Theory Fix

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theory Fix
imports Cfun
`(*  Title:      HOL/HOLCF/Fix.thy    Author:     Franz Regensburger    Author:     Brian Huffman*)header {* Fixed point operator and admissibility *}theory Fiximports Cfunbegindefault_sort pcposubsection {* Iteration *}primrec iterate :: "nat => ('a::cpo -> 'a) -> ('a -> 'a)" where    "iterate 0 = (Λ F x. x)"  | "iterate (Suc n) = (Λ F x. F·(iterate n·F·x))"text {* Derive inductive properties of iterate from primitive recursion *}lemma iterate_0 [simp]: "iterate 0·F·x = x"by simplemma iterate_Suc [simp]: "iterate (Suc n)·F·x = F·(iterate n·F·x)"by simpdeclare iterate.simps [simp del]lemma iterate_Suc2: "iterate (Suc n)·F·x = iterate n·F·(F·x)"by (induct n) simp_alllemma iterate_iterate:  "iterate m·F·(iterate n·F·x) = iterate (m + n)·F·x"by (induct m) simp_alltext {* The sequence of function iterations is a chain. *}lemma chain_iterate [simp]: "chain (λi. iterate i·F·⊥)"by (rule chainI, unfold iterate_Suc2, rule monofun_cfun_arg, rule minimal)subsection {* Least fixed point operator *}definition  "fix" :: "('a -> 'a) -> 'a" where  "fix = (Λ F. \<Squnion>i. iterate i·F·⊥)"text {* Binder syntax for @{term fix} *}abbreviation  fix_syn :: "('a => 'a) => 'a"  (binder "FIX " 10) where  "fix_syn (λx. f x) ≡ fix·(Λ x. f x)"notation (xsymbols)  fix_syn  (binder "μ " 10)text {* Properties of @{term fix} *}text {* direct connection between @{term fix} and iteration *}lemma fix_def2: "fix·F = (\<Squnion>i. iterate i·F·⊥)"unfolding fix_def by simplemma iterate_below_fix: "iterate n·f·⊥ \<sqsubseteq> fix·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·F = F·(fix·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 simpdonelemma fix_least_below: "F·x \<sqsubseteq> x ==> fix·F \<sqsubseteq> x"apply (simp add: fix_def2)apply (rule lub_below)apply (rule chain_iterate)apply (induct_tac i)apply simpapply simpapply (erule rev_below_trans)apply (erule monofun_cfun_arg)donelemma fix_least: "F·x = x ==> fix·F \<sqsubseteq> x"by (rule fix_least_below, simp)lemma fix_eqI:  assumes fixed: "F·x = x" and least: "!!z. F·z = z ==> x \<sqsubseteq> z"  shows "fix·F = x"apply (rule below_antisym)apply (rule fix_least [OF fixed])apply (rule least [OF fix_eq [symmetric]])donelemma fix_eq2: "f ≡ fix·F ==> f = F·f"by (simp add: fix_eq [symmetric])lemma fix_eq3: "f ≡ fix·F ==> f·x = F·f·x"by (erule fix_eq2 [THEN cfun_fun_cong])lemma fix_eq4: "f = fix·F ==> f = F·f"apply (erule ssubst)apply (rule fix_eq)donelemma fix_eq5: "f = fix·F ==> f·x = F·f·x"by (erule fix_eq4 [THEN cfun_fun_cong])text {* strictness of @{term fix} *}lemma fix_bottom_iff: "(fix·F = ⊥) = (F·⊥ = ⊥)"apply (rule iffI)apply (erule subst)apply (rule fix_eq [symmetric])apply (erule fix_least [THEN bottomI])donelemma fix_strict: "F·⊥ = ⊥ ==> fix·F = ⊥"by (simp add: fix_bottom_iff)lemma fix_defined: "F·⊥ ≠ ⊥ ==> fix·F ≠ ⊥"by (simp add: fix_bottom_iff)text {* @{term