src/HOL/Complete_Partial_Order.thy
 author wenzelm Fri Dec 17 17:43:54 2010 +0100 (2010-12-17) changeset 41229 d797baa3d57c parent 40252 029400b6c893 child 46041 1e3ff542e83e permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
```     1 (* Title:    HOL/Complete_Partial_Order.thy
```
```     2    Author:   Brian Huffman, Portland State University
```
```     3    Author:   Alexander Krauss, TU Muenchen
```
```     4 *)
```
```     5
```
```     6 header {* Chain-complete partial orders and their fixpoints *}
```
```     7
```
```     8 theory Complete_Partial_Order
```
```     9 imports Product_Type
```
```    10 begin
```
```    11
```
```    12 subsection {* Monotone functions *}
```
```    13
```
```    14 text {* Dictionary-passing version of @{const Orderings.mono}. *}
```
```    15
```
```    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))"
```
```    18
```
```    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
```
```    22
```
```    23 lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
```
```    24 unfolding monotone_def by iprover
```
```    25
```
```    26
```
```    27 subsection {* Chains *}
```
```    28
```
```    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 *}
```
```    32
```
```    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)"
```
```    37
```
```    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
```
```    42
```
```    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
```
```    47
```
```    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
```
```    52
```
```    53 subsection {* Chain-complete partial orders *}
```
```    54
```
```    55 text {*
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```    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 *}
```
```    59
```
```    60 class ccpo = order +
```
```    61   fixes lub :: "'a set \<Rightarrow> 'a"
```
```    62   assumes lub_upper: "chain (op \<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> lub A"
```
```    63   assumes lub_least: "chain (op \<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> lub A \<le> z"
```
```    64 begin
```
```    65
```
```    66 subsection {* Transfinite iteration of a function *}
```
```    67
```
```    68 inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"
```
```    69 for f :: "'a \<Rightarrow> 'a"
```
```    70 where
```
```    71   step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
```
```    72 | lub: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> lub M \<in> iterates f"
```
```    73
```
```    74 lemma iterates_le_f:
```
```    75   "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
```
```    76 by (induct x rule: iterates.induct)
```
```    77   (force dest: monotoneD intro!: lub_upper lub_least)+
```
```    78
```
```    79 lemma chain_iterates:
```
```    80   assumes f: "monotone (op \<le>) (op \<le>) f"
```
```    81   shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")
```
```    82 proof (rule chainI)
```
```    83   fix x y assume "x \<in> ?C" "y \<in> ?C"
```
```    84   then show "x \<le> y \<or> y \<le> x"
```
```    85   proof (induct x arbitrary: y rule: iterates.induct)
```
```    86     fix x y assume y: "y \<in> ?C"
```
```    87     and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
```
```    88     from y show "f x \<le> y \<or> y \<le> f x"
```
```    89     proof (induct y rule: iterates.induct)
```
```    90       case (step y) with IH f show ?case by (auto dest: monotoneD)
```
```    91     next
```
```    92       case (lub M)
```
```    93       then have chM: "chain (op \<le>) M"
```
```    94         and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
```
```    95       show "f x \<le> lub M \<or> lub M \<le> f x"
```
```    96       proof (cases "\<exists>z\<in>M. f x \<le> z")
```
```    97         case True then have "f x \<le> lub M"
```
```    98           apply rule
```
```    99           apply (erule order_trans)
```
```   100           by (rule lub_upper[OF chM])
```
```   101         thus ?thesis ..
```
```   102       next
```
```   103         case False with IH'
```
```   104         show ?thesis by (auto intro: lub_least[OF chM])
```
```   105       qed
```
```   106     qed
```
```   107   next
```
```   108     case (lub M y)
```
```   109     show ?case
```
```   110     proof (cases "\<exists>x\<in>M. y \<le> x")
```
```   111       case True then have "y \<le> lub M"
```
```   112         apply rule
```
```   113         apply (erule order_trans)
```
```   114         by (rule lub_upper[OF lub(1)])
```
```   115       thus ?thesis ..
```
```   116     next
```
```   117       case False with lub
```
```   118       show ?thesis by (auto intro: lub_least)
```
```   119     qed
```
```   120   qed
```
```   121 qed
```
```   122
```
```   123 subsection {* Fixpoint combinator *}
```
```   124
```
```   125 definition
```
```   126   fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
```
```   127 where
```
```   128   "fixp f = lub (iterates f)"
```
```   129
```
```   130 lemma iterates_fixp:
```
```   131   assumes f: "monotone (op \<le>) (op \<le>) f" shows "fixp f \<in> iterates f"
```
```   132 unfolding fixp_def
```
```   133 by (simp add: iterates.lub chain_iterates f)
```
```   134
```
```   135 lemma fixp_unfold:
```
```   136   assumes f: "monotone (op \<le>) (op \<le>) f"
```
```   137   shows "fixp f = f (fixp f)"
```
```   138 proof (rule antisym)
```
```   139   show "fixp f \<le> f (fixp f)"
```
```   140     by (intro iterates_le_f iterates_fixp f)
```
```   141   have "f (fixp f) \<le> lub (iterates f)"
```
```   142     by (intro lub_upper chain_iterates f iterates.step iterates_fixp)
```
```   143   thus "f (fixp f) \<le> fixp f"
```
```   144     unfolding fixp_def .
```
```   145 qed
```
```   146
```
```   147 lemma fixp_lowerbound:
```
```   148   assumes f: "monotone (op \<le>) (op \<le>) f" and z: "f z \<le> z" shows "fixp f \<le> z"
```
```   149 unfolding fixp_def
```
```   150 proof (rule lub_least[OF chain_iterates[OF f]])
```
```   151   fix x assume "x \<in> iterates f"
```
```   152   thus "x \<le> z"
```
```   153   proof (induct x rule: iterates.induct)
```
```   154     fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD)
```
```   155     also note z finally show "f x \<le> z" .
