src/HOLCF/Fixrec.thy
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(*  Title:      HOLCF/Fixrec.thy
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    ID:         $Id$
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    Author:     Amber Telfer and Brian Huffman
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*)
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header "Package for defining recursive functions in HOLCF"
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theory Fixrec
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imports Sprod Ssum Up One Tr Fix
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uses ("Tools/fixrec_package.ML")
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begin
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subsection {* Maybe monad type *}
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defaultsort cpo
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pcpodef (open) 'a maybe = "UNIV::(one ++ 'a u) set"
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by simp
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constdefs
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  fail :: "'a maybe"
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  "fail \<equiv> Abs_maybe (sinl\<cdot>ONE)"
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constdefs
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  return :: "'a \<rightarrow> 'a maybe" where
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  "return \<equiv> \<Lambda> x. Abs_maybe (sinr\<cdot>(up\<cdot>x))"
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definition
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  maybe_when :: "'b \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a maybe \<rightarrow> 'b::pcpo" where
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  "maybe_when = (\<Lambda> f r m. sscase\<cdot>(\<Lambda> x. f)\<cdot>(fup\<cdot>r)\<cdot>(Rep_maybe m))"
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lemma maybeE:
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  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = return\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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apply (unfold fail_def return_def)
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apply (cases p, rename_tac r)
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apply (rule_tac p=r in ssumE, simp add: Abs_maybe_strict)
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apply (rule_tac p=x in oneE, simp, simp)
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apply (rule_tac p=y in upE, simp, simp add: cont_Abs_maybe)
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done
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lemma return_defined [simp]: "return\<cdot>x \<noteq> \<bottom>"
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by (simp add: return_def cont_Abs_maybe Abs_maybe_defined)
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lemma fail_defined [simp]: "fail \<noteq> \<bottom>"
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by (simp add: fail_def Abs_maybe_defined)
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lemma return_eq [simp]: "(return\<cdot>x = return\<cdot>y) = (x = y)"
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by (simp add: return_def cont_Abs_maybe Abs_maybe_inject)
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lemma return_neq_fail [simp]:
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  "return\<cdot>x \<noteq> fail" "fail \<noteq> return\<cdot>x"
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by (simp_all add: return_def fail_def cont_Abs_maybe Abs_maybe_inject)
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lemma maybe_when_rews [simp]:
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  "maybe_when\<cdot>f\<cdot>r\<cdot>\<bottom> = \<bottom>"
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  "maybe_when\<cdot>f\<cdot>r\<cdot>fail = f"
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  "maybe_when\<cdot>f\<cdot>r\<cdot>(return\<cdot>x) = r\<cdot>x"
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by (simp_all add: return_def fail_def maybe_when_def cont_Rep_maybe
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                  cont_Abs_maybe Abs_maybe_inverse Rep_maybe_strict)
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translations
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  "case m of fail \<Rightarrow> t1 | return\<cdot>x \<Rightarrow> t2" == "CONST maybe_when\<cdot>t1\<cdot>(\<Lambda> x. t2)\<cdot>m"
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subsubsection {* Monadic bind operator *}
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definition
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  bind :: "'a maybe \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> 'b maybe" where
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  "bind = (\<Lambda> m f. case m of fail \<Rightarrow> fail | return\<cdot>x \<Rightarrow> f\<cdot>x)"
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text {* monad laws *}
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lemma bind_strict [simp]: "bind\<cdot>\<bottom>\<cdot>f = \<bottom>"
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by (simp add: bind_def)
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lemma bind_fail [simp]: "bind\<cdot>fail\<cdot>f = fail"
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by (simp add: bind_def)
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lemma left_unit [simp]: "bind\<cdot>(return\<cdot>a)\<cdot>k = k\<cdot>a"
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by (simp add: bind_def)
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lemma right_unit [simp]: "bind\<cdot>m\<cdot>return = m"
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by (rule_tac p=m in maybeE, simp_all)
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lemma bind_assoc:
