(*  Title:      Pure/Isar/proof.ML

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

Proof states and methods.

*)

signature PROOF =

sig

  type context

  type state

  exception STATE of string * state

  val check_result: string -> state -> 'a Seq.seq -> 'a Seq.seq

  val context_of: state -> context

  val theory_of: state -> theory

  val sign_of: state -> Sign.sg

  val the_facts: state -> thm list

  val the_fact: state -> thm

  val goal_facts: (state -> thm list) -> state -> state

  val use_facts: state -> state

  val reset_facts: state -> state

  val assert_forward: state -> state

  val assert_backward: state -> state

  val enter_forward: state -> state

  val verbose: bool ref

  val print_state: state -> unit

  val level: state -> int

  type method

  val method: (thm list -> tactic) -> method

  val refine: (context -> method) -> state -> state Seq.seq

  val bind: (indexname * string) list -> state -> state

  val bind_i: (indexname * term) list -> state -> state

  val match_bind: (string list * string) list -> state -> state

  val match_bind_i: (term list * term) list -> state -> state

  val have_thmss: thm list -> string -> context attribute list ->

    (thm list * context attribute list) list -> state -> state

  val simple_have_thms: string -> thm list -> state -> state

  val assume: (int -> tactic) -> string -> context attribute list -> (string * string list) list

    -> state -> state

  val assume_i: (int -> tactic) -> string -> context attribute list -> (term * term list) list

    -> state -> state

  val fix: (string * string option) list -> state -> state

  val fix_i: (string * typ) list -> state -> state

  val theorem: bstring -> theory attribute list -> string * string list -> theory -> state

  val theorem_i: bstring -> theory attribute list -> term * term list -> theory -> state

  val lemma: bstring -> theory attribute list -> string * string list -> theory -> state

  val lemma_i: bstring -> theory attribute list -> term * term list -> theory -> state

  val chain: state -> state

  val from_facts: thm list -> state -> state

  val show: string -> context attribute list -> string * string list -> state -> state

  val show_i: string -> context attribute list -> term * term list -> state -> state

  val have: string -> context attribute list -> string * string list -> state -> state

  val have_i: string -> context attribute list -> term * term list -> state -> state

  val at_bottom: state -> bool

  val local_qed: (state -> state Seq.seq) -> ({kind: string, name: string, thm: thm} -> unit)

    -> state -> state Seq.seq

  val global_qed: (state -> state Seq.seq) -> bstring option -> theory attribute list option

    -> state -> (theory * {kind: string, name: string, thm: thm}) Seq.seq

  val begin_block: state -> state

  val end_block: state -> state

  val next_block: state -> state

end;

signature PROOF_PRIVATE =

sig

  include PROOF

  val put_data: Object.kind -> ('a -> Object.T) -> 'a -> state -> state

end;

structure Proof: PROOF_PRIVATE =

struct

(** proof state **)

type context = ProofContext.context;

(* type goal *)

datatype kind =

  Theorem of theory attribute list |    (*top-level theorem*)

  Lemma of theory attribute list |      (*top-level lemma*)

  Goal of context attribute list |      (*intermediate result, solving subgoal*)

  Aux of context attribute list ;       (*intermediate result*)

val kind_name =

  fn Theorem _ => "theorem" | Lemma _ => "lemma" | Goal _ => "show" | Aux _ => "have";

type goal =

 (kind *               		(*result kind*)

  string *                      (*result name*)

  cterm list *			(*result assumptions*)

  (int -> tactic) list *      	(*tactics to solve result assumptions*)

  term) *                       (*result statement*)

 (thm list *                    (*use facts*)

  thm);                         (*goal: subgoals ==> statement*)

(* type mode *)

datatype mode = Forward | ForwardChain | Backward;

val mode_name =

  enclose "`" "'" o

    (fn Forward => "state" | ForwardChain => "chain" | Backward => "prove");

(* type node *)

type node =

 {context: context,

  facts: thm list option,

  mode: mode,

  goal: goal option};

fun make_node (context, facts, mode, goal) =

  {context = context, facts = facts, mode = mode, goal = goal}: node;

