(*  Title:      Pure/Isar/element.ML

     Author:     Makarius

 Explicit data structures for some Isar language elements, with derived

 logical operations.

 *)

 signature ELEMENT =

 sig

   datatype ('typ, 'term) stmt =

     Shows of (Attrib.binding * ('term * 'term list) list) list |

     Obtains of (binding * ((binding * 'typ option) list * 'term list)) list

   type statement = (string, string) stmt

   type statement_i = (typ, term) stmt

   datatype ('typ, 'term, 'fact) ctxt =

     Fixes of (binding * 'typ option * mixfix) list |

     Constrains of (string * 'typ) list |

     Assumes of (Attrib.binding * ('term * 'term list) list) list |

     Defines of (Attrib.binding * ('term * 'term list)) list |

     Notes of string * (Attrib.binding * ('fact * Attrib.src list) list) list

   type context = (string, string, Facts.ref) ctxt

   type context_i = (typ, term, thm list) ctxt

   val facts_map: (('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt) ->

    (Attrib.binding * ('fact * Attrib.src list) list) list ->

    (Attrib.binding * ('c * Attrib.src list) list) list

   val map_ctxt: {binding: binding -> binding, typ: 'typ -> 'a, term: 'term -> 'b,

     pattern: 'term -> 'b, fact: 'fact -> 'c, attrib: Attrib.src -> Attrib.src} ->

     ('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt

   val map_ctxt_attrib: (Attrib.src -> Attrib.src) ->

     ('typ, 'term, 'fact) ctxt -> ('typ, 'term, 'fact) ctxt

   val morph_ctxt: morphism -> context_i -> context_i

   val pretty_stmt: Proof.context -> statement_i -> Pretty.T list

   val pretty_ctxt: Proof.context -> context_i -> Pretty.T list

   val pretty_statement: Proof.context -> string -> thm -> Pretty.T

   type witness

   val prove_witness: Proof.context -> term -> tactic -> witness

   val witness_proof: (witness list list -> Proof.context -> Proof.context) ->

     term list list -> Proof.context -> Proof.state

   val witness_proof_eqs: (witness list list -> thm list -> Proof.context -> Proof.context) ->

     term list list -> term list -> Proof.context -> Proof.state

   val witness_local_proof: (witness list list -> Proof.state -> Proof.state) ->

     string -> term list list -> Proof.context -> bool -> Proof.state -> Proof.state

   val morph_witness: morphism -> witness -> witness

   val conclude_witness: witness -> thm

   val pretty_witness: Proof.context -> witness -> Pretty.T

   val instT_type: typ Symtab.table -> typ -> typ

   val instT_term: typ Symtab.table -> term -> term

   val instT_thm: theory -> typ Symtab.table -> thm -> thm

   val instT_morphism: theory -> typ Symtab.table -> morphism

   val inst_term: typ Symtab.table * term Symtab.table -> term -> term

   val inst_thm: theory -> typ Symtab.table * term Symtab.table -> thm -> thm

   val inst_morphism: theory -> typ Symtab.table * term Symtab.table -> morphism

   val satisfy_thm: witness list -> thm -> thm

   val satisfy_morphism: witness list -> morphism

   val satisfy_facts: witness list ->

     (Attrib.binding * (thm list * Attrib.src list) list) list ->

     (Attrib.binding * (thm list * Attrib.src list) list) list

   val generalize_facts: Proof.context -> Proof.context ->

     (Attrib.binding * (thm list * Attrib.src list) list) list ->

     (Attrib.binding * (thm list * Attrib.src list) list) list

   val eq_morphism: theory -> thm list -> morphism

   val transfer_morphism: theory -> morphism

   val init: context_i -> Context.generic -> Context.generic

   val activate_i: context_i -> Proof.context -> context_i * Proof.context

   val activate: (typ, term, Facts.ref) ctxt -> Proof.context -> context_i * Proof.context

 end;

 structure Element: ELEMENT =

 struct

 (** language elements **)

