(* Title: HOLCF/Fixrec.thy
Author: Amber Telfer and Brian Huffman
*)
header "Package for defining recursive functions in HOLCF"
theory Fixrec
imports Cprod Sprod Ssum Up One Tr Fix
uses
("Tools/holcf_library.ML")
("Tools/fixrec.ML")
begin
subsection {* Maybe monad type *}
default_sort cpo
pcpodef (open) 'a maybe = "UNIV::(one ++ 'a u) set"
by simp_all
definition
fail :: "'a maybe" where
"fail = Abs_maybe (sinl\<cdot>ONE)"
definition
return :: "'a \<rightarrow> 'a maybe" where
"return = (\<Lambda> x. Abs_maybe (sinr\<cdot>(up\<cdot>x)))"
definition
maybe_when :: "'b \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a maybe \<rightarrow> 'b::pcpo" where
"maybe_when = (\<Lambda> f r m. sscase\<cdot>(\<Lambda> x. f)\<cdot>(fup\<cdot>r)\<cdot>(Rep_maybe m))"
lemma maybeE:
"\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = return\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
apply (unfold fail_def return_def)
apply (cases p, rename_tac r)
apply (rule_tac p=r in ssumE, simp add: Abs_maybe_strict)
apply (rule_tac p=x in oneE, simp, simp)
apply (rule_tac p=y in upE, simp, simp add: cont_Abs_maybe)
done
lemma return_defined [simp]: "return\<cdot>x \<noteq> \<bottom>"
by (simp add: return_def cont_Abs_maybe Abs_maybe_defined)
lemma fail_defined [simp]: "fail \<noteq> \<bottom>"
by (simp add: fail_def Abs_maybe_defined)
lemma return_eq [simp]: "(return\<cdot>x = return\<cdot>y) = (x = y)"
by (simp add: return_def cont_Abs_maybe Abs_maybe_inject)
lemma return_neq_fail [simp]:
"return\<cdot>x \<noteq> fail" "fail \<noteq> return\<cdot>x"
by (simp_all add: return_def fail_def cont_Abs_maybe Abs_maybe_inject)
lemma maybe_when_rews [simp]:
"maybe_when\<cdot>f\<cdot>r\<cdot>\<bottom> = \<bottom>"
"maybe_when\<cdot>f\<cdot>r\<cdot>fail = f"
"maybe_when\<cdot>f\<cdot>r\<cdot>(return\<cdot>x) = r\<cdot>x"
by (simp_all add: return_def fail_def maybe_when_def cont_Rep_maybe
cont2cont_LAM
cont_Abs_maybe Abs_maybe_inverse Rep_maybe_strict)
translations
"case m of XCONST fail \<Rightarrow> t1 | XCONST return\<cdot>x \<Rightarrow> t2"
== "CONST maybe_when\<cdot>t1\<cdot>(\<Lambda> x. t2)\<cdot>m"
subsubsection {* Monadic bind operator *}
definition
bind :: "'a maybe \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> 'b maybe" where
"bind = (\<Lambda> m f. case m of fail \<Rightarrow> fail | return\<cdot>x \<Rightarrow> f\<cdot>x)"
text {* monad laws *}
lemma bind_strict [simp]: "bind\<cdot>\<bottom>\<cdot>f = \<bottom>"
by (simp add: bind_def)
lemma bind_fail [simp]: "bind\<cdot>fail\<cdot>f = fail"
by (simp add: bind_def)
lemma left_unit [simp]: "bind\<cdot>(return\<cdot>a)\<cdot>k = k\<cdot>a"
by (simp add: bind_def)
lemma right_unit [simp]: "bind\<cdot>m\<cdot>return = m"
by (rule_tac p=m in maybeE, simp_all)
lemma bind_assoc:
"bind\<cdot>(bind\<cdot>m\<cdot>k)\<cdot>h = bind\<cdot>m\<cdot>(\<Lambda> a. bind\<cdot>(k\<cdot>a)\<cdot>h)"
by (rule_tac p=m in maybeE, simp_all)
subsubsection {* Run operator *}