(* datatype state *)

datatype state =

  State of

    node *              (*current*)

    node list;          (*parents wrt. block structure*)

exception STATE of string * state;

fun err_malformed name state =

  raise STATE (name ^ ": internal error -- malformed proof state", state);

fun check_result msg state sq =

  (case Seq.pull sq of

    None => raise STATE (msg, state)

  | Some s_sq => Seq.cons s_sq);

fun map_current f (State ({context, facts, mode, goal}, nodes)) =

  State (make_node (f (context, facts, mode, goal)), nodes);

fun init_state thy =

  State (make_node (ProofContext.init thy, None, Forward, None), []);

(** basic proof state operations **)

(* context *)

fun context_of (State ({context, ...}, _)) = context;

val theory_of = ProofContext.theory_of o context_of;

val sign_of = ProofContext.sign_of o context_of;

fun map_context f = map_current (fn (ctxt, facts, mode, goal) => (f ctxt, facts, mode, goal));

fun map_context_result f (state as State ({context, facts, mode, goal}, nodes)) =

  let val (context', result) = f context

  in (State (make_node (context', facts, mode, goal), nodes), result) end;

fun put_data kind f = map_context o ProofContext.put_data kind f;

val declare_term = map_context o ProofContext.declare_term;

val add_binds = map_context o ProofContext.add_binds_i;

val auto_bind_goal = map_context o ProofContext.auto_bind_goal;

val auto_bind_facts = map_context oo ProofContext.auto_bind_facts;

val put_thms = map_context o ProofContext.put_thms;

val put_thmss = map_context o ProofContext.put_thmss;

(* facts *)

fun the_facts (State ({facts = Some facts, ...}, _)) = facts

  | the_facts state = raise STATE ("No current facts available", state);

fun the_fact state =

  (case the_facts state of

    [fact] => fact

  | _ => raise STATE ("Single fact expected", state));

fun assert_facts state = (the_facts state; state);

fun put_facts facts state =

  state

  |> map_current (fn (ctxt, _, mode, goal) => (ctxt, facts, mode, goal))

  |> put_thms ("facts", if_none facts []);

val reset_facts = put_facts None;

fun have_facts (name, facts) state =

  state

  |> put_facts (Some facts)

  |> put_thms (name, facts);

fun these_facts (state, ths) = have_facts ths state;

fun fetch_facts (State ({facts, ...}, _)) = put_facts facts;

(* goal *)

fun find_goal i (State ({goal = Some goal, ...}, _)) = (i, goal)

  | find_goal i (State ({goal = None, ...}, node :: nodes)) =

      find_goal (i + 1) (State (node, nodes))

  | find_goal _ (state as State (_, [])) = err_malformed "find_goal" state;

fun put_goal goal = map_current (fn (ctxt, facts, mode, _) => (ctxt, facts, mode, goal));

fun map_goal f (State ({context, facts, mode, goal = Some goal}, nodes)) =

      State (make_node (context, facts, mode, Some (f goal)), nodes)

  | map_goal f (State (nd, node :: nodes)) =

      let val State (node', nodes') = map_goal f (State (node, nodes))

      in State (nd, node' :: nodes') end

  | map_goal _ state = state;

fun goal_facts get state =

  state

  |> map_goal (fn (result, (_, thm)) => (result, (get state, thm)));

fun use_facts state =

  state

  |> goal_facts the_facts

  |> reset_facts;

(* mode *)

fun get_mode (State ({mode, ...}, _)) = mode;

fun put_mode mode = map_current (fn (ctxt, facts, _, goal) => (ctxt, facts, mode, goal));

val enter_forward = put_mode Forward;

val enter_forward_chain = put_mode ForwardChain;

val enter_backward = put_mode Backward;

fun assert_mode pred state =

  let val mode = get_mode state in

    if pred mode then state

    else raise STATE ("Illegal application of proof command in " ^ mode_name mode ^ " mode", state)

  end;

fun is_chain state = get_mode state = ForwardChain;

val assert_forward = assert_mode (equal Forward);

val assert_forward_or_chain = assert_mode (equal Forward orf equal ForwardChain);

val assert_backward = assert_mode (equal Backward);