 (* statement *)

 datatype ('typ, 'term) stmt =

   Shows of (Attrib.binding * ('term * 'term list) list) list |

   Obtains of (binding * ((binding * 'typ option) list * 'term list)) list;

 type statement = (string, string) stmt;

 type statement_i = (typ, term) stmt;

 (* context *)

 datatype ('typ, 'term, 'fact) ctxt =

   Fixes of (binding * 'typ option * mixfix) list |

   Constrains of (string * 'typ) list |

   Assumes of (Attrib.binding * ('term * 'term list) list) list |

   Defines of (Attrib.binding * ('term * 'term list)) list |

   Notes of string * (Attrib.binding * ('fact * Attrib.src list) list) list;

 type context = (string, string, Facts.ref) ctxt;

 type context_i = (typ, term, thm list) ctxt;

 fun facts_map f facts = Notes ("", facts) |> f |> (fn Notes (_, facts') => facts');

 fun map_ctxt {binding, typ, term, pattern, fact, attrib} =

   fn Fixes fixes => Fixes (fixes |> map (fn (x, T, mx) => (binding x, Option.map typ T, mx)))

    | Constrains xs => Constrains (xs |> map (fn (x, T) =>

       (Name.of_binding (binding (Binding.name x)), typ T)))

    | Assumes asms => Assumes (asms |> map (fn ((a, atts), propps) =>

       ((binding a, map attrib atts), propps |> map (fn (t, ps) => (term t, map pattern ps)))))

    | Defines defs => Defines (defs |> map (fn ((a, atts), (t, ps)) =>

       ((binding a, map attrib atts), (term t, map pattern ps))))

    | Notes (kind, facts) => Notes (kind, facts |> map (fn ((a, atts), bs) =>

       ((binding a, map attrib atts), bs |> map (fn (ths, btts) => (fact ths, map attrib btts)))));

 fun map_ctxt_attrib attrib =

   map_ctxt {binding = I, typ = I, term = I, pattern = I, fact = I, attrib = attrib};

 fun morph_ctxt phi = map_ctxt

  {binding = Morphism.binding phi,

   typ = Morphism.typ phi,

   term = Morphism.term phi,

   pattern = Morphism.term phi,

   fact = Morphism.fact phi,

   attrib = Args.morph_values phi};

 (** pretty printing **)

 fun pretty_items _ _ [] = []

   | pretty_items keyword sep (x :: ys) =

       Pretty.block [Pretty.keyword keyword, Pretty.brk 1, x] ::

         map (fn y => Pretty.block [Pretty.str "  ", Pretty.keyword sep, Pretty.brk 1, y]) ys;

 fun pretty_name_atts ctxt (b, atts) sep =

   if Binding.is_empty b andalso null atts then []

   else [Pretty.block (Pretty.breaks

     (Pretty.str (Binding.str_of b) :: Attrib.pretty_attribs ctxt atts @ [Pretty.str sep]))];

 (* pretty_stmt *)

 fun pretty_stmt ctxt =

   let

     val prt_typ = Pretty.quote o Syntax.pretty_typ ctxt;

     val prt_term = Pretty.quote o Syntax.pretty_term ctxt;

     val prt_terms = separate (Pretty.keyword "and") o map prt_term;

     val prt_name_atts = pretty_name_atts ctxt;

     fun prt_show (a, ts) =

       Pretty.block (Pretty.breaks (prt_name_atts a ":" @ prt_terms (map fst ts)));

     fun prt_var (x, SOME T) = Pretty.block

           [Pretty.str (Binding.name_of x ^ " ::"), Pretty.brk 1, prt_typ T]

       | prt_var (x, NONE) = Pretty.str (Binding.name_of x);

     val prt_vars = separate (Pretty.keyword "and") o map prt_var;

     fun prt_obtain (_, ([], ts)) = Pretty.block (Pretty.breaks (prt_terms ts))

       | prt_obtain (_, (xs, ts)) = Pretty.block (Pretty.breaks

           (prt_vars xs @ [Pretty.keyword "where"] @ prt_terms ts));

   in

     fn Shows shows => pretty_items "shows" "and" (map prt_show shows)

      | Obtains obtains => pretty_items "obtains" "|" (map prt_obtain obtains)

   end;

