(*  Title:      Pure/Isar/element.ML

    ID:         $Id$

    Author:     Makarius

Explicit data structures for some Isar language elements, with derived

logical operations.

*)

signature ELEMENT =

sig

  datatype ('typ, 'term) stmt =

    Shows of ((string * Attrib.src list) * ('term * 'term list) list) list |

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

  type statement  (*= (string, string) stmt*)

  type statement_i  (*= (typ, term) stmt*)

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

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

    Constrains of (string * 'typ) list |

    Assumes of ((string * Attrib.src list) * ('term * 'term list) list) list |

    Defines of ((string * Attrib.src list) * ('term * 'term list)) list |

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

  type context (*= (string, string, thmref) ctxt*)

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

  val map_ctxt: {name: string -> string,

    var: string * mixfix -> string * mixfix,

    typ: 'typ -> 'a, term: 'term -> 'b, fact: 'fact -> 'c,

    attrib: Attrib.src -> Attrib.src} -> ('typ, 'term, 'fact) ctxt -> ('a, 'b, 'c) ctxt

  val map_ctxt_values: (typ -> typ) -> (term -> term) -> (thm -> thm) -> context_i -> context_i

  val params_of: context_i -> (string * typ) list

  val prems_of: context_i -> term list

  val facts_of: theory -> context_i ->

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

  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 map_witness: (term * thm -> term * thm) -> witness -> witness

  val witness_prop: witness -> term

  val witness_hyps: witness -> term list

  val assume_witness: theory -> term -> witness

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

  val conclude_witness: witness -> thm

  val mark_witness: term -> term

  val make_witness: term -> thm -> witness

  val dest_witness: witness -> term * thm

  val transfer_witness: theory -> witness -> witness

  val refine_witness: Proof.state -> Proof.state Seq.seq

  val rename: (string * (string * mixfix option)) list -> string -> string

  val rename_var: (string * (string * mixfix option)) list -> string * mixfix -> string * mixfix

  val rename_term: (string * (string * mixfix option)) list -> term -> term

  val rename_thm: (string * (string * mixfix option)) list -> thm -> thm

  val rename_witness: (string * (string * mixfix option)) list -> witness -> witness

  val rename_ctxt: (string * (string * mixfix option)) list -> context_i -> context_i

  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_witness: theory -> typ Symtab.table -> witness -> witness

  val instT_ctxt: theory -> typ Symtab.table -> context_i -> context_i

  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_witness: theory -> typ Symtab.table * term Symtab.table -> witness -> witness

  val inst_ctxt: theory -> typ Symtab.table * term Symtab.table -> context_i -> context_i

  val satisfy_thm: witness list -> thm -> thm

  val satisfy_witness: witness list -> witness -> witness

  val satisfy_ctxt: witness list -> context_i -> context_i

  val satisfy_facts: witness list ->

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

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

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

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

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

end;

structure Element: ELEMENT =

struct

(** language elements **)

(* statement *)

datatype ('typ, 'term) stmt =

  Shows of ((string * Attrib.src list) * ('term * 'term list) list) list |

  Obtains of (string * ((string * '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 (string * 'typ option * mixfix) list |

  Constrains of (string * 'typ) list |

  Assumes of ((string * Attrib.src list) * ('term * 'term list) list) list |

  Defines of ((string * Attrib.src list) * ('term * 'term list)) list |

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

type context = (string, string, thmref) ctxt;

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

fun map_ctxt {name, var, typ, term, fact, attrib} =

  fn Fixes fixes => Fixes (fixes |> map (fn (x, T, mx) =>

       let val (x', mx') = var (x, mx) in (x', Option.map typ T, mx') end))

   | Constrains xs => Constrains (xs |> map (fn (x, T) => (#1 (var (x, NoSyn)), typ T)))

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

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

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

      ((name a, map attrib atts), (term t, map term ps))))

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

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

fun map_ctxt_values typ term thm = map_ctxt

  {name = I, var = I, typ = typ, term = term, fact = map thm,

    attrib = Args.map_values I typ term thm};

(* logical content *)

fun params_of (Fixes fixes) = fixes |> map

    (fn (x, SOME T, _) => (x, T)

      | (x, _, _) => raise TERM ("Untyped context element parameter " ^ quote x, []))

  | params_of _ = [];

fun prems_of (Assumes asms) = maps (map fst o snd) asms

  | prems_of (Defines defs) = map (fst o snd) defs

  | prems_of _ = [];

fun assume thy t = Assumption.assume (Thm.cterm_of thy t);

fun facts_of thy (Assumes asms) = map (apsnd (map (fn (t, _) => ([assume thy t], [])))) asms

  | facts_of thy (Defines defs) = map (apsnd (fn (t, _) => [([assume thy t], [])])) defs

  | facts_of _ (Notes facts) = facts

  | facts_of _ _ = [];

(** 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 (name, atts) sep =

  if name = "" andalso null atts then []

  else [Pretty.block

    (Pretty.breaks (Pretty.str name :: Args.pretty_attribs ctxt atts @ [Pretty.str sep]))];

(* pretty_stmt *)

fun pretty_stmt ctxt =

  let

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

    val prt_term = Pretty.quote o ProofContext.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 (x ^ " ::"), Pretty.brk 1, prt_typ T]

      | prt_var (x, NONE) = Pretty.str 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 ProofContext.pretty_typ ctxt;

    val prt_term = Pretty.quote o ProofContext.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 (x ^ " ::") :: Pretty.brk 1 ::

