Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
authorkrauss
Sat Oct 23 23:39:37 2010 +0200 (2010-10-23)
changeset 40106c58951943cba
parent 40105 0d579da1902a
child 40107 374f3ef9f940
Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
src/HOL/Complete_Partial_Order.thy
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Complete_Partial_Order.thy	Sat Oct 23 23:39:37 2010 +0200
     1.3 @@ -0,0 +1,263 @@
     1.4 +(* Title:    HOL/Complete_Partial_Order.thy
     1.5 +   Author:   Brian Huffman, Portland State University
     1.6 +   Author:   Alexander Krauss, TU Muenchen
     1.7 +*)
     1.8 +
     1.9 +header {* Chain-complete partial orders and their fixpoints *}
    1.10 +
    1.11 +theory Complete_Partial_Order
    1.12 +imports Product_Type
    1.13 +begin
    1.14 +
    1.15 +subsection {* Monotone functions *}
    1.16 +
    1.17 +text {* Dictionary-passing version of @{const Orderings.mono}. *}
    1.18 +
    1.19 +definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
    1.20 +where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
    1.21 +
    1.22 +lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y))
    1.23 + \<Longrightarrow> monotone orda ordb f"
    1.24 +unfolding monotone_def by iprover
    1.25 +
    1.26 +lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
    1.27 +unfolding monotone_def by iprover
    1.28 +
    1.29 +
    1.30 +subsection {* Chains *}
    1.31 +
    1.32 +text {* A chain is a totally-ordered set. Chains are parameterized over
    1.33 +  the order for maximal flexibility, since type classes are not enough.
    1.34 +*}
    1.35 +
    1.36 +definition
    1.37 +  chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
    1.38 +where
    1.39 +  "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
    1.40 +
    1.41 +lemma chainI:
    1.42 +  assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x"
    1.43 +  shows "chain ord S"
    1.44 +using assms unfolding chain_def by fast
    1.45 +
    1.46 +lemma chainD:
    1.47 +  assumes "chain ord S" and "x \<in> S" and "y \<in> S"
    1.48 +  shows "ord x y \<or> ord y x"
    1.49 +using assms unfolding chain_def by fast
    1.50 +
    1.51 +lemma chainE:
    1.52 +  assumes "chain ord S" and "x \<in> S" and "y \<in> S"
    1.53 +  obtains "ord x y" | "ord y x"
    1.54 +using assms unfolding chain_def by fast
    1.55 +
    1.56 +subsection {* Chain-complete partial orders *}
    1.57 +
    1.58 +text {*
    1.59 +  A ccpo has a least upper bound for any chain.  In particular, the
    1.60 +  empty set is a chain, so every ccpo must have a bottom element.
    1.61 +*}
    1.62 +
    1.63 +class ccpo = order +
    1.64 +  fixes lub :: "'a set \<Rightarrow> 'a"
    1.65 +  assumes lub_upper: "chain (op \<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> lub A"
    1.66 +  assumes lub_least: "chain (op \<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> lub A \<le> z"
    1.67 +begin
    1.68 +
    1.69 +subsection {* Transfinite iteration of a function *}
    1.70 +
    1.71 +inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"
    1.72 +for f :: "'a \<Rightarrow> 'a"
    1.73 +where
    1.74 +  step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
    1.75 +| lub: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> lub M \<in> iterates f"
    1.76 +
    1.77 +lemma iterates_le_f:
    1.78 +  "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
    1.79 +by (induct x rule: iterates.induct)
    1.80 +  (force dest: monotoneD intro!: lub_upper lub_least)+
    1.81 +
    1.82 +lemma chain_iterates:
    1.83 +  assumes f: "monotone (op \<le>) (op \<le>) f"
    1.84 +  shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")
    1.85 +proof (rule chainI)
    1.86 +  fix x y assume "x \<in> ?C" "y \<in> ?C"
    1.87 +  then show "x \<le> y \<or> y \<le> x"
    1.88 +  proof (induct x arbitrary: y rule: iterates.induct)
    1.89 +    fix x y assume y: "y \<in> ?C"
    1.90 +    and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
    1.91 +    from y show "f x \<le> y \<or> y \<le> f x"
    1.92 +    proof (induct y rule: iterates.induct)
    1.93 +      case (step y) with IH f show ?case by (auto dest: monotoneD)
    1.94 +    next
    1.95 +      case (lub M)
    1.96 +      then have chM: "chain (op \<le>) M"
    1.97 +        and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
    1.98 +      show "f x \<le> lub M \<or> lub M \<le> f x"
    1.99 +      proof (cases "\<exists>z\<in>M. f x \<le> z")
   1.100 +        case True then have "f x \<le> lub M"
   1.101 +          apply rule
   1.102 +          apply (erule order_trans)
   1.103 +          by (rule lub_upper[OF chM])
   1.104 +        thus ?thesis ..
