author krauss Sat Oct 23 23:41:19 2010 +0200 (2010-10-23) changeset 40107 374f3ef9f940 parent 40106 c58951943cba child 40108 dbab949c2717
first version of partial_function package
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Partial_Function.thy	Sat Oct 23 23:41:19 2010 +0200
1.3 @@ -0,0 +1,247 @@
1.4 +(* Title:    HOL/Partial_Function.thy
1.5 +   Author:   Alexander Krauss, TU Muenchen
1.6 +*)
1.7 +
1.8 +header {* Partial Function Definitions *}
1.9 +
1.10 +theory Partial_Function
1.11 +imports Complete_Partial_Order Option
1.12 +uses
1.13 +  "Tools/Function/function_lib.ML"
1.14 +  "Tools/Function/partial_function.ML"
1.15 +begin
1.16 +
1.17 +setup Partial_Function.setup
1.18 +
1.19 +subsection {* Axiomatic setup *}
1.20 +
1.21 +text {* This techical locale constains the requirements for function
1.22 +  definitions with ccpo fixed points.  *}
1.23 +
1.24 +definition "fun_ord ord f g \<longleftrightarrow> (\<forall>x. ord (f x) (g x))"
1.25 +definition "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
1.26 +definition "img_ord f ord = (\<lambda>x y. ord (f x) (f y))"
1.27 +definition "img_lub f g Lub = (\<lambda>A. g (Lub (f ` A)))"
1.28 +
1.29 +lemma call_mono[partial_function_mono]: "monotone (fun_ord ord) ord (\<lambda>f. f t)"
1.30 +by (rule monotoneI) (auto simp: fun_ord_def)
1.31 +
1.32 +lemma if_mono[partial_function_mono]: "monotone orda ordb F
1.33 +\<Longrightarrow> monotone orda ordb G
1.34 +\<Longrightarrow> monotone orda ordb (\<lambda>f. if c then F f else G f)"
1.35 +unfolding monotone_def by simp
1.36 +
1.37 +definition "mk_less R = (\<lambda>x y. R x y \<and> \<not> R y x)"
1.38 +
1.39 +locale partial_function_definitions =
1.40 +  fixes leq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
1.41 +  fixes lub :: "'a set \<Rightarrow> 'a"
1.42 +  assumes leq_refl: "leq x x"
1.43 +  assumes leq_trans: "leq x y \<Longrightarrow> leq y z \<Longrightarrow> leq x z"
1.44 +  assumes leq_antisym: "leq x y \<Longrightarrow> leq y x \<Longrightarrow> x = y"
1.45 +  assumes lub_upper: "chain leq A \<Longrightarrow> x \<in> A \<Longrightarrow> leq x (lub A)"
1.46 +  assumes lub_least: "chain leq A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> leq x z) \<Longrightarrow> leq (lub A) z"
1.47 +
1.48 +lemma partial_function_lift:
1.49 +  assumes "partial_function_definitions ord lb"
1.50 +  shows "partial_function_definitions (fun_ord ord) (fun_lub lb)" (is "partial_function_definitions ?ordf ?lubf")
1.51 +proof -
1.52 +  interpret partial_function_definitions ord lb by fact
1.53 +
1.54 +  { fix A a assume A: "chain ?ordf A"
1.55 +    have "chain ord {y. \<exists>f\<in>A. y = f a}" (is "chain ord ?C")
1.56 +    proof (rule chainI)
1.57 +      fix x y assume "x \<in> ?C" "y \<in> ?C"
1.58 +      then obtain f g where fg: "f \<in> A" "g \<in> A"
