(* Title: HOLCF/Fix.thy ID: $Id: Fix.thy,v 1.36 2005/09/22 17:06:05 huffman Exp $ Author: Franz Regensburger Definitions for fixed point operator and admissibility. *) header {* Fixed point operator and admissibility *} theory Fix imports Cfun Cprod Adm begin defaultsort pcpo subsection {* Definitions *} consts iterate :: "nat => ('a -> 'a) => 'a => 'a" Ifix :: "('a -> 'a) => 'a" "fix" :: "('a -> 'a) -> 'a" admw :: "('a => bool) => bool" primrec iterate_0: "iterate 0 F x = x" iterate_Suc: "iterate (Suc n) F x = F·(iterate n F x)" defs Ifix_def: "Ifix ≡ λF. \<Squnion>i. iterate i F ⊥" fix_def: "fix ≡ Λ F. Ifix F" admw_def: "admw P ≡ ∀F. (∀n. P (iterate n F ⊥)) --> P (\<Squnion>i. iterate i F ⊥)" subsection {* Binder syntax for @{term fix} *} syntax "@FIX" :: "('a => 'a) => 'a" (binder "FIX " 10) "@FIXP" :: "[patterns, 'a] => 'a" ("(3FIX <_>./ _)" [0, 10] 10) syntax (xsymbols) "FIX " :: "[idt, 'a] => 'a" ("(3μ_./ _)" [0, 10] 10) "@FIXP" :: "[patterns, 'a] => 'a" ("(3μ()<_>./ _)" [0, 10] 10) translations "FIX x. LAM y. t" == "fix·(LAM x y. t)" "FIX x. t" == "fix·(LAM x. t)" "FIX <xs>. t" == "fix·(LAM <xs>. t)" subsection {* Properties of @{term iterate} and @{term fix} *} text {* derive inductive properties of iterate from primitive recursion *} lemma iterate_Suc2: "iterate (Suc n) F x = iterate n F (F·x)" by (induct_tac n, auto) text {* The sequence of function iterations is a chain. This property is essential since monotonicity of iterate makes no sense. *} lemma chain_iterate2: "x \<sqsubseteq> F·x ==> chain (λi. iterate i F x)" by (rule chainI, induct_tac i, auto elim: monofun_cfun_arg) lemma chain_iterate: "chain (λi. iterate i F ⊥)" by (rule chain_iterate2 [OF minimal]) text {* Kleene's fixed point theorems for continuous functions in pointed omega cpo's *} lemma Ifix_eq: "Ifix F = F·(Ifix F)" apply (unfold Ifix_def) 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 Ifix_least: "F·x = x ==> Ifix F \<sqsubseteq> x" apply (unfold Ifix_def) apply (rule is_lub_thelub) apply (rule chain_iterate) apply (rule ub_rangeI) apply (induct_tac i) apply simp apply simp apply (erule subst) apply (erule monofun_cfun_arg) done text {* continuity of @{term iterate} *} lemma cont_iterate1: "cont (λF. iterate n F x)" by (induct_tac n, simp_all) lemma cont_iterate2: "cont (λx. iterate n F x)" by (induct_tac n, simp_all) lemma cont_iterate: "cont (iterate n)" by (rule cont_iterate1 [THEN cont2cont_lambda]) lemmas monofun_iterate2 = cont_iterate2 [THEN cont2mono, standard] lemmas contlub_iterate2 = cont_iterate2 [THEN cont2contlub, standard] text {* continuity of @{term Ifix} *} lemma cont_Ifix: "cont Ifix" apply (unfold Ifix_def) apply (rule cont2cont_lub) apply (rule ch2ch_fun_rev) apply (rule chain_iterate) apply (rule cont_iterate1) done text {* propagate properties of @{term Ifix} to its continuous counterpart *} lemma fix_eq: "fix·F = F·(fix·F)" apply (unfold fix_def) apply (simp add: cont_Ifix) apply (rule Ifix_eq) done lemma fix_least: "F·x = x ==> fix·F \<sqsubseteq> x" apply (unfold fix_def) apply (simp add: cont_Ifix) apply (erule Ifix_least) done lemma fix_eqI: "[|F·x = x; ∀z. F·z = z --> x \<sqsubseteq> z|] ==> x = fix·F" apply (rule antisym_less) apply (erule allE) apply (erule mp) apply (rule fix_eq [symmetric]) apply (erule fix_least) done lemma 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) done lemma fix_eq5: "f = fix·F ==> f·x = F·f·x" by (erule fix_eq4 [THEN cfun_fun_cong]) text {* direct connection between @{term fix} and iteration without @{term Ifix} *} lemma fix_def2: "fix·F = (\<Squnion>i. iterate i F ⊥)" apply (unfold fix_def) apply (simp add: cont_Ifix) apply (simp add: Ifix_def) done text {* strictness of @{term fix} *} lemma fix_defined_iff: "(fix·F = ⊥) = (F·⊥ = ⊥)" apply (rule iffI) apply (erule subst) apply (rule fix_eq [symmetric]) apply (erule fix_least [THEN UU_I]) done lemma fix_strict: "F·⊥ = ⊥ ==> fix·F = ⊥" by (simp add: fix_defined_iff) lemma fix_defined: "F·⊥ ≠ ⊥ ==> fix·F ≠ ⊥" by (simp add: fix_defined_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 (rule fix_eq [THEN trans], simp) subsection {* Admissibility and fixed point induction *} text {* an admissible formula is also weak admissible *} lemma adm_impl_admw: "adm P ==> admw P" apply (unfold admw_def) apply (intro strip) apply (erule admD) apply (rule chain_iterate) apply assumption done text {* some lemmata for functions with flat/chfin domain/range types *} lemma adm_chfindom: "adm (λ(u::'a::cpo -> 'b::chfin). P(u·s))" apply (unfold adm_def) apply (intro strip) apply (drule chfin_Rep_CFunR) apply (erule_tac x = "s" in allE) apply clarsimp done (* adm_flat not needed any more, since it is a special case of adm_chfindom *) text {* fixed point induction *} lemma fix_ind: "[|adm P; P ⊥; !!x. P x ==> P (F·x)|] ==> P (fix·F)" apply (subst fix_def2) apply (erule admD) apply (rule chain_iterate) apply (rule allI) apply (induct_tac "i") apply simp apply simp done lemma def_fix_ind: "[|f ≡ fix·F; adm P; P ⊥; !!x. P x ==> P (F·x)|] ==> P f" apply simp apply (erule fix_ind) apply assumption apply fast done text {* computational induction for weak admissible formulae *} lemma wfix_ind: "[|admw P; ∀n. P (iterate n F ⊥)|] ==> P (fix·F)" by (simp add: fix_def2 admw_def) lemma def_wfix_ind: "[|f ≡ fix·F; admw P; ∀n. P (iterate n F ⊥)|] ==> P f" by (simp, rule wfix_ind) end
lemma iterate_Suc2:
iterate (Suc n) F x = iterate n F (F·x)
lemma chain_iterate2:
x << F·x ==> chain (%i. iterate i F x)
lemma chain_iterate:
chain (%i. iterate i F UU)
lemma Ifix_eq:
Ifix F = F·(Ifix F)
lemma Ifix_least:
F·x = x ==> Ifix F << x
lemma cont_iterate1:
cont (%F. iterate n F x)
lemma cont_iterate2:
cont (iterate n F)
lemma cont_iterate:
cont (iterate n)
lemmas monofun_iterate2:
monofun (iterate n F)
lemmas monofun_iterate2:
monofun (iterate n F)
lemmas contlub_iterate2:
contlub (iterate n F)
lemmas contlub_iterate2:
contlub (iterate n F)
lemma cont_Ifix:
cont Ifix
lemma fix_eq:
fix·F = F·(fix·F)
lemma fix_least:
F·x = x ==> fix·F << x
lemma fix_eqI:
[| F·x = x; ∀z. F·z = z --> x << z |] ==> x = fix·F
lemma fix_eq2:
f == fix·F ==> f = F·f
lemma fix_eq3:
f == fix·F ==> f·x = F·f·x
lemma fix_eq4:
f = fix·F ==> f = F·f
lemma fix_eq5:
f = fix·F ==> f·x = F·f·x
lemma fix_def2:
fix·F = (LUB i. iterate i F UU)
lemma fix_defined_iff:
(fix·F = UU) = (F·UU = UU)
lemma fix_strict:
F·UU = UU ==> fix·F = UU
lemma fix_defined:
F·UU ≠ UU ==> fix·F ≠ UU
lemma fix_id:
(FIX x. x) = UU
lemma fix_const:
(FIX x. c) = c
lemma adm_impl_admw:
adm P ==> admw P
lemma adm_chfindom:
adm (%u. P (u·s))
lemma fix_ind:
[| adm P; P UU; !!x. P x ==> P (F·x) |] ==> P (fix·F)
lemma def_fix_ind:
[| f == fix·F; adm P; P UU; !!x. P x ==> P (F·x) |] ==> P f
lemma wfix_ind:
[| admw P; ∀n. P (iterate n F UU) |] ==> P (fix·F)
lemma def_wfix_ind:
[| f == fix·F; admw P; ∀n. P (iterate n F UU) |] ==> P f