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 "bind\<cdot>(bind\<cdot>m\<cdot>k)\<cdot>h = bind\<cdot>m\<cdot>(\<Lambda> a. bind\<cdot>(k\<cdot>a)\<cdot>h)"
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by (rule_tac p=m in maybeE, simp_all)
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subsubsection {* Run operator *}
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definition
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  run:: "'a maybe \<rightarrow> 'a::pcpo" where
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  "run = maybe_when\<cdot>\<bottom>\<cdot>ID"
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text {* rewrite rules for run *}
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lemma run_strict [simp]: "run\<cdot>\<bottom> = \<bottom>"
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by (simp add: run_def)
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lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
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by (simp add: run_def)
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lemma run_return [simp]: "run\<cdot>(return\<cdot>x) = x"
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by (simp add: run_def)
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subsubsection {* Monad plus operator *}
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definition
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  mplus :: "'a maybe \<rightarrow> 'a maybe \<rightarrow> 'a maybe" where
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  "mplus = (\<Lambda> m1 m2. case m1 of fail \<Rightarrow> m2 | return\<cdot>x \<Rightarrow> m1)"
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abbreviation
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  mplus_syn :: "['a maybe, 'a maybe] \<Rightarrow> 'a maybe"  (infixr "+++" 65)  where
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  "m1 +++ m2 == mplus\<cdot>m1\<cdot>m2"
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text {* rewrite rules for mplus *}
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lemma mplus_strict [simp]: "\<bottom> +++ m = \<bottom>"
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by (simp add: mplus_def)
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lemma mplus_fail [simp]: "fail +++ m = m"
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by (simp add: mplus_def)
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lemma mplus_return [simp]: "return\<cdot>x +++ m = return\<cdot>x"
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by (simp add: mplus_def)
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lemma mplus_fail2 [simp]: "m +++ fail = m"
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by (rule_tac p=m in maybeE, simp_all)
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lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
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by (rule_tac p=x in maybeE, simp_all)
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subsubsection {* Fatbar combinator *}
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definition
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  fatbar :: "('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe) \<rightarrow> ('a \<rightarrow> 'b maybe)" where
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  "fatbar = (\<Lambda> a b x. a\<cdot>x +++ b\<cdot>x)"
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abbreviation
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  fatbar_syn :: "['a \<rightarrow> 'b maybe, 'a \<rightarrow> 'b maybe] \<Rightarrow> 'a \<rightarrow> 'b maybe" (infixr "\<parallel>" 60)  where
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  "m1 \<parallel> m2 == fatbar\<cdot>m1\<cdot>m2"
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lemma fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> (m \<parallel> ms)\<cdot>x = \<bottom>"
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by (simp add: fatbar_def)
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lemma fatbar2: "m\<cdot>x = fail \<Longrightarrow> (m \<parallel> ms)\<cdot>x = ms\<cdot>x"
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by (simp add: fatbar_def)
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lemma fatbar3: "m\<cdot>x = return\<cdot>y \<Longrightarrow> (m \<parallel> ms)\<cdot>x = return\<cdot>y"
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by (simp add: fatbar_def)
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lemmas fatbar_simps = fatbar1 fatbar2 fatbar3
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lemma run_fatbar1: "m\<cdot>x = \<bottom> \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = \<bottom>"
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by (simp add: fatbar_def)
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lemma run_fatbar2: "m\<cdot>x = fail \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = run\<cdot>(ms\<cdot>x)"
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by (simp add: fatbar_def)
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lemma run_fatbar3: "m\<cdot>x = return\<cdot>y \<Longrightarrow> run\<cdot>((m \<parallel> ms)\<cdot>x) = y"
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by (simp add: fatbar_def)
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lemmas run_fatbar_simps [simp] = run_fatbar1 run_fatbar2 run_fatbar3
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subsection {* Case branch combinator *}
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constdefs
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  branch :: "('a \<rightarrow> 'b maybe) \<Rightarrow> ('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'c maybe)"
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  "branch p \<equiv> \<Lambda> r x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> y. return\<cdot>(r\<cdot>y))"
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lemma branch_rews:
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  "p\<cdot>x = \<bottom> \<Longrightarrow> branch p\<cdot>r\<cdot>x = \<bottom>"