(* blocks *)

fun level (State (_, nodes)) = length nodes;

fun open_block (State (node, nodes)) = State (node, node :: nodes);

fun new_block state =

  state

  |> open_block

  |> put_goal None;

fun close_block (state as State (_, node :: nodes)) =

      State (node, nodes)

      |> map_context (ProofContext.transfer_used_names (context_of state))

  | close_block state = raise STATE ("Unbalanced block parentheses", state);

(** print_state **)

val verbose = ProofContext.verbose;

fun print_facts _ None = ()

  | print_facts s (Some ths) = Pretty.writeln (Pretty.big_list (s ^ " facts:")

      (map (setmp Display.show_hyps false Display.pretty_thm) ths));

fun print_state (state as State ({context, facts, mode, goal = _}, nodes)) =

  let

    val ref (_, _, begin_goal) = Goals.current_goals_markers;

    fun levels_up 0 = ""

      | levels_up i = " (" ^ string_of_int i ^ " levels up)";

    fun print_goal (i, ((kind, name, _, _, _), (goal_facts, thm))) =

      (print_facts "Using" (if null goal_facts then None else Some goal_facts);

        writeln (kind_name kind ^ " " ^ quote name ^ levels_up (i div 2) ^ ":");

        Locale.print_goals_marker begin_goal (! goals_limit) thm);

    val ctxt_strings = ProofContext.strings_of_context context;

  in

    if ! verbose then writeln ("Nesting level: " ^ string_of_int (length nodes div 2)) else ();

    writeln "";

    writeln (mode_name mode ^ " mode");

    writeln "";

    if ! verbose orelse mode = Forward then

      (if null ctxt_strings then () else (seq writeln ctxt_strings; writeln "");

        print_facts "Current" facts;

        print_goal (find_goal 0 state))

    else if mode = ForwardChain then print_facts "Picking" facts

    else print_goal (find_goal 0 state)

  end;

(** proof steps **)

(* datatype method *)

datatype method = Method of thm list -> tactic;

val method = Method;

(* refine goal *)

fun check_sign sg state =

  if Sign.subsig (sg, sign_of state) then state

  else raise STATE ("Bad signature of result: " ^ Sign.str_of_sg sg, state);

fun refine meth_fun (state as State ({context, ...}, _)) =

  let

    val Method meth = meth_fun context;

    val (_, (result, (facts, thm))) = find_goal 0 state;

    fun refn thm' =

      state

      |> check_sign (Thm.sign_of_thm thm')

      |> map_goal (K (result, (facts, thm')));

  in Seq.map refn (meth facts thm) end;

(* prepare result *)

fun varify_frees names thm =

  let

    fun get_free x (None, t as Free (y, _)) = if x = y then Some t else None

      | get_free _ (opt, _) = opt;

    fun find_free t x = foldl_aterms (get_free x) (None, t);

    val {sign, prop, ...} = Thm.rep_thm thm;

    val frees = map (Thm.cterm_of sign) (mapfilter (find_free prop) names);

  in

    thm

    |> Drule.forall_intr_list frees

    |> Drule.forall_elim_vars 0

  end;

fun varify_tfrees thm =

  let

    val {hyps, prop, ...} = Thm.rep_thm thm;

    val frees = foldr Term.add_term_frees (prop :: hyps, []);

    val leave_tfrees = foldr Term.add_term_tfree_names (frees, []);

  in thm |> Thm.varifyT' leave_tfrees end;

fun implies_elim_hyps thm =

  foldl (uncurry Thm.implies_elim) (thm, map Thm.assume (Drule.cprems_of thm));

fun prep_result state asms t raw_thm =

  let

    val ctxt = context_of state;

    fun err msg = raise STATE (msg, state);

    val ngoals = Thm.nprems_of raw_thm;

    val _ =

      if ngoals = 0 then ()

      else (Locale.print_goals ngoals raw_thm; err (string_of_int ngoals ^ " unsolved goal(s)!"));

    val thm =

      raw_thm RS Drule.rev_triv_goal

      |> implies_elim_hyps

      |> Drule.implies_intr_list asms

      |> varify_frees (ProofContext.fixed_names ctxt)

      |> varify_tfrees;

    val {hyps, prop, sign, ...} = Thm.rep_thm thm;

    val tsig = Sign.tsig_of sign;

  in

(* FIXME

    if not (Pattern.matches tsig (t, Logic.skip_flexpairs prop)) then

      warning ("Proved a different theorem: " ^ Sign.string_of_term sign prop)

    else ();

