(*  Title:      HOLCF/Fix.thy

     Author:     Franz Regensburger

     Author:     Brian Huffman

 *)

 header {* Fixed point operator and admissibility *}

 theory Fix

 imports Cfun

 begin

 default_sort pcpo

 subsection {* Iteration *}

 primrec iterate :: "nat \<Rightarrow> ('a::cpo \<rightarrow> 'a) \<rightarrow> ('a \<rightarrow> 'a)" where

     "iterate 0 = (\<Lambda> F x. x)"

   | "iterate (Suc n) = (\<Lambda> F x. F\<cdot>(iterate n\<cdot>F\<cdot>x))"

 text {* Derive inductive properties of iterate from primitive recursion *}

 lemma iterate_0 [simp]: "iterate 0\<cdot>F\<cdot>x = x"

 by simp

 lemma iterate_Suc [simp]: "iterate (Suc n)\<cdot>F\<cdot>x = F\<cdot>(iterate n\<cdot>F\<cdot>x)"

 by simp

 declare iterate.simps [simp del]

 lemma iterate_Suc2: "iterate (Suc n)\<cdot>F\<cdot>x = iterate n\<cdot>F\<cdot>(F\<cdot>x)"

 by (induct n) simp_all

 lemma iterate_iterate:

   "iterate m\<cdot>F\<cdot>(iterate n\<cdot>F\<cdot>x) = iterate (m + n)\<cdot>F\<cdot>x"

 by (induct m) simp_all

 text {* The sequence of function iterations is a chain. *}

 lemma chain_iterate [simp]: "chain (\<lambda>i. iterate i\<cdot>F\<cdot>\<bottom>)"

 by (rule chainI, unfold iterate_Suc2, rule monofun_cfun_arg, rule minimal)

 subsection {* Least fixed point operator *}

 definition

   "fix" :: "('a \<rightarrow> 'a) \<rightarrow> 'a" where

   "fix = (\<Lambda> F. \<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"

 text {* Binder syntax for @{term fix} *}

 abbreviation

   fix_syn :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"  (binder "FIX " 10) where

   "fix_syn (\<lambda>x. f x) \<equiv> fix\<cdot>(\<Lambda> x. f x)"

 notation (xsymbols)

   fix_syn  (binder "\<mu> " 10)

 text {* Properties of @{term fix} *}

 text {* direct connection between @{term fix} and iteration *}

 lemma fix_def2: "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"

 unfolding fix_def by simp

 lemma iterate_below_fix: "iterate n\<cdot>f\<cdot>\<bottom> \<sqsubseteq> fix\<cdot>f"

   unfolding fix_def2

   using chain_iterate by (rule is_ub_thelub)

 text {*

   Kleene's fixed point theorems for continuous functions in pointed

   omega cpo's

 *}

 lemma fix_eq: "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"

 apply (simp add: fix_def2)

 apply (subst lub_range_shift [of _ 1, symmetric])

 apply (rule chain_iterate)

 apply (subst contlub_cfun_arg)

 apply (rule chain_iterate)

 apply simp

 done

 lemma fix_least_below: "F\<cdot>x \<sqsubseteq> x \<Longrightarrow> fix\<cdot>F \<sqsubseteq> x"

 apply (simp add: fix_def2)

 apply (rule lub_below)

 apply (rule chain_iterate)

 apply (induct_tac i)

 apply simp

 apply simp

 apply (erule rev_below_trans)

 apply (erule monofun_cfun_arg)

 done

 lemma fix_least: "F\<cdot>x = x \<Longrightarrow> fix\<cdot>F \<sqsubseteq> x"

 by (rule fix_least_below, simp)

 lemma fix_eqI:

   assumes fixed: "F\<cdot>x = x" and least: "\<And>z. F\<cdot>z = z \<Longrightarrow> x \<sqsubseteq> z"

   shows "fix\<cdot>F = x"

 apply (rule below_antisym)

 apply (rule fix_least [OF fixed])

 apply (rule least [OF fix_eq [symmetric]])

 done

 lemma fix_eq2: "f \<equiv> fix\<cdot>F \<Longrightarrow> f = F\<cdot>f"

 by (simp add: fix_eq [symmetric])

 lemma fix_eq3: "f \<equiv> fix\<cdot>F \<Longrightarrow> f\<cdot>x = F\<cdot>f\<cdot>x"

 by (erule fix_eq2 [THEN cfun_fun_cong])

 lemma fix_eq4: "f = fix\<cdot>F \<Longrightarrow> f = F\<cdot>f"

 apply (erule ssubst)

 apply (rule fix_eq)

 done

 lemma fix_eq5: "f = fix\<cdot>F \<Longrightarrow> f\<cdot>x = F\<cdot>f\<cdot>x"

 by (erule fix_eq4 [THEN cfun_fun_cong])

 text {* strictness of @{term fix} *}

 lemma fix_bottom_iff: "(fix\<cdot>F = \<bottom>) = (F\<cdot>\<bottom> = \<bottom>)"

 apply (rule iffI)

 apply (erule subst)

 apply (rule fix_eq [symmetric])

 apply (erule fix_least [THEN UU_I])