fix} applied to identity and constant functions *}lemma fix_id: "(μ x. x) = ⊥"by (simp add: fix_strict)lemma fix_const: "(μ x. c) = c"by (subst fix_eq, simp)subsection {* Fixed point induction *}lemma fix_ind: "[|adm P; P ⊥; !!x. P x ==> P (F·x)|] ==> P (fix·F)"unfolding fix_def2apply (erule admD)apply (rule chain_iterate)apply (rule nat_induct, simp_all)donelemma cont_fix_ind:  "[|cont F; adm P; P ⊥; !!x. P x ==> P (F x)|] ==> P (fix·(Abs_cfun F))"by (simp add: fix_ind)lemma def_fix_ind:  "[|f ≡ fix·F; adm P; P ⊥; !!x. P x ==> P (F·x)|] ==> P f"by (simp add: fix_ind)lemma fix_ind2:  assumes adm: "adm P"  assumes 0: "P ⊥" and 1: "P (F·⊥)"  assumes step: "!!x. [|P x; P (F·x)|] ==> P (F·(F·x))"  shows "P (fix·F)"unfolding fix_def2apply (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)donelemma parallel_fix_ind:  assumes adm: "adm (λx. P (fst x) (snd x))"  assumes base: "P ⊥ ⊥"  assumes step: "!!x y. P x y ==> P (F·x) (G·y)"  shows "P (fix·F) (fix·G)"proof -  from adm have adm': "adm (split P)"    unfolding split_def .  have "!!i. P (iterate i·F·⊥) (iterate i·G·⊥)"    by (induct_tac i, simp add: base, simp add: step)  hence "!!i. split P (iterate i·F·⊥, iterate i·G·⊥)"    by simp  hence "split P (\<Squnion>i. (iterate i·F·⊥, iterate i·G·⊥))"    by - (rule admD [OF adm'], simp, assumption)  hence "split P (\<Squnion>i. iterate i·F·⊥, \<Squnion>i. iterate i·G·⊥)"    by (simp add: lub_Pair)  hence "P (\<Squnion>i. iterate i·F·⊥) (\<Squnion>i. iterate i·G·⊥)"    by simp  thus "P (fix·F) (fix·G)"    by (simp add: fix_def2)qedlemma cont_parallel_fix_ind:  assumes "cont F" and "cont G"  assumes "adm (λx. P (fst x) (snd x))"  assumes "P ⊥ ⊥"  assumes "!!x y. P x y ==> P (F x) (G y)"  shows "P (fix·(Abs_cfun F)) (fix·(Abs_cfun G))"by (rule parallel_fix_ind, simp_all add: assms)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·(F::'a × 'b -> 'a × 'b) =   (μ x. fst (F·(x, μ y. snd (F·(x, y)))),    μ y. snd (F·(μ x. fst (F·(x, μ y. snd (F·(x, y)))), y)))"  (is "fix·F = (?x, ?y)")proof (rule fix_eqI)  have 1: "fst (F·(?x, ?y)) = ?x"    by (rule trans [symmetric, OF fix_eq], simp)  have 2: "snd (F·(?x, ?y)) = ?y"    by (rule trans [symmetric, OF fix_eq], simp)  from 1 2 show "F·(?x, ?y) = (?x, ?y)" by (simp add: prod_eq_iff)next  fix z assume F_z: "F·z = z"  obtain x y where z: "z = (x,y)" by (rule prod.exhaust)  from F_z z have F_x: "fst (F·(x, y)) = x" by simp  from F_z z have F_y: "snd (F·(x, y)) = y" by simp  let ?y1 = "μ y. snd (F·(x, y))"  have "?y1 \<sqsubseteq> y" by (rule fix_least, simp add: F_y)  hence "fst (F·(x, ?y1)) \<sqsubseteq> fst (F·(x, y))"    by (simp add: fst_monofun monofun_cfun)  hence "fst (F·(x, ?y1)) \<sqsubseteq> x" using F_x by simp  hence 1: "?x \<sqsubseteq> x" by (simp add: fix_least_below)  hence "snd (F·(?x, y)) \<sqsubseteq> snd (F·(x, y))"    by (simp add: snd_monofun monofun_cfun)  hence "snd (F·(?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 simpqedend`