definition
run :: "'a maybe \<rightarrow> 'a::pcpo" where
"run = maybe_when\<cdot>\<bottom>\<cdot>ID"
text {* rewrite rules for run *}
lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
by (simp add: run_def)
lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
by (simp add: run_def)
lemma run_return [simp]: "run\<cdot>(return\<cdot>x) = x"
by (simp add: run_def)
subsubsection {* Monad plus operator *}
definition
mplus :: "'a maybe \<rightarrow> 'a maybe \<rightarrow> 'a maybe" where
"mplus = (\<Lambda> m1 m2. case m1 of fail \<Rightarrow> m2 | return\<cdot>x \<Rightarrow> m1)"
abbreviation
mplus_syn :: "['a maybe, 'a maybe] \<Rightarrow> 'a maybe" (infixr "+++" 65) where
"m1 +++ m2 == mplus\<cdot>m1\<cdot>m2"
text {* rewrite rules for mplus *}
lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
by (simp add: mplus_def)
lemma mplus_fail [simp]: "fail +++ m = m"
by (simp add: mplus_def)
lemma mplus_return [simp]: "return\<cdot>x +++ m = return\<cdot>x"
by (simp add: mplus_def)
lemma mplus_fail2 [simp]: "m +++ fail = m"
by (rule_tac p=m in maybeE, simp_all)
lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
by (rule_tac p=x in maybeE, simp_all)
subsubsection {* Fatbar combinator *}
definition
fatbar :: "('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe)" where
"fatbar = (\<Lambda> a b x. a\<cdot>x +++ b\<cdot>x)"
abbreviation
fatbar_syn :: "['a \<rightarrow> 'b maybe, 'a \<rightarrow> 'b maybe] \<Rightarrow> 'a \<rightarrow> 'b maybe" (infixr "\<parallel>" 60) where
"m1 \<parallel> m2 == fatbar\<cdot>m1\<cdot>m2"
lemma fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> (m \<parallel> ms)\<cdot>x = \<bottom>"
by (simp add: fatbar_def)
lemma fatbar2: "m\<cdot>x = fail \<Longrightarrow> (m \<parallel> ms)\<cdot>x = ms\<cdot>x"
by (simp add: fatbar_def)
lemma fatbar3: "m\<cdot>x = return\<cdot>y \<Longrightarrow> (m \<parallel> ms)\<cdot>x = return\<cdot>y"
by (simp add: fatbar_def)
lemmas fatbar_simps = fatbar1 fatbar2 fatbar3
lemma run_fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = \<bottom>"
by (simp add: fatbar_def)
lemma run_fatbar2: "m\<cdot>x = fail \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = run\<cdot>(ms\<cdot>x)"
by (simp add: fatbar_def)
lemma run_fatbar3: "m\<cdot>x = return\<cdot>y \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = y"
by (simp add: fatbar_def)
lemmas run_fatbar_simps [simp] = run_fatbar1 run_fatbar2 run_fatbar3
subsection {* Case branch combinator *}
definition
branch :: "('a \<rightarrow> 'b maybe) \<Rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'c maybe)" where
"branch p \<equiv> \<Lambda> r x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> y. return\<cdot>(r\<cdot>y))"
lemma branch_rews:
"p\<cdot>x = \<bottom> \<Longrightarrow> branch p\<cdot>r\<cdot>x = \<bottom>"
"p\<cdot>x = fail \<Longrightarrow> branch p\<cdot>r\<cdot>x = fail"
"p\<cdot>x = return\<cdot>y \<Longrightarrow> branch p\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>y)"