 done

 lemma fix_strict: "F\<cdot>\<bottom> = \<bottom> \<Longrightarrow> fix\<cdot>F = \<bottom>"

 by (simp add: fix_bottom_iff)

 lemma fix_defined: "F\<cdot>\<bottom> \<noteq> \<bottom> \<Longrightarrow> fix\<cdot>F \<noteq> \<bottom>"

 by (simp add: fix_bottom_iff)

 text {* @{term fix} applied to identity and constant functions *}

 lemma fix_id: "(\<mu> x. x) = \<bottom>"

 by (simp add: fix_strict)

 lemma fix_const: "(\<mu> x. c) = c"

 by (subst fix_eq, simp)

 subsection {* Fixed point induction *}

 lemma fix_ind: "\<lbrakk>adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (fix\<cdot>F)"

 unfolding fix_def2

 apply (erule admD)

 apply (rule chain_iterate)

 apply (rule nat_induct, simp_all)

 done

 lemma def_fix_ind:

   "\<lbrakk>f \<equiv> fix\<cdot>F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P f"

 by (simp add: fix_ind)

 lemma fix_ind2:

   assumes adm: "adm P"

   assumes 0: "P \<bottom>" and 1: "P (F\<cdot>\<bottom>)"

   assumes step: "\<And>x. \<lbrakk>P x; P (F\<cdot>x)\<rbrakk> \<Longrightarrow> P (F\<cdot>(F\<cdot>x))"

   shows "P (fix\<cdot>F)"

 unfolding fix_def2

 apply (rule admD [OF adm chain_iterate])

 apply (rule nat_less_induct)

 apply (case_tac n)

 apply (simp add: 0)

 apply (case_tac nat)

 apply (simp add: 1)

 apply (frule_tac x=nat in spec)

 apply (simp add: step)

 done

 lemma parallel_fix_ind:

   assumes adm: "adm (\<lambda>x. P (fst x) (snd x))"

   assumes base: "P \<bottom> \<bottom>"

   assumes step: "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)"

   shows "P (fix\<cdot>F) (fix\<cdot>G)"

 proof -

   from adm have adm': "adm (split P)"

     unfolding split_def .

   have "\<And>i. P (iterate i\<cdot>F\<cdot>\<bottom>) (iterate i\<cdot>G\<cdot>\<bottom>)"

     by (induct_tac i, simp add: base, simp add: step)

   hence "\<And>i. split P (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>)"

     by simp

   hence "split P (\<Squnion>i. (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>))"

     by - (rule admD [OF adm'], simp, assumption)

   hence "split P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>, \<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"

     by (simp add: lub_Pair)

   hence "P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>) (\<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"

     by simp

   thus "P (fix\<cdot>F) (fix\<cdot>G)"

     by (simp add: fix_def2)

 qed

 subsection {* Fixed-points on product types *}

 text {*

   Bekic's Theorem: Simultaneous fixed points over pairs

   can be written in terms of separate fixed points.

 *}

 lemma fix_cprod:

   "fix\<cdot>(F::'a \<times> 'b \<rightarrow> 'a \<times> 'b) =

    (\<mu> x. fst (F\<cdot>(x, \<mu> y. snd (F\<cdot>(x, y)))),

     \<mu> y. snd (F\<cdot>(\<mu> x. fst (F\<cdot>(x, \<mu> y. snd (F\<cdot>(x, y)))), y)))"

   (is "fix\<cdot>F = (?x, ?y)")

 proof (rule fix_eqI)

   have 1: "fst (F\<cdot>(?x, ?y)) = ?x"

     by (rule trans [symmetric, OF fix_eq], simp)

   have 2: "snd (F\<cdot>(?x, ?y)) = ?y"

     by (rule trans [symmetric, OF fix_eq], simp)

   from 1 2 show "F\<cdot>(?x, ?y) = (?x, ?y)" by (simp add: Pair_fst_snd_eq)

 next

   fix z assume F_z: "F\<cdot>z = z"

   obtain x y where z: "z = (x,y)" by (rule prod.exhaust)

   from F_z z have F_x: "fst (F\<cdot>(x, y)) = x" by simp

   from F_z z have F_y: "snd (F\<cdot>(x, y)) = y" by simp

   let ?y1 = "\<mu> y. snd (F\<cdot>(x, y))"

   have "?y1 \<sqsubseteq> y" by (rule fix_least, simp add: F_y)

   hence "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> fst (F\<cdot>(x, y))"

     by (simp add: fst_monofun monofun_cfun)

   hence "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> x" using F_x by simp

   hence 1: "?x \<sqsubseteq> x" by (simp add: fix_least_below)

   hence "snd (F\<cdot>(?x, y)) \<sqsubseteq> snd (F\<cdot>(x, y))"

     by (simp add: snd_monofun monofun_cfun)

   hence "snd (F\<cdot>(?x, y)) \<sqsubseteq> y" using F_y by simp

   hence 2: "?y \<sqsubseteq> y" by (simp add: fix_least_below)

   show "(?x, ?y) \<sqsubseteq> z" using z 1 2 by simp

 qed

 end