1.59 +        and [simp]: "x = f a" "y = g a" by blast
1.60 +      from chainD[OF A fg]
1.61 +      show "ord x y \<or> ord y x" unfolding fun_ord_def by auto
1.62 +    qed }
1.63 +  note chain_fun = this
1.64 +
1.65 +  show ?thesis
1.66 +  proof
1.67 +    fix x show "?ordf x x"
1.68 +      unfolding fun_ord_def by (auto simp: leq_refl)
1.69 +  next
1.70 +    fix x y z assume "?ordf x y" "?ordf y z"
1.71 +    thus "?ordf x z" unfolding fun_ord_def
1.72 +      by (force dest: leq_trans)
1.73 +  next
1.74 +    fix x y assume "?ordf x y" "?ordf y x"
1.75 +    thus "x = y" unfolding fun_ord_def
1.76 +      by (force intro!: ext dest: leq_antisym)
1.77 +  next
1.78 +    fix A f assume f: "f \<in> A" and A: "chain ?ordf A"
1.79 +    thus "?ordf f (?lubf A)"
1.80 +      unfolding fun_lub_def fun_ord_def
1.81 +      by (blast intro: lub_upper chain_fun[OF A] f)
1.82 +  next
1.83 +    fix A :: "('b \<Rightarrow> 'a) set" and g :: "'b \<Rightarrow> 'a"
1.84 +    assume A: "chain ?ordf A" and g: "\<And>f. f \<in> A \<Longrightarrow> ?ordf f g"
1.85 +    show "?ordf (?lubf A) g" unfolding fun_lub_def fun_ord_def
1.86 +      by (blast intro: lub_least chain_fun[OF A] dest: g[unfolded fun_ord_def])
1.87 +   qed
1.88 +qed
1.89 +
1.90 +lemma ccpo: assumes "partial_function_definitions ord lb"
1.91 +  shows "class.ccpo ord (mk_less ord) lb"
1.92 +using assms unfolding partial_function_definitions_def mk_less_def
1.93 +by unfold_locales blast+
1.94 +
1.95 +lemma partial_function_image:
1.96 +  assumes "partial_function_definitions ord Lub"
1.97 +  assumes inj: "\<And>x y. f x = f y \<Longrightarrow> x = y"
1.98 +  assumes inv: "\<And>x. f (g x) = x"
1.99 +  shows "partial_function_definitions (img_ord f ord) (img_lub f g Lub)"
1.100 +proof -
1.101 +  let ?iord = "img_ord f ord"
1.102 +  let ?ilub = "img_lub f g Lub"
1.103 +
1.104 +  interpret partial_function_definitions ord Lub by fact
1.105 +  show ?thesis
1.106 +  proof
1.107 +    fix A x assume "chain ?iord A" "x \<in> A"
1.108 +    then have "chain ord (f ` A)" "f x \<in> f ` A"
1.109 +      by (auto simp: img_ord_def intro: chainI dest: chainD)
1.110 +    thus "?iord x (?ilub A)"
1.111 +      unfolding inv img_lub_def img_ord_def by (rule lub_upper)
1.112 +  next
1.113 +    fix A x assume "chain ?iord A"
1.114 +      and 1: "\<And>z. z \<in> A \<Longrightarrow> ?iord z x"
1.115 +    then have "chain ord (f ` A)"
1.116 +      by (auto simp: img_ord_def intro: chainI dest: chainD)
1.117 +    thus "?iord (?ilub A) x"
1.118 +      unfolding inv img_lub_def img_ord_def
1.119 +      by (rule lub_least) (auto dest: 1[unfolded img_ord_def])
1.120 +  qed (auto simp: img_ord_def intro: leq_refl dest: leq_trans leq_antisym inj)
1.121 +qed
1.122 +
1.123 +context partial_function_definitions