 (* pretty_ctxt *)

 fun pretty_ctxt ctxt =

   let

     val prt_typ = Pretty.quote o Syntax.pretty_typ ctxt;

     val prt_term = Pretty.quote o Syntax.pretty_term ctxt;

     val prt_thm = Pretty.backquote o ProofContext.pretty_thm ctxt;

     val prt_name_atts = pretty_name_atts ctxt;

     fun prt_mixfix NoSyn = []

       | prt_mixfix mx = [Pretty.brk 2, Syntax.pretty_mixfix mx];

     fun prt_fix (x, SOME T, mx) = Pretty.block (Pretty.str (Binding.name_of x ^ " ::") ::

           Pretty.brk 1 :: prt_typ T :: Pretty.brk 1 :: prt_mixfix mx)

       | prt_fix (x, NONE, mx) = Pretty.block (Pretty.str (Binding.name_of x) ::

           Pretty.brk 1 :: prt_mixfix mx);

     fun prt_constrain (x, T) = prt_fix (Binding.name x, SOME T, NoSyn);

     fun prt_asm (a, ts) =

       Pretty.block (Pretty.breaks (prt_name_atts a ":" @ map (prt_term o fst) ts));

     fun prt_def (a, (t, _)) =

       Pretty.block (Pretty.breaks (prt_name_atts a ":" @ [prt_term t]));

     fun prt_fact (ths, []) = map prt_thm ths

       | prt_fact (ths, atts) = Pretty.enclose "(" ")"

           (Pretty.breaks (map prt_thm ths)) :: Attrib.pretty_attribs ctxt atts;

     fun prt_note (a, ths) =

       Pretty.block (Pretty.breaks (flat (prt_name_atts a "=" :: map prt_fact ths)));

   in

     fn Fixes fixes => pretty_items "fixes" "and" (map prt_fix fixes)

      | Constrains xs => pretty_items "constrains" "and" (map prt_constrain xs)

      | Assumes asms => pretty_items "assumes" "and" (map prt_asm asms)

      | Defines defs => pretty_items "defines" "and" (map prt_def defs)

      | Notes ("", facts) => pretty_items "notes" "and" (map prt_note facts)

      | Notes (kind, facts) => pretty_items ("notes " ^ kind) "and" (map prt_note facts)

   end;

 (* pretty_statement *)

 local

 fun thm_name kind th prts =

   let val head =

     if Thm.has_name_hint th then

       Pretty.block [Pretty.command kind,

         Pretty.brk 1, Pretty.str (Long_Name.base_name (Thm.get_name_hint th) ^ ":")]

     else Pretty.command kind

   in Pretty.block (Pretty.fbreaks (head :: prts)) end;

 fun fix (x, T) = (Binding.name x, SOME T);

 fun obtain prop ctxt =

   let

     val ((xs, prop'), ctxt') = Variable.focus prop ctxt;

     val As = Logic.strip_imp_prems (Thm.term_of prop');

   in ((Binding.empty, (map (fix o Term.dest_Free o Thm.term_of) xs, As)), ctxt') end;

 in

 fun pretty_statement ctxt kind raw_th =

   let

     val thy = ProofContext.theory_of ctxt;

     val cert = Thm.cterm_of thy;

     val th = MetaSimplifier.norm_hhf raw_th;

     val is_elim = ObjectLogic.is_elim th;

     val ((_, [th']), ctxt') = Variable.import true [th] (Variable.set_body false ctxt);

     val prop = Thm.prop_of th';

     val (prems, concl) = Logic.strip_horn prop;

     val concl_term = ObjectLogic.drop_judgment thy concl;

     val fixes = fold_aterms (fn v as Free (x, T) =>

         if Variable.newly_fixed ctxt' ctxt x andalso not (v aconv concl_term)

         then insert (op =) (x, T) else I | _ => I) prop [] |> rev;

     val (assumes, cases) = take_suffix (fn prem =>

       is_elim andalso concl aconv Logic.strip_assums_concl prem) prems;

   in

     pretty_ctxt ctxt' (Fixes (map (fn (x, T) => (Binding.name x, SOME T, NoSyn)) fixes)) @

     pretty_ctxt ctxt' (Assumes (map (fn t => (Attrib.empty_binding, [(t, [])])) assumes)) @

