src/HOLCF/Fixrec.thy

tightened notion of function equations

(*  Title:      HOLCF/Fixrec.thy
    ID:         $Id$
    Author:     Amber Telfer and Brian Huffman
*)

header "Package for defining recursive functions in HOLCF"

theory Fixrec
imports Sprod Ssum Up One Tr Fix
uses ("fixrec_package.ML")
begin

subsection {* Maybe monad type *}

defaultsort cpo

pcpodef (open) 'a maybe = "UNIV::(one ++ 'a u) set"
by simp

constdefs
  fail :: "'a maybe"
  "fail \<equiv> Abs_maybe (sinl\<cdot>ONE)"

  return :: "'a \<rightarrow> 'a maybe"
  "return \<equiv> \<Lambda> x. Abs_maybe (sinr\<cdot>(up\<cdot>x))"

  maybe_when :: "'b \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a maybe \<rightarrow> 'b::pcpo"
  "maybe_when \<equiv> \<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
                  cont_Abs_maybe Abs_maybe_inverse Rep_maybe_strict)

translations
  "case m of fail \<Rightarrow> t1 | return\<cdot>x \<Rightarrow> t2" == "maybe_when\<cdot>t1\<cdot>(\<Lambda> x. t2)\<cdot>m"


subsubsection {* Monadic bind operator *}

constdefs
  bind :: "'a maybe \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> 'b maybe"
  "bind \<equiv> \<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 *}

constdefs
  run:: "'a maybe \<rightarrow> 'a::pcpo"
  "run \<equiv> 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 *}

constdefs
  mplus :: "'a maybe \<rightarrow> 'a maybe \<rightarrow> 'a maybe"
  "mplus \<equiv> \<Lambda> m1 m2. case m1 of fail \<Rightarrow> m2 | return\<cdot>x \<Rightarrow> m1"

syntax "+++" :: "['a maybe, 'a maybe] \<Rightarrow> 'a maybe" (infixr "+++" 65)
translations "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 *}

constdefs
  fatbar :: "('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe)"
  "fatbar \<equiv> \<Lambda> a b x. a\<cdot>x +++ b\<cdot>x"

syntax
  "\<parallel>" :: "['a \<rightarrow> 'b maybe, 'a \<rightarrow> 'b maybe] \<Rightarrow> 'a \<rightarrow> 'b maybe" (infixr "\<parallel>" 60)
translations
  "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 *}

constdefs
  branch :: "('a \<rightarrow> 'b maybe) \<Rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'c maybe)"
  "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)


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" == "Fixrec.run\<cdot>(ms\<cdot>x)"
  "_Case2 m ms" == "m \<parallel> ms"

text {* Parsing Case expressions *}

syntax
  "_pat" :: "'a"
  "_var" :: "'a"

translations
  "_Case1 p r" => "branch (_pat p)\<cdot>(_var p r)"
  "_var (_args x y) r" => "csplit\<cdot>(_var x (_var y r))"
  "_var () r" => "unit_when\<cdot>r"

parse_translation {*
(* rewrites (_pat x) => (return) *)
(* rewrites (_var x t) => (Abs_CFun (%x. t)) *)
  [("_pat", K (Syntax.const "Fixrec.return")),
   mk_binder_tr ("_var", "Abs_CFun")];
*}

text {* Printing Case expressions *}

syntax
  "_match" :: "'a"

print_translation {*
  let
    fun dest_LAM (Const ("Rep_CFun",_) $ Const ("unit_when",_) $ t) =
          (Syntax.const "Unity", t)
    |   dest_LAM (Const ("Rep_CFun",_) $ Const ("csplit",_) $ t) =
          let
            val (v1, t1) = dest_LAM t;
            val (v2, t2) = dest_LAM t1;
          in (Syntax.const "_args" $ v1 $ v2, t2) end 
    |   dest_LAM (Const ("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 "_var" $ x, t') end
    |   dest_LAM _ = raise Match; (* too few vars: abort translation *)

    fun Case1_tr' [Const("branch",_) $ p, r] =
          let val (v, t) = dest_LAM r;
          in Syntax.const "_Case1" $ (Syntax.const "_match" $ p $ v) $ t end;

  in [("Rep_CFun", Case1_tr')] end;
*}

translations
  "x" <= "_match Fixrec.return (_var x)"


subsection {* Pattern combinators for data constructors *}

types ('a, 'b) pat = "'a \<rightarrow> 'b maybe"

constdefs
  cpair_pat :: "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a \<times> 'b, 'c \<times> 'd) pat"
  "cpair_pat p1 p2 \<equiv> \<Lambda>\<langle>x, y\<rangle>.
    bind\<cdot>(p1\<cdot>x)\<cdot>(\<Lambda> a. bind\<cdot>(p2\<cdot>y)\<cdot>(\<Lambda> b. return\<cdot>\<langle>a, b\<rangle>))"