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  "p\<cdot>x = fail \<Longrightarrow> branch p\<cdot>r\<cdot>x = fail"
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  "p\<cdot>x = return\<cdot>y \<Longrightarrow> branch p\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>y)"
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by (simp_all add: branch_def)
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lemma branch_return [simp]: "branch return\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>x)"
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by (simp add: branch_def)
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subsection {* Case syntax *}
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nonterminals
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  Case_syn  Cases_syn
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syntax
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  "_Case_syntax":: "['a, Cases_syn] => 'b"               ("(Case _ of/ _)" 10)
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  "_Case1"      :: "['a, 'b] => Case_syn"                ("(2_ =>/ _)" 10)
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  ""            :: "Case_syn => Cases_syn"               ("_")
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  "_Case2"      :: "[Case_syn, Cases_syn] => Cases_syn"  ("_/ | _")
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syntax (xsymbols)
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  "_Case1"      :: "['a, 'b] => Case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
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translations
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  "_Case_syntax x ms" == "Fixrec.run\<cdot>(ms\<cdot>x)"
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  "_Case2 m ms" == "m \<parallel> ms"
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text {* Parsing Case expressions *}
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syntax
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  "_pat" :: "'a"
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  "_var" :: "'a"
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translations
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  "_Case1 p r" => "CONST branch (_pat p)\<cdot>(_var p r)"
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  "_var (_args x y) r" => "CONST csplit\<cdot>(_var x (_var y r))"
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  "_var () r" => "CONST unit_when\<cdot>r"
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parse_translation {*
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(* rewrites (_pat x) => (return) *)
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(* rewrites (_var x t) => (Abs_CFun (%x. t)) *)
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  [("_pat", K (Syntax.const "Fixrec.return")),
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   mk_binder_tr ("_var", @{const_syntax Abs_CFun})];
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*}
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text {* Printing Case expressions *}
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syntax
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  "_match" :: "'a"
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print_translation {*
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  let
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    fun dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax unit_when},_) $ t) =
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          (Syntax.const @{const_syntax Unity}, t)
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    |   dest_LAM (Const (@{const_syntax Rep_CFun},_) $ Const (@{const_syntax csplit},_) $ t) =
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          let
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            val (v1, t1) = dest_LAM t;
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            val (v2, t2) = dest_LAM t1;
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          in (Syntax.const "_args" $ v1 $ v2, t2) end 
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    |   dest_LAM (Const (@{const_syntax Abs_CFun},_) $ t) =
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          let
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            val abs = case t of Abs abs => abs
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                | _ => ("x", dummyT, incr_boundvars 1 t $ Bound 0);
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            val (x, t') = atomic_abs_tr' abs;
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          in (Syntax.const "_var" $ x, t') end
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    |   dest_LAM _ = raise Match; (* too few vars: abort translation *)
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    fun Case1_tr' [Const(@{const_syntax branch},_) $ p, r] =
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          let val (v, t) = dest_LAM r;
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          in Syntax.const "_Case1" $ (Syntax.const "_match" $ p $ v) $ t end;
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  in [(@{const_syntax Rep_CFun}, Case1_tr')] end;
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*}
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translations
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  "x" <= "_match Fixrec.return (_var x)"
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subsection {* Pattern combinators for data constructors *}
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types ('a, 'b) pat = "'a \<rightarrow> 'b maybe"
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definition
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  cpair_pat :: "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a \<times> 'b, 'c \<times> 'd) pat" where
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  "cpair_pat p1 p2 = (\<Lambda>\<langle>x, y\<rangle>.