   1.105 +      next
   1.106 +        case False with IH'
   1.107 +        show ?thesis by (auto intro: lub_least[OF chM])
   1.108 +      qed
   1.109 +    qed
   1.110 +  next
   1.111 +    case (lub M y)
   1.112 +    show ?case
   1.113 +    proof (cases "\<exists>x\<in>M. y \<le> x")
   1.114 +      case True then have "y \<le> lub M"
   1.115 +        apply rule
   1.116 +        apply (erule order_trans)
   1.117 +        by (rule lub_upper[OF lub(1)])
   1.118 +      thus ?thesis ..
   1.119 +    next
   1.120 +      case False with lub
   1.121 +      show ?thesis by (auto intro: lub_least)
   1.122 +    qed
   1.123 +  qed
   1.124 +qed
   1.125 +
   1.126 +subsection {* Fixpoint combinator *}
   1.127 +
   1.128 +definition
   1.129 +  fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
   1.130 +where
   1.131 +  "fixp f = lub (iterates f)"
   1.132 +
   1.133 +lemma iterates_fixp:
   1.134 +  assumes f: "monotone (op \<le>) (op \<le>) f" shows "fixp f \<in> iterates f"
   1.135 +unfolding fixp_def
   1.136 +by (simp add: iterates.lub chain_iterates f)
   1.137 +
   1.138 +lemma fixp_unfold:
   1.139 +  assumes f: "monotone (op \<le>) (op \<le>) f"
   1.140 +  shows "fixp f = f (fixp f)"
   1.141 +proof (rule antisym)
   1.142 +  show "fixp f \<le> f (fixp f)"
   1.143 +    by (intro iterates_le_f iterates_fixp f)
   1.144 +  have "f (fixp f) \<le> lub (iterates f)"
   1.145 +    by (intro lub_upper chain_iterates f iterates.step iterates_fixp)
   1.146 +  thus "f (fixp f) \<le> fixp f"
   1.147 +    unfolding fixp_def .
   1.148 +qed
   1.149 +
   1.150 +lemma fixp_lowerbound:
   1.151 +  assumes f: "monotone (op \<le>) (op \<le>) f" and z: "f z \<le> z" shows "fixp f \<le> z"
   1.152 +unfolding fixp_def
   1.153 +proof (rule lub_least[OF chain_iterates[OF f]])
   1.154 +  fix x assume "x \<in> iterates f"
   1.155 +  thus "x \<le> z"
   1.156 +  proof (induct x rule: iterates.induct)
   1.157 +    fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD)
   1.158 +    also note z finally show "f x \<le> z" .