1.124 +begin
1.125 +
1.126 +abbreviation "le_fun \<equiv> fun_ord leq"
1.127 +abbreviation "lub_fun \<equiv> fun_lub lub"
1.128 +abbreviation "fixp_fun == ccpo.fixp le_fun lub_fun"
1.129 +abbreviation "mono_body \<equiv> monotone le_fun leq"
1.130 +
1.131 +text {* Interpret manually, to avoid flooding everything with facts about
1.132 +  orders *}
1.133 +
1.134 +lemma ccpo: "class.ccpo le_fun (mk_less le_fun) lub_fun"
1.135 +apply (rule ccpo)
1.136 +apply (rule partial_function_lift)
1.137 +apply (rule partial_function_definitions_axioms)
1.138 +done
1.139 +
1.140 +text {* The crucial fixed-point theorem *}
1.141 +
1.142 +lemma mono_body_fixp:
1.143 +  "(\<And>x. mono_body (\<lambda>f. F f x)) \<Longrightarrow> fixp_fun F = F (fixp_fun F)"
1.144 +by (rule ccpo.fixp_unfold[OF ccpo]) (auto simp: monotone_def fun_ord_def)
1.145 +
1.146 +text {* Version with curry/uncurry combinators, to be used by package *}
1.147 +
1.148 +lemma fixp_rule_uc:
1.149 +  fixes F :: "'c \<Rightarrow> 'c" and
1.150 +    U :: "'c \<Rightarrow> 'b \<Rightarrow> 'a" and
1.151 +    C :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c"
1.152 +  assumes mono: "\<And>x. mono_body (\<lambda>f. U (F (C f)) x)"
1.153 +  assumes eq: "f \<equiv> C (fixp_fun (\<lambda>f. U (F (C f))))"
1.154 +  assumes inverse: "\<And>f. C (U f) = f"
1.155 +  shows "f = F f"
1.156 +proof -
1.157 +  have "f = C (fixp_fun (\<lambda>f. U (F (C f))))" by (simp add: eq)
1.158 +  also have "... = C (U (F (C (fixp_fun (\<lambda>f. U (F (C f)))))))"
1.159 +    by (subst mono_body_fixp[of "%f. U (F (C f))", OF mono]) (rule refl)
1.160 +  also have "... = F (C (fixp_fun (\<lambda>f. U (F (C f)))))" by (rule inverse)
1.161 +  also have "... = F f" by (simp add: eq)
1.162 +  finally show "f = F f" .
1.163 +qed
1.164 +
1.165 +text {* Rules for @{term mono_body}: *}
1.166 +
1.167 +lemma const_mono[partial_function_mono]: "monotone ord leq (\<lambda>f. c)"
1.168 +by (rule monotoneI) (rule leq_refl)
1.169 +
1.170 +declaration {* Partial_Function.init @{term fixp_fun}
1.171 +  @{term mono_body} @{thm fixp_rule_uc} *}
1.172 +
1.173 +end
1.174 +
1.175 +
1.176 +subsection {* Flat interpretation: tailrec and option *}
1.177 +
1.178 +definition
1.179 +  "flat_ord b x y \<longleftrightarrow> x = b \<or> x = y"
1.180 +
1.181 +definition
1.182 +  "flat_lub b A = (if A \<subseteq> {b} then b else (THE x. x \<in> A - {b}))"
1.183 +
1.184 +lemma flat_interpretation:
1.185 +  "partial_function_definitions (flat_ord b) (flat_lub b)"
1.186 +proof
1.187 +  fix A x assume 1: "chain (flat_ord b) A" "x \<in> A"
1.188 +  show "flat_ord b x (flat_lub b A)"
1.189 +  proof cases
1.190 +    assume "x = b"
1.191 +    thus ?thesis by (simp add: flat_ord_def)
1.192 +  next
1.193 +    assume "x \<noteq> b"
1.194 +    with 1 have "A - {b} = {x}"