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    bind\<cdot>(p1\<cdot>x)\<cdot>(\<Lambda> a. bind\<cdot>(p2\<cdot>y)\<cdot>(\<Lambda> b. return\<cdot>\<langle>a, b\<rangle>)))"
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definition
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  spair_pat ::
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  "('a, 'c) pat \<Rightarrow> ('b, 'd) pat \<Rightarrow> ('a::pcpo \<otimes> 'b::pcpo, 'c \<times> 'd) pat" where
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  "spair_pat p1 p2 = (\<Lambda>(:x, y:). cpair_pat p1 p2\<cdot>\<langle>x, y\<rangle>)"
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definition
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  sinl_pat :: "('a, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where
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  "sinl_pat p = sscase\<cdot>p\<cdot>(\<Lambda> x. fail)"
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definition
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  sinr_pat :: "('b, 'c) pat \<Rightarrow> ('a::pcpo \<oplus> 'b::pcpo, 'c) pat" where
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  "sinr_pat p = sscase\<cdot>(\<Lambda> x. fail)\<cdot>p"
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definition
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  up_pat :: "('a, 'b) pat \<Rightarrow> ('a u, 'b) pat" where
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  "up_pat p = fup\<cdot>p"
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definition
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  TT_pat :: "(tr, unit) pat" where
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  "TT_pat = (\<Lambda> b. If b then return\<cdot>() else fail fi)"
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definition
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  FF_pat :: "(tr, unit) pat" where
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  "FF_pat = (\<Lambda> b. If b then fail else return\<cdot>() fi)"
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definition
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  ONE_pat :: "(one, unit) pat" where
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  "ONE_pat = (\<Lambda> ONE. return\<cdot>())"
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text {* Parse translations (patterns) *}
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translations
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  "_pat (CONST cpair\<cdot>x\<cdot>y)" => "CONST cpair_pat (_pat x) (_pat y)"
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  "_pat (CONST spair\<cdot>x\<cdot>y)" => "CONST spair_pat (_pat x) (_pat y)"
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  "_pat (CONST sinl\<cdot>x)" => "CONST sinl_pat (_pat x)"
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  "_pat (CONST sinr\<cdot>x)" => "CONST sinr_pat (_pat x)"
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  "_pat (CONST up\<cdot>x)" => "CONST up_pat (_pat x)"
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  "_pat (CONST TT)" => "CONST TT_pat"
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  "_pat (CONST FF)" => "CONST FF_pat"
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  "_pat (CONST ONE)" => "CONST ONE_pat"
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text {* Parse translations (variables) *}
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translations
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  "_var (CONST cpair\<cdot>x\<cdot>y) r" => "_var (_args x y) r"
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  "_var (CONST spair\<cdot>x\<cdot>y) r" => "_var (_args x y) r"
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  "_var (CONST sinl\<cdot>x) r" => "_var x r"
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  "_var (CONST sinr\<cdot>x) r" => "_var x r"
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  "_var (CONST up\<cdot>x) r" => "_var x r"
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  "_var (CONST TT) r" => "_var () r"
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  "_var (CONST FF) r" => "_var () r"
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  "_var (CONST ONE) r" => "_var () r"
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text {* Print translations *}
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  "CONST cpair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)"
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      <= "_match (CONST cpair_pat p1 p2) (_args v1 v2)"
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  "CONST spair\<cdot>(_match p1 v1)\<cdot>(_match p2 v2)"
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      <= "_match (CONST spair_pat p1 p2) (_args v1 v2)"
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  "CONST sinl\<cdot>(_match p1 v1)" <= "_match (CONST sinl_pat p1) v1"
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  "CONST sinr\<cdot>(_match p1 v1)" <= "_match (CONST sinr_pat p1) v1"
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  "CONST up\<cdot>(_match p1 v1)" <= "_match (CONST up_pat p1) v1"
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  "CONST TT" <= "_match (CONST TT_pat) ()"
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  "CONST FF" <= "_match (CONST FF_pat) ()"
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  "CONST ONE" <= "_match (CONST ONE_pat) ()"
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lemma cpair_pat1:
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  "branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = \<bottom>"
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apply (simp add: branch_def cpair_pat_def)
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apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
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done
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lemma cpair_pat2:
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  "branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = fail"
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apply (simp add: branch_def cpair_pat_def)
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apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
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done
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lemma cpair_pat3:
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  "branch p\<cdot>r\<cdot>x = return\<cdot>s \<Longrightarrow>
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   branch (cpair_pat p q)\<cdot>(csplit\<cdot>r)\<cdot>\<langle>x, y\<rangle> = branch q\<cdot>s\<cdot>y"
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apply (simp add: branch_def cpair_pat_def)
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apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
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   340
apply (rule_tac p="q\<cdot>y" in maybeE, simp_all)
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done