   1.159 +  qed (auto intro: lub_least)
   1.160 +qed
   1.161 +
   1.162 +
   1.163 +subsection {* Fixpoint induction *}
   1.164 +
   1.165 +definition
   1.166 +  admissible :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
   1.167 +where
   1.168 +  "admissible P = (\<forall>A. chain (op \<le>) A \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
   1.169 +
   1.170 +lemma admissibleI:
   1.171 +  assumes "\<And>A. chain (op \<le>) A \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)"
   1.172 +  shows "admissible P"
   1.173 +using assms unfolding admissible_def by fast
   1.174 +
   1.175 +lemma admissibleD:
   1.176 +  assumes "admissible P"
   1.177 +  assumes "chain (op \<le>) A"
   1.178 +  assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
   1.179 +  shows "P (lub A)"
   1.180 +using assms by (auto simp: admissible_def)
   1.181 +
   1.182 +lemma fixp_induct:
   1.183 +  assumes adm: "admissible P"
   1.184 +  assumes mono: "monotone (op \<le>) (op \<le>) f"
   1.185 +  assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
   1.186 +  shows "P (fixp f)"
   1.187 +unfolding fixp_def using adm chain_iterates[OF mono]
   1.188 +proof (rule admissibleD)
   1.189 +  fix x assume "x \<in> iterates f"
   1.190 +  thus "P x"
   1.191 +    by (induct rule: iterates.induct)
   1.192 +      (auto intro: step admissibleD adm)
   1.193 +qed
   1.194 +
   1.195 +lemma admissible_True: "admissible (\<lambda>x. True)"
   1.196 +unfolding admissible_def by simp
   1.197 +
   1.198 +lemma admissible_False: "\<not> admissible (\<lambda>x. False)"
   1.199 +unfolding admissible_def chain_def by simp
   1.200 +
   1.201 +lemma admissible_const: "admissible (\<lambda>x. t) = t"
   1.202 +by (cases t, simp_all add: admissible_True admissible_False)
   1.203 +
   1.204 +lemma admissible_conj:
   1.205 +  assumes "admissible (\<lambda>x. P x)"
   1.206 +  assumes "admissible (\<lambda>x. Q x)"
   1.207 +  shows "admissible (\<lambda>x. P x \<and> Q x)"
   1.208 +using assms unfolding admissible_def by simp
   1.209 +
   1.210 +lemma admissible_all:
   1.211 +  assumes "\<And>y. admissible (\<lambda>x. P x y)"
   1.212 +  shows "admissible (\<lambda>x. \<forall>y. P x y)"
   1.213 +using assms unfolding admissible_def by fast
   1.214 +
   1.215 +lemma admissible_ball:
   1.216 +  assumes "\<And>y. y \<in> A \<Longrightarrow> admissible (\<lambda>x. P x y)"
   1.217 +  shows "admissible (\<lambda>x. \<forall>y\<in>A. P x y)"
   1.218 +using assms unfolding admissible_def by fast
   1.219 +
   1.220 +lemma chain_compr: "chain (op \<le>) A \<Longrightarrow> chain (op \<le>) {x \<in> A. P x}"
   1.221 +unfolding chain_def by fast
   1.222 +
   1.223 +lemma admissible_disj_lemma:
   1.224 +  assumes A: "chain (op \<le>)A"
   1.225 +  assumes P: "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y"
   1.226 +  shows "lub A = lub {x \<in> A. P x}"
   1.227 +proof (rule antisym)
   1.228 +  have *: "chain (op \<le>) {x \<in> A. P x}"
   1.229 +    by (rule chain_compr [OF A])
   1.230 +  show "lub A \<le> lub {x \<in> A. P x}"
   1.231 +    apply (rule lub_least [OF A])
   1.232 +    apply (drule P [rule_format], clarify)
   1.233 +    apply (erule order_trans)
   1.234 +    apply (simp add: lub_upper [OF *])
   1.235 +    done
   1.236 +  show "lub {x \<in> A. P x} \<le> lub A"
   1.237 +    apply (rule lub_least [OF *])
   1.238 +    apply clarify
   1.239 +    apply (simp add: lub_upper [OF A])
   1.240 +    done
   1.241 +qed
   1.242 +
   1.243 +lemma admissible_disj:
   1.244 +  fixes P Q :: "'a \<Rightarrow> bool"
   1.245 +  assumes P: "admissible (\<lambda>x. P x)"
   1.246 +  assumes Q: "admissible (\<lambda>x. Q x)"
   1.247 +  shows "admissible (\<lambda>x. P x \<or> Q x)"
   1.248 +proof (rule admissibleI)
   1.249 +  fix A :: "'a set" assume A: "chain (op \<le>) A"
   1.250 +  assume "\<forall>x\<in>A. P x \<or> Q x"
   1.251 +  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)"
   1.252 +    using chainD[OF A] by blast
   1.253 +  hence "lub A = lub {x \<in> A. P x} \<or> lub A = lub {x \<in> A. Q x}"
   1.254 +    using admissible_disj_lemma [OF A] by fast
   1.255 +  thus "P (lub A) \<or> Q (lub A)"
   1.256 +    apply (rule disjE, simp_all)
   1.257 +    apply (rule disjI1, rule admissibleD [OF P chain_compr [OF A]], simp)
   1.258 +    apply (rule disjI2, rule admissibleD [OF Q chain_compr [OF A]], simp)
   1.259 +    done
   1.260 +qed
   1.261 +
   1.262 +end
   1.263 +
   1.264 +
   1.265 +
   1.266 +end