1.195 +      by (auto elim: chainE simp: flat_ord_def)
1.196 +    then have "flat_lub b A = x"
1.197 +      by (auto simp: flat_lub_def)
1.198 +    thus ?thesis by (auto simp: flat_ord_def)
1.199 +  qed
1.200 +next
1.201 +  fix A z assume A: "chain (flat_ord b) A"
1.202 +    and z: "\<And>x. x \<in> A \<Longrightarrow> flat_ord b x z"
1.203 +  show "flat_ord b (flat_lub b A) z"
1.204 +  proof cases
1.205 +    assume "A \<subseteq> {b}"
1.206 +    thus ?thesis
1.207 +      by (auto simp: flat_lub_def flat_ord_def)
1.208 +  next
1.209 +    assume nb: "\<not> A \<subseteq> {b}"
1.210 +    then obtain y where y: "y \<in> A" "y \<noteq> b" by auto
1.211 +    with A have "A - {b} = {y}"
1.212 +      by (auto elim: chainE simp: flat_ord_def)
1.213 +    with nb have "flat_lub b A = y"
1.214 +      by (auto simp: flat_lub_def)
1.215 +    with z y show ?thesis by auto
1.216 +  qed
1.217 +qed (auto simp: flat_ord_def)
1.218 +
1.219 +interpretation tailrec!:
1.220 +  partial_function_definitions "flat_ord undefined" "flat_lub undefined"
1.221 +by (rule flat_interpretation)
1.222 +
1.223 +interpretation option!:
1.224 +  partial_function_definitions "flat_ord None" "flat_lub None"
1.225 +by (rule flat_interpretation)
1.226 +
1.227 +abbreviation "option_ord \<equiv> flat_ord None"
1.228 +abbreviation "mono_option \<equiv> monotone (fun_ord option_ord) option_ord"
1.229 +
1.230 +lemma bind_mono[partial_function_mono]:
1.231 +assumes mf: "mono_option B" and mg: "\<And>y. mono_option (\<lambda>f. C y f)"
1.232 +shows "mono_option (\<lambda>f. Option.bind (B f) (\<lambda>y. C y f))"
1.233 +proof (rule monotoneI)
1.234 +  fix f g :: "'a \<Rightarrow> 'b option" assume fg: "fun_ord option_ord f g"
1.235 +  with mf
1.236 +  have "option_ord (B f) (B g)" by (rule monotoneD[of _ _ _ f g])
1.237 +  then have "option_ord (Option.bind (B f) (\<lambda>y. C y f)) (Option.bind (B g) (\<lambda>y. C y f))"
1.238 +    unfolding flat_ord_def by auto
1.239 +  also from mg
1.240 +  have "\<And>y'. option_ord (C y' f) (C y' g)"
1.241 +    by (rule monotoneD) (rule fg)
1.242 +  then have "option_ord (Option.bind (B g) (\<lambda>y'. C y' f)) (Option.bind (B g) (\<lambda>y'. C y' g))"
1.243 +    unfolding flat_ord_def by (cases "B g") auto
1.244 +  finally (option.leq_trans)
1.245 +  show "option_ord (Option.bind (B f) (\<lambda>y. C y f)) (Option.bind (B g) (\<lambda>y'. C y' g))" .
1.246 +qed
1.247 +
1.248 +
1.249 +end
1.250 +
```
```     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/src/HOL/Tools/Function/partial_function.ML	Sat Oct 23 23:41:19 2010 +0200
2.3 @@ -0,0 +1,238 @@
2.4 +(*  Title:      HOL/Tools/Function/partial_function.ML
2.5 +    Author:     Alexander Krauss, TU Muenchen
2.6 +
2.7 +Partial function definitions based on least fixed points in ccpos.