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lemmas cpair_pat [simp] =
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  cpair_pat1 cpair_pat2 cpair_pat3
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lemma spair_pat [simp]:
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  "branch (spair_pat p1 p2)\<cdot>r\<cdot>\<bottom> = \<bottom>"
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  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk>
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     \<Longrightarrow> branch (spair_pat p1 p2)\<cdot>r\<cdot>(:x, y:) =
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         branch (cpair_pat p1 p2)\<cdot>r\<cdot>\<langle>x, y\<rangle>"
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by (simp_all add: branch_def spair_pat_def)
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   352
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lemma sinl_pat [simp]:
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  "branch (sinl_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>"
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  "x \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = branch p\<cdot>r\<cdot>x"
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   356
  "y \<noteq> \<bottom> \<Longrightarrow> branch (sinl_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = fail"
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   357
by (simp_all add: branch_def sinl_pat_def)
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   358
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lemma sinr_pat [simp]:
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  "branch (sinr_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>"
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   361
  "x \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinl\<cdot>x) = fail"
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   362
  "y \<noteq> \<bottom> \<Longrightarrow> branch (sinr_pat p)\<cdot>r\<cdot>(sinr\<cdot>y) = branch p\<cdot>r\<cdot>y"
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   363
by (simp_all add: branch_def sinr_pat_def)
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   364
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lemma up_pat [simp]:
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  "branch (up_pat p)\<cdot>r\<cdot>\<bottom> = \<bottom>"
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   367
  "branch (up_pat p)\<cdot>r\<cdot>(up\<cdot>x) = branch p\<cdot>r\<cdot>x"
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   368
by (simp_all add: branch_def up_pat_def)
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   369
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lemma TT_pat [simp]:
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   371
  "branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>"
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   372
  "branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = return\<cdot>r"
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   373
  "branch TT_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = fail"
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   374
by (simp_all add: branch_def TT_pat_def)
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   375
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lemma FF_pat [simp]:
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   377
  "branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>"
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   378
  "branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>TT = fail"
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   379
  "branch FF_pat\<cdot>(unit_when\<cdot>r)\<cdot>FF = return\<cdot>r"
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   380
by (simp_all add: branch_def FF_pat_def)
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   381
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   382
lemma ONE_pat [simp]:
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   383
  "branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>\<bottom> = \<bottom>"
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   384
  "branch ONE_pat\<cdot>(unit_when\<cdot>r)\<cdot>ONE = return\<cdot>r"
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   385
by (simp_all add: branch_def ONE_pat_def)
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   386
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   387
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   388
subsection {* Wildcards, as-patterns, and lazy patterns *}
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   389
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   390
syntax
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   391
  "_as_pat" :: "[idt, 'a] \<Rightarrow> 'a" (infixr "\<as>" 10)
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   392
  "_lazy_pat" :: "'a \<Rightarrow> 'a" ("\<lazy> _" [1000] 1000)
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   393
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   394
definition
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   395
  wild_pat :: "'a \<rightarrow> unit maybe" where
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   396
  "wild_pat = (\<Lambda> x. return\<cdot>())"
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   397
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   398
definition
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   399
  as_pat :: "('a \<rightarrow> 'b maybe) \<Rightarrow> 'a \<rightarrow> ('a \<times> 'b) maybe" where
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   400
  "as_pat p = (\<Lambda> x. bind\<cdot>(p\<cdot>x)\<cdot>(\<Lambda> a. return\<cdot>\<langle>x, a\<rangle>))"
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   401
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   402
definition
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   403
  lazy_pat :: "('a \<rightarrow> 'b::pcpo maybe) \<Rightarrow> ('a \<rightarrow> 'b maybe)" where
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   404
  "lazy_pat p = (\<Lambda> x. return\<cdot>(run\<cdot>(p\<cdot>x)))"
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   405
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   406
text {* Parse translations (patterns) *}
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diff changeset
   407
translations
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   408
  "_pat _" => "CONST wild_pat"
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   409
  "_pat (_as_pat x y)" => "CONST as_pat (_pat y)"
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   410
  "_pat (_lazy_pat x)" => "CONST lazy_pat (_pat x)"
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   411
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   412
text {* Parse translations (variables) *}
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   413
translations
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   414
  "_var _ r" => "_var () r"