```
```   156   qed (auto intro: lub_least)
```
```   157 qed
```
```   158
```
```   159
```
```   160 subsection {* Fixpoint induction *}
```
```   161
```
```   162 definition
```
```   163   admissible :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
```
```   164 where
```
```   165   "admissible P = (\<forall>A. chain (op \<le>) A \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
```
```   166
```
```   167 lemma admissibleI:
```
```   168   assumes "\<And>A. chain (op \<le>) A \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)"
```
```   169   shows "admissible P"
```
```   170 using assms unfolding admissible_def by fast
```
```   171
```
```   172 lemma admissibleD:
```
```   173   assumes "admissible P"
```
```   174   assumes "chain (op \<le>) A"
```
```   175   assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
```
```   176   shows "P (lub A)"
```
```   177 using assms by (auto simp: admissible_def)
```
```   178
```
```   179 lemma fixp_induct:
```
```   180   assumes adm: "admissible P"
```
```   181   assumes mono: "monotone (op \<le>) (op \<le>) f"
```
```   182   assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
```
```   183   shows "P (fixp f)"
```
```   184 unfolding fixp_def using adm chain_iterates[OF mono]
```
```   185 proof (rule admissibleD)
```
```   186   fix x assume "x \<in> iterates f"
```
```   187   thus "P x"
```
```   188     by (induct rule: iterates.induct)
```
```   189       (auto intro: step admissibleD adm)
```
```   190 qed
```
```   191
```
```   192 lemma admissible_True: "admissible (\<lambda>x. True)"
```
```   193 unfolding admissible_def by simp
```
```   194
```
```   195 lemma admissible_False: "\<not> admissible (\<lambda>x. False)"
```
```   196 unfolding admissible_def chain_def by simp
```
```   197
```
```   198 lemma admissible_const: "admissible (\<lambda>x. t) = t"
```
```   199 by (cases t, simp_all add: admissible_True admissible_False)
```
```   200
```
```   201 lemma admissible_conj:
```
```   202   assumes "admissible (\<lambda>x. P x)"
```
```   203   assumes "admissible (\<lambda>x. Q x)"
```
```   204   shows "admissible (\<lambda>x. P x \<and> Q x)"
```
```   205 using assms unfolding admissible_def by simp
```
```   206
```
```   207 lemma admissible_all:
```
```   208   assumes "\<And>y. admissible (\<lambda>x. P x y)"
```
```   209   shows "admissible (\<lambda>x. \<forall>y. P x y)"
```
```   210 using assms unfolding admissible_def by fast
```
```   211
```
```   212 lemma admissible_ball:
```
```   213   assumes "\<And>y. y \<in> A \<Longrightarrow> admissible (\<lambda>x. P x y)"
```
```   214   shows "admissible (\<lambda>x. \<forall>y\<in>A. P x y)"
```
```   215 using assms unfolding admissible_def by fast
```
```   216
```
```   217 lemma chain_compr: "chain (op \<le>) A \<Longrightarrow> chain (op \<le>) {x \<in> A. P x}"
```
```   218 unfolding chain_def by fast
```
```   219
```
```   220 lemma admissible_disj_lemma:
```
```   221   assumes A: "chain (op \<le>)A"
```
```   222   assumes P: "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y"
```
```   223   shows "lub A = lub {x \<in> A. P x}"
```
```   224 proof (rule antisym)
```
```   225   have *: "chain (op \<le>) {x \<in> A. P x}"
```
```   226     by (rule chain_compr [OF A])
```
```   227   show "lub A \<le> lub {x \<in> A. P x}"
```
```   228     apply (rule lub_least [OF A])
```
```   229     apply (drule P [rule_format], clarify)
```
```   230     apply (erule order_trans)
```
```   231     apply (simp add: lub_upper [OF *])
```
```   232     done
```
```   233   show "lub {x \<in> A. P x} \<le> lub A"
```
```   234     apply (rule lub_least [OF *])
```
```   235     apply clarify
```
```   236     apply (simp add: lub_upper [OF A])
```
```   237     done
```
```   238 qed
```
```   239
```
```   240 lemma admissible_disj:
```
```   241   fixes P Q :: "'a \<Rightarrow> bool"
```
```   242   assumes P: "admissible (\<lambda>x. P x)"
```
```   243   assumes Q: "admissible (\<lambda>x. Q x)"
```
```   244   shows "admissible (\<lambda>x. P x \<or> Q x)"
```
```   245 proof (rule admissibleI)
```
```   246   fix A :: "'a set" assume A: "chain (op \<le>) A"
```
```   247   assume "\<forall>x\<in>A. P x \<or> Q x"
```
```   248   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)"
```
```   249     using chainD[OF A] by blast
```
```   250   hence "lub A = lub {x \<in> A. P x} \<or> lub A = lub {x \<in> A. Q x}"
```
```   251     using admissible_disj_lemma [OF A] by fast
```
```   252   thus "P (lub A) \<or> Q (lub A)"
```
```   253     apply (rule disjE, simp_all)
```
```   254     apply (rule disjI1, rule admissibleD [OF P chain_compr [OF A]], simp)
```
```   255     apply (rule disjI2, rule admissibleD [OF Q chain_compr [OF A]], simp)
```
```   256     done
```
```   257 qed
```
```   258
```
```   259 end
```
```   260
```
```   261 hide_const (open) lub iterates fixp admissible
```
```   262
```
```   263 end
```