by (simp_all add: branch_def)
lemma branch_return [simp]: "branch return\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>x)"
by (simp add: branch_def)
subsubsection {* Cases operator *}
definition
cases :: "'a maybe \<rightarrow> 'a::pcpo" where
"cases = maybe_when\<cdot>\<bottom>\<cdot>ID"
text {* rewrite rules for cases *}
lemma cases_strict [simp]: "cases\<cdot>\<bottom> = \<bottom>"
by (simp add: cases_def)
lemma cases_fail [simp]: "cases\<cdot>fail = \<bottom>"
by (simp add: cases_def)
lemma cases_return [simp]: "cases\<cdot>(return\<cdot>x) = x"
by (simp add: cases_def)
subsection {* Case syntax *}
nonterminals
Case_syn Cases_syn
syntax
"_Case_syntax":: "['a, Cases_syn] => 'b" ("(Case _ of/ _)" 10)
"_Case1" :: "['a, 'b] => Case_syn" ("(2_ =>/ _)" 10)
"" :: "Case_syn => Cases_syn" ("_")
"_Case2" :: "[Case_syn, Cases_syn] => Cases_syn" ("_/ | _")
syntax (xsymbols)
"_Case1" :: "['a, 'b] => Case_syn" ("(2_ \<Rightarrow>/ _)" 10)
translations
"_Case_syntax x ms" == "CONST Fixrec.cases\<cdot>(ms\<cdot>x)"
"_Case2 m ms" == "m \<parallel> ms"
text {* Parsing Case expressions *}
syntax
"_pat" :: "'a"
"_variable" :: "'a"
"_noargs" :: "'a"
translations
"_Case1 p r" => "CONST branch (_pat p)\<cdot>(_variable p r)"
"_variable (_args x y) r" => "CONST csplit\<cdot>(_variable x (_variable y r))"
"_variable _noargs r" => "CONST unit_when\<cdot>r"
parse_translation {*
(* rewrite (_pat x) => (return) *)
(* rewrite (_variable x t) => (Abs_CFun (%x. t)) *)
[(@{syntax_const "_pat"}, fn _ => Syntax.const @{const_syntax Fixrec.return}),
mk_binder_tr (@{syntax_const "_variable"}, @{const_syntax Abs_CFun})];
*}
text {* Printing Case expressions *}
syntax
"_match" :: "'a"
print_translation {*
let
fun dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax unit_when},_) $ t) =
(Syntax.const @{syntax_const "_noargs"}, t)
| dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax csplit},_) $ t) =
let
val (v1, t1) = dest_LAM t;
val (v2, t2) = dest_LAM t1;
in (Syntax.const @{syntax_const "_args"} $ v1 $ v2, t2) end
| dest_LAM (Const (@{const_syntax Abs_CFun},_) $ t) =
let
val abs =
case t of Abs abs => abs
| _ => ("x", dummyT, incr_boundvars 1 t $ Bound 0);
val (x, t') = atomic_abs_tr' abs;
in (Syntax.const @{syntax_const "_variable"} $ x, t') end
| dest_LAM _ = raise Match; (* too few vars: abort translation *)
fun Case1_tr' [Const(@{const_syntax branch},_) $ p, r] =
let val (v, t) = dest_LAM r in
Syntax.const @{syntax_const "_Case1"} $
(Syntax.const @{syntax_const "_match"} $ p $ v) $ t
end;
in [(@{const_syntax Rep_CFun}, Case1_tr')] end;
*}
translations
"x" <= "_match (CONST Fixrec.return) (_variable x)"
subsection {* Pattern combinators for data constructors *}
types ('a, 'b) pat = "'a \<rightarrow> 'b maybe"
definition
cpair_pat :: "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a \<times> 'b, 'c \<times> 'd) pat" where
"cpair_pat p1 p2 = (\<Lambda>(x, y).