      (if null cases then pretty_stmt ctxt' (Shows [(Attrib.empty_binding, [(concl, [])])])

       else

         let val (clauses, ctxt'') = fold_map (obtain o cert) cases ctxt'

         in pretty_stmt ctxt'' (Obtains clauses) end)

   end |> thm_name kind raw_th;

 end;

 (** logical operations **)

 (* witnesses -- hypotheses as protected facts *)

 datatype witness = Witness of term * thm;

 val mark_witness = Logic.protect;

 fun witness_prop (Witness (t, _)) = t;

 fun witness_hyps (Witness (_, th)) = #hyps (Thm.rep_thm th);

 fun map_witness f (Witness witn) = Witness (f witn);

 fun morph_witness phi = map_witness (fn (t, th) => (Morphism.term phi t, Morphism.thm phi th));

 fun prove_witness ctxt t tac =

   Witness (t, Thm.close_derivation (Goal.prove ctxt [] [] (mark_witness t) (fn _ =>

     Tactic.rtac Drule.protectI 1 THEN tac)));

 local

 val refine_witness =

   Proof.refine (Method.Basic (K (RAW_METHOD

     (K (ALLGOALS

       (CONJUNCTS (ALLGOALS

         (CONJUNCTS (TRYALL (Tactic.rtac Drule.protectI)))))))), Position.none));

 fun gen_witness_proof proof after_qed wit_propss eq_props =

   let

     val propss = (map o map) (fn prop => (mark_witness prop, [])) wit_propss

       @ [map (rpair []) eq_props];

     fun after_qed' thmss =

       let val (wits, eqs) = split_last ((map o map) Thm.close_derivation thmss);

       in after_qed ((map2 o map2) (curry Witness) wit_propss wits) eqs end;

   in proof after_qed' propss #> refine_witness #> Seq.hd end;

 in

 fun witness_proof after_qed wit_propss =

   gen_witness_proof (Proof.theorem_i NONE) (fn wits => fn _ => after_qed wits)

     wit_propss [];

 val witness_proof_eqs = gen_witness_proof (Proof.theorem_i NONE);

 fun witness_local_proof after_qed cmd wit_propss goal_ctxt int =

   gen_witness_proof (fn after_qed' => fn propss =>

     Proof.map_context (K goal_ctxt)

     #> Proof.local_goal (ProofDisplay.print_results int) (K I) ProofContext.bind_propp_i

       cmd NONE after_qed' (map (pair Thm.empty_binding) propss))

     (fn wits => fn _ => after_qed wits) wit_propss [];

 end;

 fun compose_witness (Witness (_, th)) r =

   let

     val th' = Goal.conclude th;

     val A = Thm.cprem_of r 1;

   in

     Thm.implies_elim

       (Conv.gconv_rule Drule.beta_eta_conversion 1 r)

       (Conv.fconv_rule Drule.beta_eta_conversion

         (Thm.instantiate (Thm.match (Thm.cprop_of th', A)) th'))

   end;

 fun conclude_witness (Witness (_, th)) =

   Thm.close_derivation (MetaSimplifier.norm_hhf_protect (Goal.conclude th));

 fun pretty_witness ctxt witn =

   let val prt_term = Pretty.quote o Syntax.pretty_term ctxt in

     Pretty.block (prt_term (witness_prop witn) ::

       (if ! show_hyps then [Pretty.brk 2, Pretty.list "[" "]"

          (map prt_term (witness_hyps witn))] else []))

   end;

