src/HOL/HOLCF/Fixrec.thy
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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/HOLCF/Fixrec.thy
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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 Plain_HOLCF
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keywords "fixrec" :: thy_decl
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uses
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  ("Tools/holcf_library.ML")
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  ("Tools/fixrec.ML")
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begin
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subsection {* Pattern-match monad *}
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default_sort cpo
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pcpodef (open) 'a match = "UNIV::(one ++ 'a u) set"
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by simp_all
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definition
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  fail :: "'a match" where
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  "fail = Abs_match (sinl\<cdot>ONE)"
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definition
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  succeed :: "'a \<rightarrow> 'a match" where
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  "succeed = (\<Lambda> x. Abs_match (sinr\<cdot>(up\<cdot>x)))"
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lemma matchE [case_names bottom fail succeed, cases type: match]:
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  "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; p = fail \<Longrightarrow> Q; \<And>x. p = succeed\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
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unfolding fail_def succeed_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_match_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_match)
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done
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lemma succeed_defined [simp]: "succeed\<cdot>x \<noteq> \<bottom>"
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by (simp add: succeed_def cont_Abs_match Abs_match_bottom_iff)
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lemma fail_defined [simp]: "fail \<noteq> \<bottom>"
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by (simp add: fail_def Abs_match_bottom_iff)
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lemma succeed_eq [simp]: "(succeed\<cdot>x = succeed\<cdot>y) = (x = y)"
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by (simp add: succeed_def cont_Abs_match Abs_match_inject)
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lemma succeed_neq_fail [simp]:
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  "succeed\<cdot>x \<noteq> fail" "fail \<noteq> succeed\<cdot>x"
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by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)
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subsubsection {* Run operator *}
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definition
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  run :: "'a match \<rightarrow> 'a::pcpo" where
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  "run = (\<Lambda> m. sscase\<cdot>\<bottom>\<cdot>(fup\<cdot>ID)\<cdot>(Rep_match m))"
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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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unfolding run_def
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by (simp add: cont_Rep_match Rep_match_strict)
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lemma run_fail [simp]: "run\<cdot>fail = \<bottom>"
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by (simp add: cont_Rep_match Abs_match_inverse)
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lemma run_succeed [simp]: "run\<cdot>(succeed\<cdot>x) = x"
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by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
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subsubsection {* Monad plus operator *}
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definition
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  mplus :: "'a match \<rightarrow> 'a match \<rightarrow> 'a match" where
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  "mplus = (\<Lambda> m1 m2. sscase\<cdot>(\<Lambda> _. m2)\<cdot>(\<Lambda> _. m1)\<cdot>(Rep_match m1))"
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abbreviation
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  mplus_syn :: "['a match, 'a match] \<Rightarrow> 'a match"  (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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unfolding mplus_def
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by (simp add: cont_Rep_match Rep_match_strict)
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lemma mplus_fail [simp]: "fail +++ m = m"
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unfolding mplus_def fail_def
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by (simp add: cont_Rep_match Abs_match_inverse)
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lemma mplus_succeed [simp]: "succeed\<cdot>x +++ m = succeed\<cdot>x"
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unfolding mplus_def succeed_def
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by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
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lemma mplus_fail2 [simp]: "m +++ fail = m"
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by (cases m, simp_all)
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lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
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by (cases x, simp_all)
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subsection {* Match functions for built-in types *}
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default_sort pcpo
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definition
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  match_bottom :: "'a \<rightarrow> 'c match \<rightarrow> 'c match"
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where
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  "match_bottom = (\<Lambda> x k. seq\<cdot>x\<cdot>fail)"
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definition
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  match_Pair :: "'a::cpo \<times> 'b::cpo \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
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where
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  "match_Pair = (\<Lambda> x k. csplit\<cdot>k\<cdot>x)"
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definition
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  match_spair :: "'a \<otimes> 'b \<rightarrow> ('a \<rightarrow> 'b \<rightarrow> 'c match) \<rightarrow> 'c match"
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where
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  "match_spair = (\<Lambda> x k. ssplit\<cdot>k\<cdot>x)"
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definition
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  match_sinl :: "'a \<oplus> 'b \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match"
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where
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  "match_sinl = (\<Lambda> x k. sscase\<cdot>k\<cdot>(\<Lambda> b. fail)\<cdot>x)"
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definition
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  match_sinr :: "'a \<oplus> 'b \<rightarrow> ('b \<rightarrow> 'c match) \<rightarrow> 'c match"
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where
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  "match_sinr = (\<Lambda> x k. sscase\<cdot>(\<Lambda> a. fail)\<cdot>k\<cdot>x)"
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definition
