chain (op \<le>) M"
93         and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
94       show "f x \<le> Sup M \<or> Sup M \<le> f x"
95       proof (cases "\<exists>z\<in>M. f x \<le> z")
96         case True then have "f x \<le> Sup M"
97           apply rule
98           apply (erule order_trans)
99           by (rule ccpo_Sup_upper[OF chM])
100         thus ?thesis ..
101       next
102         case False with IH'
103         show ?thesis by (auto intro: ccpo_Sup_least[OF chM])
104       qed
105     qed
106   next
107     case (Sup M y)
108     show ?case
109     proof (cases "\<exists>x\<in>M. y \<le> x")
110       case True then have "y \<le> Sup M"
111         apply rule
112         apply (erule order_trans)
113         by (rule ccpo_Sup_upper[OF Sup(1)])
114       thus ?thesis ..
115     next
116       case False with Sup
117       show ?thesis by (auto intro: ccpo_Sup_least)
118     qed
119   qed
120 qed
122 subsection {* Fixpoint combinator *}
124 definition
125   fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
126 where
127   "fixp f = Sup (iterates f)"
129 lemma iterates_fixp:
130   assumes f: "monotone (op \<le>) (op \<le>) f" shows "fixp f \<in> iterates f"
131 unfolding fixp_def
132 by (simp add: iterates.Sup chain_iterates f)
134 lemma fixp_unfold:
135   assumes f: "monotone (op \<le>) (op \<le>) f"
136   shows "fixp f = f (fixp f)"
137 proof (rule antisym)
138   show "fixp f \<le> f (fixp f)"
139     by (intro iterates_le_f iterates_fixp f)
140   have "f (fixp f) \<le> Sup (iterates f)"
141     by (intro ccpo_Sup_upper chain_iterates f iterates.step iterates_fixp)
142   thus "f (fixp f) \<le> fixp f"
143     unfolding fixp_def .
144 qed
146 lemma fixp_lowerbound:
147   assumes f: "monotone (op \<le>) (op \<le>) f" and z: "f z \<le> z" shows "fixp f \<le> z"
148 unfolding fixp_def
149 proof (rule ccpo_Sup_least[OF chain_iterates[OF f]])
150   fix x assume "x \<in> iterates f"
151   thus "x \<le> z"
152   proof (induct x rule: iterates.induct)
153     fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD)
154     also note z finally show "f x \<le> z" .
155   qed (auto intro: ccpo_Sup_least)
156 qed
159 subsection {* Fixpoint induction *}
161 definition
162   admissible :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
163 where
164   "admissible P = (\<forall>A. chain (op \<le>) A \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (Sup A))"
167   assumes "\<And>A. chain (op \<le>) A \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (Sup A)"
169 using assms unfolding admissible_def by fast
173   assumes "chain (op \<le>) A"
174   assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
175   shows "P (Sup A)"
176 using assms by (auto simp: admissible_def)
178 lemma fixp_induct:
180   assumes mono: "monotone (op \<le>) (op \<le>) f"
181   assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
182   shows "P (fixp f)"
183 unfolding fixp_def using adm chain_iterates[OF mono]
185   fix x assume "x \<in> iterates f"
186   thus "P x"
187     by (induct rule: iterates.induct)
189 qed
195 unfolding admissible_def chain_def by simp
201   assumes "admissible (\<lambda>x. P x)"
202   assumes "admissible (\<lambda>x. Q x)"
203   shows "admissible (\<lambda>x. P x \<and> Q x)"
204 using assms unfolding admissible_def by simp
207   assumes "\<And>y. admissible (\<lambda>x. P x y)"
208   shows "admissible (\<lambda>x. \<forall>y. P x y)"
209 using assms unfolding admissible_def by fast
212   assumes "\<And>y. y \<in> A \<Longrightarrow> admissible (\<lambda>x. P x y)"
213   shows "admissible (\<lambda>x. \<forall>y\<in>A. P x y)"
214 using assms unfolding admissible_def by fast
216 lemma chain_compr: "chain (op \<le>) A \<Longrightarrow> chain (op \<le>) {x \<in> A. P x}"
217 unfolding chain_def by fast
220   assumes A: "chain (op \<le>)A"
221   assumes P: "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y"
222   shows "Sup A = Sup {x \<in> A. P x}"
223 proof (rule antisym)
224   have *: "chain (op \<le>) {x \<in> A. P x}"
225     by (rule chain_compr [OF A])
226   show "Sup A \<le> Sup {x \<in> A. P x}"
227     apply (rule ccpo_Sup_least [OF A])
228     apply (drule P [rule_format], clarify)
229     apply (erule order_trans)
230     apply (simp add: ccpo_Sup_upper [OF *])
231     done
232   show "Sup {x \<in> A. P x} \<le> Sup A"
233     apply (rule ccpo_Sup_least [OF *])
234     apply clarify
235     apply (simp add: ccpo_Sup_upper [OF A])
236     done
237 qed
240   fixes P Q :: "'a \<Rightarrow> bool"
241   assumes P: "admissible (\<lambda>x. P x)"
242   assumes Q: "admissible (\<lambda>x. Q x)"
243   shows "admissible (\<lambda>x. P x \<or> Q x)"
245   fix A :: "'a set" assume A: "chain (op \<le>) A"
246   assume "\<forall>x\<in>A. P x \<or> Q x"
247   hence "(\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"
248     using chainD[OF A] by blast
249   hence "Sup A = Sup {x \<in> A. P x} \<or> Sup A = Sup {x \<in> A. Q x}"
250     using admissible_disj_lemma [OF A] by fast
251   thus "P (Sup A) \<or> Q (Sup A)"
252     apply (rule disjE, simp_all)
253     apply (rule disjI1, rule admissibleD [OF P chain_compr [OF A]], simp)
254     apply (rule disjI2, rule admissibleD [OF Q chain_compr [OF A]], simp)
255     done
256 qed
258 end
260 instance complete_lattice \<subseteq> ccpo
261   by default (fast intro: Sup_upper Sup_least)+
263 lemma lfp_eq_fixp:
264   assumes f: "mono f" shows "lfp f = fixp f"
265 proof (rule antisym)
266   from f have f': "monotone (op \<le>) (op \<le>) f"
267     unfolding mono_def monotone_def .
268   show "lfp f \<le> fixp f"
269     by (rule lfp_lowerbound, subst fixp_unfold [OF f'], rule order_refl)
270   show "fixp f \<le> lfp f"
271     by (rule fixp_lowerbound [OF f'], subst lfp_unfold [OF f], rule order_refl)
272 qed
274 hide_const (open) iterates fixp admissible
276 end