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diff changeset
   415
  "_var (_as_pat x y) r" => "_var (_args x y) r"
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   416
  "_var (_lazy_pat x) r" => "_var x r"
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   417
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   418
text {* Print translations *}
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   419
translations
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   420
  "_" <= "_match (CONST wild_pat) ()"
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   421
  "_as_pat x (_match p v)" <= "_match (CONST as_pat p) (_args (_var x) v)"
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   422
  "_lazy_pat (_match p v)" <= "_match (CONST lazy_pat p) v"
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diff changeset
   423
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   424
text {* Lazy patterns in lambda abstractions *}
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diff changeset
   425
translations
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   426
  "_cabs (_lazy_pat p) r" == "CONST run oo (_Case1 (_lazy_pat p) r)"
19327
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diff changeset
   427
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diff changeset
   428
lemma wild_pat [simp]: "branch wild_pat\<cdot>(unit_when\<cdot>r)\<cdot>x = return\<cdot>r"
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diff changeset
   429
by (simp add: branch_def wild_pat_def)
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diff changeset
   430
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huffman
parents: 18112
diff changeset
   431
lemma as_pat [simp]:
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   432
  "branch (as_pat p)\<cdot>(csplit\<cdot>r)\<cdot>x = branch p\<cdot>(r\<cdot>x)\<cdot>x"
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   433
apply (simp add: branch_def as_pat_def)
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   434
apply (rule_tac p="p\<cdot>x" in maybeE, simp_all)
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   435
done
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   436
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   437
lemma lazy_pat [simp]:
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   438
  "branch p\<cdot>r\<cdot>x = \<bottom> \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>\<bottom>)"
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   439
  "branch p\<cdot>r\<cdot>x = fail \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>(r\<cdot>\<bottom>)"
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   440
  "branch p\<cdot>r\<cdot>x = return\<cdot>s \<Longrightarrow> branch (lazy_pat p)\<cdot>r\<cdot>x = return\<cdot>s"
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   441
apply (simp_all add: branch_def lazy_pat_def)
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   442
apply (rule_tac [!] p="p\<cdot>x" in maybeE, simp_all)
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   443
done
4eaa654c92f2 reimplement Case expression pattern matching to support lazy patterns
huffman
parents: 18112
diff changeset
   444
18112
dc1d6f588204 reimplemented Case syntax using print/parse translations; moved as-patterns to separate section
huffman
parents: 18110
diff changeset
   445
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   446
subsection {* Match functions for built-in types *}
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   447
16776
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   448
defaultsort pcpo
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   449
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   450
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   451
  match_UU :: "'a \<rightarrow> unit maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   452
  "match_UU = (\<Lambda> x. fail)"
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   453
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   454
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   455
  match_cpair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<times> 'b) maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   456
  "match_cpair = csplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)"
16776
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   457
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   458
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   459
  match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<times> 'b) maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   460
  "match_spair = ssplit\<cdot>(\<Lambda> x y. return\<cdot><x,y>)"
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   461
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   462
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   463
  match_sinl :: "'a \<oplus> 'b \<rightarrow> 'a maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   464
  "match_sinl = sscase\<cdot>return\<cdot>(\<Lambda> y. fail)"
16551
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   465
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   466
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   467
  match_sinr :: "'a \<oplus> 'b \<rightarrow> 'b maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   468
  "match_sinr = sscase\<cdot>(\<Lambda> x. fail)\<cdot>return"
16551
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   469
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   470
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   471
  match_up :: "'a::cpo u \<rightarrow> 'a maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   472
  "match_up = fup\<cdot>return"
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   473
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   474
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   475
  match_ONE :: "one \<rightarrow> unit maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   476
  "match_ONE = (\<Lambda> ONE. return\<cdot>())"
18094
404f298220af simplify definitions
huffman
parents: 16779
diff changeset
   477
 
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   478
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   479
  match_TT :: "tr \<rightarrow> unit maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   480
  "match_TT = (\<Lambda> b. If b then return\<cdot>() else fail fi)"
18094
404f298220af simplify definitions
huffman
parents: 16779
diff changeset
   481
 
25131
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   482
definition
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   483
  match_FF :: "tr \<rightarrow> unit maybe" where
2c8caac48ade modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents: 23152
diff changeset
   484
  "match_FF = (\<Lambda> b. If b then fail else return\<cdot>() fi)"
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   485
16776
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   486
lemma match_UU_simps [simp]:
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   487