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  match_up :: "'a::cpo u \<rightarrow> ('a \<rightarrow> 'c match) \<rightarrow> 'c match"
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where
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  "match_up = (\<Lambda> x k. fup\<cdot>k\<cdot>x)"
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definition
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  match_ONE :: "one \<rightarrow> 'c match \<rightarrow> 'c match"
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where
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  "match_ONE = (\<Lambda> ONE k. k)"
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definition
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  match_TT :: "tr \<rightarrow> 'c match \<rightarrow> 'c match"
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where
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  "match_TT = (\<Lambda> x k. If x then k else fail)"
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definition
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  match_FF :: "tr \<rightarrow> 'c match \<rightarrow> 'c match"
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where
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  "match_FF = (\<Lambda> x k. If x then fail else k)"
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lemma match_bottom_simps [simp]:
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  "match_bottom\<cdot>x\<cdot>k = (if x = \<bottom> then \<bottom> else fail)"
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by (simp add: match_bottom_def)
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lemma match_Pair_simps [simp]:
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  "match_Pair\<cdot>(x, y)\<cdot>k = k\<cdot>x\<cdot>y"
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by (simp_all add: match_Pair_def)
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lemma match_spair_simps [simp]:
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  "\<lbrakk>x \<noteq> \<bottom>; y \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> match_spair\<cdot>(:x, y:)\<cdot>k = k\<cdot>x\<cdot>y"
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  "match_spair\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_spair_def)
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lemma match_sinl_simps [simp]:
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  "x \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinl\<cdot>x)\<cdot>k = k\<cdot>x"
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  "y \<noteq> \<bottom> \<Longrightarrow> match_sinl\<cdot>(sinr\<cdot>y)\<cdot>k = fail"
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  "match_sinl\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_sinl_def)
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lemma match_sinr_simps [simp]:
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  "x \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinl\<cdot>x)\<cdot>k = fail"
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  "y \<noteq> \<bottom> \<Longrightarrow> match_sinr\<cdot>(sinr\<cdot>y)\<cdot>k = k\<cdot>y"
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  "match_sinr\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_sinr_def)
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lemma match_up_simps [simp]:
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  "match_up\<cdot>(up\<cdot>x)\<cdot>k = k\<cdot>x"
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  "match_up\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_up_def)
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lemma match_ONE_simps [simp]:
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  "match_ONE\<cdot>ONE\<cdot>k = k"
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  "match_ONE\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_ONE_def)
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lemma match_TT_simps [simp]:
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  "match_TT\<cdot>TT\<cdot>k = k"
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  "match_TT\<cdot>FF\<cdot>k = fail"
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  "match_TT\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_TT_def)
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lemma match_FF_simps [simp]:
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  "match_FF\<cdot>FF\<cdot>k = k"
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  "match_FF\<cdot>TT\<cdot>k = fail"
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  "match_FF\<cdot>\<bottom>\<cdot>k = \<bottom>"
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by (simp_all add: match_FF_def)
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subsection {* Mutual recursion *}
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text {*
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  The following rules are used to prove unfolding theorems from
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  fixed-point definitions of mutually recursive functions.
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*}
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lemma Pair_equalI: "\<lbrakk>x \<equiv> fst p; y \<equiv> snd p\<rbrakk> \<Longrightarrow> (x, y) \<equiv> p"
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by simp
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lemma Pair_eqD1: "(x, y) = (x', y') \<Longrightarrow> x = x'"
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by simp
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lemma Pair_eqD2: "(x, y) = (x', y') \<Longrightarrow> y = y'"
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by simp
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lemma def_cont_fix_eq:
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  "\<lbrakk>f \<equiv> fix\<cdot>(Abs_cfun F); cont F\<rbrakk> \<Longrightarrow> f = F f"
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by (simp, subst fix_eq, simp)
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lemma def_cont_fix_ind:
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  "\<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"
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by (simp add: fix_ind)
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text {* lemma for proving rewrite rules *}
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lemma ssubst_lhs: "\<lbrakk>t = s; P s = Q\<rbrakk> \<Longrightarrow> P t = Q"
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by simp
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subsection {* Initializing the fixrec package *}
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use "Tools/holcf_library.ML"
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use "Tools/fixrec.ML"
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setup {* Fixrec.setup *}
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setup {*
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  Fixrec.add_matchers
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    [ (@{const_name up}, @{const_name match_up}),
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      (@{const_name sinl}, @{const_name match_sinl}),
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      (@{const_name sinr}, @{const_name match_sinr}),
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      (@{const_name spair}, @{const_name match_spair}),
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      (@{const_name Pair}, @{const_name match_Pair}),
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      (@{const_name ONE}, @{const_name match_ONE}),
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      (@{const_name TT}, @{const_name match_TT}),
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      (@{const_name FF}, @{const_name match_FF}),
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      (@{const_name bottom}, @{const_name match_bottom}) ]
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*}
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hide_const (open) succeed fail run
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end