  spair_pat ::
  "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a::pcpo \<otimes> 'b::pcpo, 'c \<times> 'd) pat"
  "spair_pat p1 p2 \<equiv> \<Lambda>(:x, y:). cpair_pat p1 p2\<cdot>\<langle>x, y\<rangle>"

  sinl_pat :: "('a, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat"
  "sinl_pat p \<equiv> sscase\<cdot>p\<cdot>(\<Lambda> x. fail)"

  sinr_pat :: "('b, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat"
  "sinr_pat p \<equiv> sscase\<cdot>(\<Lambda> x. fail)\<cdot>p"

  up_pat :: "('a, 'b) pat \<Rightarrow> ('a u, 'b) pat"
  "up_pat p \<equiv> fup\<cdot>p"

  TT_pat :: "(tr, unit) pat"
  "TT_pat \<equiv> \<Lambda> b. If b then return\<cdot>() else fail fi"

  FF_pat :: "(tr, unit) pat"
  "FF_pat \<equiv> \<Lambda> b. If b then fail else return\<cdot>() fi"

  ONE_pat :: "(one, unit) pat"
  "ONE_pat \<equiv> \<Lambda> ONE. return\<cdot>()"

text {* Parse translations (patterns) *}
translations
  "_pat (cpair\<cdot>x\<cdot>y)" => "cpair_pat (_pat x) (_pat y)"
  "_pat (spair\<cdot>x\<cdot>y)" => "spair_pat (_pat x) (_pat y)"
  "_pat (sinl\<cdot>x)" => "sinl_pat (_pat x)"
  "_pat (sinr\<cdot>x)" => "sinr_pat (_pat x)"
  "_pat (up\<cdot>x)" => "up_pat (_pat x)"
  "_pat TT" => "TT_pat"
  "_pat FF" => "FF_pat"
  "_pat ONE" => "ONE_pat"

text {* Parse translations (variables) *}
translations
  "_var (cpair\<cdot>x\<cdot>y) r" => "_var (_args x y) r"
  "_var (spair\<cdot>x\<cdot>y) r" => "_var (_args x y) r"
  "_var (sinl\<cdot>x) r" => "_var x r"
  "_var (sinr\<cdot>x) r" => "_var x r"
  "_var (up\<cdot>x) r" => "_var x r"
  "_var TT r" => "_var () r"
  "_var FF r" => "_var () r"
  "_var ONE r" => "_var () r"

text {* Print translations *}
translations
  "cpair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)"
      <= "_match (cpair_pat p1 p2) (_args v1 v2)"
  "spair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)"
      <= "_match (spair_pat p1 p2) (_args v1 v2)"
  "sinl\<cdot>(_match p1 v1)" <= "_match (sinl_pat p1) v1"
  "sinr\<cdot>(_match p1 v1)" <= "_match (sinr_pat p1) v1"
  "up\<cdot>(_match p1 v1)" <= "_match (up_pat p1) v1"
  "TT" <= "_match TT_pat ()"
  "FF" <= "_match FF_pat ()"
  "ONE" <= "_match ONE_pat ()"

lemma cpair_pat1:
  "branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = \<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>\<langle>x, y\<rangle> = 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>\<langle>x, y\<rangle> = 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>\<langle>x, y\<rangle>"
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 *}

syntax
  "_as_pat" :: "[idt, 'a] \<Rightarrow> 'a" (infixr "\<as>" 10)
  "_lazy_pat" :: "'a \<Rightarrow> 'a" ("\<lazy> _" [1000] 1000)

constdefs
  wild_pat :: "'a \<rightarrow> unit maybe"
  "wild_pat \<equiv> \<Lambda> x. return\<cdot>()"

  as_pat :: "('a \<rightarrow> 'b maybe) \<Rightarrow> 'a \<rightarrow> ('a \<times> 'b) maybe"
  "as_pat p \<equiv> \<Lambda> x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> a. return\<cdot>\<langle>x, a\<rangle>)"

  lazy_pat :: "('a \<rightarrow> 'b::pcpo maybe) \<Rightarrow> ('a \<rightarrow> 'b maybe)"
  "lazy_pat p \<equiv> \<Lambda> x. return\<cdot>(run\<cdot>(p\<cdot>x))"

text {* Parse translations (patterns) *}
translations
  "_pat _" => "wild_pat"
  "_pat (_as_pat x y)" => "as_pat (_pat y)"
  "_pat (_lazy_pat x)" => "lazy_pat (_pat x)"

text {* Parse translations (variables) *}
translations
  "_var _ r" => "_var () r"
  "_var (_as_pat x y) r" => "_var (_args x y) r"
  "_var (_lazy_pat x) r" => "_var x r"

text {* Print translations *}
translations
  "_" <= "_match wild_pat ()"
  "_as_pat x (_match p v)" <= "_match (as_pat p) (_args (_var x) v)"
  "_lazy_pat (_match p v)" <= "_match (lazy_pat p) v"

text {* Lazy patterns in lambda abstractions *}
translations
  "_cabs (_lazy_pat p) r" == "run oo (_Case1 (_lazy_pat p) r)"