*)

    if not (null hyps) then

      err ("Additional hypotheses:\n" ^ cat_lines (map (Sign.string_of_term sign) hyps))

(* FIXME    else if not (Pattern.matches tsig (t, Logic.skip_flexpairs prop)) then

      err ("Proved a different theorem: " ^ Sign.string_of_term sign prop) *)

    else thm

  end;

(* prepare final result *)

fun strip_flexflex thm =

  Seq.hd (Thm.flexflex_rule thm) handle THM _ => thm;

fun final_result state pre_thm =

  let

    val thm =

      pre_thm

      |> strip_flexflex

      |> Thm.strip_shyps

      |> Drule.standard;

    val str_of_sort = Sign.str_of_sort (Thm.sign_of_thm thm);

    val xshyps = Thm.extra_shyps thm;

  in

    if not (null xshyps) then

      raise STATE ("Extra sort hypotheses: " ^ commas (map str_of_sort xshyps), state)

    else thm

  end;

(* solve goal *)

fun RANGE [] _ = all_tac

  | RANGE (tac :: tacs) i = RANGE tacs (i + 1) THEN tac i;

fun solve_goal rule tacs state =

  let

    val (_, (result, (facts, thm))) = find_goal 0 state;

    val thms' = FIRSTGOAL (rtac rule THEN' RANGE tacs) thm;

  in Seq.map (fn thm' => map_goal (K (result, (facts, thm'))) state) thms' end;

(*** structured proof commands ***)

(** context **)

(* bind *)

fun gen_bind f x state =

  state

  |> assert_forward

  |> map_context (f x)

  |> reset_facts;

val bind = gen_bind ProofContext.add_binds;

val bind_i = gen_bind ProofContext.add_binds_i;

val match_bind = gen_bind ProofContext.match_binds;

val match_bind_i = gen_bind ProofContext.match_binds_i;

(* have_thmss *)

fun have_thmss ths name atts ths_atts state =

  state

  |> assert_forward

  |> map_context_result (ProofContext.have_thmss ths (PureThy.default_name name) atts ths_atts)

  |> these_facts;

fun simple_have_thms name thms = have_thmss [] name [] [(thms, [])];

(* fix *)

fun gen_fix f xs state =

  state

  |> assert_forward

  |> map_context (f xs)

  |> reset_facts;

val fix = gen_fix ProofContext.fix;

val fix_i = gen_fix ProofContext.fix_i;

(* assume *)

fun gen_assume f tac name atts props state =

  state

  |> assert_forward

  |> map_context_result (f tac (PureThy.default_name name) atts props)

  |> (fn (st, (facts, prems)) =>

    (st, facts)

    |> these_facts

    |> put_thms ("prems", prems));

val assume = gen_assume ProofContext.assume;

val assume_i = gen_assume ProofContext.assume_i;

(** goals **)

(* forward chaining *)

fun chain state =

  state

  |> assert_forward

  |> assert_facts

  |> enter_forward_chain;

fun from_facts facts state =

  state

  |> put_facts (Some facts)

  |> chain;

(* setup goals *)

fun setup_goal opt_block prepp kind name atts raw_propp state =

  let

    val (state', concl) =

      state

      |> assert_forward_or_chain

      |> enter_forward

      |> opt_block

      |> map_context_result (fn c => prepp (c, raw_propp));

    val cterm_of = Thm.cterm_of (sign_of state);

    val (casms, asm_tacs) = Library.split_list (ProofContext.assumptions (context_of state'));

    val cprems = map cterm_of (Logic.strip_imp_prems concl);

    val prem_tacs = replicate (length cprems) (K all_tac);

    val prop = Logic.list_implies (map Thm.term_of casms, concl);

    val cprop = cterm_of prop;

    val thm = Drule.mk_triv_goal cprop;

  in

    state'

    |> put_goal (Some ((kind atts, (PureThy.default_name name), casms @ cprems,

        asm_tacs @ prem_tacs, prop), ([], thm)))

    |> auto_bind_goal prop

    |> (if is_chain state then use_facts else reset_facts)

    |> new_block

    |> enter_backward

  end;