2.8 +*)
2.9 +
2.10 +signature PARTIAL_FUNCTION =
2.11 +sig
2.12 +  val setup: theory -> theory
2.13 +  val init: term -> term -> thm -> declaration
2.14 +
2.15 +  val add_partial_function: string -> (binding * typ option * mixfix) list ->
2.16 +    Attrib.binding * term -> local_theory -> local_theory
2.17 +
2.18 +  val add_partial_function_cmd: string -> (binding * string option * mixfix) list ->
2.19 +    Attrib.binding * string -> local_theory -> local_theory
2.20 +end;
2.21 +
2.22 +
2.23 +structure Partial_Function: PARTIAL_FUNCTION =
2.24 +struct
2.25 +
2.26 +(*** Context Data ***)
2.27 +
2.28 +structure Modes = Generic_Data
2.29 +(
2.30 +  type T = ((term * term) * thm) Symtab.table;
2.31 +  val empty = Symtab.empty;
2.32 +  val extend = I;
2.33 +  fun merge (a, b) = Symtab.merge (K true) (a, b);
2.34 +)
2.35 +
2.36 +fun init fixp mono fixp_eq phi =
2.37 +  let
2.38 +    val term = Morphism.term phi;
2.39 +    val data' = ((term fixp, term mono), Morphism.thm phi fixp_eq);
2.40 +    val mode = (* extract mode identifier from morphism prefix! *)
2.41 +      Binding.prefix_of (Morphism.binding phi (Binding.empty))
2.42 +      |> map fst |> space_implode ".";
2.43 +  in
2.44 +    if mode = "" then I
2.45 +    else Modes.map (Symtab.update (mode, data'))
2.46 +  end
2.47 +
2.48 +val known_modes = Symtab.keys o Modes.get o Context.Proof;
2.49 +val lookup_mode = Symtab.lookup o Modes.get o Context.Proof;
2.50 +
2.51 +
2.52 +structure Mono_Rules = Named_Thms
2.53 +(
2.54 +  val name = "partial_function_mono";
2.55 +  val description = "monotonicity rules for partial function definitions";
2.56 +);
2.57 +
2.58 +
2.59 +(*** Automated monotonicity proofs ***)
2.60 +
2.61 +fun strip_cases ctac = ctac #> Seq.map snd;
2.62 +
2.63 +(*rewrite conclusion with k-th assumtion*)
2.64 +fun rewrite_with_asm_tac ctxt k =
2.65 +  Subgoal.FOCUS (fn {context=ctxt', prems, ...} =>
2.66 +    Local_Defs.unfold_tac ctxt' [nth prems k]) ctxt;
2.67 +
2.68 +fun dest_case thy t =
2.69 +  case strip_comb t of
2.70 +    (Const (case_comb, _), args) =>
2.71 +      (case Datatype.info_of_case thy case_comb of
2.72 +         NONE => NONE
2.73 +       | SOME {case_rewrites, ...} =>
2.74 +           let
2.75 +             val lhs = prop_of (hd case_rewrites)
2.76 +               |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst;
2.77 +             val arity = length (snd (strip_comb lhs));
2.78 +             val conv = funpow (length args - arity) Conv.fun_conv
2.79 +               (Conv.rewrs_conv (map mk_meta_eq case_rewrites));
2.80 +           in
2.81 +             SOME (nth args (arity - 1), conv)
2.82 +           end)
2.83 +  | _ => NONE;
2.84 +
2.85 +(*split on case expressions*)
2.86 +val split_cases_tac = Subgoal.FOCUS_PARAMS (fn {context=ctxt, ...} =>
2.87 +  SUBGOAL (fn (t, i) => case t of
2.88 +    _ \$ (_ \$ Abs (_, _, body)) =>
2.89 +      (case dest_case (ProofContext.theory_of ctxt) body of
2.90 +         NONE => no_tac
2.91 +       | SOME (arg, conv) =>
2.92 +           let open Conv in
2.93 +              if not (null (loose_bnos arg)) then no_tac
2.94 +              else ((DETERM o strip_cases o Induct.cases_tac ctxt false [[SOME arg]] NONE [])
2.95 +                THEN_ALL_NEW (rewrite_with_asm_tac ctxt 0)
2.96 +                THEN_ALL_NEW etac @{thm thin_rl}
2.97 +                THEN_ALL_NEW (CONVERSION
2.98 +                  (params_conv ~1 (fn ctxt' =>
2.99 +                    arg_conv (arg_conv (abs_conv (K conv) ctxt'))) ctxt))) i
2.100 +           end)
2.101 +  | _ => no_tac) 1);
2.102 +
2.103 +(*monotonicity proof: apply rules + split case expressions*)
2.104 +fun mono_tac ctxt =
2.105 +  K (Local_Defs.unfold_tac ctxt [@{thm curry_def}])
2.106 +  THEN' (TRY o REPEAT_ALL_NEW
2.107 +   (resolve_tac (Mono_Rules.get ctxt)
2.108 +     ORELSE' split_cases_tac ctxt));
2.109 +
2.110 +
2.111 +(*** Auxiliary functions ***)
2.112 +
2.113 +(*positional instantiation with computed type substitution.