 (* derived rules *)

 fun instantiate_tfrees thy subst th =

   let

     val certT = Thm.ctyp_of thy;

     val idx = Thm.maxidx_of th + 1;

     fun cert_inst (a, (S, T)) = (certT (TVar ((a, idx), S)), certT T);

     fun add_inst (a, S) insts =

       if AList.defined (op =) insts a then insts

       else (case AList.lookup (op =) subst a of NONE => insts | SOME T => (a, (S, T)) :: insts);

     val insts =

       Term.fold_types (Term.fold_atyps (fn TFree v => add_inst v | _ => I))

         (Thm.full_prop_of th) [];

   in

     th

     |> Thm.generalize (map fst insts, []) idx

     |> Thm.instantiate (map cert_inst insts, [])

   end;

 fun instantiate_frees thy subst =

   let val cert = Thm.cterm_of thy in

     Drule.forall_intr_list (map (cert o Free o fst) subst) #>

     Drule.forall_elim_list (map (cert o snd) subst)

   end;

 fun hyps_rule rule th =

   let val {hyps, ...} = Thm.crep_thm th in

     Drule.implies_elim_list

       (rule (Drule.implies_intr_list hyps th))

       (map (Thm.assume o Drule.cterm_rule rule) hyps)

   end;

 (* instantiate types *)

 fun instT_type env =

   if Symtab.is_empty env then I

   else Term.map_type_tfree (fn (x, S) => the_default (TFree (x, S)) (Symtab.lookup env x));

 fun instT_term env =

   if Symtab.is_empty env then I

   else Term.map_types (instT_type env);

 fun instT_subst env th = (Thm.fold_terms o Term.fold_types o Term.fold_atyps)

   (fn T as TFree (a, _) =>

     let val T' = the_default T (Symtab.lookup env a)

     in if T = T' then I else insert (op =) (a, T') end

   | _ => I) th [];

 fun instT_thm thy env th =

   if Symtab.is_empty env then th

   else

     let val subst = instT_subst env th

     in if null subst then th else th |> hyps_rule (instantiate_tfrees thy subst) end;

 fun instT_morphism thy env =

   let val thy_ref = Theory.check_thy thy in

     Morphism.morphism

      {binding = I,

       typ = instT_type env,

       term = instT_term env,

       fact = map (fn th => instT_thm (Theory.deref thy_ref) env th)}

   end;

 (* instantiate types and terms *)

 fun inst_term (envT, env) =

   if Symtab.is_empty env then instT_term envT

   else

     let

       val instT = instT_type envT;

       fun inst (Const (x, T)) = Const (x, instT T)

         | inst (Free (x, T)) =

             (case Symtab.lookup env x of

               NONE => Free (x, instT T)

             | SOME t => t)

         | inst (Var (xi, T)) = Var (xi, instT T)

         | inst (b as Bound _) = b

         | inst (Abs (x, T, t)) = Abs (x, instT T, inst t)

         | inst (t $ u) = inst t $ inst u;

     in Envir.beta_norm o inst end;

 fun inst_thm thy (envT, env) th =

   if Symtab.is_empty env then instT_thm thy envT th

   else

     let

       val substT = instT_subst envT th;

       val subst = (Thm.fold_terms o Term.fold_aterms)

        (fn Free (x, T) =>

           let

             val T' = instT_type envT T;

             val t = Free (x, T');

             val t' = the_default t (Symtab.lookup env x);

           in if t aconv t' then I else insert (eq_fst (op =)) ((x, T'), t') end

        | _ => I) th [];

     in

       if null substT andalso null subst then th

       else th |> hyps_rule

        (instantiate_tfrees thy substT #>

         instantiate_frees thy subst #>

         Conv.fconv_rule (Thm.beta_conversion true))

     end;

 fun inst_morphism thy envs =

   let val thy_ref = Theory.check_thy thy in

     Morphism.morphism

      {binding = I,

       typ = instT_type (#1 envs),

       term = inst_term envs,

       fact = map (fn th => inst_thm (Theory.deref thy_ref) envs th)}

   end;

