(* Title: HOL/HOLCF/Fixrec.thy
Author: Franz Regensburger
Author: Amber Telfer and Brian Huffman
*)
theory Fixrec
imports Cprod Sprod Ssum Up One Tr Cfun
keywords "fixrec" :: thy_defn
begin
section \<open>Fixed point operator and admissibility\<close>
subsection \<open>Iteration\<close>
primrec iterate :: "nat \<Rightarrow> ('a \<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::pcpo \<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::pcpo \<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:
fixes F :: "'a::pcpo \<times> 'b::pcpo \<rightarrow> 'a \<times> 'b"
shows
"fix\<cdot>F =
(\<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
section "Package for defining recursive functions in HOLCF"
subsection \<open>Pattern-match monad\<close>
pcpodef 'a match = "UNIV::(one ++ 'a u) set"
by simp_all
definition
fail :: "'a match" where
"fail = Abs_match (sinl\<cdot>ONE)"
definition
succeed :: "'a \<rightarrow> 'a match" where
"succeed = (\<Lambda> x. Abs_match (sinr\<cdot>(up\<cdot>x)))"
lemma matchE [case_names bottom fail succeed, cases type: match]:
"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = succeed\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding fail_def succeed_def
apply (cases p, rename_tac r)
apply (rule_tac p=r in ssumE, simp add: Abs_match_strict)
apply (rule_tac p=x in oneE, simp, simp)
apply (rule_tac p=y in upE, simp, simp add: cont_Abs_match)
done
lemma succeed_defined [simp]: "succeed\<cdot>x \<noteq> \<bottom>"
by (simp add: succeed_def cont_Abs_match Abs_match_bottom_iff)
lemma fail_defined [simp]: "fail \<noteq> \<bottom>"
by (simp add: fail_def Abs_match_bottom_iff)
lemma succeed_eq [simp]: "(succeed\<cdot>x = succeed\<cdot>y) = (x = y)"
by (simp add: succeed_def cont_Abs_match Abs_match_inject)
lemma succeed_neq_fail [simp]:
"succeed\<cdot>x \<noteq> fail" "fail \<noteq> succeed\<cdot>x"
by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)
subsubsection \<open>Run operator\<close>
definition
run :: "'a match \<rightarrow> 'a::pcpo" where
"run = (\<Lambda> m. sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)\<cdot>(Rep_match m))"
text \<open>rewrite rules for run\<close>
lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
unfolding run_def
by (simp add: cont_Rep_match Rep_match_strict)
lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
unfolding run_def fail_def
by (simp add: cont_Rep_match Abs_match_inverse)
lemma run_succeed [simp]: "run\<cdot>(succeed\<cdot>x) = x"
unfolding run_def succeed_def
by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
subsubsection \<open>Monad plus operator\<close>
definition
mplus :: "'a match \<rightarrow> 'a match \<rightarrow> 'a match" where
"mplus = (\<Lambda> m1 m2. sscase\<cdot>(\<Lambda> _. m2)\<cdot>(\<Lambda> _. m1)\<cdot>(Rep_match m1))"
abbreviation
mplus_syn :: "['a match, 'a match] \<Rightarrow> 'a match" (infixr \<open>+++\<close> 65) where
"m1 +++ m2 == mplus\<cdot>m1\<cdot>m2"
text \<open>rewrite rules for mplus\<close>
lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
unfolding mplus_def
by (simp add: cont_Rep_match Rep_match_strict)
lemma mplus_fail [simp]: "fail +++ m = m"
unfolding mplus_def fail_def
by (simp add: cont_Rep_match Abs_match_inverse)
lemma mplus_succeed [simp]: "succeed\<cdot>x +++ m = succeed\<cdot>x"
unfolding mplus_def succeed_def
by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
lemma mplus_fail2 [simp]: "m +++ fail = m"
by (cases m, simp_all)
lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
by (cases x, simp_all)
subsection \<open>Match functions for built-in types\<close>
definition
match_bottom :: "'a::pcpo \<rightarrow> 'c match \<rightarrow> 'c match"
where
"match_bottom = (\<Lambda> x k. seq\<cdot>x\<cdot>fail)"
definition
match_Pair :: "'a \<times> 'b \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
where
"match_Pair = (\<Lambda> x k. csplit\<cdot>k\<cdot>x)"
definition
match_spair :: "'a::pcpo \<otimes> 'b::pcpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c::pcpo match"
where
"match_spair = (\<Lambda> x k. ssplit\<cdot>k\<cdot>x)"
definition
match_sinl :: "'a::pcpo \<oplus> 'b::pcpo \<rightarrow> ('a \<rightarrow> 'c::pcpo match) \<rightarrow> 'c match"
where
"match_sinl = (\<Lambda> x k. sscase\<cdot>k\<cdot>(\<Lambda> b. fail)\<cdot>x)"
definition