2.114 +  internal version of  attribute "[of s t u]".*)
2.115 +fun cterm_instantiate' cts thm =
2.116 +  let
2.117 +    val thy = Thm.theory_of_thm thm;
2.118 +    val vs = rev (Term.add_vars (prop_of thm) [])
2.119 +      |> map (Thm.cterm_of thy o Var);
2.120 +  in
2.121 +    cterm_instantiate (zip_options vs cts) thm
2.122 +  end;
2.123 +
2.124 +(*Returns t \$ u, but instantiates the type of t to make the
2.125 +application type correct*)
2.126 +fun apply_inst ctxt t u =
2.127 +  let
2.128 +    val thy = ProofContext.theory_of ctxt;
2.129 +    val T = domain_type (fastype_of t);
2.130 +    val T' = fastype_of u;
2.131 +    val subst = Type.typ_match (Sign.tsig_of thy) (T, T') Vartab.empty
2.132 +      handle Type.TYPE_MATCH => raise TYPE ("apply_inst", [T, T'], [t, u])
2.133 +  in
2.134 +    map_types (Envir.norm_type subst) t \$ u
2.135 +  end;
2.136 +
2.137 +fun head_conv cv ct =
2.138 +  if can Thm.dest_comb ct then Conv.fun_conv (head_conv cv) ct else cv ct;
2.139 +
2.140 +
2.141 +(*** currying transformation ***)
2.142 +
2.143 +fun curry_const (A, B, C) =
2.144 +  Const (@{const_name Product_Type.curry},
2.145 +    [HOLogic.mk_prodT (A, B) --> C, A, B] ---> C);
2.146 +
2.147 +fun mk_curry f =
2.148 +  case fastype_of f of
2.149 +    Type ("fun", [Type (_, [S, T]), U]) =>
2.150 +      curry_const (S, T, U) \$ f
2.151 +  | T => raise TYPE ("mk_curry", [T], [f]);
2.152 +
2.153 +(* iterated versions. Nonstandard left-nested tuples arise naturally
2.154 +from "split o split o split"*)
2.155 +fun curry_n arity = funpow (arity - 1) mk_curry;
2.156 +fun uncurry_n arity = funpow (arity - 1) HOLogic.mk_split;
2.157 +
2.158 +val curry_uncurry_ss = HOL_basic_ss addsimps
2.159 +  [@{thm Product_Type.curry_split}, @{thm Product_Type.split_curry}]
2.160 +
2.161 +
2.162 +(*** partial_function definition ***)
2.163 +
2.164 +fun gen_add_partial_function prep mode fixes_raw eqn_raw lthy =
2.165 +  let
2.166 +    val ((fixp, mono), fixp_eq) = the (lookup_mode lthy mode)
2.167 +      handle Option.Option => error (cat_lines ["Unknown mode " ^ quote mode ^ ".",
2.168 +        "Known modes are " ^ commas_quote (known_modes lthy) ^ "."]);
2.169 +
2.170 +    val ((fixes, [(eq_abinding, eqn)]), _) = prep fixes_raw [eqn_raw] lthy;
2.171 +    val (_, _, plain_eqn) = Function_Lib.dest_all_all_ctx lthy eqn;
2.172 +
2.173 +    val ((f_binding, fT), mixfix) = the_single fixes;
2.174 +    val fname = Binding.name_of f_binding;
2.175 +
2.176 +    val cert = cterm_of (ProofContext.theory_of lthy);
2.177 +    val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop plain_eqn);