(*global goals*)

fun global_goal prep kind name atts x thy =

  setup_goal I prep kind name atts x (init_state thy);

val theorem = global_goal ProofContext.bind_propp Theorem;

val theorem_i = global_goal ProofContext.bind_propp_i Theorem;

val lemma = global_goal ProofContext.bind_propp Lemma;

val lemma_i = global_goal ProofContext.bind_propp_i Lemma;

(*local goals*)

fun local_goal prep kind name atts x =

  setup_goal open_block prep kind name atts x;

val show   = local_goal ProofContext.bind_propp Goal;

val show_i = local_goal ProofContext.bind_propp_i Goal;

val have   = local_goal ProofContext.bind_propp Aux;

val have_i = local_goal ProofContext.bind_propp_i Aux;

(** conclusions **)

(* current goal *)

fun current_goal (State ({goal = Some goal, ...}, _)) = goal

  | current_goal state = raise STATE ("No current goal!", state);

fun assert_current_goal true (state as State ({goal = None, ...}, _)) =

      raise STATE ("No goal in this block!", state)

  | assert_current_goal false (state as State ({goal = Some _, ...}, _)) =

      raise STATE ("Goal present in this block!", state)

  | assert_current_goal _ state = state;

fun assert_bottom true (state as State (_, _ :: _)) =

      raise STATE ("Not at bottom of proof!", state)

  | assert_bottom false (state as State (_, [])) =

      raise STATE ("Already at bottom of proof!", state)

  | assert_bottom _ state = state;

val at_bottom = can (assert_bottom true o close_block);

(* finish proof *)

fun finish_proof bot state =

  state

  |> assert_forward

  |> close_block

  |> assert_bottom bot

  |> assert_current_goal true;

(* local_qed *)

fun finish_local print_result state =

  let

    val ((kind, name, asms, tacs, t), (_, raw_thm)) = current_goal state;

    val result = prep_result state asms t raw_thm;

    val (atts, opt_solve) =

      (case kind of

        Goal atts => (atts, solve_goal result tacs)

      | Aux atts => (atts, Seq.single)

      | _ => raise STATE ("No local goal!", state));

  in

    print_result {kind = kind_name kind, name = name, thm = result};

    state

    |> close_block

    |> auto_bind_facts name [t]

    |> have_thmss [] name atts [Thm.no_attributes [result]]

    |> opt_solve

  end;

fun local_qed finalize print_result state =

  state

  |> finish_proof false

  |> finalize

  |> (Seq.flat o Seq.map (finish_local print_result));

(* global_qed *)

fun finish_global alt_name alt_atts state =

  let

    val ((kind, def_name, asms, _, t), (_, raw_thm)) = current_goal state;

    val result = final_result state (prep_result state asms t raw_thm);

    val name = if_none alt_name def_name;

    val atts =

      (case kind of

        Theorem atts => if_none alt_atts atts

      | Lemma atts => (if_none alt_atts atts) @ [Drule.tag_lemma]

      | _ => raise STATE ("No global goal!", state));

    val (thy', result') = PureThy.store_thm ((name, result), atts) (theory_of state);

  in (thy', {kind = kind_name kind, name = name, thm = result'}) end;

(*Note: should inspect first result only, backtracking may destroy theory*)

fun global_qed finalize alt_name alt_atts state =

  state

  |> finish_proof true

  |> finalize

  |> Seq.map (finish_global alt_name alt_atts);

(** blocks **)

(* begin_block *)

fun begin_block state =

  state

  |> assert_forward

  |> new_block

  |> open_block;

(* end_block *)

fun end_block state =

  state

  |> assert_forward

  |> close_block

  |> assert_current_goal false

  |> close_block

  |> fetch_facts state;          (* FIXME proper check / export (!?) *)

(* next_block *)

fun next_block state =

  state

  |> assert_forward

  |> close_block

  |> new_block;

end;