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

      | prt_fix (x, NONE, mx) = Pretty.block (Pretty.str x :: Pretty.brk 1 :: prt_mixfix mx);

    fun prt_constrain (x, T) = prt_fix (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)) :: Args.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)

  end;

(* pretty_statement *)

local

fun thm_name kind th prts =

  let val head =

    (case #1 (Thm.get_name_tags th) of

      "" => Pretty.command kind

    | a => Pretty.block [Pretty.command kind, Pretty.brk 1, Pretty.str (Sign.base_name a ^ ":")])

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

fun obtain prop ctxt =

  let

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

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

    fun var (x, T) = (ProofContext.revert_skolem ctxt' x, SOME T);

  in (("", (map (var 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 = norm_hhf raw_th;

    val is_elim = ObjectLogic.is_elim th;

    val ((_, [th']), ctxt') = Variable.import true [th] 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 |> map (apfst (ProofContext.revert_skolem ctxt'));

    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) => (x, SOME T, NoSyn)) fixes)) @

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

    pretty_stmt ctxt'

     (if null cases then Shows [(("", []), [(concl, [])])]

      else Obtains (#1 (fold_map (obtain o cert) cases ctxt')))

  end |> thm_name kind raw_th;

end;

(** logical operations **)

(* witnesses -- hypotheses as protected facts *)

datatype witness = Witness of term * thm;

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

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

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

fun assume_witness thy t =

  Witness (t, Goal.protect (Thm.assume (Thm.cterm_of thy t)));

fun prove_witness ctxt t tac =

  Witness (t, Goal.prove ctxt [] [] (Logic.protect t) (fn _ =>

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

fun conclude_witness (Witness (_, th)) = norm_hhf (Goal.conclude th);

val mark_witness = Logic.protect;

fun make_witness t th = Witness (t, th);

fun dest_witness (Witness w) = w;

fun transfer_witness thy (Witness (t, th)) = Witness (t, Thm.transfer thy th);

val refine_witness =

  Proof.refine (Method.Basic (K (Method.RAW_METHOD

    (K (ALLGOALS

      (PRECISE_CONJUNCTS ~1 (ALLGOALS

        (PRECISE_CONJUNCTS ~1 (TRYALL (Tactic.rtac Drule.protectI))))))))));

(* 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 cterm_rule = Drule.mk_term #> rule #> Drule.dest_term;

    val {hyps, ...} = Thm.crep_thm th;

  in

    Drule.implies_elim_list

      (rule (Drule.implies_intr_list hyps th))

      (map (Thm.assume o cterm_rule) hyps)

  end;

(* rename *)

fun rename ren x =

  (case AList.lookup (op =) ren (x: string) of

    NONE => x

  | SOME (x', _) => x');

fun rename_var ren (x, mx) =

  (case (AList.lookup (op =) ren (x: string), mx) of

    (NONE, _) => (x, mx)

  | (SOME (x', NONE), Structure) => (x', mx)

  | (SOME (x', SOME _), Structure) =>

      error ("Attempt to change syntax of structure parameter " ^ quote x)

  | (SOME (x', NONE), _) => (x', NoSyn)

  | (SOME (x', SOME mx'), _) => (x', mx'));

fun rename_term ren (Free (x, T)) = Free (rename ren x, T)

  | rename_term ren (t $ u) = rename_term ren t $ rename_term ren u

  | rename_term ren (Abs (x, T, t)) = Abs (x, T, rename_term ren t)

  | rename_term _ a = a;

fun rename_thm ren th =

  let

    val thy = Thm.theory_of_thm th;

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

      (fn Free (x, T) =>

        let val x' = rename ren x

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

      | _ => I) th [];

  in

    if null subst then th

    else th |> hyps_rule (instantiate_frees thy subst)

  end;

fun rename_witness ren =

  map_witness (fn (t, th) => (rename_term ren t, rename_thm ren th));

fun rename_ctxt ren =

  map_ctxt_values I (rename_term ren) (rename_thm ren)

  #> map_ctxt {name = I, typ = I, term = I, fact = I, attrib = I, var = rename_var ren};

(* 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 = (Drule.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_witness thy env =

  map_witness (fn (t, th) => (instT_term env t, instT_thm thy env th));

fun instT_ctxt thy env =

  map_ctxt_values (instT_type env) (instT_term env) (instT_thm thy env);

(* 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 = (Drule.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 #>

        Drule.fconv_rule (Thm.beta_conversion true))

    end;

fun inst_witness thy envs =

  map_witness (fn (t, th) => (inst_term envs t, inst_thm thy envs th));

fun inst_ctxt thy envs =

  map_ctxt_values (instT_type (#1 envs)) (inst_term envs) (inst_thm thy envs);

(* 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 (Witness (_, th)) => Drule.implies_intr_protected [hyp] #> Goal.comp_hhf th))

  (#hyps (Thm.crep_thm thm));

fun satisfy_witness witns = map_witness (apsnd (satisfy_thm witns));

fun satisfy_ctxt witns = map_ctxt_values I I (satisfy_thm witns);

fun satisfy_facts witns facts =

  satisfy_ctxt witns (Notes facts) |> (fn Notes facts' => facts');

(* 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 gen_thm = Thm.adjust_maxidx_thm maxidx #> singleton (Variable.export inner outer);

    val gen_term = Thm.cterm_of thy #> Drule.mk_term #> gen_thm #> Drule.dest_term #> Thm.term_of;

    val gen_typ = Logic.mk_type #> gen_term #> Logic.dest_type;

    val Notes facts' = map_ctxt_values gen_typ gen_term gen_thm (Notes facts);

  in facts' end;

end;