  "match_UU\<cdot>x = fail"
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   488
by (simp add: match_UU_def)
a3899ac14a1c generalized types of monadic operators to class cpo; added match function for UU
huffman
parents: 16758
diff changeset
   489
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   490
lemma match_cpair_simps [simp]:
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   491
  "match_cpair\<cdot><x,y> = return\<cdot><x,y>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   492
by (simp add: match_cpair_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   493
16551
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   494
lemma match_spair_simps [simp]:
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   495
  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x,y:) = return\<cdot><x,y>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   496
  "match_spair\<cdot>\<bottom> = \<bottom>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   497
by (simp_all add: match_spair_def)
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   498
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   499
lemma match_sinl_simps [simp]:
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   500
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x) = return\<cdot>x"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   501
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>x) = fail"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   502
  "match_sinl\<cdot>\<bottom> = \<bottom>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   503
by (simp_all add: match_sinl_def)
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   504
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   505
lemma match_sinr_simps [simp]:
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   506
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>x) = return\<cdot>x"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   507
  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x) = fail"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   508
  "match_sinr\<cdot>\<bottom> = \<bottom>"
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   509
by (simp_all add: match_sinr_def)
7abf8a713613 added match functions for spair, sinl, sinr
huffman
parents: 16463
diff changeset
   510
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   511
lemma match_up_simps [simp]:
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   512
  "match_up\<cdot>(up\<cdot>x) = return\<cdot>x"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   513
  "match_up\<cdot>\<bottom> = \<bottom>"
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   514
by (simp_all add: match_up_def)
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   515
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   516
lemma match_ONE_simps [simp]:
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   517
  "match_ONE\<cdot>ONE = return\<cdot>()"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   518
  "match_ONE\<cdot>\<bottom> = \<bottom>"
18094
404f298220af simplify definitions
huffman
parents: 16779
diff changeset
   519
by (simp_all add: match_ONE_def)
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   520
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   521
lemma match_TT_simps [simp]:
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   522
  "match_TT\<cdot>TT = return\<cdot>()"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   523
  "match_TT\<cdot>FF = fail"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   524
  "match_TT\<cdot>\<bottom> = \<bottom>"
18094
404f298220af simplify definitions
huffman
parents: 16779
diff changeset
   525
by (simp_all add: match_TT_def)
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   526
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   527
lemma match_FF_simps [simp]:
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   528
  "match_FF\<cdot>FF = return\<cdot>()"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   529
  "match_FF\<cdot>TT = fail"
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   530
  "match_FF\<cdot>\<bottom> = \<bottom>"
18094
404f298220af simplify definitions
huffman
parents: 16779
diff changeset
   531
by (simp_all add: match_FF_def)
16460
72a08d509d62 added match functions for ONE, TT, and FF; added theorem mplus_fail2
huffman
parents: 16417
diff changeset
   532
16401
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   533
subsection {* Mutual recursion *}
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   534
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   535
text {*
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   536
  The following rules are used to prove unfolding theorems from
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   537
  fixed-point definitions of mutually recursive functions.
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   538
*}
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   539
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   540
lemma cpair_equalI: "\<lbrakk>x \<equiv> cfst\<cdot>p; y \<equiv> csnd\<cdot>p\<rbrakk> \<Longrightarrow> <x,y> \<equiv> p"
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   541
by (simp add: surjective_pairing_Cprod2)
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   542
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   543
lemma cpair_eqD1: "<x,y> = <x',y'> \<Longrightarrow> x = x'"
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   544
by simp
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   545
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   546
lemma cpair_eqD2: "<x,y> = <x',y'> \<Longrightarrow> y = y'"
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   547
by simp
57c35ede00b9 fixrec package now handles mutually-recursive definitions
huffman
parents: 16229
diff changeset
   548
16463
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   549
text {* lemma for proving rewrite rules *}
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   550
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   551
lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   552
by simp
342d74ca8815 fixrec shows unsolved subgoals when proofs of rewrites fail
huffman
parents: 16460
diff changeset
   553
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   554
16758
c32334d98fcd fix typo
huffman
parents: 16754
diff changeset
   555
subsection {* Initializing the fixrec package *}
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   556
23152
9497234a2743 moved HOLCF tools to canonical place;
wenzelm
parents: 19439
diff changeset
   557
use "Tools/fixrec_package.ML"
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   558
19439
27c2e4cd634b hide common name of constant 'run'
huffman
parents: 19396
diff changeset
   559
hide (open) const return bind fail run
19104
7d69b6d7b8f1 use qualified name for return
huffman
parents: 19092
diff changeset
   560
16221
879400e029bf New theory with lemmas for the fixrec package
huffman
parents:
diff changeset
   561
end