match_sinr :: "'a::pcpo \<oplus> 'b::pcpo \<rightarrow> ('b \<rightarrow> 'c::pcpo match) \<rightarrow> 'c match"
where
"match_sinr = (\<Lambda> x k. sscase\<cdot>(\<Lambda> a. fail)\<cdot>k\<cdot>x)"
definition
match_up :: "'a u \<rightarrow> ('a \<rightarrow> 'c::pcpo match) \<rightarrow> 'c match"
where
"match_up = (\<Lambda> x k. fup\<cdot>k\<cdot>x)"
definition
match_ONE :: "one \<rightarrow> 'c::pcpo match \<rightarrow> 'c match"
where
"match_ONE = (\<Lambda> ONE k. k)"
definition
match_TT :: "tr \<rightarrow> 'c::pcpo match \<rightarrow> 'c match"
where
"match_TT = (\<Lambda> x k. If x then k else fail)"
definition
match_FF :: "tr \<rightarrow> 'c::pcpo match \<rightarrow> 'c match"
where
"match_FF = (\<Lambda> x k. If x then fail else k)"
lemma match_bottom_simps [simp]:
"match_bottom\<cdot>x\<cdot>k = (if x = \<bottom> then \<bottom> else fail)"
by (simp add: match_bottom_def)
lemma match_Pair_simps [simp]:
"match_Pair\<cdot>(x, y)\<cdot>k = k\<cdot>x\<cdot>y"
by (simp_all add: match_Pair_def)
lemma match_spair_simps [simp]:
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x, y:)\<cdot>k = k\<cdot>x\<cdot>y"
"match_spair\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_spair_def)
lemma match_sinl_simps [simp]:
"x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x)\<cdot>k = k\<cdot>x"
"y \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>y)\<cdot>k = fail"
"match_sinl\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_sinl_def)
lemma match_sinr_simps [simp]:
"x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x)\<cdot>k = fail"
"y \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>y)\<cdot>k = k\<cdot>y"
"match_sinr\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_sinr_def)
lemma match_up_simps [simp]:
"match_up\<cdot>(up\<cdot>x)\<cdot>k = k\<cdot>x"
"match_up\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_up_def)
lemma match_ONE_simps [simp]:
"match_ONE\<cdot>ONE\<cdot>k = k"
"match_ONE\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_ONE_def)
lemma match_TT_simps [simp]:
"match_TT\<cdot>TT\<cdot>k = k"
"match_TT\<cdot>FF\<cdot>k = fail"
"match_TT\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_TT_def)
lemma match_FF_simps [simp]:
"match_FF\<cdot>FF\<cdot>k = k"
"match_FF\<cdot>TT\<cdot>k = fail"
"match_FF\<cdot>\<bottom>\<cdot>k = \<bottom>"
by (simp_all add: match_FF_def)
subsection \<open>Mutual recursion\<close>
text \<open>
The following rules are used to prove unfolding theorems from
fixed-point definitions of mutually recursive functions.
\<close>
lemma Pair_equalI: "\<lbrakk>x \<equiv> fst p; y \<equiv> snd p\<rbrakk> \<Longrightarrow> (x, y) \<equiv> p"
by simp
lemma Pair_eqD1: "(x, y) = (x', y') \<Longrightarrow> x = x'"
by simp
lemma Pair_eqD2: "(x, y) = (x', y') \<Longrightarrow> y = y'"
by simp
lemma def_cont_fix_eq:
"\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F\<rbrakk> \<Longrightarrow> f = F f"
by (simp, subst fix_eq, simp)
lemma def_cont_fix_ind:
"\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F; adm P; P \<bottom>; \<And>x. P x \<Longrightarrow> P (F x)\<rbrakk> \<Longrightarrow> P f"
by (simp add: fix_ind)
text \<open>lemma for proving rewrite rules\<close>
lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
by simp
subsection \<open>Initializing the fixrec package\<close>
ML_file \<open>Tools/holcf_library.ML\<close>
ML_file \<open>Tools/fixrec.ML\<close>
method_setup fixrec_simp = \<open>
Scan.succeed (SIMPLE_METHOD' o Fixrec.fixrec_simp_tac)
\<close> "pattern prover for fixrec constants"
setup \<open>
Fixrec.add_matchers
[ (\<^const_name>\<open>up\<close>, \<^const_name>\<open>match_up\<close>),
(\<^const_name>\<open>sinl\<close>, \<^const_name>\<open>match_sinl\<close>),
(\<^const_name>\<open>sinr\<close>, \<^const_name>\<open>match_sinr\<close>),
(\<^const_name>\<open>spair\<close>, \<^const_name>\<open>match_spair\<close>),
(\<^const_name>\<open>Pair\<close>, \<^const_name>\<open>match_Pair\<close>),
(\<^const_name>\<open>ONE\<close>, \<^const_name>\<open>match_ONE\<close>),
(\<^const_name>\<open>TT\<close>, \<^const_name>\<open>match_TT\<close>),
(\<^const_name>\<open>FF\<close>, \<^const_name>\<open>match_FF\<close>),
(\<^const_name>\<open>bottom\<close>, \<^const_name>\<open>match_bottom\<close>) ]
\<close>
hide_const (open) succeed fail run
end