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 *}

defaultsort pcpo

constdefs
  match_UU :: "'a \<rightarrow> unit maybe"
  "match_UU \<equiv> \<Lambda> x. fail"

  match_cpair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<times> 'b) maybe"
  "match_cpair \<equiv> csplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)"

  match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<times> 'b) maybe"
  "match_spair \<equiv> ssplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)"

  match_sinl :: "'a \<oplus> 'b \<rightarrow> 'a maybe"
  "match_sinl \<equiv> sscase\<cdot>return\<cdot>(\<Lambda> y. fail)"

  match_sinr :: "'a \<oplus> 'b \<rightarrow> 'b maybe"
  "match_sinr \<equiv> sscase\<cdot>(\<Lambda> x. fail)\<cdot>return"

  match_up :: "'a::cpo u \<rightarrow> 'a maybe"
  "match_up \<equiv> fup\<cdot>return"

  match_ONE :: "one \<rightarrow> unit maybe"
  "match_ONE \<equiv> \<Lambda> ONE. return\<cdot>()"
 
  match_TT :: "tr \<rightarrow> unit maybe"
  "match_TT \<equiv> \<Lambda> b. If b then return\<cdot>() else fail fi"
 
  match_FF :: "tr \<rightarrow> unit maybe"
  "match_FF \<equiv> \<Lambda> b. If b then fail else return\<cdot>() fi"

lemma match_UU_simps [simp]:
  "match_UU\<cdot>x = fail"
by (simp add: match_UU_def)

lemma match_cpair_simps [simp]:
  "match_cpair\<cdot><x,y> = return\<cdot><x,y>"
by (simp add: match_cpair_def)

lemma match_spair_simps [simp]:
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x,y:) = return\<cdot><x,y>"
  "match_spair\<cdot>\<bottom> = \<bottom>"
by (simp_all add: match_spair_def)

lemma match_sinl_simps [simp]:
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x) = return\<cdot>x"
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>x) = fail"
  "match_sinl\<cdot>\<bottom> = \<bottom>"
by (simp_all add: match_sinl_def)

lemma match_sinr_simps [simp]:
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>x) = return\<cdot>x"
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x) = fail"
  "match_sinr\<cdot>\<bottom> = \<bottom>"
by (simp_all add: match_sinr_def)

lemma match_up_simps [simp]:
  "match_up\<cdot>(up\<cdot>x) = return\<cdot>x"
  "match_up\<cdot>\<bottom> = \<bottom>"
by (simp_all add: match_up_def)

lemma match_ONE_simps [simp]:
  "match_ONE\<cdot>ONE = return\<cdot>()"
  "match_ONE\<cdot>\<bottom> = \<bottom>"
by (simp_all add: match_ONE_def)

lemma match_TT_simps [simp]:
  "match_TT\<cdot>TT = return\<cdot>()"
  "match_TT\<cdot>FF = fail"
  "match_TT\<cdot>\<bottom> = \<bottom>"
by (simp_all add: match_TT_def)

lemma match_FF_simps [simp]:
  "match_FF\<cdot>FF = return\<cdot>()"
  "match_FF\<cdot>TT = fail"
  "match_FF\<cdot>\<bottom> = \<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 cpair_equalI: "\<lbrakk>x \<equiv> cfst\<cdot>p; y \<equiv> csnd\<cdot>p\<rbrakk> \<Longrightarrow> <x,y> \<equiv> p"
by (simp add: surjective_pairing_Cprod2)

lemma cpair_eqD1: "<x,y> = <x',y'> \<Longrightarrow> x = x'"
by simp

lemma cpair_eqD2: "<x,y> = <x',y'> \<Longrightarrow> y = y'"
by simp

text {* lemma for proving rewrite rules *}

lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
by simp

ML {*
val cpair_equalI = thm "cpair_equalI";
val cpair_eqD1 = thm "cpair_eqD1";
val cpair_eqD2 = thm "cpair_eqD2";
val ssubst_lhs = thm "ssubst_lhs";
val branch_def = thm "branch_def";
*}

subsection {* Initializing the fixrec package *}

use "fixrec_package.ML"

hide (open) const return bind fail run

end