bind\<cdot>(p1\<cdot>x)\<cdot>(\<Lambda> a. bind\<cdot>(p2\<cdot>y)\<cdot>(\<Lambda> b. return\<cdot>(a, b))))"
definition
spair_pat ::
"('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a::pcpo \<otimes> 'b::pcpo, 'c \<times> 'd) pat" where
"spair_pat p1 p2 = (\<Lambda>(:x, y:). cpair_pat p1 p2\<cdot>(x, y))"
definition
sinl_pat :: "('a, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where
"sinl_pat p = sscase\<cdot>p\<cdot>(\<Lambda> x. fail)"
definition
sinr_pat :: "('b, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where
"sinr_pat p = sscase\<cdot>(\<Lambda> x. fail)\<cdot>p"
definition
up_pat :: "('a, 'b) pat \<Rightarrow> ('a u, 'b) pat" where
"up_pat p = fup\<cdot>p"
definition
TT_pat :: "(tr, unit) pat" where
"TT_pat = (\<Lambda> b. If b then return\<cdot>() else fail fi)"
definition
FF_pat :: "(tr, unit) pat" where
"FF_pat = (\<Lambda> b. If b then fail else return\<cdot>() fi)"
definition
ONE_pat :: "(one, unit) pat" where
"ONE_pat = (\<Lambda> ONE. return\<cdot>())"
text {* Parse translations (patterns) *}
translations
"_pat (XCONST Pair x y)" => "CONST cpair_pat (_pat x) (_pat y)"
"_pat (XCONST spair\<cdot>x\<cdot>y)" => "CONST spair_pat (_pat x) (_pat y)"
"_pat (XCONST sinl\<cdot>x)" => "CONST sinl_pat (_pat x)"
"_pat (XCONST sinr\<cdot>x)" => "CONST sinr_pat (_pat x)"
"_pat (XCONST up\<cdot>x)" => "CONST up_pat (_pat x)"
"_pat (XCONST TT)" => "CONST TT_pat"
"_pat (XCONST FF)" => "CONST FF_pat"
"_pat (XCONST ONE)" => "CONST ONE_pat"
text {* CONST version is also needed for constructors with special syntax *}
translations
"_pat (CONST Pair x y)" => "CONST cpair_pat (_pat x) (_pat y)"
"_pat (CONST spair\<cdot>x\<cdot>y)" => "CONST spair_pat (_pat x) (_pat y)"
text {* Parse translations (variables) *}
translations
"_variable (XCONST Pair x y) r" => "_variable (_args x y) r"
"_variable (XCONST spair\<cdot>x\<cdot>y) r" => "_variable (_args x y) r"
"_variable (XCONST sinl\<cdot>x) r" => "_variable x r"
"_variable (XCONST sinr\<cdot>x) r" => "_variable x r"
"_variable (XCONST up\<cdot>x) r" => "_variable x r"
"_variable (XCONST TT) r" => "_variable _noargs r"
"_variable (XCONST FF) r" => "_variable _noargs r"
"_variable (XCONST ONE) r" => "_variable _noargs r"
translations
"_variable (CONST Pair x y) r" => "_variable (_args x y) r"
"_variable (CONST spair\<cdot>x\<cdot>y) r" => "_variable (_args x y) r"
text {* Print translations *}
translations
"CONST Pair (_match p1 v1) (_match p2 v2)"
<= "_match (CONST cpair_pat p1 p2) (_args v1 v2)"
"CONST spair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)"
<= "_match (CONST spair_pat p1 p2) (_args v1 v2)"
"CONST sinl\<cdot>(_match p1 v1)" <= "_match (CONST sinl_pat p1) v1"
"CONST sinr\<cdot>(_match p1 v1)" <= "_match (CONST sinr_pat p1) v1"
"CONST up\<cdot>(_match p1 v1)" <= "_match (CONST up_pat p1) v1"
"CONST TT" <= "_match (CONST TT_pat) _noargs"
"CONST FF" <= "_match (CONST FF_pat) _noargs"
"CONST ONE" <= "_match (CONST ONE_pat) _noargs"
lemma cpair_pat1:
"branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>(x, y) = \<bottom>"
apply (simp add: branch_def cpair_pat_def)
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
done
lemma cpair_pat2:
"branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>(x, y) = fail"
apply (simp add: branch_def cpair_pat_def)
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
done
lemma cpair_pat3:
"branch p\<cdot>r\<cdot>x = return\<cdot>s \<Longrightarrow>
branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>(x, y) = branch q\<cdot>s\<cdot>y"
apply (simp add: branch_def cpair_pat_def)
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
apply (rule_tac p="q\<cdot>y" in maybeE, simp_all)
done
lemmas cpair_pat [simp] =
cpair_pat1 cpair_pat2 cpair_pat3
lemma spair_pat [simp]:
"branch (spair_pat p1 p2)\<cdot>r\<cdot>\<bottom> = \<bottom>"
"\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk>
\<Longrightarrow> branch (spair_pat p1 p2)\<cdot>r\<cdot>(:x, y:) =
branch (cpair_pat p1 p2)\<cdot>r\<cdot>(x, y)"
by (simp_all add: branch_def spair_pat_def)
lemma sinl_pat [simp]:
"branch (sinl_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>"
"x \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = branch p\<cdot>r\<cdot>x"
"y \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = fail"
by (simp_all add: branch_def sinl_pat_def)
lemma sinr_pat [simp]:
"branch (sinr_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>"
"x \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = fail"
"y \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = branch p\<cdot>r\<cdot>y"
by (simp_all add: branch_def sinr_pat_def)
lemma up_pat [simp]:
"branch (up_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>"
"branch (up_pat p)\<cdot>r\<cdot>(up\<cdot>x) = branch p\<cdot>r\<cdot>x"
by (simp_all add: branch_def up_pat_def)
lemma TT_pat [simp]:
"branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>"
"branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = return\<cdot>r"
"branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = fail"
by (simp_all add: branch_def TT_pat_def)
lemma FF_pat [simp]:
"branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>"
"branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = fail"
"branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = return\<cdot>r"
by (simp_all add: branch_def FF_pat_def)
lemma ONE_pat [simp]:
"branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>"
"branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>ONE = return\<cdot>r"
by (simp_all add: branch_def ONE_pat_def)
subsection {* Wildcards, as-patterns, and lazy patterns *}
definition
wild_pat :: "'a \<rightarrow> unit maybe" where
"wild_pat = (\<Lambda> x. return\<cdot>())"
definition
as_pat :: "('a \<rightarrow> 'b maybe) \<Rightarrow> 'a \<rightarrow> ('a \<times> 'b) maybe" where
"as_pat p = (\<Lambda> x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> a. return\<cdot>(x, a)))"
definition
lazy_pat :: "('a \<rightarrow> 'b::pcpo maybe) \<Rightarrow> ('a \<rightarrow> 'b maybe)" where
"lazy_pat p = (\<Lambda> x. return\<cdot>(cases\<cdot>(p\<cdot>x)))"
text {* Parse translations (patterns) *}
translations
"_pat _" => "CONST wild_pat"
text {* Parse translations (variables) *}
translations
"_variable _ r" => "_variable _noargs r"
text {* Print translations *}
translations
"_" <= "_match (CONST wild_pat) _noargs"
lemma wild_pat [simp]: "branch wild_pat\<cdot>(unit_when\<cdot>r)\<cdot>x = return\<cdot>r"
by (simp add: branch_def wild_pat_def)
lemma as_pat [simp]:
"branch (as_pat p)\<cdot>(csplit\<cdot>r)\<cdot>x = branch p\<cdot>(r\<cdot>x)\<cdot>x"
apply (simp add: branch_def as_pat_def)
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
done
lemma lazy_pat [simp]:
"branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>\<bottom>)"
"branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>\<bottom>)"
"branch p\<cdot>r\<cdot>x = return\<cdot>s \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>s"
apply (simp_all add: branch_def lazy_pat_def)
apply (rule_tac [!] p="p\<cdot>x" in maybeE, simp_all)
done
subsection {* Match functions for built-in types *}
default_sort pcpo
definition
match_UU :: "'a \<rightarrow> 'c maybe \<rightarrow> 'c maybe"
where
"match_UU = strictify\<cdot>(\<Lambda> x k. fail)"
definition
match_cpair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c maybe) \<rightarrow> 'c maybe"
where
"match_cpair = (\<Lambda> x k. csplit\<cdot>k\<cdot>x)"
definition
match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c maybe) \<rightarrow> 'c maybe"
where
"match_spair = (\<Lambda> x k. ssplit\<cdot>k\<cdot>x)"
definition
match_sinl :: "'a \<oplus> 'b \<rightarrow> ('a \<rightarrow> 'c maybe) \<rightarrow> 'c maybe"
where
"match_sinl = (\<Lambda> x k. sscase\<cdot>k\<cdot>(\<Lambda> b. fail)\<cdot>x)"
definition
match_sinr :: "'a \<oplus> 'b \<rightarrow> ('b \<rightarrow> 'c maybe) \<rightarrow> 'c maybe"
where
"match_sinr = (\<Lambda> x k. sscase\<cdot>(\<Lambda> a. fail)\<cdot>k\<cdot>x)"
definition
match_up :: "'a::cpo u \<rightarrow> ('a \<rightarrow> 'c maybe) \<rightarrow> 'c maybe"
where
"match_up = (\<Lambda> x k. fup\<cdot>k\<cdot>x)"
definition
match_ONE :: "one \<rightarrow> 'c maybe \<rightarrow> 'c maybe"
where
"match_ONE = (\<Lambda> ONE k. k)"
definition
match_TT :: "tr \<rightarrow> 'c maybe \<rightarrow> 'c maybe"
where
"match_TT = (\<Lambda> x k. If x then k else fail fi)"
definition
match_FF :: "tr \<rightarrow> 'c maybe \<rightarrow> 'c maybe"
where
"match_FF = (\<Lambda> x k. If x then fail else k fi)"
lemma match_UU_simps [simp]:
"match_UU\<cdot>\<bottom>\<cdot>k = \<bottom>"
"x \<noteq> \<bottom> \<Longrightarrow> match_UU\<cdot>x\<cdot>k = fail"
by (simp_all add: match_UU_def)
lemma match_cpair_simps [simp]:
"match_cpair\<cdot>(x, y)\<cdot>k = k\<cdot>x\<cdot>y"
by (simp_all add: match_cpair_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 {* Mutual recursion *}
text {*
The following rules are used to prove unfolding theorems from
fixed-point definitions of mutually recursive functions.
*}
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 {* lemma for proving rewrite rules *}
lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
by simp
subsection {* Initializing the fixrec package *}
use "Tools/holcf_library.ML"
use "Tools/fixrec.ML"
setup {* Fixrec.setup *}
setup {*
Fixrec.add_matchers
[ (@{const_name up}, @{const_name match_up}),
(@{const_name sinl}, @{const_name match_sinl}),
(@{const_name sinr}, @{const_name match_sinr}),
(@{const_name spair}, @{const_name match_spair}),
(@{const_name Pair}, @{const_name match_cpair}),
(@{const_name ONE}, @{const_name match_ONE}),
(@{const_name TT}, @{const_name match_TT}),
(@{const_name FF}, @{const_name match_FF}),
(@{const_name UU}, @{const_name match_UU}) ]
*}
hide_const (open) return bind fail run cases
lemmas [fixrec_simp] =
run_strict run_fail run_return
mplus_strict mplus_fail mplus_return
spair_strict_iff
sinl_defined_iff
sinr_defined_iff
up_defined
ONE_defined
dist_eq_tr(1,2)
match_UU_simps
match_cpair_simps
match_spair_simps
match_sinl_simps
match_sinr_simps
match_up_simps
match_ONE_simps
match_TT_simps
match_FF_simps
end