--- a/src/HOL/HOLCF/Fix.thy Tue Dec 10 21:43:04 2024 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,247 +0,0 @@
-(* Title: HOL/HOLCF/Fix.thy
- Author: Franz Regensburger
- Author: Brian Huffman
-*)
-
-section \<open>Fixed point operator and admissibility\<close>
-
-theory Fix
- imports Cfun
-begin
-
-default_sort pcpo
-
-
-subsection \<open>Iteration\<close>
-
-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 \<open>Derive inductive properties of iterate from primitive recursion\<close>
-
-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 \<open>The sequence of function iterations is a chain.\<close>
-
-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 \<open>Least fixed point operator\<close>
-
-definition "fix" :: "('a \<rightarrow> 'a) \<rightarrow> 'a"
- where "fix = (\<Lambda> F. \<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
-
-text \<open>Binder syntax for \<^term>\<open>fix\<close>\<close>
-
-abbreviation fix_syn :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" (binder \<open>\<mu> \<close> 10)
- where "fix_syn (\<lambda>x. f x) \<equiv> fix\<cdot>(\<Lambda> x. f x)"
-
-notation (ASCII)
- fix_syn (binder \<open>FIX \<close> 10)
-
-text \<open>Properties of \<^term>\<open>fix\<close>\<close>
-
-text \<open>direct connection between \<^term>\<open>fix\<close> and iteration\<close>
-
-lemma fix_def2: "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
- by (simp add: fix_def)
-
-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 \<open>
- Kleene's fixed point theorems for continuous functions in pointed
- omega cpo's
-\<close>
-
-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"
- by (erule ssubst) (rule fix_eq)
-
-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 \<open>strictness of \<^term>\<open>fix\<close>\<close>
-
-lemma fix_bottom_iff: "fix\<cdot>F = \<bottom> \<longleftrightarrow> F\<cdot>\<bottom> = \<bottom>"
- apply (rule iffI)
- apply (erule subst)
- apply (rule fix_eq [symmetric])
- apply (erule fix_least [THEN bottomI])
- 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 \<open>\<^term>\<open>fix\<close> applied to identity and constant functions\<close>
-
-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 \<open>Fixed point induction\<close>
-
-lemma fix_ind: "adm P \<Longrightarrow> P \<bottom> \<Longrightarrow> (\<And>x. P x \<Longrightarrow> P (F\<cdot>x)) \<Longrightarrow> P (fix\<cdot>F)"
- unfolding fix_def2
- apply (erule admD)
- apply (rule chain_iterate)
- apply (rule nat_induct, simp_all)
- done
-
-lemma cont_fix_ind: "cont F \<Longrightarrow> adm P \<Longrightarrow> P \<bottom> \<Longrightarrow> (\<And>x. P x \<Longrightarrow> P (F x)) \<Longrightarrow> P (fix\<cdot>(Abs_cfun F))"
- by (simp add: fix_ind)
-
-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 (case_prod P)"
- unfolding split_def .
- have "P (iterate i\<cdot>F\<cdot>\<bottom>) (iterate i\<cdot>G\<cdot>\<bottom>)" for i
- by (induct i) (simp add: base, simp add: step)
- then have "\<And>i. case_prod P (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>)"
- by simp
- then have "case_prod P (\<Squnion>i. (iterate i\<cdot>F\<cdot>\<bottom>, iterate i\<cdot>G\<cdot>\<bottom>))"
- by - (rule admD [OF adm'], simp, assumption)
- then have "case_prod P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>, \<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"
- by (simp add: lub_Pair)
- then have "P (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>) (\<Squnion>i. iterate i\<cdot>G\<cdot>\<bottom>)"
- by simp
- then show "P (fix\<cdot>F) (fix\<cdot>G)"
- by (simp add: fix_def2)
-qed
-
-lemma cont_parallel_fix_ind:
- assumes "cont F" and "cont G"
- assumes "adm (\<lambda>x. P (fst x) (snd x))"
- assumes "P \<bottom> \<bottom>"
- assumes "\<And>x y. P x y \<Longrightarrow> P (F x) (G y)"
- shows "P (fix\<cdot>(Abs_cfun F)) (fix\<cdot>(Abs_cfun G))"
- by (rule parallel_fix_ind) (simp_all add: assms)
-
-
-subsection \<open>Fixed-points on product types\<close>
-
-text \<open>
- Bekic's Theorem: Simultaneous fixed points over pairs
- can be written in terms of separate fixed points.
-\<close>
-
-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 *: "fst (F\<cdot>(?x, ?y)) = ?x"
- by (rule trans [symmetric, OF fix_eq], simp)
- have "snd (F\<cdot>(?x, ?y)) = ?y"
- by (rule trans [symmetric, OF fix_eq], simp)
- with * show "F\<cdot>(?x, ?y) = (?x, ?y)"
- by (simp add: prod_eq_iff)
-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)
- then have "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> fst (F\<cdot>(x, y))"
- by (simp add: fst_monofun monofun_cfun)
- with F_x have "fst (F\<cdot>(x, ?y1)) \<sqsubseteq> x"
- by simp
- then have *: "?x \<sqsubseteq> x"
- by (simp add: fix_least_below)
- then have "snd (F\<cdot>(?x, y)) \<sqsubseteq> snd (F\<cdot>(x, y))"
- by (simp add: snd_monofun monofun_cfun)
- with F_y have "snd (F\<cdot>(?x, y)) \<sqsubseteq> y"
- by simp
- then have "?y \<sqsubseteq> y"
- by (simp add: fix_least_below)
- with z * show "(?x, ?y) \<sqsubseteq> z"
- by simp
-qed
-
-end