2.178 +    val (head, args) = strip_comb lhs;
2.179 +    val F = fold_rev lambda (head :: args) rhs;
2.180 +
2.181 +    val arity = length args;
2.182 +    val (aTs, bTs) = chop arity (binder_types fT);
2.183 +
2.184 +    val tupleT = foldl1 HOLogic.mk_prodT aTs;
2.185 +    val fT_uc = tupleT :: bTs ---> body_type fT;
2.186 +    val f_uc = Var ((fname, 0), fT_uc);
2.187 +    val x_uc = Var (("x", 0), tupleT);
2.189 +    val curry = lambda f_uc (curry_n arity f_uc);
2.190 +
2.191 +    val F_uc =
2.192 +      lambda f_uc (uncurry_n arity (F \$ curry_n arity f_uc));
2.193 +
2.194 +    val mono_goal = apply_inst lthy mono (lambda f_uc (F_uc \$ f_uc \$ x_uc))
2.195 +      |> HOLogic.mk_Trueprop
2.196 +      |> Logic.all x_uc;
2.197 +
2.198 +    val mono_thm = Goal.prove_internal [] (cert mono_goal)
2.199 +        (K (mono_tac lthy 1))
2.200 +      |> Thm.forall_elim (cert x_uc);
2.201 +
2.202 +    val f_def_rhs = curry_n arity (apply_inst lthy fixp F_uc);
2.203 +    val f_def_binding = Binding.conceal (Binding.name (Thm.def_name fname));
2.204 +    val ((f, (_, f_def)), lthy') = Local_Theory.define
2.205 +      ((f_binding, mixfix), ((f_def_binding, []), f_def_rhs)) lthy;
2.206 +
2.207 +    val eqn = HOLogic.mk_eq (list_comb (f, args),
2.208 +        Term.betapplys (F, f :: args))
2.209 +      |> HOLogic.mk_Trueprop;
2.210 +
2.211 +    val unfold =
2.212 +      (cterm_instantiate' (map (SOME o cert) [uncurry, F, curry]) fixp_eq
2.213 +        OF [mono_thm, f_def])
2.214 +      |> Tactic.rule_by_tactic lthy (Simplifier.simp_tac curry_uncurry_ss 1);
2.215 +
2.216 +    val rec_rule = let open Conv in
2.217 +      Goal.prove lthy' (map (fst o dest_Free) args) [] eqn (fn _ =>
2.218 +        CONVERSION ((arg_conv o arg1_conv o head_conv o rewr_conv) (mk_meta_eq unfold)) 1
2.219 +        THEN rtac @{thm refl} 1) end;
2.220 +  in
2.221 +    lthy'
2.222 +    |> Local_Theory.note (eq_abinding, [rec_rule])
2.223 +    |-> (fn (_, rec') =>
2.224 +      Local_Theory.note ((Binding.qualify true fname (Binding.name "rec"), []), rec'))
2.225 +    |> snd
2.226 +  end;
2.227 +
2.230 +
2.231 +val mode = Parse.\$\$\$ "(" |-- Parse.xname --| Parse.\$\$\$ ")";
2.232 +
2.233 +val _ = Outer_Syntax.local_theory
2.234 +  "partial_function" "define partial function" Keyword.thy_goal
2.235 +  ((mode -- (Parse.fixes -- (Parse.where_ |-- Parse_Spec.spec)))
2.236 +     >> (fn (mode, (fixes, spec)) => add_partial_function_cmd mode fixes spec));
2.237 +
2.238 +
2.239 +val setup = Mono_Rules.setup;
2.240 +
2.241 +end
```