 (* satisfy hypotheses *)

 fun satisfy_thm witns thm = thm |> fold (fn hyp =>

     (case find_first (fn Witness (t, _) => Thm.term_of hyp aconv t) witns of

       NONE => I

     | SOME w => Thm.implies_intr hyp #> compose_witness w)) (#hyps (Thm.crep_thm thm));

 val satisfy_morphism = Morphism.thm_morphism o satisfy_thm;

 val satisfy_facts = facts_map o morph_ctxt o satisfy_morphism;

 (* rewriting with equalities *)

 fun eq_morphism thy thms = Morphism.morphism

  {binding = I,

   typ = I,

   term = MetaSimplifier.rewrite_term thy thms [],

   fact = map (MetaSimplifier.rewrite_rule thms)};

 (* generalize type/term parameters *)

 val maxidx_atts = fold Args.maxidx_values;

 fun generalize_facts inner outer facts =

   let

     val thy = ProofContext.theory_of inner;

     val maxidx =

       fold (fn ((_, atts), bs) => maxidx_atts atts #> fold (maxidx_atts o #2) bs) facts ~1;

     val exp_fact = map (Thm.adjust_maxidx_thm maxidx) #> Variable.export inner outer;

     val exp_term = Drule.term_rule thy (singleton exp_fact);

     val exp_typ = Logic.type_map exp_term;

     val morphism = Morphism.morphism {binding = I, typ = exp_typ, term = exp_term, fact = exp_fact};

   in facts_map (morph_ctxt morphism) facts end;

 (* transfer to theory using closure *)

 fun transfer_morphism thy =

   let val thy_ref = Theory.check_thy thy

   in Morphism.thm_morphism (fn th => transfer (Theory.deref thy_ref) th) end;

 (** activate in context **)

 (* init *)

 fun init (Fixes fixes) = Context.map_proof (ProofContext.add_fixes fixes #> #2)

   | init (Constrains _) = I

   | init (Assumes asms) = Context.map_proof (fn ctxt =>

       let

         val asms' = Attrib.map_specs (Attrib.attribute_i (ProofContext.theory_of ctxt)) asms;

         val (_, ctxt') = ctxt

           |> fold Variable.auto_fixes (maps (map #1 o #2) asms')

           |> ProofContext.add_assms_i Assumption.assume_export asms';

       in ctxt' end)

   | init (Defines defs) = Context.map_proof (fn ctxt =>

       let

         val defs' = Attrib.map_specs (Attrib.attribute_i (ProofContext.theory_of ctxt)) defs;

         val asms = defs' |> map (fn ((name, atts), (t, ps)) =>

             let val ((c, _), t') = LocalDefs.cert_def ctxt t  (* FIXME adapt ps? *)

             in (t', ((Thm.def_binding_optional (Binding.name c) name, atts), [(t', ps)])) end);

         val (_, ctxt') = ctxt

           |> fold Variable.auto_fixes (map #1 asms)

           |> ProofContext.add_assms_i LocalDefs.def_export (map #2 asms);

       in ctxt' end)

   | init (Notes (kind, facts)) = (fn context =>

       let

         val facts' = Attrib.map_facts (Attrib.attribute_i (Context.theory_of context)) facts;

         val context' = context |> Context.mapping

           (PureThy.note_thmss kind facts' #> #2)

           (ProofContext.note_thmss kind facts' #> #2);

       in context' end);

 (* activate *)

 fun activate_i elem ctxt =

   let

     val elem' = map_ctxt_attrib Args.assignable elem;

     val ctxt' = Context.proof_map (init elem') ctxt;

   in (map_ctxt_attrib Args.closure elem', ctxt') end;

 fun activate raw_elem ctxt =

   let val elem = raw_elem |> map_ctxt

    {binding = tap Name.of_binding,

     typ = I,

     term = I,

     pattern = I,

     fact = ProofContext.get_fact ctxt,

     attrib = Attrib.intern_src (ProofContext.theory_of ctxt)}

   in activate_i elem ctxt end;

 end;